ON THE MATHEMATICS OF SELF-ASSEMBLY

by

Dustin Reishus

A Dissertation Presented to theFACULTY OF THE GRADUATE SCHOOL

UNIVERSITY OF SOUTHERN CALIFORNIAIn Partial Fulfillment of theRequirements for the Degree

DOCTOR OF PHILOSOPHY(COMPUTER SCIENCE)

August 2009

Copyright 2009 Dustin Reishus

Acknowledgments

It would not have been possible for me to get a PhD without the love, support, guidance,

and friendship I received from many, many people along the way. It is also not possible

for me to adequately express the gratitude I feel for everyone who helped me along the

way, nor is it possible for me to list everyone who deserves that gratitude, but that won’t

stop me from trying.

First, of course, comes my family; my father Allan, my mother Donna, my sister

Anna, and my brother-in-law Nick: From my earliest memories and throughout my life,

you’ve instilled in me the desire to learn, to know, to achieve. You are the reason I am

where I am today. I love you all.

My advisor Len: You’ve taught me more than just how to do experiments, how to

write papers, and how to do math. You’ve taught me how to think. You’ve helped me

with problems, both academic and personal. The mentor/mentee relationship may be

how we began, but now, and for all times, you are and will be my cherished friend.

My best friends and fellow PhD students, Yuriy and Dave: I literally don’t know

where I would be without you guys. We have had more good times than I can count. We

have had more fun than anyone could reasonably expect, but there have also been hard

times. Thank you guys for sharing with me the good times; thank you for being there for

ii

me through the bad times; thank you for helping me with research; thank you for being

my friends.

My good friends The Robots; Brian, Justin, David, Mike, Josh, the other Mike, and

Dominic: Living with you guys through college and in the Robot House is one of the best

things I’ve ever done in my life. You made my undergraduate experience the happiest

period of my life and you all will be my life-long friends.

My friends and colleagues, Nickolas, Manoj, Bilal, Joe, Henry, and Urmila: For the

countless hours we spent in the lab doing experiments, or in meetings writing papers

and doing math; for the midnight conversations discussing the trickier points of algebraic

complex analysis; and for the times we’ve just hung out and discussed life; thank you.

My mentors, Paul, Ming, Steve, Lila, Ashish, and many others: You’ve given your

time, your expertise, your advice, and your guidance. I am forever grateful.

And finally, my teachers: To all my teachers and professors, thank you for the gift

of knowledge that you have given me. I will spend the rest of my life trying to pay it

forward.

iii

Table of Contents

Acknowledgments ii

List Of Figures vi

Abstract viii

Chapter 1: Introduction 1

Chapter 2: DNA Self-Assembly 52.1 Double-Double Crossover Complex . . . . . . . . . . . . . . . . . . . . 62.2 Triangular Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Chapter 3: Tile Assembly Model 163.1 Basic Definitions and Notation . . . . . . . . . . . . . . . . . . . . . . 193.2 A Directed Tile System with the Strong Plane-Filling Property . . . . 23

3.2.1 Generalized tile systems . . . . . . . . . . . . . . . . . . . . . . 253.2.2 Microtiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.3 Minitiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.4 Macrotiles: The 3x3 block construction . . . . . . . . . . . . . 39

3.3 Undecidability of Existence of Infinite Ribbons . . . . . . . . . . . . . 493.4 Undecidability of Self-Assembly of Arbitrarily Large Supertiles . . . . 57

Chapter 4: Event-Systems Model 594.1 Basic Definitions and Notation . . . . . . . . . . . . . . . . . . . . . . 644.2 Finite Event-Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.3 Finite Physical Event-Systems . . . . . . . . . . . . . . . . . . . . . . . 774.4 Finite Natural Event-Systems . . . . . . . . . . . . . . . . . . . . . . . 944.5 Finite Natural Atomic Event-Systems . . . . . . . . . . . . . . . . . . 124

Chapter 5: Zeta Event-Systems 1305.1 Riemann Zeta Function . . . . . . . . . . . . . . . . . . . . . . . . . . 1325.2 Basic Definitions and Notation . . . . . . . . . . . . . . . . . . . . . . 135

5.2.1 Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385.2.2 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395.2.3 Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

5.3 Zeta Event-Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1435.4 Visualizing Event-Processes . . . . . . . . . . . . . . . . . . . . . . . . 151

iv

5.5 Numerical Simulations of Event Processes . . . . . . . . . . . . . . . . 1555.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

References 158

v

List Of Figures

Figure 2.1: DNA double-double crossover (DDX) complex . . . . . . . . . . 7

Figure 2.2: Atomic force micrographs of DDX structures . . . . . . . . . . . 9

Figure 2.3: Diagram of an alternative DDX complex . . . . . . . . . . . . . 9

Figure 2.4: Diagram of the four-helix bundle complex . . . . . . . . . . . . 10

Figure 2.5: Diagram of a triangular complex . . . . . . . . . . . . . . . . . . 12

Figure 2.6: Atomic force micrographs of triangular complexes . . . . . . . . 13

Figure 3.1: The five types of microtiles . . . . . . . . . . . . . . . . . . . . . 28

Figure 3.2: Labeling mixed arms . . . . . . . . . . . . . . . . . . . . . . . . 28

Figure 3.3: The (2n+1 − 1)-XY -square . . . . . . . . . . . . . . . . . . . . . 30

Figure 3.4: The 7-XY -square . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Figure 3.5: Paths through 2n+1 × 2n+1 minitiles . . . . . . . . . . . . . . . 34

Figure 3.6: The paths A4 and A16 . . . . . . . . . . . . . . . . . . . . . . . 34

Figure 3.7: The labeling of mixed arms . . . . . . . . . . . . . . . . . . . . 36

Figure 3.8: The labeling of mixed arms (continued) . . . . . . . . . . . . . . 36

Figure 3.9: Directions assigned to minitiles . . . . . . . . . . . . . . . . . . 37

Figure 3.10: Microtiles, minitiles, and macrotiles . . . . . . . . . . . . . . . 39

Figure 3.11: Macrotile glues . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

vi

Figure 3.12: The (2n+1 − 1)-square used in the proof of Lemma 3.2.6 . . . . 45

Figure 3.13: Pairs of macrotiles attached to S(1) and S(2) . . . . . . . . . . 47

Figure 3.14: A sandwich tile . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Figure 3.15: A ribbon motif . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Figure 3.16: Variant ribbon motifs . . . . . . . . . . . . . . . . . . . . . . . 53

Figure 3.17: The bump-and-dent pair that simulate a glue . . . . . . . . . . 54

Figure 5.1: Visualizing complex-valued functions of a complex variable usingdomain coloring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Figure 5.2: Images of a simplified Eζ,2(s)-process . . . . . . . . . . . . . . . 153

Figure 5.3: Images of the numerical simulation of a Eζ,2(s)-process . . . . . 159

vii

Abstract

Self-assembly is the ubiquitous process by which simple objects come together under

simple rules to form more complex objects. Self-assembly occurs in nature to produce

structures of extraordinary complexity. In the future, it may be possible to harness the

power of self-assembly to manufacture useful devices in enormous quantities at little cost.

In order to do so, it would be valuable to have a deep understanding of self-assembly, at

both theoretical and practical levels.

I first describe experimental work with DNA self-assembly. DNA is an ideal substance

to use in experimental self-assembly: It has well-understood structure; it has readily-

available tools to synthesize, manipulate, and visualize it; and it has “programmable”

interactions with other molecules of DNA. I describe two self-assembling DNA complexes

that can further self-assemble into regular lattices.

Mathematical models of self-assembly have been created to aid in the analysis of the

power and limits of self-assembly. I explore decidability questions in a mathematical

model of self-assembly known as the tile assembly model. I prove the undecidability of

distinguishing self-assembling systems in which infinite structures can be assembled from

systems in which only finite structures can be assembled.

viii

Many self-assembly processes are rooted in chemistry. The event-systems model gen-

eralizes the classical theory of chemical thermodynamics and places the kinetic theory of

chemical reactions on a firm mathematical foundation. I prove that many of the expec-

tations acquired through empirical study are warranted.

Finally, I use the event-systems model to explore questions in pure mathematics. The

atomic hypothesis in chemistry (the theory that every substance is composed of a unique

set of atoms) is analogous to the fundamental theorem of arithmetic in mathematics

(the theory that every natural number is the product of a unique set of primes). I

exploit this analogy by creating event-systems in which the basic components are natural

numbers that can “react” through multiplication. Important thermodynamic properties

such as temperature and pressure have purely mathematical implications in these systems.

In particular, the pressure at equilibrium is the Riemann zeta function’s value of the

temperature of the system.

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Chapter 1

Introduction

Self-assembly is the ubiquitous process in which simple objects come together under

simple rules to form more complex objects. Examples of natural self-assembling systems

abound on all scales, from the microscopic to the cosmologic. At the tiniest scales,

subatomic particles self-assemble to form atoms, atoms self-assemble to form molecules,

and molecules self-assemble to form crystals. The simple rules these objects are following

are consequences of natural forces: the strong nuclear force and the electromagnetic force.

At the largest scales, stars self-assemble to form galaxies and galaxies self-assemble to

form galaxy clusters; consequences of the gravitational force.

In nature, self-assembly processes create intricate structures (such as the crystal struc-

ture of a snowflake) and practical structures (such as virus capsids). Recently it has been

suggested that complex self-assembly processes will ultimately be used in integrated cir-

cuit fabrication, nanorobotics, DNA computation and amorphous computing. Indeed, in

electronics, engineering, medicine, material science, manufacturing and other disciplines,

there is a continuous drive toward miniaturization. Unfortunately, current “top-down”

techniques such as lithography may not be capable of efficiently creating structures with

1

features the size of molecules or atoms. Self-assembly provides a “bottom-up” approach

by which such fine structures might be created.

Experimental research on self-assembly includes creation of Penrose tilings using

macroscopic plastic tiles that assemble at an oil/water interface [71], studies of self-

directed growth of hybrid organic molecules on a silicon substrate [58], self-assembly of

regular lipid hollow icosahedra [23], self-assembly of patterns of lead on a copper surface

with the goal of creating templates for fabricating nanostructures [64], formation of elec-

trical networks by self-assembly of small plastic and metal objects [37, 84], self-assembly

of molecular machines [35], the construction by self-assembly of a molecular-thickness

transistor [76], self-assembly of DNA nanomachines (e.g., molecular switches [55], tweez-

ers [91, 61], walkers [78], autonomous DNA motors [82, 11, 38]), as well as algorithmic

self-assembly of Sierpinski triangles [74], self-assembled cloneable DNA octahedra [79],

stiff self-assembled tetrahedra for three-dimensional electronic circuits [36], and DNA

origami [72].

Investigations into DNA Computing [2] and amorphous computing [1] have demon-

strated a strong connection between self-assembly and computation. While [88, 53, 67, 85]

have shown the potential of DNA self-assembly for computation, [59] has demonstrated

the execution of logical operations (cumulative XOR) by self-assembly of DNA triple

cross-over molecules and [12] has demonstrated the potential for programmable self-

assembled computing devices.

A deep understanding of self-assembly would be valuable in designing self-assembling

systems with desired properties. Mathematical models of self-assembly could help provide

this understanding. Hopefully these models of self-assembly processes abstract away

2

irrelevant details, leaving only the phenomena we wish to explore. Currently there are

several models of self-assembly, each looking at the processes in a slightly different way,

each abstracting out different details.

In the chapters that follow, I will explore both theoretical and experimental aspects

of self-assembly.

In Chapter 2, I will describe experimental work in DNA self-assembly that I have done

with my colleagues. DNA, as a material, has many properties that make it ideal to use

when exploring self-assembling systems. Its structure is well-understood. There are many

tools from molecular biology for creating, manipulating, and measuring it. It is cheap

to obtain in abundance. Most importantly, it has programmable interactions with other

molecules. By changing the sequence of nucleotides, it is possible to design a molecule

that sticks to another molecule in a particular way and with a particular strength.

In Chapter 3, I will explore a formal model of self-assembly known as the tile assembly

model (TAM). TAM was initially created to model the self-assembly of DNA complexes

such as the ones discussed in Chapter 2. Using a mathematical model of self-assembly

allows us to study the limits and power of self-assembly processes. In particular, I will

describe the work I have done with colleagues proving the undecidability of the problem

of distinguishing self-assembling systems in which infinite structures can be assembled

from systems in which only finite structures can be assembled.

In Chapter 4, I will explore a different model of self-assembly known as the event-

systems model (ESM). I, along with my colleagues, created ESM, which generalizes the

classical theory of chemical thermodynamics. Given a set of chemical reactions, the

theory of event-systems describes how to use the law of mass action to obtain a set of

3

differential equations that predict the concentrations of self-assembling species through

time. Using the model, I prove that, under certain conditions or assumptions, many of

the expectations acquired through empirical studies are justified.

In Chapter 5, I will describe a particular class of event-systems that I call zeta event-

systems. Unlike in classical thermodynamics, in the event-systems model the reaction

rates, concentrations, temperature, and even time can be complex-valued. These systems

are given in terms of a parameter s, and have the property that the total pressure or mass

of the system is ζ(s), where ζ is the Riemann zeta function. By studying the dynamics of

these systems, I hope to be able to better understand this very important mathematical

function.

4

Chapter 2

DNA Self-Assembly

In this chapter, we describe the double-double crossover complex and the triangular

complex. Each of these complexes self-assemble from molecules of DNA. The complexes

themselves serve as building blocks which can further self-assemble into regular lattices.

A preliminary description of this work appeared in [15] and later, in more detail, in [68]

and [16].

The field of human-designed, self-assembled DNA structures was initiated by Seeman

et al. [18]. In 1993, Fu et al. [31] described the double crossover (DX) complex consisting

of two DNA double helices connected by two reciprocal exchanges (crossovers). In 1998,

Winfree et al. [86] used DX complexes to self-assemble regular planar structures. Sub-

sequently, several other DNA complexes have been designed and used to self-assemble

organized structures [52, 89].

5

2.1 Double-Double Crossover Complex

In this section, we present a new complex based on the DX, the double-double crossover

(DDX), consisting of four DNA double helices connected by a total of six reciprocal ex-

changes. The work presented in this section is the result of collaboration between myself,

Bilal Shaw, Yuriy Brun, Nickolas Chelyapov, and Leonard Adleman and was published

in [68]. The first person plural pronouns in this section refer to those authors. We use

DDX complexes to self-assemble high-density, doubly-connected, planar structures. We

also briefly describe a potential modification of the DDX complex that may permit the

self-assembly of three-dimensional structures.

Structures self-assembled by Winfree et al. [86] and LaBean et al. [52] are singly-

connected (adjacent complexes are held together by a single pair of complementary sticky

ends), while structures self-assembled by Yan et al. [89] and Ding et al. [22] are doubly-

connected (adjacent complexes are held together by two pairs of complementary sticky

ends). Doubly-connected structures may have greater structural integrity than singly-

connected ones. However, the doubly-connected structures that have been created to date

lack the high density (DNA mass per unit area) that is found in structures self-assembled

using DX complexes. Watson-Crick base pairing endows DNA with the capacity to self-

assemble into a wide variety of shapes and patterns [74]. This capacity, together with

double connectedness and high density, may make structures created with DDX complexes

desirable. For example, such structures might be useful as scaffoldings for the deposition

of nanomaterials in the creation of high-density electrical and quantum devices.

6

Figure 2.1a shows the expected structure of the DDX complex, with two reciprocal

exchanges connecting each pair of adjacent helices. Figure 2.1b shows how the complexes

might tile the plane with individual complexes connected to neighbors via two pairs of

sticky ends.

(a) (b) (c) (d)

Figure 2.1: (a) Diagram of the DNA double-double crossover (DDX) complex consist-ing of eight strands forming four double helices connected via six reciprocal exchanges.(b) Diagram of the DDX complexes tiling the plane while connecting every pair of adja-cent complexes via two pairs of sticky ends. (c) Diagram of the alternative DDX complex.(d) Diagram of the alternative DDX complexes tiling the plane.

Complexes that are capable of periodically tiling a plane are also theoretically capable

of forming tubular structures. Rothemund et al. [73] presented evidence that one of the

factors that affects the curvature of the structures formed by DX complexes is the number

of bases between reciprocal exchanges. Certain choices favor planar structures, while

others favor tubular structures.

In order to design DDX complexes that would self-assemble into planar structures, we

wrote an algorithm that searches for a set of crossover positions that minimize interhelix

strain in an idealized planar complex consisting of four parallel DNA double helices. Based

7

on the output of the algorithm, we generated DNA sequences with 16 bases between outer

reciprocal exchanges and 26 bases between inner ones (see Figure 2.1a). We designed

sticky ends so that complexes of a single type would be capable of tiling the plane.

DNA strands were synthesized and PAGE purified by Integrated DNA Technologies.

A solution of 0.2µM of each strand was prepared in TAE/Mg2+ buffer (40mM Tris-HCl,

pH 8.0; 1mM EDTA; 12.5mM Magnesium Acetate). The solution was heated to 90C

in an insulated water bath and left to cool to room temperature for no less than one day

and no more than three days. A 5µL aliquot of the complexes was spotted onto freshly

cleaved mica (Ted Pella), left for 30 seconds and then topped with 25µL of TAE/Mg2+

buffer. Imaging was performed on a Multimode Nanoscope IIIa atomic force microscope

(Digital Instruments) in tapping mode, using a fluid cell, J scanner and 200µm cantilevers

with Si3N4 tips.

Figures 2.2a and 2.2b show atomic force micrographs of DDX complexes tiling the

plane. We frequently observed crystals of this size and regularity. Each unit in the crystal

is approximately 14 nm by 10 nm in size, which is consistent with our predictions of the

size of the DDX complex.

While the correct placement of the reciprocal exchange points may be important in

creating planar crystals, it appears that other factors are involved. When we generated

an alternative DDX complex with the same distances between the reciprocal exchange

points but different DNA strand lengths and sequences (see Figures 2.1c and 2.1d), the

complexes sometimes assembled into planar structures (Figure 2.1d) and sometimes into

tubular structures (Figure 2.3b). A preliminary report on this alternative complex can

be found in [15].

8

(a) 600 nm × 600 nm (b) 500 nm × 500 nm

Figure 2.2: Atomic force micrographs of double-double crossover (DDX) complexes as-sembling into high-density, planar structures. Some F faces are labeled.

Notice that in Figure 2.1b, there are flat (F) faces and stepped (S) faces [41]. A new

complex accretes to an S face via four pairs of sticky ends, but accretes to an F face via

only two pairs of sticky ends. In theory, this suggests that F faces should be more stable

than S faces of the same length and should be abundant at equilibrium. This may explain

(a) 250 nm × 25 nm (b) 800 nm × 800 nm

Figure 2.3: (a) Atomic force micrograph of a planar structure self-assembled from thealternative DDX complexes. (b) Atomic force micrograph of tubular structures self-assembled from the alternative DDX complexes.

9

the F faces in Figures 2.2a and 2.2b. However, it is also possible that these faces are the

result of tearing along minimum-energy faults during the destructive processes of AFM

sample preparation and imaging.

Several groups have considered the problem of using DNA to self-assemble regular

3-D structures: Winfree et al. [87] using two-helix bundles (DX), Park et al. [63] using

three-helix bundles, and Mathieu et al. [60] using six-helix bundles. As a future direction,

it may be possible to modify the DDX complex to form the four-helix bundle complex

diagrammed in Figure 2.4a. Such complexes might self-assemble to form high-density,

doubly-connected, three-dimensional crystals as shown in Figure 2.4b and are described

in greater detail in [15].

(a) (b)

Figure 2.4: (a) Diagram of the four-helix bundle complex theoretically obtainable byconnecting the leftmost and rightmost helices of the double-double crossover complex.(b) Diagram of a three-dimensional structure that, in theory, can self-assemble fromthese complexes. Colors added to distinguish individual units.

10

2.2 Triangular Complexes

The work presented in this section is the result of collaboration between Nickolas Chel-

yapov, Yuriy Brun, Manoj Gopalkrishnan, myself, Bilal Shaw, and Leonard Adleman

and was published in [16]. The first person plural pronouns in this section refer to those

authors.

There are exactly three regular tilings of the plane: one composed of triangles, one

of squares, and one of hexagons. We have constructed DNA complexes in the form of

triangles that self-assemble into planar structures in the form of regular hexagonal tilings.

In addition to the DNA complexes described in Section 2.1, LaBean et al. [52] extended

the double crossover motif to create triple crossover complexes that assemble into planar

structures in the form of rectangular tilings, Yan et al. [89] created 4× 4 complexes that

assemble into planar structures in the form of square tilings, and Liu et al. [57] created

triangular complexes that can assemble into orderly structures of several different forms.

Ding et al. [22] has created hexagonal structures using an approach different from the

one presented here.

We were inspired to explore triangular complexes by Yang et al. [90], who used them as

markers on structures formed from double crossover complexes. We created freestanding

triangular complexes composed of seven strands of DNA. We designed two such complexes

that stick to one another at their vertices. The type-a complex, Figure 2.5a, has a 90-

mer core strand, three 52-mer side strands with identical sequences and three 14-mer

horseshoe strands with identical sequences. The type-b complex, Figure 2.5b, has a 90-

mer core strand with sequence identical to that used in the type-a complex, three 52-mer

11

(a) (b) (c) (d) (e)

Figure 2.5: (a) Type-a triangular complex with core strand (black), side strands (red),horseshoe strands (purple), Watson-Crick pairing (gray). (b) Type-b triangular complexwith core strand (black), side strands (green), horseshoe strands (orange), Watson-Crickpairing (gray). (c) Hexagonal structure composed of six triangular complexes. (d) Hexag-onal tiling composed of hexagonal structures. (e) A pair of overlapping hexagonal tilings.Top layer shown gray; bottom layer shown black. (see also Figure 2.6b).

side strands with identical sequences and three 30-mer horseshoe strands with identical

sequences. The unpaired bases at the ends of the side strands in the type-a complex are

complementary to the unpaired bases at the ends of the horseshoe strands in the type-b

complex, allowing triangles to connect at these sticky ends. In theory, a pair of triangles

can form a hexagon as shown in Figure 2.5c and hexagons can form a tiling as shown in

Figure 2.5d.

After assembling the two types of triangular complexes in separate tubes, we combined

them at room temperature. The annealing and imaging protocols we used were the same

as described in Section 2.1. Atomic force microscope images of the resulting structures

are shown in Figure 2.6.

Figure 2.6a shows six triangular complexes assembled into a single hexagonal struc-

ture. The distance between opposing sides is approximately 35 nm, which is in good

agreement with expectations based on number of base pairs in the structure. Figure 2.6b

(see also Figure 2.5e) shows two hexagonal tilings, one lying on top of the other. One half

12

(a) 100 nm × 100 nm (b) 133 nm × 133 nm (c) 470 nm × 470 nm

Figure 2.6: (a) Atomic force micrograph of hexagonal structure composed of six triangularcomplexes. (b) Atomic force micrograph of a pair of overlapping hexagonal tilings (seealso Figure 2.5e). (c) Atomic force micrograph of structures composed of hexagonal andnon-hexagonal rings.

of the triangles of the top tiling lie in the centers of the hexagons of the bottom tiling.

The remaining triangles of the top tiling lie directly on top of the triangles of the bottom

tiling. Where triangles overlap, the structure has greater height as revealed by bright

spots in Figure 2.6b. In our experience, tilings layered in this way are common. It is

possible that such layering is a manifestation of solution phase tube formation followed by

tube flattening when structures adhere to mica prior to AFM imaging. Why the triangles

of the tilings align as they do is not understood.

We designed our complexes to form equilateral triangles and to stick to each other

but not to themselves. Figure 2.6a suggests that they do have this form and stick in

this way. However, sticking in this way is also consistent with the formation of ring

structures containing any even number of triangular complexes. Such structures are

perhaps energetically less favorable that a hexagonal structure; nonetheless, they do

form. Figure 2.6c shows a broad view of structures consisting of hexagons together with

13

ring structures with more or fewer than six vertices. It seems likely that the number of

such non-hexagonal ring structures could be reduced by using more than two types of

triangular complexes.

Occasionally, rings with an odd number of triangles also form. These may result from

our use of the same core strand in the two types of triangular complexes. A common core

strand allows for the creation of chimeric triangular complexes containing side strands

from complexes of different types. It seems likely that the number of such rings could be

reduced by using distinct strands in triangular complexes of different types.

In many published works on DNA self-assembly, the assembled structures are planar

and composed of double helices running in parallel [77, 86, 52, 56, 66, 31]. While our

structures are planar, they do not have this form. In the case of hexagons as shown

in Figure 2.5c, helices run parallel where sticky ends come together, but meet at angles

of 150 or 60 where no sticky ends are present. Hence, some helices in our structures

are apparently bent. We suspect that the use of long side strands with a 30-mer com-

plementarity to the core strand in our triangular complexes was critical in allowing the

required bending to occur. In fact, when triangular complexes (data not shown) em-

ploying shorter side strands with only a 21-mer complementarity to the core strand were

attempted, AFM imaging revealed no structures. Bending helices may be a useful design

option in the creation of future self-assembled DNA structures.

Using the 4 × 4 complex produces planar structures with helices intersecting at 90

angles [89]. The helices in the 4 × 4 complex have unpaired stretches of polyT. Images

show that helices make right angle turns, apparently at these sites [89]. Thus, unlike

14

our structures where helices apparently bend, helices in structures created with the 4× 4

complex apparently hinge.

Liu et al. [57] also described structures with non-parallel helices. However these

structures are not planar, and helices are allowed to cross each other without bending.

Like our structures, these structures are created from triangular complexes designed to

stick to one another at vertices. While the triangular complexes of Liu et al. stick to one

another via one helix, our complexes stick via two. This may provide greater integrity.

In addition, our structures may be useful when planarity is desired.

Using the design concepts described here, it seems possible, in principle, to create

complexes with arbitrary polygonal shapes [15]. Of immediate interest would be the

creation of squares, pentagons and non-equilateral triangles.

15

Chapter 3

Tile Assembly Model

Mathematical models can be used to understand the limits of what structures can and

cannot be self-assembled efficiently. In this chapter, we use a model known as the tile

assembly model to investigate decidability questions. A preliminary description of this

work appeared in [7]; a more detailed description appeared in [8]. The work presented in

this chapter is the result of collaboration between Leonard Adleman, Jarkko Kari, Lila

Kari, myself, and Petr Sosik, and the first person plural pronouns in this chapter refer to

those authors.

A systematic study of self-assembly as a computational process was initiated in [3].

The individual components are modeled as square tiles on the infinite two-dimensional

plane. Each side of a tile is covered by a specific “glue” and two adjacent tiles will stick iff

they have matching glues on their abutting edges. This model of self-assembly is known as

the tile assembly model, and formal definitions will be given in Section 3.1 As such, tiles

have been previously studied in the context of classic questions about tilings of the plane

[83]. The Tiling Problem, for example (proved undecidable in [13, 70]), asks whether a

given set of tiles (supply of each tile type unlimited) can be used to correctly tile the

16

entire plane. However, the new requirements of experimental self-assembly of tiles into

required nano-scale shapes poses new types of questions. What is the minimal number

of tiles required for the self-assembly of a desired shape [5, 34, 19, 80, 48]? What is the

running time of such a self-assembly [5, 4]? Do the results still hold if one generalizes

the model by making the “sticking” reversible, or by defining different bond strengths

and requiring more than one bond for sticking to occur [75, 4, 6, 5]? Do reversible self-

assemblies achieve equilibrium [6]? What is the optimal initial concentration of tile types

that guarantees the fastest assembly of a required shape [5]? Can a self-assembled shape

recover from the loss of arbitrary many tiles [17]? What are possible computational

primitives for algorithmic self-assembly [10]?

One of the interesting problems of self-assembly is whether or not a given set of

tiles allows uncontrollable growth, that is, whether arbitrarily large structures can be

produced. It turns out that this question is equivalent to asking whether or not, given

a set of tiles, there exists an infinite ribbon that can be formed with tiles from this

set. A ribbon is a non-self-crossing sequence of tiles on the plane, in which successive

tiles are adjacent along an edge, and abutting edges of consecutive tiles have matching

glues. If a given “seed” tile is specified, the problem of existence of an infinite ribbon

starting at that tile has been proved undecidable [26, 24, 27]. However, when no such

seed is specified, existing proof techniques failed to produce an answer and the problem

was declared open in [27]. The unseeded problem is even more relevant given that,

in physical simulations of self-assembly, the growth mechanism was deemed incompatible

with computations that use a chosen input [71]. Here we prove that the unseeded problem

17

is also undecidable. This result settles the decade-old problem known as the “unlimited

infinite snake problem”, [27].

The chapter is organized as follows. Section 3.1 introduces the notions of tile, glue, tile

system, sticking, ribbon, zipper, valid tiling of the plane, and directed tiles. In particular,

a zipper is a ribbon with the additional requirement that even non-consecutive tiles that

touch must have matching glues at their abutting edges.

Section 3.2 begins with the definition of the strong plane-filling property in a directed

tile system: in a system with this property, every infinite directed zipper is forced to

follow a specific plane-filling self-similar path and, moreover, an infinite directed zipper

is always guaranteed to exist. The construction of a directed tile system with the above

property uses a 3 × 3 block-construction based on directed tiles defined in [50, 49, 7],

which, in turn, resemble tiles devised by Robinson in [70] to only produce aperiodic

tilings of the plane. These tiles were augmented in [49, 50] with directions that force

every directed tiled path to form a self-similar plane-filling Hilbert curve that recursively

fills arbitrarily large squares by filling each of their quadrants. The original construction

used the condition that no glue mismatch be present in the 3 × 3 neighborhood of any

tile on the path, to force the path to follow the desired curve. Here, the additional 3× 3

block construction is needed to ensure that the weaker condition of no glue mismatch

between any two tiles on the directed tiled path is enough to force its course.

Section 3.3 then proves the undecidability of the existence of an infinite directed

zipper by reducing the Tiling Problem to it. The reduction is based on a construction

of sandwich tiles, that have directed tiles from the directed tile system with the strong

plane-filling property on top and undirected tiles from a given tile system on the bottom.

18

The next result of the section proves the undecidability of the existence of an infinite

ribbon by reducing the undecidable infinite directed-zipper problem to it. The proof

uses a “ribbon-motif” construction that simulates each directed zipper tile by a ribbon of

undirected tiles following its contours, and uses geometry (bump and dents) to simulate

zipper-tile glues.

Section 3.4 points to the implications of the undecidability of existence of infinite

ribbons for the problem of self-assembly of arbitrarily large supertiles.

3.1 Basic Definitions and Notation

A tile is an oriented unit square. The north, east, south and west edges of the tile are

each labeled with a symbol called a glue from a finite alphabet X. Tiles can be placed

on the plane but can not be rotated. The positions of the tiles on the plane are indexed

by Z2, the set of pairs of integers. More formally,

Definition 3.1.1. Let X be a finite set of symbols called glues. A tile is a quadruple

t = 〈tN , tE , tS , tW 〉 ∈ X4. A tile system is a finite subset of X4.

The components tN , tE , tS , tW will be called the glues on the north, east, south, and

west edges of the tile, respectively. Except in cases which may lead to confusion, we will

assume the set X of glues is understood from context and we will not refer to it explicitly.

Definition 3.1.2. For all tiles t = 〈tN , tE , tS , tW 〉 and t′ = 〈t′N , t′E , t′S , t′W 〉, t sticks on

the north to t′ iff tN = t′S . Sticking on the east, south, and west is defined similarly.

19

Definition 3.1.3. The directions D = N,E, S,W are functions from Z2 to Z2:

N(x, y) = (x, y + 1) E(x, y) = (x+ 1, y)

S(x, y) = (x, y − 1) W (x, y) = (x− 1, y).

The positions (x, y) and (x′, y′) are adjacent iff

(x′, y′) ∈ N(x, y), E(x, y), S(x, y),W (x, y).

In addition, (x, y) abuts (x′, y′) on the north iff (x′, y′) = N(x, y), and similarly for the

other directions.

Definition 3.1.4. Given a tile system T , a T -tiling of the plane is a total function f :

Z2 → T . The tiling f is valid at position (x, y) iff f(x, y) sticks on the north to f(N(x, y)),

f(x, y) sticks on the east to f(E(x, y)), f(x, y) sticks on the south to f(S(x, y)), and

f(x, y) sticks on the west to f(W (x, y)). The tiling is valid iff for all (x, y) ∈ Z2, f is

valid at (x, y).

That is to say, f is valid at (x, y) iff the tile at position (x, y) sticks on the appropriate

sides to all the tiles that are at positions adjacent to it, and f is valid iff it is valid at

every position in the plane. If there is a position (x, y) and an adjacent position (x′, y′)

such that f(x, y) does not stick to f(x′, y′) on the appropriate side, we will say there is a

glue mismatch between those two tiles.

The well-known Tiling Problem posed by Wang asks: Given a tile system T , does there

exists a valid T -tiling of the plane? The Tiling Problem was first proven undecidable by

20

Berger [13], and the result improved by Robinson [70]. One can generalize the notion of

T -tiling of the entire plane by defining a partial T -tiling as a partial function g : Z2 o−→ T .

(Unless noted otherwise, an arrow f : A → B will be used to indicate a total function

from A to B, while an arrow with a circle f : A o−→ B will be used to indicate a partial

function from A to B.) A partial tiling g is valid on a region R ⊆ dom(g) iff, for all

(x1, y1), (x2, y2) ∈ R, if (x1, y1) abuts (x2, y2) on the north (east, south, west), then

g(x1, y1) sticks to g(x2, y2) on the north (east, south, west). The partial tiling g is called

valid iff it is valid on dom(g).

Definition 3.1.5. A path is a function P : I → Z2 where I is a set of consecutive integers

and for all i, i+ 1 ∈ I, P (i) and P (i+ 1) are adjacent.

That is, a path is a sequence of adjacent positions on the plane. When P and I are

understood from the context, for all i ∈ I we will denote P (i) as (xi, yi) and we will say

that (x, y) is on P iff (x, y) ∈ range(P ).

Definition 3.1.6. For all tile systems T , a T -tiled path is a pair (P, r) where P is a path

and r is a function r : range(P )→ T .

Definition 3.1.7. A T -tiled path (P, r) is a T -ribbon iff

1. P is one-to-one and

2. For all i, i + 1 ∈ dom(P ), if (xi+1, yi+1) abuts (xi, yi) on the north (south, east,

west) then r(xi+1, yi+1) sticks to r(xi, yi) on the north (south, east, west).

Informally, a tiled path is a ribbon iff it does not cross itself and the glue between

tiles at consecutive positions along the path match.

21

Definition 3.1.8. A T -ribbon (P, r) is a T -zipper iff for all (x, y), (x′, y′) on P if (x′, y′)

abuts (x, y) on the north (south, east, west) then r(x′, y′) sticks to r(x, y) on the north

(south, east, west).

Note that the notion of a zipper is more restrictive than the notion of a ribbon in that

a zipper requires all of its tiles (consecutive or not) in adjacent positions to stick.

Definition 3.1.9. A directed tile system is a pair (T, d) where T is a tile system and

d : T → N,E, S,W is a function from the tile system to the direction functions.

A directed tile system can be thought of as a tile system in which each tile has an

arrow painted on it pointing north, east, south, or west.

Definition 3.1.10. A (T, d)-directed tiled path is a pair (P, r), where (P, r) is a T -tiled

path and for all i, i + 1 ∈ dom(P ), (xi+1, yi+1) = d(r(xi, yi))(xi, yi). If (P, r) is a (T, d)-

directed tiled path and (P, r) is a T -ribbon (zipper) then (P, r) is a (T, d)-directed ribbon

(zipper).

A directed tiled path is a tiled path in which each tile points to its successor on the

path. A path, (directed) tiled path, (directed) ribbon, (directed) zipper is finite, semi-

infinite, infinite, iff the domain of the associated path is finite, has a least or greatest

element and is infinite, or is Z respectively.

To prove the undecidability of existence of infinite ribbons, we will first prove the un-

decidability of existence of infinite directed zippers. Then, a “ribbon-motif” construction

will be employed to prove the result for ribbons: the construction simulates a directed

zipper by a ribbon-motif of smaller tiles that goes around the contours of the zipper tiles,

and uses geometry to simulate zipper-tile glues.

22

In order to prove the undecidability of existence of infinite directed zippers, we will use

the undecidability of the Tiling Problem in conjunction with a sandwich construction that

makes use of the existence of a directed tile system with the so-called strong plane-filling

property.

Definition 3.1.11. A directed tile system (T, d) has the strong plane-filling property iff:

1. There exists an infinite (T, d)-directed zipper and

2. For all infinite (T, d)-directed zippers (P, r), for all n ∈ Z≥0, there exists (x, y) ∈ Z2

such that (x+ i, y + j) is on P for all i = 0, 1, 2, . . . , n and j = 0, 1, 2, . . . , n.

If a directed zipper (or ribbon or path) satisfies the property in Definition 3.1.11.2,

we also say that it covers arbitrarily large squares.

Definition 3.1.11 states that in a directed tile system with the strong plane-filling

property every infinite directed zipper is forced to be plane-filling (by filling arbitrarily

large squares) and, moreover, that such an infinite directed zipper always exists.

3.2 A Directed Tile System with the Strong Plane-Filling

Property

An essential element of our subsequent proofs will be the main result of this section,

Theorem 3.2.7, namely the construction of a directed tile system that satisfies the strong

plane-filling property of Definition 3.1.11. In fact, the directed tile system we will con-

struct in this section satisfies an even stronger requirement than part 2 of Definition 3.1.11:

every infinite directed tiled path is either plane-filling or it has a glue mismatch between

23

two of its tiles. In particular, using our tiles, an infinite directed tiled path may not

form a loop without encountering a glue mismatch along the way. Consequently, every

non-plane-filling infinite directed tiled path must necessarily contain two neighboring tiles

with a glue mismatch. The constructed tiles satisfy therefore a stronger version of Defi-

nition 3.1.11, satisfying thus a stronger property than the original plane-filling property

defined in [49, 50].

In [50], every infinite directed tiled path was forced to be plane-filling unless there was

a mismatch of glues inside the 3× 3 neighborhood of some tile of the path. As described

in Subsections 3.2.2 and 3.2.3, directed tiles with this weaker plane-filling property were

explicitly constructed by augmenting Robinson’s aperiodic tiles [70] with directions. With

these tiles, all directed tiled paths consisting of tiles without errors in their 3× 3 neigh-

borhood formed self-similar plane-filling curves of the kind used by Hilbert [42] to show

that the unit square is a continuous image of the unit segment. The constructed tiles

filled arbitrarily large squares recursively, by first filling the individual quadrants of a

square. The process was guided by the direction associated to each tile, which forced the

construction to proceed along the self-similar Hilbert curve that traversed each quadrant

of a square and linked the quadrants to each other.

In the present application we need the stronger variant of the plane-filling property

introduced in this paper in Definition 3.1.11. In our case the infinite directed tiled path

must be forced to be plane-filling even under the weak assumption that there is no glue

mismatch between two tiles belonging to it. It turns out that such a directed tile system

is obtained by taking 3×3 blocks of the tiles in [50], as described in Subsection 3.2.4. For

this particular set of tiles, our 3×3 block-construction ensures that the weak requirement

24

of absence of glue mismatches between any two tiles belonging to the directed tiled path

is sufficient to determine its uniqueness. Namely, we can prove that the constructed

directed tile system has the property that, starting with any arbitrary tile, the only way

to build an infinite error-free directed zipper is by forming a Hilbert curve that covers

arbitrarily large squares.

3.2.1 Generalized tile systems

We begin by generalizing the notion of neighborhood from Section 3.1, where the “neigh-

bors” of a position were the positions at its north, east, south and west.

Definition 3.2.1. A neighborhood vector is a k-tuple

V = 〈x1, x2, . . . , xk〉, xi ∈ Z2, 1 ≤ i ≤ k.

The neighbors of a position x ∈ Z2 are the positions x+ xi for 1 ≤ i ≤ k.

The classical von Neumann neighborhood is defined by the vector

VvN = 〈(0, 0), (1, 0), (0, 1), (−1, 0), (0,−1)〉.

In the von Neumann neighborhood, each position has five neighbors: itself and its four

adjacent positions, as defined in Section 3.1. In fact, all definitions in Section 3.1 are

based on the von Neumann neighborhood.

25

Another particular neighborhood of interest, the Moore neighborhood, is defined as

follows:

VM = 〈(−1, 1), (0, 1), (1, 1), (−1, 0), (0, 0), (1, 0), (−1,−1), (0,−1), (1,−1)〉.

The Moore neighborhood of a position is a 3×3 block centered at that position, where the

eight directions of the compass are used when we refer to the eight surrounding positions.

In contrast, in the generalized sense of Definition 3.2.1, the neighbors of a position need

not be adjacent to it. Under this generalized definition, the neighborhood of a position

may even have holes in it, i.e., adjacent positions may not belong to its neighborhood,

while positions further away may. For example,

V =⟨(−2, 2), (0, 2), (2, 2), (−1, 1), (1, 1), (−2, 0), (0, 0),

(2, 0), (−1,−1), (1,−1), (−2,−2), (0,−2), (2,−2)⟩

defines a checkerboard type of neighborhood, where the neighbors of the center “white”

position are all the “white” positions in the 5× 5 square centered at it.

We shall now use the generalized notion of neighborhood to define a generalized tile

system and its corresponding notion of valid tiling.

Definition 3.2.2. A generalized tile system is a triple 〈T, V,R〉 where T is a finite set,

V = 〈x1, x2, . . . , xk〉, xi ∈ Z2, 1 ≤ i ≤ k, is a neighborhood vector, and R ⊆ T k is a k-ary

relation on T .

26

As usual, a (partial) 〈T, V,R〉-tiling (shortly T -tiling if the other elements are obvious

from the context) is a (partial) function ψ : Z2 → T.

Definition 3.2.3. A 〈T, V,R〉-tiling ψ is valid at x ∈ dom(ψ) iff there exists r ∈ R such

that, for all i, 1 ≤ i ≤ k, either (x + xi) ∈ dom(ψ) and ψ(x + xi) = r(i) or ψ(x + xi) is

undefined.

A 〈T, V,R〉-tiling ψ is valid on a region S ⊆ dom(ψ) iff ψ|S is valid at every point

x ∈ S. We say that ψ is valid iff it is valid on dom(ψ). If ψ is valid on Z2 then ψ is called

a valid 〈T, V,R〉-tiling of the plane.

As before, we can define a generalized directed tile system 〈T, V,R, d〉 where 〈T, V,R〉

is a generalized tile system and d : T → N,E, S,W is a function from the generalized

directed tile system to the direction functions.

3.2.2 Microtiles

The starting point of our construction of a directed tile system (T0, d0) with the strong

plane-filling property, will be a generalized directed tile system 〈Tµ, Vµ, Rµ, dµ〉 con-

structed in [50]1. The elements of Tµ are called microtiles. They use labeled arrows

to define their glues so that arrow heads match arrow tails on the edges of tiles in order

to stick. A Tµ-tiling will sometimes be called microtiling.

There are five different types of microtiles in Tµ (Figure 3.1): (a) single and (b) double

crosses, (c) single, (d) double and (e) mixed arms. The arms can be horizontal or vertical

arms and can point North or South (vertical arms) or East or West (horizontal arms).

Only one direction is shown in the figure for clarity, the other tiles are 90 degree rotations1Figures 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.9, and 3.12 are from [50].

27

of the ones shown. There are also labeled diagonal arrows assigned to microtiles in

[50]. They play a role in proofs of Lemmas 3.2.1 and 3.2.2. Since these proofs are not

reproduced here, we omit the description of the diagonal arrows.

(a) single cross (b) double cross (c) single arm (d) double arm (e) mixed arm

Figure 3.1: The five types of microtiles. Each type can be rotated to give up to fourdifferent tiles. Diagonal arrows are not shown.

A cross, either single or double, has its arrow heads labeled from SE ,SW ,NE ,NW ,

but all four arrow heads must have the same label. The principal arrow of an arm (single,

double or mixed) has a label from SE ,SW ,NE ,NW , with the same label on both

ends. The side arrows of mixed arms have labels as described in Figure 3.2. A single

or double horizontal arm has its upper (lower) arrow labeled SX (NX, respectively), for

X ∈ E,W. A single or double vertical arm has its left (right) arrow labeled YE (YW,

respectively), for Y ∈ N,S. To conclude the description of microtiles, we note that

more labels will be described in the next section.

NE

SE

NW

SW

NENW SESW

Figure 3.2: The possible labels of the two side arrows on a mixed arm.

28

Definition 3.2.4. (from [50]) A microtile system is a generalized directed tile system

〈Tµ, Vµ, Rµ, dµ〉, where

• Tµ is the set of tiles with labeled arrows described above;

• Vµ is the Moore neighborhood;

• Rµ is the set of all v ∈ T 9µ such that in the 3× 3 tiling (Vµ(i), v(i)) | 1 ≤ i ≤ 9 all

arrow heads meet on the edges of tiles matching arrow tails with the same labels

and vice versa.

The labelling of microtiles described above allows that, for each n ∈ N, four special

squares called (2n − 1)-XY -squares can be defined recursively.

Definition 3.2.5. A (2n − 1)-XY -square, n ≥ 1, XY ∈ NE ,NW ,SE ,SW , is a mi-

crotiling fXY : Sn → Tµ where Sn = (x+i, y+j) | 1 ≤ i, j ≤ 2n−1 for some (x, y) ∈ Z2.

It is iteratively constructed in the following way.

A single cross labeled XY is a 1-XY -square. A (2n+1 − 1)-XY -square consists of

four (2n − 1)-X ′Y ′-squares, X ′Y ′ ∈ NW ,NE ,SW ,SE, ordered as in Fig. 3.3. These

squares are separated by a double cross labeled XY at the position (x+2n, y+2n), called

the central cross of the square, and rows of arms radiating from it. The arms are double

near the central cross, but halfway the mixed arms make them single.

The iterative construction is illustrated in Figure 3.3. By the construction of a mi-

crotile system, all (2n − 1)-XY -squares are valid microtilings. Figure 3.4 provides an

example of a 7-XY -square. We may sometimes refer to a (2n−1)-XY -square as (2n−1)-

square if XY is arbitrary. Proofs of the following results can be found in [50].

29

XY

(2 – 1)-NE-squaren(2 – 1)-NW-squaren

(2 – 1)-SW-squaren (2 – 1)-SE-squaren

Figure 3.3: Constructing the (2n+1 − 1)-XY -square.

30

XY

NW

SE

NENW

SW

NE

SE

NENW

SW

SE

SE

NENW

SW

SW

SE

NENW

SW

Figure 3.4: The 7-XY -square with the labels on the crosses.

31

Lemma 3.2.1. ([50]) Let f : S(1) → Tµ, g : S(2) → Tµ be two (2n − 1)-squares with the

same n. If f |S(1)∩S(2) = g|S(1)∩S(2) and there exists (x, y) ∈ S(1) ∩ S(2) such that f(x, y)

is a single cross, then S(1) = S(2).

In other words, if S(1) and S(2) have a nonempty common overlap containing a single

cross, then they are identical.

Lemma 3.2.2. ([50]) Let f : Z2 → Tµ be a microtiling of the plane and assume there

exists a (2n − 1)-XY -square fXY : Sn → Tµ such that f |Sn = fXY and moreover f is

valid on Sn. Let XY = SW (NW, NE, SE). Then

(i) the microtile just outside the NE (SE, SW, NW, respectively) corner of the square

is a double-cross;

(ii) the row of arms radiating Y -wise (X-wise) from this double-cross has the length

2n−1, consists of 2n−1−1 horizontal (vertical) double arms, followed by a horizontal

(vertical) mixed arm and then by 2n−1 − 1 horizontal (vertical) single arms.

(iii) the microtile just outside the SE (NE, NW , SW , respectively) corner is a hor-

izontal arm and the microtile just outside the NW (SW , SE, NE, respectively)

corner is a vertical arm.

In Figure 3.4 for example, the double cross right outside the NE corner of the 3-

SW-square is the double cross microtile labeled XY , with the row of microtiles radiating

west-wise and south-wise as required.

32

3.2.3 Minitiles

Another type of tiles needed in our construction are minitiles [50]. Each minitile will be

a 2× 2 block of microtiles that has exactly one single cross in its upper-right corner, see

Figure 3.10 (b).

Definition 3.2.6. Consider the microtile system 〈Tµ, Vµ, Rµ, dµ〉. A minitile is a quadru-

ple a = 〈µa1, µa2, µa3, µa4〉 ∈ T 4µ such that µa1 is a single cross, µa2, µ

a3, µ

a4 are not, and the

microtiling ((1, 1), µa1), ((1, 0), µa2), ((0, 0), µa3), ((0, 1), µa4) is valid.

Therefore, by the above definition, all the arrow heads and tails between µa1, µa2, µ

a3, µ

a4

match. Let Tm be a set of minitiles, then a Tm-tiling will be called minitiling. If f :

Z2 o−→ Tm is a minitiling, then its corresponding microtiling m(f) : Z2 o−→ Tµ is defined

as follows. If f(x, y) = a = (µa1, µa2, µ

a3, µ

a4) then:

m(f)(2x+ 1, 2y + 1) = µa1,

m(f)(2x+ 1, 2y) = µa2,

m(f)(2x, 2y) = µa3,

m(f)(2x, 2y + 1) = µa4.

Next, directions are added to minitiles. As we restricted ourselves to 2 × 2 blocks

having exactly one single-cross located in the upper right corner, attaching directions to

minitiles will amount to attaching directions to single-cross microtiles. Let us describe

first the types of directed minitiled paths (or simply paths, if no danger of confusion exists)

33

these directions will define. The paths are constructed recursively through squares of size

2n × 2n minitiles.

Definition 3.2.7. For each n ≥ 0 we define recursively A2n (B2n , C2n , D2n) -minitiled

path as follows. For n = 0, a minitile with its single cross labeled X is an X1-path, for

X ∈ A,B,C,D.

For each n ≥ 0, a A2n+1 (B2n+1 , C2n+1 , D2n+1)-path is built recursively from four

X2n-paths, X ∈ A,B,C,D, as described in Figure 3.5.

A2n A2n

D2n C2n

6

?

D2n C2n

B2n B2n

6-?

C2n B2n

C2n A2n

6

-

B2n D2n

A2n D2n

?

-

A2n+1 B2n+1 C2n+1 D2n+1

Figure 3.5: Constructing paths through squares of 2n+1 × 2n+1 minitiles.

Figure 3.6: The paths A4 and A16.

34

For example, the A4-path starts in the lower right corner of the square, visits all its

minitiles, and ends in the lower left corner (see Figure 3.6). Paths created using Definition

3.2.7 form the familiar Hilbert curve. One can easily verify the following result:

Lemma 3.2.3. For each n ≥ 0 the A2n (B2n , C2n , D2n)-path starts in the SE (NW, SE,

NW, respectively) corner, ends in the SW (NE, NE, SW, respectively) corner, visits all

minitiles of the underlying 2n×2n minitile square and does not visit any minitiles outside

of the underlying square.

Let fXY : Sn+1 → Tµ be a (2n+1 − 1)-XY -square of microtiles where Sn+1 = (x +

i, y+j) | 1 ≤ i, j ≤ 2n+1−1. Then, by the iterative construction in Definition 3.2.5, those

(and only those) microtiles at positions (x+ 2i− 1, y + 2j − 1), 1 ≤ i, j ≤ 2n, are single

crosses. Since moreover fXY is a valid microtiling, all 2 × 2 squares of Sn+1 with single

crosses in their upper right corner form minitiles. The single crosses in the leftmost

column and the bottommost row of Sn+1 can be easily completed by adding another

column and a row of microtiles to fXY so that they again form valid 2 × 2 squares of

microtiles. Therefore, by Definition 3.2.6, we obtain a (partial) minitiling f : Z2 o−→ Tm

from which the square fXY arose, i.e., m(f)|Sn+1 = fXY .

If a minitile has its single cross in Sn+1, we say that it is attached to Sn+1. Recall

that there are exactly 2n× 2n single crosses in Sn+1, therefore there will be 2n× 2n = 4n

minitiles attached to it. The directions will be defined in such a way that the path these

minitiles form is exactly an A2n , B2n , C2n or D2n-path.

In order to control which of the four possible paths the directions define in a specific

square, additional labels are assigned to arrows in microtiles [50]. Each arrow (single or

35

Direction of the p.a. E W N S E W N S E W N S E W N SLabel of the p.a. A A A A B B B B C C C C D D D DLabel of the c.a. C A A D B D C B A C B C D B D ALabel of the c.c.a. A D A C C B D B B C C A D A B D

Figure 3.7: The labeling of mixed arms. The abbreviations p.a. (c.a., c.c.a.) denote theprincipal arrow (the arrow situated clockwise, counterclockwise, respectively, relative tothe principal arrow) of the arm.

A A

A

AA

C

A

D

A A D C-6 6

? ?6

- -

?

Figure 3.8: The labeling of mixed arms whose principal arrow has the label A.

double) can be given any label from the set A,B,C,D, the only restrictions we impose

on crosses and mixed arms. As usual, in each cross all four arrow heads must have the

same label. The central cross of each (2n+1 − 1)-square will determine the type of path

the directions define on the square (see Lemma 3.2.6). The labelling of mixed arms is

crucial for the composition of A-, B-, C-, or D- squares and is specified in Figure 3.7. As

usual, in a valid microtiling the meeting arrow heads and arrow tails must have the same

label.

One can verify that the labels in Figure 3.7 correspond exactly to the construction

of the paths in Fig. 3.5. For example, the mixed arms in the first four columns in

Figure 3.7 are illustrated in Fig. 3.8. The first of these mixed arms has its principal

arrow labeled A. This indicates that the mixed arm is a connector in an (2n+1 − 1)-

square whose central cross is labeled A, and which therefore contains an A2n+1-path. The

vertical arrow incoming from the north (south) is labeled A (respectively C). These facts

indicate that the (2n− 1)-NE-square on top of the mixed arm contains an A2n-path, and

36

that the (2n − 1)-SE-square below it contains a C2n-path, as they should, according to

the recursive construction of A2n-paths in Fig. 3.5.

The direction of a single cross is:N if its NE neighbor is a double cross with label C or a vertical arm whose left

edge has a side arrow with label A or B.E if its NE neighbor is a double cross with label B or a horizontal arm whose

lower edge has a side arrow with label C or D.S if its SW neighbor is a double cross with label D or a vertical arm whose

right edge has a side arrow with label A or B.W if its SW neighbor is a double cross with label A or a horizontal arm whose

upper edge has a side arrow with label C or D.

Figure 3.9: The definition of directions assigned to minitiles (or to their respective singlecrosses).

The definitions of directions are summarized in Figure 3.9. It has been shown in

[50] that if fXY : Sn → Tµ is a (2n − 1)-XY -square, then for each single cross whose

Moore neighborhood is contained in Sn there exists exactly one rule in Figure 3.9 that

can be applied. All the above defined elements forming the directed paths, i.e., the

new labels A,B,C,D and the directions of minitiles, are unified together by means of

the neighborhood relation Rm. Now we can complete the formal definition of a minitile

system, based on the above described microtile system 〈Tµ, Vµ, Rµ, dµ〉.

Definition 3.2.8. A minitile system is a generalized directed tile system

〈Tm, Vm, Rm, dm〉 such that

(i) Tm is the set of all minitiles as defined in Definition 3.2.6;

(ii) Vm is the Moore neighborhood;

(iii) Rm is the set of all 9-tuples r ∈ T 9m such that all the following conditions are met:

37

(a) the underlying 6× 6 microtiling m((Vm(i), r(i)) | 1 ≤ i ≤ 9) is valid;

(b) exactly one rule in Figure 3.9 is applicable to the single cross µr(5)1 of the

central minitile r(5);

(c) exactly one of the following four conditions holds: dm(r(2)) = S, dm(r(4)) = E,

dm(r(6)) = W, dm(r(8)) = N ;

(iv) dm(a) = dµ(µa1) for each a = 〈µa1, µa2, µa3, µa4〉 ∈ Tm.

Recall that a 9-tuple r ∈ Rm corresponds to a 3× 3 block of minitiles. The condition

(a) states that a necessary condition for the minitiling f to be valid at (x, y) ∈ Z2 is

that its corresponding microtiling m(f) be valid at (2x+ i, 2y + j) for all −1 ≤ i, j ≤ 2.

In other words, inside the 6 × 6 block of microtiles corresponding to the 3 × 3 Moore

neighborhood of (x, y), all arrow heads match meeting arrow tails and their labels match

as well, including the new labels A,B,C,D.

The condition (b) adds another necessary requirement that the direction of the minitile

at (x, y) must be uniquely defined via Figure 3.9. Finally, by the condition (c) a minitiling

is not valid at (x, y) unless exactly one of the four adjacent minitiles at its N, S, E, W

has its direction pointing towards it.

Recall that, due to Definition 3.2.3, if some minitiles in the Moore neighborhood of

(x, y) are missing, we consider the minitiling valid at (x, y) if the missing positions can

be filled with minitiles such that the conditions (iii) (a)–(c) would be met.

38

3.2.4 Macrotiles: The 3x3 block construction

In this section we finalize the construction of a directed tile system (T0, d0) with the

strong plane-filling property. We will use the generalized directed tile system of mini-

tiles 〈Tm, Vm, Rm, dm〉 constructed so far to obtain the directed tile system of macrotiles

(T0, d0). When dealing with (T0, d0) we revert to the usual notion of tiles sticking by

glues as defined in Section 3.1.

The idea of the construction is as follows. Consider all the validly minitiled Moore

neighborhoods as defined by Rm. To each such an 9-tuple, which intuitively is a 3 × 3

block of minitiles, we associate a macrotile (see Figure 3.10). The position of the macrotile

in a plane is identical with the position of its center minitile. The glues of a macrotile will

be the 2× 3 and 3× 2 sub-blocks of minitiles corresponding to each side. Two adjacent

macrotiles will stick if they have the same glues at their abutting edges (see Figure 3.11).

(a) (b) (c)

Figure 3.10: (a) A microtile; (b) A minitile is a 2×2 block of microtiles having exactly onesingle cross, in its upper-right corner; (c) A macrotile (shaded); the non-shaded minitilesdefine the “glues” of the macrotile.

39

m(5) m(6)m(4)

m(8) m(9)m(7)

m(2) m(3)m(1)

n(5) n(6)n(4)

n(8) n(9)n(7)

n(2) n(3)n(1)

m* n*

Figure 3.11: Two adjacent macrotiles m∗ and n∗ (shaded) stick iff their glues at theabutting edges are equal. This means that the overlapping 2× 3 portions of their Mooreneighborhoods coincide, i.e., m(2) = n(1), m(3) = n(2), m(5) = n(4), etc.

Definition 3.2.9. Consider the minitile system 〈Tm, Vm, Rm, dm〉. A macrotile system is

a directed tile system (T0, d0), T0 ⊆ X40 , where

(i) X0 = T 6m;

(ii) T0 is constructed as follows. For each validly minitiled Moore neighborhood t ∈ Rm,

there exists a macrotile t∗ ∈ T0 with glues:

t∗N = 〈t(1), t(2), t(3), t(4), t(5), t(6)〉,

t∗S = 〈t(4), t(5), t(6), t(7), t(8), t(9)〉,

t∗E = 〈t(2), t(3), t(5), t(6), t(8), t(9)〉,

t∗W = 〈t(1), t(2), t(4), t(5), t(7), t(8)〉;

(iii) d0(t∗) = dm(t(5)).

Intuitively, a macrotile corresponds to a 3× 3 block of minitiles, its north glue is the

2× 3 top sub-block, the south glue is the bottom 2× 3 sub-block, the west glue the left

3× 2 sub-block, and the east glue the right 3× 2 sub-block of minitiles.

40

The direction of a macrotile t∗ is the direction of its center minitile t(5) which we

will denote also by c(t∗). Similarly as with minitiles, we say that t∗ is attached to a

(2n − 1)-square Sn −→ Tµ if the single cross of c(t∗) is located within Sn.

Two adjacent macrotiles will stick iff the glues on their abutting edges are identical.

For example, two macrotiles at positions (x, y), (x + 1, y) will stick if, when viewed as

3× 3 blocks centered at (x, y) respectively (x+ 1, y), their overlapping minitiles coincide.

(See Figure 3.11).

Definition 3.2.10. Consider the macrotile system (T0, d0) and a T0-tiling f : Z2 o−→ T0.

(i) The overlap map o(f) : Z2 o−→ 2Tm is a function defined as follows. For all x ∈

dom(f), if f(x) = t∗, then t(i) ∈ o(f)(x+ Vm(i)).

(ii) The projection map p(f) : Z2 o−→ Tm is a function defined as follows: If (x, y) ∈

dom(f) and f(x, y) = t∗, then p(f)(x, y) = c(t∗).

In other words, the overlap map of a set of macrotiles situated on the plane will place

at each position all the minitiles that would occupy it if we were to view each macrotile

at (x, y) as a 3× 3 block centered at (x, y) and covering a 3× 3 neighborhood around it.

The projection map replaces each macrotile at (x, y) with the minitile c(t∗) at its center.

Note that, while dom(p(f)) = dom(f),dom(f) ⊆ dom(o(f)).

Consider the following special case of the overlap map: for all (x, y) ∈ Z2, either o(f)

is undefined or card(o(f)) = 1. Such a one-to-one overlap map is called a perfect overlap.

The following results are straightforward:

41

Lemma 3.2.4. Let f : Z2 o−→ T0 be a T0-tiling with dom(f) = (x, y), (x′, y′) and the

positions (x, y) and (x′, y′) are adjacent, then f(x, y) will stick to f(x′, y′) iff o(f) is a

perfect overlap.

In other words, two macrotiles situated at adjacent positions will stick (in the glue-

match-at-abutting-edges sense defined in Section 3.1) iff the overlap map they define is

one-to-one.

Let us introduce the following notation for the set of positions on the plane belonging

to the Moore neighborhood of x ∈ Z2 : NM (x) = x+ VM (i) | 1 ≤ i ≤ 9.

Lemma 3.2.5. Let f : Z2 o−→ T0 be a T0-tiling and let x, y ∈ dom(f). Let further f

contain a T0-ribbon (P, r) such that x, y ⊆ range(P ) and

NM (x) ∩NM (y) ⊆⋂

z∈range(P )

NM (z).

Then the overlap map o(f |x,y) is perfect.

Informally, let two macrotiles x∗ and y∗ be connected with a (short) ribbon such that

the intersection of the Moore neighborhoods of x∗ and y∗ is included in (and hence is

equal to) the intersection of the Moore neighborhoods of all tiles in the ribbon. Then, by

Lemma 3.2.4, each two adjacent tiles in the ribbon have perfect overlap. By transitivity,

we deduce that the intersection of overlap maps of all the tiles in the ribbon is one-to-one

and hence so is the common overlap map of x∗ and y∗.

The following lemma states that each infinite directed zipper of macrotiles covers

arbitrarily large squares.

42

Lemma 3.2.6. Let n ∈ N and let (P, r) be a T0-directed zipper, P : I → Z2, r :

range(P )→ T0, such that there exists i ∈ I with min(I) + 4n ≤ i ≤ max(I)− 4n. Denote

P (i) = (xi, yi) and r(xi, yi) = t∗. Then,

(i) There exists a (2n+1−1)-square fXY : Sn+1 → Tµ such that (2xi+1, 2yi+1) ∈ Sn+1,

Sn+1 ⊆ dom(m(p(r))) and fXY = m(p(r))|Sn+1 ;

(ii) Let Q = (k, l) ∈ Z2 | (2k+ 1, 2l+ 1) ∈ Sn+1. If the central cross of Sn+1 has label

A (B, C, D), then p(r|Q) is an A2n (B2n , C2n , D2n , respectively)-path;

(iii) The overlap map o(r|Q) is a perfect overlap.

Informally, consider a directed zipper of macrotiles that passes through a macrotile

t∗ and extends for at least 4n positions preceding and the 4n positions succeeding the

position of t∗. Then (i), the single cross of the minitile c(t∗) belongs to a (2n+1 − 1)-

square fXY : Sn+1 → Tµ.Moreover, (ii) the projection of the portion of the directed zipper

consisting of the macrotiles attached to Sn+1 is an A2n , B2n , C2n or D2n-directed minitiled

path if the central cross microtile of the square has label A, B, C or D, respectively.

Lastly, (iii) the overlap map of that portion of the directed zipper is a perfect overlap.

Proof. Induction on n.

n = 0. Assume there is no zipper-error involving 40 = 1 macrotile before and one

macrotile after t∗. Then the required 2n+1 − 1 = 21 − 1 = 1-square consists of only the

single cross at the upper right corner of the minitile c(t∗) and the statement is vacuously

true.

43

Assuming the statement holds for n − 1, we prove it for n. Assume there are no zipper-

errors involving any of the 4n macrotiles that preceed or succeed t∗. By I.H. (i), the

minitile c(t∗) is attached to a (2n − 1)-square S(1) −→ Tµ with the required properties.

Assume that it is a (2n − 1)-SW-square (the other cases are analogous.)

Properties of the square S(1)

Let Q1 = (k, l) ∈ Z2 | (2k+1, 2l+1) ∈ S(1) be the square of macrotiles attached to S(1).

By I.H. (iii), the overlap map of r|Q1 is perfect (one-to-one). The underlying microtiling

m(o(r|Q1)) covers S(1) and the Moore neighborhood of all its tiles (actually, it is even

larger) and, by Definitions 3.2.8 and 3.2.9, is valid. Then, by Lemma 3.2.2, the microtile

just outside the upper right corner of S(1) is a double-cross, denoted by X in Fig. 3.12.

The double-cross can have any of the four labels A, B, C, D. Assume it has label C.

By Lemma 3.2.2 again, the microtiles exactly on top of S(1) are horizontal arms (first

double arms, then one mixed arm, then single arms) radiating west-wise, with the same

label C. Then, by Figure 3.7, the lower edge of the mixed arm has a side arrow labeled

C (column 10, row 4 of the figure). Then the central cross of S(1) has the same label C,

due to the column of arms radiating from it north-wise to the mentioned mixed arm (see

Definition 3.2.5). This means the path through S(1) is, by I.H. (ii), a C-path which, by

Lemma 3.2.3, starts at a single cross f and ends at a single cross a (Figure 3.12).

Adding the square S(2)

As a has at its NE a double cross labeled C, according to Figure 3.9, the direction of a

is N. This means that the path of macrotiles, passing through the macrotile a∗ with the

44

Y

f g h

a

X

b

c d e

Z

S(1) S(4)

S(2) S(3)

Figure 3.12: The (2n+1 − 1)-square constructed during the proof of Lemma 3.2.6.

center single cross a, will proceed to b∗. Denote by b the single cross of c(b∗) (Figure

3.12).

Recall that for the macrotile t∗ attached to S(1) we assumed that it is preceeded and

succeeded by at least 4n macrotiles on the path P. Since there are exactly 4n−1 macrotiles

attached to S(1), the distance of t∗ and b∗ on the path P is at most 4n−1, by Lemma 3.2.3.

Hence the induction hypothesis can be applied to b∗ because the number of macrotiles

preceeding and succeeding b∗ on P is at least 4n − 4n−1 = 3 · 4n−1 > 4n−1. By I.H. (i), b

belongs to a (2n − 1)-square S(2) → Tµ.

Observe that a is not in S(2) (otherwise, by Lemma 3.2.1, S(1) and S(2) would coincide,

which is impossible as b is in S(2) − S(1)). Hence S(1) and S(2) are disjoint. By I.H. (ii),

S(2) contains an A-, B-, C-, or D-path. As c(b∗) is the first minitile of the path, and b

45

is located just above a, by Lemma 3.2.3 the path must be either A-path or C-path. In

both cases b is located in the SE corner of S(2) and we conclude that S(2) is above S(1).

As S(2) has all the properties conferred by I.H., we repeat for S(2) the steps previously

made for S(1). In particular, let Q2 = (k, l) ∈ Z2 | (2k + 1, 2l + 1) ∈ S(2). By I.H. (iii),

the overlap map o(r|Q2) of the macrotiles attached to S(2) is one-to-one. Hence, the

microtiling m(o(r|Q2)) covering S(2) and its Moore neighborhood is valid.

By definition of mixed arms in Figure 3.2, the mixed arm under S(2) has an upper

edge with a side arrow labeled NW. As there is a column of arms connecting this arrow

with the central microtile of S(2), we have that S(2) is a (2n− 1)-NW-square. By Lemma

3.2.2 we have to the right of S(2) a column of vertical arms with a mixed arm in the

middle. By Figure 3.7, the left edge of this mixed arm has a side arrow labeled C, and

hence the central cross of S(2) is also labeled C. By I.H. (ii), the path through S(2) is a

C-path. By Lemma 3.2.3, the path enters through b∗ and exits through c∗ (Figure 3.12).

We know already that, when taken separately, the overlap maps of the partial tilings

with macrotiles attached to S(1), respectively S(2), are one-to-one. When considering

the overlap map associated with S(1) and S(2), we have to inspect the cases when there

are two macrotiles, x∗ attached to S(1) and y∗ attached to S(2), such that their Moore

neighborhoods overlap. The possible cases are those described in Figure 3.13 and the

symmetrical ones. Recall that, by Lemma 3.2.3, the zipper (P, r) visits all the macrotiles

attached to S(1) ∪ S(2). Hence any adjacent pair of them stick and any tiled path forms

a ribbon. Then, in each of the cases in Fig. 3.13 the figure also shows the short ribbon

required by Lemma 3.2.5. (These ribbons have nothing to do with the zipper (P, r).)

Hence, by Lemma 3.2.5, the common overlap map of x∗ and y∗ is perfect. This further

46

implies that the overlap map o(r|Q1∪Q2) of all the macrotiles attached to S(1) and S(2) is

one-to-one.

y∗y∗

y∗y∗

y∗y∗

x∗ x∗x∗ x∗x∗ x∗

Figure 3.13: Pairs of macrotiles attached to S(1) and S(2) with nonempty common overlap.

Adding the square S(3)

By Lemma 3.2.2 (ii) applied to S(2), there is a row of vertical arms labeled C to the right

of S(2) which ends just to the right of the single cross c. By Lemma 3.2.2 (iii) Z must be a

horizontal arm whose lower edge has its side arrow labeled C. As Z is the NE neighbor of

c, by Figure 3.9, the direction of c is E. We can reason similarly as in the case of tile b in

S(2) to conclude that d is the first single cross in a B-path through a (2n− 1)-NE-square

S(3) → Tµ situated at the right side of S(2). The path ends at tile e (Figure 3.12).

Reasoning as in the case of S(1) and S(2) we can conclude that the overlap map of

macrotiles associated to S(2) and S(3) is one-to-one. To conclude that the overlap map

associated to S(1) and S(2) and S(3) is one-to-one, we have to consider pairs of macrotiles,

x∗ attached to S(1) and y∗ attached to S(3), such that their Moore neighborhoods overlap.

One can easily verify that there are at most nine such pairs. Similarly as in the case of

common overlap of S(1) and S(2), for each of these pairs exists a short ribbon satisfying

requirements of Lemma 3.2.5. By reasoning similar to the previous case, the overlap map

of x∗ and y∗ is one-to-one and so is the common overlap map of macrotiles attached to

S(1) ∪ S(2) ∪ S(3).

47

Adding the square S(4)

Consider the overlap map of macrotiles attached to S(1), S(2) and S(3) which forms a

valid minitiling. Let f∗ be the macrotile with the center single cross f , where the C-path

through S(1) starts. The overlap map extends one minitile to the east and south of f∗,

hence covering both microtiles g and Y. By Lemma 3.2.2, Y is a horizontal arm whose

upper edge has a side arrow labeled C. As Y is the SW neighbor of the single cross g, it

implies, by Figure 3.9, than g has to have direction W.

Moreover, by the nonexistence of a zipper-error on the path segment considered, and

the definition of the overlap map, there is no tiling error in the overlap map of S(1), S(2)

and S(3). Recall that, by Definition 3.2.8, in a valid minitiling each single cross has only

one other single cross pointing towards it. Consequently, g is the unique predecessor of

f on the path, therefore g∗ uniquely preceeds f∗.

According to I.H. (i), g is known to belong to a (2n−1)-square S(4) → Tµ. In the same

way as before, we conclude that S(4) is situated at the right side of S(1), and the path

through S(4) is an A-path starting at h and ending at g that visits all its single crosses.

We can also reason as before to prove that the overlap map of macrotiles involved in S(1),

S(2), S(3) and S(4) is one-to-one.

Conclusion

Altogether, by Definition 3.2.5 and Lemma 3.2.2, the squares S(1), S(2), S(3) and S(4)

form a (2n+1 − 1)-square fXY : Sn+1 −→ Tµ. All its single crosses are visited by the

projection of the path and the macrotile t∗ is on this path, as required in the statement

(i) of the Lemma.

48

Furthermore, by Definition 3.2.7, the A-, C-, C- and B-paths through fXY form a

C2n+1-path. As we have started with the assumption that the central cross X of Sn+1 is

labeled C, the statement (ii) of the Lemma holds, too.

Finally, we have also shown that the common overlap map of the partial macrotiling

with macrotiles attached to S(1), S(2), S(3) and S(4) is a perfect overlap. This verifies

the statement (iii) of the Lemma and concludes the induction step. The other cases (X

labeled by A, B, D and S(1) having other label than SW) are analogous.

Theorem 3.2.7. There exists a directed macrotile system (T0, d0) with the strong plane-

filling property.

Proof. Consider the macrotile system (T0, d0) from Definition 3.2.9. Observe that there

exists an infinite directed zipper of macrotiles from T0, namely the zipper (P, r) con-

structed inductively in the proof of Lemma 3.2.6. Moreover, by the statement of Lemma

3.2.6 (i), any infinite path following the directions, which in addition has no zipper-errors,

covers arbitrarily large squares.

3.3 Undecidability of Existence of Infinite Ribbons

We shall now use Theorem 3.2.7 to prove the undecidability of the existence of an infinite

directed zipper.

Theorem 3.3.1. The set (T, d) | (T, d) is a directed tile system and there exists an

infinite (T, d)-directed zipper is undecidable.

Proof. Reduce the undecidable Tiling Problem to our problem.

49

b

a

Figure 3.14: A sandwich tile σ(a, b) consists of a directed tile a ∈ T0 placed on top of atile b ∈ T1. The direction of σ(a, b) is d(σ(a, b)) = d0(a), the direction of its top tile.

By Theorem 3.2.7 there exists a directed tile system (T0, d0) with the strong plane-

filling property.

Let T1 be a tile system. Consider the following “sandwich tiles”, each consisting

of a tile from T0 placed on top of a tile from T1. That is, create a new directed tile

system (T, d) of sandwich tiles such that T = σ(a, b) | a ∈ T0 and b ∈ T1 where

for all a = 〈aN , aE , aS , aW 〉 ∈ T0 and b = 〈bN , bE , bS , bW 〉 ∈ T1, the sandwich tile

σ(a, b) = 〈(aN , bN ), (aE , bE), (aS , bS), (aW , bW )〉 and the direction of the sandwich tile

is d(σ(a, b)) = d0(a) (see Figure 3.14).

Hence, a sandwich tile σ(a, b) sticks on the north (east, south, west) to σ(a′, b′) iff a

sticks on the north (east, south, west) to a′ and b sticks on the north (east, south, west)

to b′.

We will show now that there exists a valid T1-tiling of the plane iff there exists an

infinite (T, d)-directed zipper.

“⇒” Assume that there exists a valid T1-tiling of the plane f : Z2 → T1. Consider

an infinite (T0, d0)-directed zipper (P, r) (such exists, since (T0, d0) has the strong plane-

filling property). Consider the T -tiled path (P, q) of sandwich tiles such that, for all

(x, y) on P , q(x, y) = σ(r(x, y), f(x, y)). Informally, the directed tiled path of sandwich

tiles consists of the infinite (T0, d0)-directed zipper on its top, and the corresponding

50

partial valid T1-tiling f |range(P ) on the bottom. It is clear that in fact (P, q) is an infinite

(T, d)-directed zipper of sandwich tiles.

“⇐” Assume that there exists an infinite (T, d)-directed zipper (P, q) of sandwich tiles.

Then, let (P, r) be its top layer, i.e., for all (x, y) on P , r(x, y) = a iff q(x, y) = σ(a, b)

for some b ∈ T1. Then clearly, (P, r) is an infinite (T0, d0)-directed zipper. Hence, as

(T0, d0) has the strong plane-filling property, P contains “arbitrarily large squares”: for

all n ∈ Z≥0 there exist (x, y) ∈ Z2 such that (x + i, y + j) is on P for i = 0, 1, 2, . . . , n

and j = 0, 1, 2, . . . , n. Let (P, z) be the bottom layer of (P, q), i.e., the T1-tiled path such

that for all (x, y) on P , z(x, y) = b iff q(x, y) = σ(a, b). Then clearly, (P, z) is in fact an

infinite T1-zipper. It follows that range(P ) = dom(z) contains arbitrarily large squares

and moreover, for all n, z : range(P ) → T1 is a partial tiling valid on a square of size n.

It now follows from the Konig infinity lemma [51] that there exists a valid T1-tiling of the

plane.

Having proved the undecidability of existence of an infinite directed zipper, we shall

use this result to show that the existence of an infinite ribbon is also undecidable.

Theorem 3.3.2. The set T | T is a tile system and there exists an infinite T -ribbon

is undecidable.

Proof. Reduce the set of Theorem 3.3.1 to this set.

Given a directed tile system (T, d), we will construct a tile system T ′ such that there

exists an infinite (T, d)-directed zipper iff there exists an infinite T ′-ribbon. Recall that a

zipper differs from a ribbon in that in a ribbon only consecutive tiles are required to have

51

Figure 3.15: A ribbon motif with input tile on the west and output tile on the north.

matching glues on abutting edges, while on a zipper even non-consecutive neighboring

tiles have to have matching glues on their abutting edges.

The construction is as follows. For each directed tile t ∈ T , construct three T ′-motifs.

A motif is a finite T ′-ribbon (P, r) of special form. We construct these ribbon motifs as

follows (see Figure 3.15).

Each motif is essentially a tiled path that outlines the contours of a square. There

are dents and bumps on the north, east, south and west sides of the motif. In addition,

if (P, r) is a motif, then the first position on P is midway along one side of the motif.

52

Figure 3.16: The two remaining variant motifs corresponding to a directed zipper-tilewith direction north (the first one is depicted in Figure 3.15). The bumps and dents areomitted from the variant motifs for clarity.

This side is called the input side of the motif and the tile at the first position is called

the input tile of the motif. The last position on P is midway along a side of the motif

different than the input side. This side is called the output side of the motif and the tile

at the last position is called the output tile of the motif.

Given a directed tile t ∈ T , we construct the three possible motifs with output side

d(t). We call these three motifs the “variants” of t (see Figure 3.16). On each variant, we

put a bump on the east (south) side, to encode the glue on the east (south) side of t. We

also put a dent on the west (north) side, to encode the glue on the west (north) side of

t. The dents and bumps are designed so that if, for example, t1 sticks on the north to t2,

then the dents on the north of t1 variant motifs fit the bumps on the south of t2 variants

(see Figure 3.17). If however, t1 does not stick on the north to t2, then the north sides of

t1 variant motifs will overlap the bumps on the south of t2 variants. Since overlaps are

not allowed in ribbons, if t1 does not stick on the north to t2, no T ′-ribbon can have a t2

variant motif directly north of a t1 variant motif.

53

Bump

Dent

Figure 3.17: The bump-and-dent pair that simulate a glue. Each glue is assigned aunique position on the side of a motif. If the glues on two directed zipper tiles situatedat adjacent positions match, then the bump on the first motif will be placed exactly atthe necessary position to fit in the dent of the second motif.

The glues on tiles in T ′ are chosen so that each tile can occur in exactly one variant

motif, and in each variant motif it occurs exactly once. To this aim, the two edges of

each ribbon tile connecting it with the preceding respectively succeeding tile on a variant

motif are labeled so that this is the only sticking that can form along those edges. The

only exception to this rule are the input (output) tiles of motifs, which have the edges

connecting them to the preceding (succeeding) tiles labeled differently, as detailed below.

So far, the bumps and dents were used to simulate the glues of zipper-tiles. We now

simulate the direction of a zipper-tile by incorporating it in the glues of the input and

output tiles of its variant motifs. We namely use four new glues, West-to-East, East-

to-West, North-to-South and South-to-North as follows. If the direction of a directed

zipper-tile t1 is north, i.e. d(t1) = N , then:

(a) the output ribbon-tiles of all three variant motifs of t1 will have their north edge

labeled South-to-North, and

(b) the input tile of the variant motif of t1 with a west (east, south) input side will

have its west (east, south) edge labeled West-to-East (East-to-West, South-to-North).

54

We label in a similar way the appropriate edges of the input and output tiles of other

motifs, namely those edges that are meant to connect the motifs to each other. This

labeling ensures that the variant motifs will be connected to each other in the proper

order dictated by the direction of the originating directed zipper-tiles.

Lastly, we want to ensure that only motifs of the kind described above can form, and

that motifs can connect to each other only through their input and output tiles. To this

aim, we have two new different “null” glues: null(1) and null(2). We label, for all the

above constructed ribbon tiles, the so-far-unlabeled north and west edges with null(1) and

the unlabeled east and south edges with null(2). Because null(1) only matches null(1),

and null(1) is only used on the west and north edges, the edges of tiles labeled with these

glues will not stick to any other tiles along those edges. The same is true for null(2).

These “non-stick” glues ensure that the ribbon will only follow the intended motif and

will not fill it in, or stick to anything outside it, except at the input and output tiles.

To summarize, given the directed tile system (T, d) we can construct the tile system

T ′ consisting of the ribbon tiles used in all variant motifs of tiles in T , as described above.

Assume that there exists an infinite directed (T, d)-zipper (P, r). One can easily

construct an infinite T ′-ribbon (P ′, r′). The idea is that, for each position (xi, yi) on P

the tile r(xi, yi) is replaced by one of its variant motifs. The variant chosen is one with

input side opposite d(r(xi−1, yi−1)).

Conversely, assume that there exists an infinite T ′-ribbon, (P ′, r′). It is clear from

our choice of ribbon tiles and glues that (P ′, r′) must consist of an infinite sequence of

motifs. Hence we can construct an infinite (T, d)-directed zipper by replacing each motif

with the tile t of which it is a variant.

55

Theorem 3.3.2 proves the undecidability of existence of an infinite ribbon. This settles

the open problem [27], stating: “Problem 4.1. Given a tiling system T , is there an infinite

T -snake within the infinite grid G = Z× Z?”

In the terminology of [27], a tiling system T is exactly as defined in Section 3.1, a

finite set of tiles, i.e., of squares with colored edges that cannot be rotated and with

infinitely many copies of each tile available; The grid G is the integer grid of positions

in the plane; An infinite T -snake is a sequence of tiles on the plane in which successive

tiles are adjacent along an edge and touching edges have the same color, i.e., infinite

T -snakes are (possibly self-crossing) 2-way infinite ribbons where identical tiles must be

present at the crossing sites. In general, an infinite snake problem asks, given a tiling

system T , and some portion P of the plane, whether there is an infinite T -snake that lies

entirely within P . [27] proves that, given T and a strip of width k ∈ N , the existence

of an infinite T -snake that lies entirely within the strip is decidable. Given a tile system

T , and a specific tile t0 ∈ T , the problem of whether there exists an infinite T -snake

that contains t0 is proved undecidable in [27] using methods in [26] (the case of a 1-way

infinite snake starting at t0 was proven undecidable in [24]). If the special “seed” tile

is not specified, like in Problem 4.1, [27] conjectures the problem undecidable but states

that “[...] it seems that this would be difficult to prove. We have not been able to adjust

the proof techniques of other undecidability results for this purpose”.

Theorem 3.3.2, by showing the undecidability of existence of infinite non-self-crossing

snakes, proves Problem 4.1 undecidable.

56

3.4 Undecidability of Self-Assembly of Arbitrarily Large

Supertiles

Let us return to the discussion of self-assembly. Supertiles are constructed by an incre-

mental process starting from a single tile and proceeding by addition of single tiles that

“stick” to the hitherto built structure. The problem we are addressing is whether or not,

given a tile system, an infinite supertile can self-assemble with tiles from that system. To

formalize our notions,

Definition 3.4.1. A shape is a function f : I → Z2 such that I is a set of consecutive

natural numbers, 0 ∈ I and for all i ∈ I, if i > 0 then there exists j ∈ I such that j < i

and f(i) and f(j) are adjacent.

A shape describes thus a connected region of the plane. The size of a shape f is the

cardinality of its range, i.e., the number of positions it contains, regardless of how many

times they are visited. If we fill in each position of a shape with tiles from a given tile

system T , we obtain the notion of a T -supertile.

Definition 3.4.2. For all tile systems T , a T -supertile is a pair (f, g) where f is a shape

and g : range(f)→ T is a function such that for all i ∈ dom(f), if i > 0 then there exists

j ∈ dom(f) such that j < i and f(i) abuts f(j) on the north (east, south, west) and

g(f(i)) sticks on the north (east, south, west) to g(f(j))].

The size of a T -supertile (f, g) is the size of its corresponding shape f . Two T -

supertiles (f, g) and (f ′, g′) are equivalent iff there exists i, j ∈ Z such that for all (x, y) ∈

Z2, (x, y) ∈ range(f) iff (x+ i, y + j) ∈ range(f ′) and for all (x, y) ∈ range(f), g(x, y) =

57

g′(x + i, y + j). That is, the T -supertile (f ′, g′) is equivalent to the T -supertile (f, g) if

and only if (f ′, g′) can be obtained by translating (f, g) on the plane.

Note that the definitions of shape and supertile may differ works by other authors.

The ideas are similar, but the details may not be the same. The problem stated at the

beginning of this section is settled by the following result.

Theorem 3.4.1. The following sets are undecidable:

S1 = T | T is a tile system and there exists an infinite T -supertile, and

S2 = T | T is a tile system and there exists infinitely many non-equivalent finite

T -supertiles.

Proof. It is easily shown that sets S1 and S2 are identical to the set T | T is a tile system

and there exists an infinite T -ribbon . Hence Theorem 3.4.1 follows from Theorem 3.3.2.

58

Chapter 4

Event-Systems Model

Chemistry provides many examples of self-assembling systems. Crystal growth is one

such example: atoms spontaneously align themselves into lattices. The DNA systems

presented in Chapter 2 are examples of engineered systems of crystal growth. The tile

assembly model presented in Chapter 3 was originally intended to be a formal model of

such growth. There are many other examples of self-assembly in chemistry, however, from

the simple reaction that forms water from gaseous hydrogen and oxygen, to the complex

system of interactions that form proteins from RNA templates in living cells.

In this chapter, we will explore a formal model of chemistry based on mass action

kinetics. The work presented in this chapter is the result of collaboration between Leonard

Adleman, Manoj Gopalkrishnan, Ming-Deh Huang, Pablo Moisset, and myself and the

first person plural pronouns in this chapter refer to those authors.

The assumptions and notation used in chemistry are the result of hundreds of years of

consideration. The same can be said of mathematics. In this chapter, we are attempting

to take a purely mathematical look at the foundations of chemistry. Necessarily, we

have adopted the notation, conventions, and paradigms of mathematics, but this creates

59

difficulties in conveying our findings to chemists, biologists, and others who are more

familiar with the notation, conventions, and paradigms commonly used in other fields.

Ideally, there would be a simple dictionary that translates the mathematics into chemistry,

but creating such a dictionary is a daunting task and has not been accomplished here.

Nonetheless, where possible we have attempted to indicate the meaning of our results in

a language understandable to as wide an audience as possible.

The study of mass action kinetics dates back at least to 1864, when Waage and Guld-

berg [39] formulated the “law of mass action.” Since that time, a great deal of knowledge

on the topic has been amassed in the form of empirical facts, physical theories, and math-

ematical theorems by chemists, chemical engineers, physicists and mathematicians. In

recent years, Horn and Jackson [45], and Feinberg [30] have made significant mathematical

contributions, and these have guided our work.

It is our view that a critical mass of knowledge has been obtained, sufficient to warrant

a formal mathematical consolidation. A major goal of this consolidation is to solidify the

mathematical foundations of this aspect of chemistry — to provide precise definitions,

elucidate what can now be proved, and indicate what is only conjectured. In addition,

we believe that the law of mass action is of intrinsic mathematical interest and should be

made available in a form that might transcend its application to chemistry alone.

To make the law of mass action available for consideration by researchers in areas

other than chemistry, we present mass action kinetics in a new form, which we call event-

systems. Our formulation begins with the observation that systems of chemical reactions

can be represented by sets of binomials. This gives us an opportunity to extend the law

of mass action to arbitrary sets of binomials. Once this extension is made, there is no

60

reason to restrict ourselves to binomials with real coefficients. Hence, we are led to a

dynamical theory of sets of binomials over the complex numbers. Possible mathematical

applications of this theory include:

1. Binomials are objects of intrinsic mathematical interest [25]. For example, they

occur in the study of toric varieties, and hence in string theory. With each set

of binomials over the complex numbers, we associate a corresponding system of

differential equations. Ideally, this dynamical viewpoint will help advance the theory

of binomials, and enhance our understanding of their associated algebraic sets.

2. When we extend the study of the law of mass action to sets of binomials over the

complex numbers, we can consider reactions that involve complex rates, complex

concentrations, and move through complex time. Extending to the complex num-

bers gives us direct access to the powerful theorems of complex analysis. Though

this clearly transcends conventional chemistry, it may have applications in pure

mathematics. Chapter 5 has more information on this topic.

3. Systems of linear differential equations are well understood. In contrast, systems

of ordinary non-linear differential equations can be notoriously intractable. Differ-

ential equations that arise from event-systems lie somewhere in between — more

structured than arbitrary non-linear differential equations, but more challenging

than linear differential equations. As such, they appear to be an important new

class for consideration in the theory of ordinary differential equations.

61

In addition to their use in mathematics, event-systems provide a vehicle by which

ideas in algebraic geometry may be made readily available to the study of mass action

kinetics. As such, they may help solidify the foundations of this aspect of chemistry.

This chapter is organized as follows:

In Section 4.1, we present the basic mathematical notations and definitions for the

study of event-systems.

In Section 4.2, and all of the sections that follow in this chapter, we restrict to finite

event-systems. Theorem 4.2.3 demonstrates that the stoichiometric coefficients give rise

to flow-invariant affine subspaces — “conservation classes”.

In Section 4.3, and all of the sections that follow in this chapter, we restrict to “physical

event-systems”. Though we have defined event-systems over the complex numbers, in

this chapter we focus on consolidating results from the mass action kinetics of reversible

chemical reactions. Physical event-systems capture the idea that the specific rates of

chemical reactions are always positive real numbers. The main result of this section

is Theorem 4.3.5, which demonstrates that for physical event-systems, if initially all

concentrations are non-negative, then they stay non-negative for all future real times

so long as the solution exists. Further, the concentration of every species whose initial

concentration is positive, stays positive.

In Section 4.4, and all the sections that follow in this chapter, we restrict to “natu-

ral event-systems”. Natural event-systems capture the concept of detailed balance from

chemistry. In Theorem 4.4.1, we give four equivalent characterizations of natural event-

systems; in particular, we show that natural event-systems are precisely those physical

62

event-systems that have no “energy cycles”. In Theorem 4.4.6, following Horn and Jack-

son [45], we show that natural event-systems have associated Lyapunov functions. This

theorem is reminiscent of the second law of thermodynamics. The main result of this

section is Theorem 4.4.15, which establishes that for natural event-systems, given non-

negative initial conditions:

1. Solutions exist for all forward real times.

2. Solutions are uniformly bounded in forward real time.

3. All positive equilibria satisfy detailed balance.

4. Every conservation class containing a positive point also contains exactly one posi-

tive equilibrium point.

5. Every positive equilibrium point is asymptotically stable relative to its conservation

class.

For systems of reversible reactions that satisfy detailed balance, must concentrations

approach equilibrium? We believe this to be the case, but are unable to prove it. In 1972,

an incorrect proof was offered [45, Lemma 4C]. This proof was retracted in 1974 [44]. To

the best of our knowledge, this question in mass action kinetics remains unresolved [81, p.

10]. We pose it formally in Open Problem 4.4.16, and consider it the fundamental open

question in the field.

In Section 4.5, we introduce the notion of “atomic event-systems”. As the name

suggests, this is an attempt to capture mathematically the atomic hypothesis that all

species are composed of atoms. The main theorem of this section is Theorem 4.5.1,

63

which establishes that for natural, atomic event-systems, solutions with positive initial

conditions asymptotically approach positive equilibria. Hence, Open Problem 4.4.16 is

resolved in the affirmative for this restricted class of event-systems.

4.1 Basic Definitions and Notation

Before formally defining event-systems, we give a very brief, informal introduction to

chemical reactions. All reactions are assumed to take place at constant temperature in a

well-stirred vessel of constant volume.

Consider

A+ 2Bσ

GGGGGBFGGGGG

τC.

This chemical equation concerns the reacting species A,B and C. In the forward direction,

one mole of A combines with two moles of B to form one mole of C. The symbol “σ”

represents a real number greater than zero. It denotes, in appropriate units, the rate

of the forward reaction when the reaction vessel contains one mole of A and one mole

of B. It is called the specific rate of the forward reaction. In the reverse direction, one

mole of C decomposes to form one mole of A and two moles of B. The symbol “τ”

represents the specific rate of the reverse reaction. Chemists typically determine specific

rates empirically by measuring concentrations at equilibrium. In this chapter, we consider

the specific rates to be constants that are given as part of a particular system and then

consider the kinetics of the system with those rates. Though irreversible reactions (those

with σ = 0 or τ = 0) have been studied, they will not be considered here. In Section 5.2.2

we will briefly explore how the specific rates change as a function of temperature.

64

Inspired by the law of mass action, we introduce a multiplicative notation for chemical

reactions, as an alternative to the chemical equation notation. In our notation, each

chemical reaction is represented by a binomial. Consider the following examples. On the

left are chemical equations. On the right are the corresponding binomials.

X2

1/3GGGGGGGBFGGGGGGG

1/2X1 → 1

3X2 −

12X1

X3

1/3GGGGGGGBFGGGGGGG

1/2X1 +X2 → 1

3X3 −

12X1X2

2X1 + 3X6

σGGGGGBFGGGGG

τ3X1 + 2X2 → σX2

1X36 − τX3

1X22

Our notation leads us to view every set of binomials over an arbitrary field F as

a formal system of reversible reactions with specific rates in F \ 0. For our present

purposes, we will restrict our attention to binomials over the complex numbers. With

this in mind, we now define our notion of event-system.

Notation 4.1.1. Let C∞ =⋃∞n=1 C[X1, X2, · · · , Xn]. A monic monomial of C∞ is a

product of the form∏∞i=1X

eii where the ei are non-negative integers all but finitely

many of which are zero. We will write M∞ to denote the set of all monic monomials of

C∞. More generally, if S ⊂ X1, X2, · · · , we let C[S] be the ring of polynomials with

indeterminants in S and we let MS = M∞ ∩ C[S] (i.e., the monic monomials in C[S]).

If n ∈ Z>0, p ∈ C[X1, X2, · · · , Xn], and a = 〈a1, a2, · · · , an〉 ∈ Cn then, as is usual,

we will let p(a) denote the value of p on argument a.

65

Given two monic monomials M =∏∞i=1X

eii and N =

∏∞i=1X

fii from M∞, we will

say M precedes N (and we will write M ≺ N) iff M 6= N and for the least i such that

ei 6= fi, ei < fi.

It follows that 1 is a monic monomial of C∞ and that each element of C∞ is a C-

linear combination of finitely many monic monomials. We will be particularly concerned

with the set of binomials B∞ = σM + τN | σ, τ ∈ C \ 0 and M,N are distinct monic

monomials of C∞.

Definition 4.1.2 (Event-system). An event-system E is a nonempty subset of B∞.

If E is an event-system, its elements will be called “E-events” or just “events”. Note

that if σM + τN is an event then M 6= N .

Our map from chemical equations to events is as follows. A chemical equation

∑i

aiXi

σGGGGGBFGGGGG

τ

∑j

bjXj goes to:

1. σ∏i

Xaii − τ

∏j

Xbjj if

∏i

Xaii ≺

∏j

Xbjj

or 2. τ∏j

Xbjj − σ

∏i

Xaii if

∏j

Xbjj ≺

∏i

Xaii

66

For example:

X1

1/2GGGGGGGBFGGGGGGG

1/3X2 → 1

3X2 −

12X1 (because X2 ≺ X1)

X2

1/3GGGGGGGBFGGGGGGG

1/2X1 → 1

3X2 −

12X1

X1

-1/2GGGGGGGGBFGGGGGGGG

-1/3X2 → −1

3X2 +

12X1

X1

-1/2GGGGGGGGBFGGGGGGGG

1/3X2 → 1

3X2 +

12X1

X1 +X2

1/2GGGGGGGBFGGGGGGG

1/3X3 → 1

3X3 −

12X1X2

3X1 + 2X2

σGGGGGBFGGGGG

τ2X1 + 3X6 → τX2

1X36 − σX3

1X22

Note that our order of monomials is arbitrary. Any linear order would do. The order

is necessary to achieve a one-to-one map from chemical reactions to events.

Our definition of event-systems allows for an infinite number of reactions, and an

infinite number of reacting species. Indeed, polymerization reactions are commonplace

in nature and, in principle, they are capable of creating arbitrarily long polymers (for

example, DNA molecules).

The next definition introduces the notion of systems of reactions for which the number

of reacting species is finite.

Definition 4.1.3 (Finite-dimensional event-system). An event-system E is finite-dimen-

sional iff there exists an n ∈ Z>0 such that E ⊂ C[X1, X2, · · · , Xn].

Definition 4.1.4 (Dimension of event-systems). Let E be a finite-dimensional event-

system. Then the least n such that E ⊂ C[X1, X2, · · · , Xn] is the dimension of E .

67

Definition 4.1.5 (Physical event, Physical event-system). A binomial e ∈ B∞ is a physi-

cal event iff there exist σ, τ ∈ R>0 and M , N ∈M∞ such that M ≺ N and e = σM−τN .

An event-system E is physical iff each e ∈ E is physical.

Chemical reaction systems typically have positive real forward and backward rates.

Physical event-systems generalize this notion.

Definition 4.1.6. Let n ∈ Z>0. Let α = 〈α1, α2, . . . , αn〉 ∈ Cn.

1. α is a non-negative point iff for i = 1, 2, . . . , n, αi ∈ R≥0.

2. α is a positive point iff for i = 1, 2, . . . , n, αi ∈ R>0.

3. α is a z-point iff there exists an i such that αi = 0.

In chemistry, a system is said to have achieved detailed balance when it is at a point

where the net flux of each reaction is zero. Given the corresponding event-system, points

of detailed balance corresponds to points where each event evaluates to zero, and vice

versa. We call such points “strong equilibrium points”.

Definition 4.1.7 (Strong equilibrium point). Let E be a finite-dimensional event-system

of dimension n. Then α ∈ Cn is a strong E-equilibrium point iff for all e ∈ E , e(α) = 0.

In the language of algebraic geometry, when E is a finite-dimensional event-system,

its corresponding algebraic set is precisely the set of its strong E-equilibrium points.

It is widely believed that all “real” chemical reactions achieve detailed balance. We

now introduce natural event-systems, a restriction of finite-dimensional, physical event-

systems to those that can achieve detailed balance.

68

Definition 4.1.8 (Natural event-system). A finite-dimensional event-system E is natural

iff it is physical and there exists a positive strong E-equilibrium point.

Our next goal is to introduce atomic event-systems: finite-dimensional event-systems

obeying the atomic hypothesis that all species are composed of atoms. Towards this

goal, we will define a graph for each finite-dimensional event-system. The vertices of this

graph are the monomials from M∞ and the edges are determined by the events. If a

weight r is assigned to an edge, then r represents the energy released when a reaction

corresponding to that edge takes place. For the purpose of defining atomic event-systems,

the reader may ignore the weights; they are included here for use elsewhere in the paper

(Definition 4.4.1).

Though graphs corresponding to systems of chemical reactions have been defined

elsewhere (e.g., [30, 81]), it is important to note that these definitions do not coincide

with ours.

Definition 4.1.9 (Event-graph). Let E be a finite-dimensional event-system. The event-

graph GE = 〈V,E,w〉 is a weighted, directed multigraph such that:

1. V = M∞

2. For all M1, M2 ∈M∞, for all r ∈ C,

〈M1,M2〉 ∈ E and r ∈ w (〈M1,M2〉) iff

there exist e ∈ E and σ, τ ∈ C and M,N, T ∈ M∞ such that e = σM + τN and

M ≺ N and either

(a) M1 = TM and M2 = TN and r = ln(−στ

)or

(b) M1 = TN and M2 = TM and r = − ln(−στ

)69

Notice that two distinct weights r1 and r2 could be assigned to a single edge. For

example, let E = X1X2−2X21 , X2−5X1. Consider the edge inGE from the monomialX2

1

to the monomial X1X2. Weight ln 2 is assigned to this edge due to the event X1X2−2X21 ,

with T = 1. Weight ln 5 is also assigned to this edge due to the event X2 − 5X1, with

T = X1.

Definition 4.1.10. Let E be a finite-dimensional event-system. For all M ∈ M∞, the

connected component of M , denoted CE(M), is the set of all N ∈M∞ such that there is

a path in GE from M to N .

It follows from the definition of “path” that every monomial belongs to its connected

component.

Definition 4.1.11 (Atomic event-system). Let E be a finite-dimensional event-system

of dimension n. Let S = X1, X2, · · · , Xn. Let AE =Xi ∈ S | CE(Xi) = Xi

. E is

atomic iff for all M ∈MS , C(M) contains a unique monomial in MAE .

If E is atomic then the members of AE will be called the atoms of E . It follows from

the definition that in atomic event-systems, atoms are not decomposable, non-atoms are

uniquely decomposable into atoms and events preserve atoms.

Since the set MX1,X2...,Xn is infinite, it is not possible to decide whether E is atomic

by exhaustively checking the connected component of every monomial in MX1,X2...,Xn.

The following is sometimes helpful in deciding whether a finite-dimensional event-system

is atomic (proof not provided).

Let E be an event-system of dimension n with no event of the form σ + τN . Let

BE = Xi | For all σ, τ ∈ C \ 0 and N ∈ M∞ : σXi + τN /∈ E. Then E is atomic

70

iff there exist M1 ∈ CE(X1) ∩MBE ,M2 ∈ CE(X2) ∩MBE , . . . ,Mn ∈ CE(Xn) ∩MBE such

that:

For all σn∏i=1

Xaii − τ

n∏i=1

Xbii ∈ E ,

n∏i=1

Maii =

n∏i=1

M bii . (4.1)

We have shown (proof not provided) that if E and BE are as above, and there exist

M1 ∈ CE(X1) ∩ MBE ,M2 ∈ CE(X2) ∩ MBE , . . . ,Mn ∈ CE(Xn) ∩ MBE and there ex-

ists σ∏ni=1X

aii − τ

∏ni=1X

bii ∈ E such that

∏ni=1M

aii 6=

∏ni=1M

bii , then E is not atomic.

Hence, to check whether an event-system with no event of the form σ+τN is atomic, it suf-

fices to examine an arbitrary choice ofM1 ∈ CE(X1)∩MBE ,M2 ∈ CE(X2)∩MBE , . . . ,Mn ∈

CE(Xn) ∩MBE , if one exists, and check whether (4.1) above holds.

Example 4.1.1. Let E = X22−X2

1. Then BE = X1, X2. Let M1 = X1 and M2 = X2.

Trivially, M1,M2 ∈MBE , M1 ∈ CE(X1) and M2 ∈ CE(X2). Consider the event X22 −X2

1 .

Since M22 = X2

2 6= X21 = M2

1 , E is not atomic. Note that the event X22 − X2

1 does not

preserve atoms.

Example 4.1.2. Let E = X24 − X2, X

25 − X3, X2X3 − X1. Then BE = X4, X5.

Let M1 = X24X

25 ,M2 = X2

4 ,M3 = X25 ,M4 = X4,M5 = X5. Clearly these are all in

MBE . X25 − X3 ∈ E implies M3 ∈ CE(X3). X2

4 − X2 ∈ E implies M2 ∈ CE(X2). Since

(X1, X2X3, X2X25 , X

24X

25 ) is a path in GE , we have M1 ∈ CE(X1). For the event X2

4 −X2,

we have M24 = X2

4 = M2. For the event X25 − X3, we have M2

5 = X25 = M3. For the

event X2X3 −X1, we have M2M3 = X24X

25 = M1. Therefore, E is atomic.

Note that it is possible to have an atomic event-system where AE is the empty set.

For example:

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Example 4.1.3. Let E = 1 − X1. In this case, S = X1 and MS is the set

1, X1, X21 , X

31 , . . . . It is clear that MS forms a single connected component C in GE .

Hence, X1 is not in AE , and AE = ∅. 1 is the only monomial in MAE . Since 1 is in C, E

is atomic.

4.2 Finite Event-Systems

In the rest of this chapter only finite event-systems (i.e., where the set E is finite) will

be considered. It is clear that all finite event-systems are finite-dimensional. Chapter 5

has definitions and results relating to certain types of infinite and infinite-dimensional

event-systems.

Definition 4.2.1 (Stoichiometric matrix). Let E = e1, e2, · · · , em be an event-system

of dimension n. Let i ≤ n and j ≤ m be positive integers. Let ej = σM + τN ,

where M ≺ N . Then γj,i is the number of times Xi divides N minus the number of

times Xi divides M . The stoichiometric matrix ΓE of E is the m × n matrix of integers

ΓE = (γj,i)m×n.

Example 4.2.1. Let e1 = 0.5X52−500X1X

32X7. Let E = e1. Then γ1,1 = 1, γ1,2 = −2,

γ1,7 = 0 and for all other i, γ1,i = 0, hence ΓE =(

1 −2 0 0 0 0 1

).

Definition 4.2.2. Let E = e1, · · · , em be a finite event-system of dimension n. Then:

1. P E is the column vector 〈P1, P2, . . . , Pn〉T = ΓTE 〈e1, e2, . . . , em〉T .

2. Let α ∈ Cn. Then α is an E-equilibrium point iff for i = 1, 2, . . . , n : Pi(α) = 0.

72

The Pi’s arise from the Law of Mass Action in chemistry. For a system of chemical

reactions, the Pi’s are the right-hand sides of the differential equations that describe the

concentration kinetics. Definition 4.2.2 extends the Law of Mass Action to arbitrary

event-systems, and hence, arbitrary sets of binomials.

It follows from the definition that for finite event-systems, all strong equilibrium points

are equilibrium points, but the converse need not be true.

Example 4.2.2. Let e1 = X2 − X1 and e2 = X2 − 2X1. Let E = e1, e2. Then

ΓE =

1 −1

1 −1

and P E =

P1

P2

=

2X2 − 3X1

3X1 − 2X2

. Therefore (2, 3) is an E-

equilibrium point. Since e1(2, 3) = 1, (2, 3) is not a strong E-equilibrium point.

Example 4.2.3. Let e1 = 6 − X1X2 and e2 = 2X22 − 9X1. Let E = e1, e2. Then

ΓE =

1 1

1 −2

and P E =

P1

P2

=

6−X1X2 + 2X22 − 9X1

6−X1X2 − 4X22 + 18X1

. The point

(2, 3) is a strong equilibrium point because e1(2, 3) = 0 and e2(2, 3) = 0. Since P1(2, 3) =

e1(2, 3) + e2(2, 3) = 0 and P2(2, 3) = e1(2, 3) − 2e2(2, 3) = 0, the point (2, 3) is also an

equilibrium point.

The event-system in Example 4.2.2 is not natural, whereas the one in Example 4.2.3

is. In Theorem 4.4.7, it is shown that if E is a finite, natural event-system then all positive

E-equilibrium points are strong E-equilibrium points.

Definition 4.2.3 (Event-process). Let E be a finite event-system of dimension n. Let

〈P1, P2, . . . , Pn〉T = P E . Let Ω ⊆ C be a non-empty simply-connected open set. Let

f = 〈f1, f2, · · · , fn〉 where for i = 1, 2, . . . , n, fi : C → C is defined on Ω. Then f is an

E-process on Ω iff for i = 1, 2, . . . , n:

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1. f ′i exists on Ω.

2. f ′i = Pi f on Ω.

Note that E-processes evolve through complex time, and hence generalize the idea of

the time-evolution of concentrations in a system of chemical reactions.

Definition 4.2.3 immediately implies that if f = 〈f1, f2, . . . , fn〉 is an E-process on Ω,

then for i = 1, 2, . . . , n, fi is holomorphic on Ω. In particular, for each i and all α ∈ Ω,

there is a power series around α that agrees with fi on a disk of non-zero radius.

Systems of chemical reactions sometimes obey certain conservation laws. For example,

they may conserve mass, or the total number of each kind of atom. Event-systems also

sometimes obey conservation laws.

Definition 4.2.4 (Conservation law, Linear conservation law). Let E be a finite event-

system of dimension n. A function g : Cn → C is a conservation law of E iff g is

holomorphic on Cn, g(〈0, 0, · · · , 0〉) = 0 and ∇g · P E is identically zero on Cn. If g is a

conservation law of E and g is linear (i.e. ∀c ∈ C, ∀α,β ∈ Cn, g(cα+β) = cg(α) + g(β)),

then g is a linear conservation law of E .

The event-system described in Example 4.2.2 has a linear conservation law

g(X1, X2) = X1 +X2.

The next theorem shows that conservation laws of E are dynamical invariants of E-

processes.

74

Theorem 4.2.1. For all finite event-systems E, for all conservation laws g of E, for all

simply-connected open sets Ω ⊆ C, for all E-processes f on Ω, there exists k ∈ C such

that g f − k is identically zero on Ω.

Proof. Let n be the dimension of E . Let 〈P1, P2, . . . , Pn〉T = P E . For all t ∈ Ω, by

Definition 4.2.3, for i = 1, 2, . . . , n, fi(t) and f ′i(t) are defined. Further, by Definition 4.2.4,

g is holomorphic on Cn. Hence, g f is holomorphic on Ω. Therefore, by the chain

rule, (g f)′(t) = (∇g|f(t)) · 〈f ′1(t), f ′2(t), . . . , f ′n(t)〉. By Definition 4.2.3, for all t ∈ Ω,

〈f ′1(t), f ′2(t), . . . , f ′n(t)〉 = 〈P1(f(t)), P2(f(t)), . . . , Pn(f(t))〉. From these, it follows that

(g f)′(t) = (∇g ·P E)(f(t)). But by Definition 4.2.4, ∇g ·P E is identically zero. Hence,

for all t ∈ Ω, (g f)′(t) = 0. In addition, Ω is a simply-connected open set. Therefore,

by [9, Theorem 11], there exists k ∈ C such that g f − k is identically zero on Ω.

The next theorem shows a way to derive linear conservation laws of an event-system

from its stoichiometric matrix.

Theorem 4.2.2. Let E be a finite event-system of dimension n. For all v ∈ ker ΓE ,

v · 〈X1, · · · , Xn〉 is a linear conservation law of E.

Proof. Let Γ = ΓE , then ker Γ is orthogonal to the image of ΓT . By the definition of

P = P E , for all w ∈ Cn, P (w) lies in the image of ΓT . Hence, for all v ∈ ker Γ, for all

w ∈ Cn, v · P (w) = 0. But v is the gradient of v · 〈X1, · · · , Xn〉. It now follows from

Definition 4.2.4 that v · 〈X1, · · · , Xn〉 is a linear conservation law of E .

Definition 4.2.5 (Primitive conservation law). Let E be a finite event-system of dimen-

sion n. For all v ∈ ker ΓE , the linear conservation law v · 〈X1, X2, · · · , Xn〉 is a primitive

conservation law.

75

Definition 4.2.6 (Conservation class, Positive conservation class). Let E be a finite

event-system of dimension n. A coset of (ker ΓE)⊥ is a conservation class of E . If a

conservation class of E contains a positive point, then the class is a positive conservation

class of E .

Equivalently, α,β ∈ Cn are in the same conservation class if and only if they agree on

all primitive conservation laws. Note that if H is a conservation class of E then it is closed

in Cn. The following theorem shows that the name “conservation class” is appropriate.

Theorem 4.2.3. Let E be a finite event-system. Let Ω ⊂ C be a simply-connected open set

containing 0. Let f be an E-process on Ω. Let H be a conservation class of E containing

f(0). Then for all t ∈ Ω, f(t) ∈ H.

Proof. Let E , Ω, f , H and t be as in the statement of this theorem. For all v ∈ ker ΓE ,

the primitive conservation law v · 〈X1, X2, · · · , Xn〉 is a dynamical invariant of f , from

Theorem 4.2.2 and Theorem 4.2.1. Hence,

v · 〈f1(0), f2(0), · · · , fn(0)〉 = v · 〈f1(t), f2(t), · · · , fn(t)〉

That is,

v · 〈f1(0)− f1(t), f2(0)− f2(t), · · · , fn(0)− fn(t)〉 = 0

Hence, f(t)− f(0) is in (ker ΓE)⊥. By Definition 4.2.6, f(t) ∈ H.

76

4.3 Finite Physical Event-Systems

In this section, we investigate finite, physical event-systems — a generalization of systems

of chemical reactions.

It is widely believed that systems of chemical reactions that begin with positive (re-

spectively, non-negative) concentrations will have positive (respectively, non-negative)

concentrations at all future times. This property has been addressed mathematically in

numerous papers [32, 30, 14, 81]. The notion of “system of chemical reactions” varies

between papers. Several papers have provided no proof, incomplete proofs or inadequate

proofs that this property holds for their systems. Sontag [81] provides a lovely proof of

this property for the systems he considers — zero deficiency reaction networks with one

linkage class. We shall prove in Theorem 4.3.5 that the property holds for finite, physical

event-systems. Finite, physical event-systems have a large intersection with the systems

considered by Sontag, but each includes a large class of systems that the other does not.

We remark that our methods of proof differ from Sontag’s, but it is possible that Sontag’s

proof might be adaptable to our setting.

Lemmas 4.3.4 and 4.3.11 are proved here because they apply to finite, physical event-

systems. However, they are only invoked in subsequent sections. Lemma 4.3.4 relates

E-processes to solutions of ordinary differential equations over the reals. Lemma 4.3.11

establishes that if an E-process defined on the positive reals starts at a real, non-negative

point, then its ω-limit set is invariant and contains only real, non-negative points.

The next lemma shows that if two E-processes evaluate to the same real point on a

real argument then they must agree and be real-valued on an open interval containing

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that argument. The proof exploits the fact that E-processes are analytic, by considering

their power series expansions.

Lemma 4.3.1. Let E be a finite, physical event-system of dimension n, let Ω,Ω′ ⊆ C

be open and simply-connected, let f = 〈f1, f2, . . . , fn〉 be an E-process on Ω and let g =

〈g1, g2, . . . , gn〉 be an E-process on Ω′. If t0 ∈ Ω∩Ω′∩R and f(t0) ∈ Rn and f(t0) = g(t0),

then there exists an open interval I ⊆ R such that t0 ∈ I and for all t ∈ I:

1. f(t) = g(t).

2. For i = 1, 2, . . . , n : if∑∞

j=0 cj(z − t0)j is the Taylor series expansion of fi at t0

then for all j ∈ Z≥0, cj ∈ R.

3. f(t) ∈ Rn.

Proof. Let k ∈ Z≥0. By Definition 4.2.3, f and g are vectors of functions analytic at t0.

For i = 1, 2, . . . , n, let f (k)i be the kth derivative of fi and let f (k) = 〈f (k)

1 , f(k)2 , . . . , f

(k)n 〉.

Define g(k)i and g(k) similarly. To prove 1, it is enough to show that for i = 1, 2, . . . , n,

fi and gi have the same Taylor series around t0. Let V 0 = 〈X1, X2, . . . , Xn〉. Let

V k = Jac(V k−1)P E (recall that if H = 〈h1(V 0), h2(V 0), . . . , hn(V 0)〉 is a vector of

functions in m variables then Jac(H) is the n×m matrix ( ∂hi∂xj), where i = 1, 2, . . . , n and

j = 1, 2, . . . ,m). Let 〈Vk,1, Vk,2, . . . , Vk,n〉 = V k. We claim that f (k) = V k f on Ω and

g(k) = V k g on Ω′ and for i = 1, 2, . . . , n, Vk,i ∈ R[X1, X2, . . . , Xn]. We prove the claim

by induction on k. If k = 0, the proof is immediate. If k ≥ 1, on Ω:

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f (k) = (f (k−1))′

= (V k−1 f)′ (Inductive hypothesis)

= (Jac(V k−1) f)f ′ (Chain-rule of derivation)

= (Jac(V k−1) f)(P E f) (f is an E-process)

= (Jac(V k−1)P E) f

= V k f

By a similar argument, we conclude that g(k) = V k g on Ω′. By the inductive hy-

pothesis, V k−1 is a vector of polynomials in R[X1, X2, . . . , Xn]. It follows that Jac(V k−1)

is an n× n matrix of polynomials in R[X1, X2, . . . , Xn]. Since E is physical, P E is a vec-

tor of polynomials in R[X1, X2, . . . , Xn]. Therefore, V k = Jac(V k−1)P E is a vector of

polynomials in R[X1, X2, . . . , Xn]. This establishes the claim.

We have proved that f (k) = V kf on Ω and g(k) = V kg on Ω′. Since, by assumption,

t0 ∈ Ω ∩ Ω′ and f(t0) = g(t0), it follows that f (k)(t0) = g(k)(t0). Therefore, for i =

1, 2, . . . , n, fi and gi have the same Taylor series around t0. For i = 1, 2, . . . , n, let ai be

the radius of convergence of the Taylor series of fi around t0. Let rf = mini∈1,2,...,n ai.

Define rg similarly. Let D ⊆ Ω ∩ Ω′ be some non-empty open disk centered at t0 with

radius r ≤ min(rf , rg). Since Ω and Ω′ are open sets and t0 ∈ Ω ∩ Ω′, such a disk must

exist. Letting I = (t0 − r, t0 + r) completes the proof of 1.

By assumption, f(t0) ∈ Rn, and we have proved that f (k) = V k f and V k is a

vector of polynomials in R[X1, X2, . . . , Xn]. It follows that f (k)(t0) ∈ Rn. Therefore, for

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i = 1, 2, . . . , n, all coefficients in the Taylor series of fi around t0 are real. It follows that

fi is real valued on I, completing the proof of 3.

The next lemma is a kind of uniqueness result. It shows that if two E-processes

evaluate to the same real point at 0 then they must agree and be real-valued on every

open interval containing 0 where both are defined. The proof uses continuity to extend

the result of Lemma 4.3.1.

Lemma 4.3.2. Let E be a finite, physical event-system of dimension n, let Ω,Ω′ ⊆ C

be open and simply-connected, let f = 〈f1, f2, . . . , fn〉 be an E-process on Ω and let g =

〈g1, g2, . . . , gn〉 be an E-process on Ω′. If 0 ∈ Ω ∩ Ω′ and f(0) ∈ Rn and f(0) = g(0),

then for all open intervals I ⊆ Ω ∩ Ω′ ∩ R such that 0 ∈ I, for all t ∈ I, f(t) = g(t) and

f(t) ∈ Rn.

Proof. Assume there exists an open interval I ⊆ Ω ∩ Ω′ ∩ R such that 0 ∈ I and B =

t ∈ I | f(t) 6= g(t) or f(t) 6∈ Rn 6= ∅. Let BP = B ∩R≥0 and let BN = B ∩R<0. Note

that B = BP ∪BN , hence, BP 6= ∅ or BN 6= ∅. Suppose BP 6= ∅ and let tP = inf(BP ).

By Lemma 4.3.1, there exists an ε ∈ R>0 such that (−ε, ε) ∩B = ∅. Hence, tP ≥ ε > 0.

By definition of tP , for all t ∈ [0, tP ), f(t) = g(t) and f(t) ∈ Rn. Since f and g are

analytic at tP , they are continuous at tP . Therefore, f(tP ) = g(tP ) and f(tP ) ∈ Rn. By

Lemma 4.3.1, there exists an ε′ ∈ R>0 such that for all t ∈ (tP − ε′, tP + ε′), f(t) = g(t)

and f(t) ∈ Rn, contradicting tP being the infimum of BP . Therefore, BP = ∅. Using

a similar agument, we can prove that BN = ∅. Therefore, B = ∅, and for all t ∈ I,

f(tP ) = g(tP ) and f(tP ) ∈ Rn.

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The next lemma is a convenient technical result that lets us ignore the choice of origin

for the time variable.

Lemma 4.3.3. Let E be a finite, physical event-system of dimension n, let Ω, Ω ⊆ C

be open and simply connected, let f = 〈f1, f2, . . . , fn〉 be an E-process on Ω and let

f = 〈f1, f2, . . . , fn〉 be an E-process on Ω. Let u ∈ Ω and u ∈ Ω and α ∈ Rn. Let I ⊆ R

be an open interval. If

1. f(u) = f(u) = α and

2. 0 ∈ I and

3. for all s ∈ I, u+ s ∈ Ω and u+ s ∈ Ω

then for all t ∈ I, f(u+ t) = f(u+ t).

Proof. Suppose f(u) = f(u) = α ∈ Rn. Let Ωu = z ∈ C | u + z ∈ Ω and Ωu =

z ∈ C | u + z ∈ Ω. Let h = 〈h1, h2, . . . , hn〉 where for i = 1, 2, . . . , n, hi : Ωu → C

is such that for all z ∈ Ωu, hi(z) = fi(u + z) and let h = 〈h1, h2, . . . , hn〉 where for

i = 1, 2, . . . , n, hi : Ωu → C is such that for all z ∈ Ωu, hi(z) = fi(u + z). Since u + z

is differentiable on Ωu and for i = 1, 2, . . . , n, fi is differentiable on Ω, it follows that

for i = 1, 2, . . . , n, hi is differentiable on Ωu. Further, for i = 1, 2, . . . , n, for all z ∈ Ωu,

h′i(z) = f ′i(u + z) = P E(fi(u + z)) = P E(hi(z)), so h is an E-process on Ωu. Similarly,

h is an E-process on Ωu. Note that 0 ∈ Ωu ∩ Ωu because u ∈ Ω and u ∈ Ω and that

h(0) = h(0) = α because f(u) = f(u) = α. By Lemma 4.3.2, for all open intervals

I ⊆ Ωu ∩ Ωu ∩ R such that 0 ∈ I, for all t ∈ I, h(t) = h(t), so f(u+ t) = f(u+ t).

81

Because event-systems are defined over the complex numbers, we have access to re-

sults from complex analysis. However, there is a considerable body of results regarding

ordinary differential equations over the reals. Definition 4.3.1 and Lemma 4.3.4 estab-

lish a relationship between E-processes and solutions to systems of ordinary differential

equations over the reals.

Definition 4.3.1 (Real event-process). Let E be a finite, physical event-system of dimen-

sion n. Let 〈P1, P2, . . . , Pn〉T = P E . Let I ⊆ R be an interval. Let h = 〈h1, h2, . . . , hn〉

where for i = 1, 2, . . . , n, hi : R → R is defined on I. Then h is a real-E-process on I iff

for i = 1, 2, . . . , n:

1. h′i exists on I.

2. h′i = Pi h on I.

Lemma 4.3.4 (All real-E-processes are restrictions of E-processes). Let E be a finite,

physical event-system of dimension n. Let I ⊆ R be an interval. Let h = 〈h1, h2, . . . , hn〉

be a real-E-process on I. Then there exist an open, simply-connected Ω ⊆ C and an

E-process f on Ω such that:

1. I ⊂ Ω

2. For all t ∈ I : f(t) = h(t).

Proof. Let P = 〈P1, P2, . . . , Pn〉 = P E . For i = 1, 2, . . . , n, Pi is a polynomial and there-

fore analytic on Cn. By Cauchy’s existence theorem for ordinary differential equations

with analytic right-hand sides [47], for all a ∈ I, there exist a non-empty open disk Da ⊆ C

centered at a and functions fa,1, fa,2, . . . , fa,n analytic on Da such that for i = 1, 2, . . . , n :

82

1. fa,i(a) = hi(a)

2. f ′a,i exists on Da and for all t ∈ Da : f ′a,i(t) = Pi(fa,1(t), fa,2(t), . . . , fa,n(t)). That

is, fa = 〈fa,1, fa,2, . . . , fa,n〉 is an E-process on Da.

Claim: For all a ∈ I, there exists δa ∈ R>0 such that for all t ∈ I ∩ (a − δa, a + δa) :

fa(t) = h(t). To see this, by Lemma 4.3.1, for all a ∈ I there exists βa ∈ R>0 such

that for all t ∈ (a − βa, a + βa) ∩ Da, fa(t) ∈ Rn. Let Ia = (a − βa, a + βa) ∩ Da.

Note that fa|Ia is a real-E-process on Ia . By the theorem of uniqueness of solutions to

differential equations with C1 right-hand sides [43], there exists γa ∈ R>0 such that for

all t ∈ (a− γa, a+ γa) ∩ Ia ∩ I, fa(t) = h(t). Clearly, we can choose δa ∈ R>0 such that

(a− δa, a+ δa) ⊆ (a− γa, a+ γa) ∩ Ia. This establishes the claim.

For all a ∈ I, let δa ∈ R>0 be such that for all t ∈ I ∩ (a− δa, a+ δa) : fa(t) = h(t).

Let Da be an open disk centered at a of radius δa.

Claim: For all a1, a2 ∈ I, for all t ∈ Da1 ∩ Da2 : fa1(t) = fa2

(t). To see this,

suppose Da1 ∩ Da2 6= ∅. Let J = Da1 ∩ Da2 ∩ R. Since Da1 and Da2 are open disks

centered on the real line, J is a non-empty open real interval. For all t ∈ J , by the claim

above, fa1(t) = h(t) and fa2

(t) = h(t). Hence, fa1(t) = fa2

(t). Since J is a non-empty

interval, J contains an accumulation point. Since fa1and fa2

are analytic on Da1 ∩ Da2

and Da1∩Da2 is simply connected, for all t ∈ Da1∩Da2 : fa1(t) = fa2

(t). This establishes

the claim.

Let Ω =⋃a∈I Da. Clearly, I ⊂ Ω. Ω is a union of open discs, and is therefore open.

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For all t ∈ Ω, there exists a ∈ I such that t ∈ Da. Since Da is a disk, t and a

are path-connected in Ω. Since I is path-connected, and I ⊆ Ω, it follows that Ω is

path-connected.

To see that Ω is simply-connected, consider the function R : [0, 1]× Ω→ Ω given by

(u, z) 7→ Re(z) + i Im(z)(1 − u). Observe that R is continuous on [0, 1] × Ω, and for all

z ∈ Ω: R(0, z) = z, R(1,Ω) ⊂ Ω, and for all u ∈ [0, 1], for all z ∈ Ω ∩ R : R(u, z) ∈ Ω.

Therefore, R is a deformation retraction. Note that R(0,Ω) = Ω and R(1,Ω) ⊆ R, and

Ω is path-connected together imply that R(1,Ω) is a real interval. Hence, R(1,Ω) is

simply-connected. Since R was a deformation retraction, Ω is simply-connected.

Let f : Ω→ Cn be the unique function such that for all a ∈ I, for all t ∈ Da : f(t) =

fa(t). By the claim above and from the definition of Ω, f is well-defined.

Observe that for all t ∈ I,

h(t) = f t(t) (Definition of f t)

= f(t) (I ⊂ Ω and definition of f).

Claim: f is an E-process on Ω. From the definitions of Ω and f , for all t ∈ Ω, there

exists a ∈ I such that t ∈ Da and for all s ∈ Da, f(s) = fa(s). Since fa is an E-process

on Da, the claim follows.

In Theorem 4.3.5, we prove that if E is a finite, physical event-system, then E-processes

that begin at positive (respectively non-negative) points remain positive (respectively

non-negative) through all forward real time where they are defined. In fact, Theorem 4.3.5

84

establishes more detail about E-processes. In particular, if at some time a species’ con-

centration is positive, then it will be positive at subsequent times.

Theorem 4.3.5. Let E be a finite, physical event-system of dimension n, let Ω ⊆ C be

open and simply-connected, and let f = 〈f1, f2, . . . , fn〉 be an E-process on Ω. If I ⊆

Ω ∩ R≥0 is connected and 0 ∈ I and f(0) is a non-negative point then for k = 1, 2, . . . , n

either:

1. For all t ∈ I, fk(t) = 0, or

2. For all t ∈ I ∩ R>0, fk(t) ∈ R>0.

The proof of Theorem 4.3.5 is highly technical, and relies on a detailed examination

of the vector of polynomials P E . This allows us to show (Lemma 4.3.8) that if f =

〈f1, f2, . . . , fn〉 is an E-process that at real time t0 is non-negative, then each fi is “right

non-negative”. That is, the Taylor series expansion of fi around t0 has real coefficients

and the first non-zero coefficient, if any, is positive. Further, (Lemma 4.3.10) if fi(t0) = 0

and its Taylor series expansion has a non-zero coefficient, then there exists k such that

fk(t0) = 0 and the first derivative of fk with respect to time is positive at t0.

Definition 4.3.2. Let n ∈ Z>0 and let k ∈ 1, 2, . . . , n. Let f ∈ R[X1, X2, . . . , Xn]

be a polynomial. Then f is non-nullifying with respect to k iff there exist m ∈ N,

c1, c2, · · · , cm ∈ R>0, M1,M2, . . . ,Mm ∈ MX1,X2,...,Xn and h ∈ R[X1, X2, . . . , Xn] such

that f =∑m

i=1 ciMi +Xkh.

Observe that for all k, the polynomial 0 is non-nullifying with respect to k.

85

Lemma 4.3.6. Let E be a finite, physical event-system of dimension n. Let P1, P2, . . . , Pn

be such that 〈P1, P2, . . . , Pn〉 = P E . Then, for all i ∈ 1, 2, . . . , n, Pi is non-nullifying

with respect to i.

Proof. Let m = |E|. Let (γj,i)m×n = ΓE . Since E is physical, there exist σ1, σ2, . . . , σm,

τ1, τ2, . . . , τm ∈ R>0 and M1,M2, . . . ,Mm, N1, N2, . . . , Nm ∈ M∞ such that for j =

1, 2, . . . ,m : Mj ≺ Nj and σ1M1 − τ1N1, σ2M2, τ2N2, . . . , σmMm − τmNm = E . Let

i ∈ 1, 2, . . . , n.

From the definition of P E , Pi =∑m

j=1 γj,i(σjMj − τjNj). It is sufficient to prove

that for j = 1, 2, . . . ,m : γj,i(σjMj − τjNj) is non-nullifying with respect to i. Let

j ∈ 1, 2, . . . ,m. If γj,i = 0 then γj,i(σjMj − τjNj) = 0 which is non-nullifying with

respect to i. If γj,i > 0 then, from the definition of ΓE , Xi | Nj and

γj,i(σjMj − τjNj) = γj,iσjMj +Xi

(−γj,iτj

Nj

Xi

)

which is non-nullifying with respect to i since γj,iσj > 0. Similarly, if γj,i < 0 then

Xi |Mj and

γj,i(σjMj − τjNj) = −γj,iτjNj +Xiγj,iσjMj

Xi

which is non-nullifying with respect to i since −γj,iτj > 0. Hence, Pi is non-nullifying

with respect to i.

Definition 4.3.3. Let t0 ∈ C, let f : C→ C be analytic at t0 and let f(t) =∑∞

k=0 ck(t−

t0)k be the Taylor series expansion of f around t0. Then O(f, t0) is the least k such that

ck 6= 0. If for all k, ck = 0, then O(f, t0) =∞.

86

Definition 4.3.4 (Right non-negative). Let t0 ∈ R, let f : C→ C be analytic at t0 and

let f(t) =∑∞

k=0 ck(t− t0)k be the Taylor series expansion of f around t0. Then f is RNN

at t0 iff both:

1. For all k ∈ N, ck ∈ R and

2. Either O(f, t0) =∞ or cO(f,t0) ∈ R>0.

Lemma 4.3.7. Let t0 ∈ C. Let f, g : C→ C be functions analytic at t0. Then:

1. O(f · g, t0) = O(f, t0) +O(g, t0).

2. If t0 ∈ R and f, g are RNN at t0 then f · g is RNN at t0.

The proof is obvious.

Lemma 4.3.8. Let E be a finite, physical event-system of dimension n, let Ω ⊆ C be

open and simply-connected and let f = 〈f1, f2, . . . , fn〉 be an E-process on Ω. For all

t0 ∈ Ω ∩ R, if f(t0) ∈ Rn≥0 then for i = 1, 2, . . . , n : fi is RNN at t0.

Proof. Suppose t0 ∈ Ω ∩ R and f(t0) ∈ Rn≥0. Let P = 〈P1, P2, . . . , Pn〉 = P E . Let

C = i | fi is not RNN at t0.

For the sake of contradiction, suppose C 6= ∅. Let m = mini∈C O(fi, t0). Let k ∈ C be

such that O(fk, t0) = m. Let fk(t) =∑∞

i=0 ai(t− t0)i be the Taylor series expansion of fk

around t0. Since E is physical and t0 ∈ R and f(t0) ∈ Rn≥0, it follows from Lemma 4.3.1.2

that for all i ∈ N, ai ∈ R. Further:

a0 = a1 = . . . = am−1 = 0 (O(fk, t0) = m.) (4.2)

am ∈ R<0 (fk is not RNN at t0.) (4.3)

87

Since f(t0) ∈ Rn≥0 and am ∈ R<0 and a0 = fk(t0), it follows that m > 0.

Consider f ′k = Pk f . By differentiation, the Taylor series expansion of f ′k at t0 is:

f ′k(t) =∞∑i=0

(i+ 1)ai+1(t− t0)i. (4.4)

From Lemma 4.3.6, Pk is non-nullifying. Hence, there exist l ∈ N, b1, b2, . . . , bl ∈ R>0,

M1,M2, . . . ,Ml ∈MX1,X2,...,Xn and h ∈ R[X1, X2, . . . , Xn] such that Pk =∑l

j=1 bjMj+

Xk · h. Then for all t ∈ Ω:

f ′k(t) = Pk f(t) =l∑

j=1

bjMj f(t) + fk(t) · (h f(t)) (4.5)

Since h is a polynomial, h f is analytic at t0. Therefore, fk · (h f) is analytic at

t0. Let∑∞

i=0 ci(t − t0)i be the Taylor series expansion of fk · (h f) at t0. Similarly,

for j = 1, 2, . . . , l, bjMj f is analytic at t0. Let∑∞

i=0 dj,i(t − t0)i be the Taylor series

expansion of bjMj f at t0. From Equations (4.4) and (4.5), equating Taylor series

coefficients, for i = 0, 1, . . . ,m− 1:

(i+ 1)ai+1 = ci +l∑

j=1

dj,i (4.6)

From Lemma 4.3.7.1,

O(fk · (h f), t0) = O(fk, t0) +O(h f , t0) ≥ O(fk, t0) = m

88

Hence,

c0 = c1 = . . . = cm−1 = 0. (4.7)

From Equations (4.2), (4.6), and (4.7), for i = 0, 1, . . . ,m− 2:

l∑j=1

dj,i = 0 (4.8)

Since m > 0, from Equations (4.3), (4.6), and (4.7):

l∑j=1

dj,m−1 = mam ∈ R<0 (4.9)

Let i0 = minj=1,2,...,lO(bjMjf , t0). From Equation (4.9), it follows that i0 ≤ m−1.

Case 1: For j = 1, 2, . . . , l : dj,i0 ∈ R≥0. From the definition of i0 it follows that∑lj=1 dj,i0 ∈ R>0. If i0 < m − 1, this contradicts Equation (4.8). If i0 = m − 1, this

contradicts Equation (4.9).

Case 2: There exists j0 ∈ 1, 2, . . . , l such that dj0,i0 ∈ R<0. From the definition of

i0, O(bj0Mj0 , t0) = i0 ≤ m−1. Therefore, for each i such that Xi |Mj0 , O(fi, t0) ≤ m−1.

From the definitions of C and m, this implies that for each i such that Xi | Mj0 , fi is

RNN at t0. Since bj0 ∈ R>0, it follows that bj0Mj0 f is a product of RNN functions.

Hence, by Lemma 4.3.7.2, bj0Mj0 f is RNN at t0 and dj0,i0 ∈ R>0, a contradiction.

Hence, for i = 1, 2, . . . , n, fi is RNN at t0.

Lemma 4.3.9. Let t0 ∈ R and let f be a function RNN at t0. There exists an ε ∈ R>0

such that either for all t ∈ (t0, t0 + ε), f(t) ∈ R>0 or for all t ∈ (t0, t0 + ε), f(t) = 0.

89

Proof. Let m = O(f, t0). If m = ∞, f is identically zero and the lemma follows im-

mediately. Otherwise, let f (m) denote the mth derivative of f . Since f is RNN at

t0 and has order m, f (m)(t0) ∈ R>0. Since f is analytic at t0, f (m) is analytic at

t0, and hence continuous at t0. By continuity, there exists ε ∈ R>0 such that for all

τ ∈ [t0, t0 + ε] : f (m)(τ) ∈ R>0. From Taylor’s theorem, for all t ∈ (t0, t0 + ε), there exists

τ ∈ [t0, t0 + ε] such that:

f(t) =(t− t0)m

m!f (m)(τ)

Therefore, f(t) ∈ R>0.

Note that Lemmas 4.3.8 and 4.3.9 together already imply that if E is a finite, physical

event-system, then E-processes that begin at non-negative points remain non-negative

through all forward real time where they are defined. This result is weaker than Theo-

rem 4.3.5.

Lemma 4.3.10. Let E be a finite, physical event-system of dimension n, let Ω ⊆ C be

open and simply-connected, let f = 〈f1, f2, . . . , fn〉 be an E-process on Ω. Let t0 ∈ Ω. If

f(t0) is non-negative and there exists j ∈ 1, 2, . . . , n such that 0 < O(fj , t0) <∞ then

there exists k ∈ 1, 2, . . . , n such that O(fk, t0) = 1.

Proof. Suppose f(t0) ∈ Rn≥0. Let C = i | 0 < O(fi, t0) < ∞. Suppose C 6= ∅. Let

m = mini∈C O(fi, t0). There exists k ∈ C such that O(fk, t0) = m.

Let P = 〈P1, P2, . . . , Pn〉 = P E . From Lemma 4.3.6, Pk is non-nullifying with respect

to k. Hence, there exist l ∈ N, b1, b2, . . . , bl ∈ R>0, M1,M2, . . . ,Ml ∈ MX1,X2,...,Xn and

h ∈ R[X1, X2, . . . , Xn] such that Pk =∑l

j=1 bjMj +Xk · h.

90

For all t ∈ Ω: f ′k(t) = Pk f(t) =∑l

j=1 bjMj f(t) + fk(t) · (h f(t)). From

Lemma 4.3.7.1, O(fk · (h f), t0) = O(fk, t0) + O(h f , t0) ≥ O(fk, t0) = m. It follows

that:

m− 1 = O(f ′k, t0) = O

l∑j=1

bjMj f , t0

(4.10)

From Lemma 4.3.7.2 and 4.3.8, for j = 1, 2, . . . , l : bjMj f is RNN at t0. It follows

that O(∑l

j=1 bjMj f , t0) = minj=1,2,...,lO(bjMj f , t0). From Equation (4.10), m− 1 =

minj=1,2,...,lO(bjMj f , t0). Hence, there exists j0 such that O(bj0Mj0 f , t0) = m − 1.

From Lemma 4.3.7.1, for all i such that Xi |Mj0 , O(fi, t0) ≤ m−1. From the definition of

m, for all i such that Xi |Mj0 , O(fi, t0) = 0. It follows that m−1 = O(bj0Mj0 f , t0) = 0.

Hence, m = 1.

We are now ready to prove Theorem 4.3.5.

Proof of Theorem 4.3.5. Suppose I ⊆ Ω ∩ R≥0 is connected and 0 ∈ I and f(0) is a

non-negative point. If I ∩ R>0 = ∅, the theorem is immediate. Suppose I ∩ R>0 6= ∅.

It is clear that for all k, O(fk, 0) = ∞ iff for all t ∈ I, fk(t) = 0. Let C = i |

O(fi, 0) 6= ∞. From Lemmas 4.3.8 and 4.3.9, for all k ∈ C, there exists εk ∈ I ∩ R>0

such that for all t ∈ (0, εk) : fk(t) ∈ R>0.

Suppose for the sake of contradiction that there exist i ∈ C and t ∈ I ∩R>0 such that

fi(t) /∈ R>0. From Lemma 4.3.2, fi(t) ∈ R. Since fi(εi/2) ∈ R>0 and fi(t) ∈ R≤0, by

continuity there exists t′ ∈ I ∩ R>0 such that fi(t′) = 0.

Let t0 = inft ∈ I ∩ R>0 | There exists i ∈ C with fi(t) = 0. It follows that:

91

1. t0 ∈ R>0 because t0 ≥ mini∈Cεi.

2. f(t0) ∈ Rn≥0, from the definition of t0.

3. There exists i1 ∈ C such that O(fi1 , t0) = 1. This follows because there exist i0 ∈ C

and T ⊆ I ∩R>0 such that t0 = inf(T ) and for all t ∈ T : fi0(t) = 0. By continuity,

fi0(t0) = 0. Hence, O(fi0 , t0) > 0. Since i0 ∈ C, O(fi0 , 0) 6= ∞. By connectedness

of I, O(fi0 , t0) 6= ∞. Therefore, 0 < O(fi0 , t0) < ∞. Since f(t0) ∈ Rn≥0, by

Lemma 4.3.10, there exists i1 ∈ 1, 2, . . . , n such that O(fi1 , t0) = 1. Assume

i1 /∈ C. Then O(fi1 , 0) =∞. By connectedness of I, O(fi1 , t0) =∞, contradicting

that O(fi1 , t0) = 1. Hence, i1 ∈ C.

Hence, fi1(t0) = 0. Since f(t0) ∈ Rn≥0, by Lemma 4.3.8 f ′i1(t0) ∈ R>0.

From the definition of t0, for all t ∈ (0, t0), fi1(t) ∈ R>0. Since t0 ∈ R>0,

f ′i1(t0) = limh→0+

fi1(t0)− fi1(t0 − h)h

= limh→0+

−fi1(t0 − h)h

∈ R≤0,

a contradiction. The theorem follows.

There is a notion in chemistry that, for systems of chemical reactions, concentrations

evolve through time to reach equilibrium. In later sections of this paper, we will investi-

gate this notion. In the remainder of this section of the paper, we will prepare for that

investigation.

Definition 4.3.5. Let E be a finite event-system of dimension n, let Ω ⊆ C be open,

simply connected and such that R≥0 ⊆ Ω, let f be an E-process on Ω, and let q ∈ Cn.

92

Then q is an ω-limit point of f iff for all ε ∈ R>0 there exists a sequence of non-negative

reals tii∈Z>0 such that ti →∞ as i→∞ and for all i ∈ Z>0, ‖f(ti)− q‖2 < ε.

Sometimes, an ω-limit is defined by the existence of a single sequence of times such

that the value approaches the limit. The above definition is easily seen to be equivalent.

Definition 4.3.6. Let E be a finite event-system of dimension n and let S ⊆ Cn. S

is an invariant set of E iff for all q ∈ S, for all open, simply-connected Ω ⊆ C, for all

E-processes f on Ω, if 0 ∈ Ω and f(0) = q then for all t ∈ R≥0 such that [0, t] ⊆ Ω,

f(t) ∈ S.

Lemma 4.3.11. Let E be a finite, physical event-system of dimension n, let Ω ⊆ C be

open and simply connected, and let f be an E-process on Ω. If R≥0 ⊆ Ω and f(0) is a

non-negative point, then the set of all ω-limit points of f is an invariant set of E and is

contained in Rn≥0.

Proof. Let S be the set of all ω-limit points of f . By Lemma 4.3.5, for all t ∈ R≥0,

f(t) ∈ Rn≥0, hence S ⊆ Rn

≥0.

Let q ∈ S, let Ω ⊆ C be open, simply-connected, and such that 0 ∈ Ω, and let h be

an E-process on Ω such that h(0) = q. Suppose u ∈ R≥0 and [0, u] ⊆ Ω. Since E is finite

and physical, P E |Rn can be viewed as a map F : Rn → Rn of class C1. By Lemma 4.3.2,

for all t ∈ [0, u], h(t) ∈ Rn, so h|[0,u] can be viewed as a map X : [0, u] → Rn such

that X ′ = F (X). By [43], there exists a neighborhood U ⊂ Rn of q and a constant

K such that for all α ∈ U , there exists a unique real-E-process ρα defined on [0, u]

with ρα(0) = α and ‖ρα(u) − h(u)‖2 ≤ K‖α − q‖2 exp(Ku). Observe that necessarily

K ∈ R≥0. By Lemma 4.3.4 for all α ∈ U there exists an open, simply-connected Ωα ⊆ C

93

and an E-process %α on Ωα such that [0, u] ⊆ Ωα and for all t ∈ [0, u], %α(t) = ρα(t).

Therefore, ‖%α(u)− h(u)‖2 ≤ K‖α− q‖2 exp(Ku).

Let ε ∈ R>0 and let δ1, δ2 ∈ R>0 be such that Kδ1 exp(Ku) ≤ ε and the open ball

centered at q of radius δ2 is contained in U . Let δ = min(δ1, δ2). Since q is an ω-limit

point of f , there exists a sequence of non-negative reals tii∈Z>0 such that ti → ∞ as

i → ∞ and for all i ∈ Z>0, ‖f(ti) − q‖2 < δ. Then for all i ∈ Z>0, f(ti) ∈ U , so by

Lemma 4.3.3 for all t ∈ [0, u], f(ti + t) = %f(ti)(t). Then

‖f(ti + u)− h(u)‖2 = ‖%f(ti)(u)− h(u)‖2

≤ K‖f(ti)− q‖2 exp(Ku)

≤ Kδ exp(Ku)

≤ ε

Thus h(u) is an ω-limit point of f , so S is an invariant set of E .

4.4 Finite Natural Event-Systems

In this section, we focus on finite, natural event-systems — a subclass of finite, physical

event-systems which has much in common with systems of chemical reactions that obey

detailed balance.

In chemical reactions, the total bond energy of the reactants minus the total bond

energy of the products is a measure of the heat released. For example, in the reaction,

σX2 − τX1, ln(στ

)is taken to be the quantity of heat released. If there are multiple

94

reaction paths that take the same reactants to the same products, then the quantity of

heat released along each path must be the same.

The finite, physical event-system E = 2X2 − X1, X2 − X1 does not behave like a

chemical reaction system since, when X2 is converted to X1 by the first reaction, ln (2)

units of heat are released; however, when X2 is converted to X1 by the second reaction,

ln (1) = 0 units of heat are released. When an event-system admits a pair of paths from

the same reactants to the same products but with different quantities of heat released,

we say that the system has an “energy cycle”.

Definition 4.4.1 (Energy cycle). Let E be a finite, physical event-system. E has an

energy cycle iff GE has a cycle of non-zero weight.

Example 4.4.1. For the physical event-system E1 = 2X2 − X1, X2 − X1, the event

X2 −X1 induces an edge 〈X2, X1〉 in the event graph with weight ln(

11

)= 0. The event

2X2 − X1 induces an edge 〈X1, X2〉 with weight − ln(

21

)= − ln (2). The weight of the

cycle from X2 to X1 and back to X2 using these two edges, is − ln (2) 6= 0. Hence, E1 has

an energy cycle by Definition 4.4.1.

Example 4.4.2. For the physical event-system E2 = X2−X1, 2X3X4−X2X3, X4X5−

X1X5, the cycle 〈X3X4X5, X2X3X5, X1X3X5, X3X4X5〉 is induced by the sequence of

events 2X3X4−X2X3, X2−X1, X4X5−X1X5 and has corresponding weight ln 21 +ln 1

1 +

ln 11 = ln (2) 6= 0. Hence, E2 has an energy cycle.

The following theorem gives multiple characterizations of natural event-systems.

Theorem 4.4.1. Let E be a finite, physical event-system of dimension n. The following

are equivalent:

95

1. E is natural.

2. E has a strong equilibrium point that is not a z-point. (i.e., there exists α ∈ Cn

such that for all i = 1 to n, αi 6= 0 and for all e ∈ E, e (α) = 0.)

3. E has no energy cycles.

4. If E = σ1M1−τ1N1, σ2M2−τ2N2, . . . , σmMm−τmNm and for all j = 1, 2, . . . ,m,

Mj ≺ Nj and σj , τj > 0 then there exists α ∈ Rn such that

ΓEα =⟨

ln(σ1

τ1

), . . . , ln

(σmτm

)⟩T.

To prove Theorem 4.4.1, we will use the following lemma.

Lemma 4.4.2. Let E = σ1M1−τ1N1, σ2M2−τ2N2, . . . , σmMm−τmNm be a finite, phys-

ical event-system of dimension n such that for all j = 1, 2, . . . ,m : σj , τj > 0 and Mj ≺ Nj.

Then for all α = 〈α1, α2, . . . , αn〉T ∈ Rn, ΓE · α =⟨

ln(σ1τ1

), ln(σ2τ2

), . . . , ln

(σmτm

)⟩Tiff

〈eα1 , · · · , eαn〉 is a positive strong E-equilibrium point.

96

Proof. Let E = σ1M1 − τ1N1, σ2M2 − τ2N2, . . . , σmMm − τmNm and for all j =

1, 2, . . . ,m : Mj ≺ Nj and σj , τj > 0. Let Γ = ΓE . For all α = 〈α1, . . . , αn〉 ∈ Rn,

Γα =⟨

ln(σ1

τ1

), ln(σ2

τ2

), . . . , ln

(σmτm

)⟩T⇔

n∑i=1

γj,iαi = ln (σj/τj) , ∀j = 1, 2, . . . ,m

⇔n∏i=1

(eαi)γj,i = σj/τj , ∀j = 1, 2, . . . ,m (Exponentiation.)

⇔Nj (〈eα1 , . . . , eαn〉) /Mj (〈eα1 , . . . , eαn〉) = σj/τj , ∀j = 1, 2, . . . ,m (Definition of Γ.)

⇔σjMj (〈eα1 , . . . , eαn〉)− τjNj (〈eα1 , . . . , eαn〉) = 0, ∀j = 1, 2, . . . ,m

⇔〈eα1 , . . . , eαn〉 is a positive strong E-equilibrium point.

Proof of Theorem 4.4.1. (4) ⇒ (1) : Follows from Lemma 4.4.2.

(1) ⇒ (2) : Follows immediately from definitions.

(2) ⇒ (3) : Consider an arbitrary cycle C in GE given by the sequence of k edges

〈v0, v1〉, 〈v1, v2〉, . . . , 〈vk−1, vk = v0〉 with corresponding weights r1, r2, . . . , rk. By Defi-

nition 4.1.9, for i = 1, 2, . . . , k, there exist Ti ∈ M∞ and ei ∈ E with ei = σiMi − τiNi

where σi, τi > 0 and Mi, Ni ∈M∞ and Mi ≺ Ni such that either

1) vi−1 = TiMi and vi = TiNi and ri = ln σiτi∈ w (〈vi−1, vi〉) or

97

2) vi−1 = TiNi and vi = TiMi and ri = − ln σiτi∈ w (〈vi−1, vi〉)

Hence, there exists a vector b = 〈b1, b2, . . . , bk〉 with bi = 0 or 1 such that:

k∏i=1

M bii N

1−bii =

k∏i=1

M1−bii N bi

i (4.11)

w (C) =k∑i=1

ri =k∑i=1

(2bi − 1) ln(σiτi

)(4.12)

Let α be a strong equilibrium point of E that is not a z-point. Then, by Definition 4.1.7,

for i = 1 to k, σiMi (α)− τiNi (α) = 0

⇒ σiMi (α) = τiNi (α) for i = 1 to k

⇒ (σiMi (α))bi = (τiNi (α))bi and (τiNi (α))1−bi = (σiMi (α))1−bi for i = 1 to k

⇒ (σiMi (α))bi (τiNi (α))1−bi = (σiMi (α))1−bi (τiNi (α))bi for i = 1 to k

⇒∏ki=1 (σiMi (α))bi (τiNi (α))1−bi =

∏ki=1 (σiMi (α))1−bi (τiNi (α))bi

⇒∏ki=1 σi

biτi1−bi =

∏ki=1 σi

1−biτibi [From Equation (1) and since α is not a z-point]

⇒∏ki=1

σibiτi

1−bi

σi1−biτibi= 1

⇒∑k

i=1 (2bi − 1) ln(σiτi

)= 0 [Taking logarithm]

⇒ w (C) = 0 [From Equation (2)]

Hence, E has no energy cycle.

(3) ⇒ (4) :

Let E = σ1M1 − τ1N1, σ2M2 − τ2N2, . . . , σmMm − τmNm and for all j = 1 to m,

Mj ≺ Nj and σj , τj > 0. Let Γ = ΓE . We shall prove that if the linear equation

Γα = 〈ln (σ1/τ1) , . . . , ln (σm/τm)〉T has no solution in Rn then E has an energy cy-

cle. For j = 1 to m, let Γj be the jth row of Γ. If the system of linear equations

Γα = 〈ln (σ1/τ1) , . . . , ln (σm/τm)〉T has no solution in Rn then, from linear algebra [62,

98

p. 164, Theorem] and the fact that Γ is a matrix of integers, it follows that there exists

l, there exist (not necessarily distinct) integers j1, j2, . . . , jl ∈ 1, 2, . . . ,m, there exist

a1, a2, . . . , al ∈ +1,−1 such that:

a1Γj1 + a2Γj2 + · · ·+ alΓjl = 0 (4.13)

a1 ln (σj1/τj1) + a2 ln (σj2/τj2) + · · ·+ al ln (σjl/τjl) 6= 0 (4.14)

Consider the sequence C of l + 1 vertices in the event-graph defined recursively by

v0 =l∏

i=1,ai=+1

Mji

l∏i=1,ai=−1

Nji

and for i = 1 to l,

vi =vi−1N

aiji

Maiji

Observe that by (3),l∏

i=1

(Nji

Mji

)ai= 1

Hence,

v0 =l∏

i=1,ai=+1

Maiji

l∏i=1,ai=−1

N−aiji=

l∏i=1,ai=+1

Naiji

l∏i=1,ai=−1

M−aiji= vl

99

Hence, C is a cycle. Further, for i = 1 to l,

ai ln σjiτji∈ w (〈vi−1, vi〉)

From Equation (4),

w (C) = a1 ln (σj1/τj1) + a2 ln (σj2/τj2) + · · ·+ al ln (σjl/τjl) 6= 0

Hence, C is an energy cycle.

Horn and Jackson [45] and Feinberg [30] have proved that chemical reaction networks

with appropriate properties admit Lyapunov functions. While finite, natural event-

systems are closely related to the chemical reaction networks considered by Horn and

Jackson and by Feinberg, they are not identical. Consequently, we will prove the exis-

tence of Lyapunov functions for finite, natural event-systems (Theorem 4.4.6).

The Lyapunov function is analogous in form and properties to “Entropy of the Uni-

verse” in thermodynamics. The Lyapunov function composed with an event-process is

monotonic with respect to time, providing an analogy to the second law of thermody-

namics.

Definition 4.4.2. Let E be a finite, natural event-system of dimension n with positive

strong E-equilibrium point c = 〈c1, c2, . . . , cn〉. Then gE,c : Rn>0 → R is given by

gE,c (x1, x2, . . . , xn) =n∑i=1

(xi (ln (xi)− 1− ln (ci)) + ci)

The function gE,c will turn out to be the desired Lyapunov function.

100

Note that if E1 and E2 are two finite natural event-systems of the same dimension and

if c is a positive strong E1-equilibrium point as well as a positive strong E2-equilibrium

point, then the functions gE1,c and gE2,c are identical.

Lemma 4.4.3. Let E = σ1M1 − τ1N1, σ2M2 − τ2N2, . . . , σmMm − τmNm be a finite,

natural event-system of dimension n with positive strong E-equilibrium point c, such that

for all j = 1 to m, σj , τj > 0 and Mj ≺ Nj. Then for all x ∈ Rn>0,

∇gE,c (x) · P E (x) =m∑j=1

(σjMj (x)− τjNj (x)) ln(τjNj (x)σjMj (x)

)

Proof. Let g = gE,c. Let x = 〈x1, x2, . . . , xn〉 ∈ Rn>0. Let P = P E .

∇g (x) · P (x) =n∑i=1

(∂g

∂xi(x) · Pi (x)

)

=n∑i=1

ln(xici

) m∑j=1

γj,i (σjMj (x)− τjNj (x))

=

m∑j=1

(σjMj (x)− τjNj (x))n∑i=1

ln((

xici

)γj,i)

=m∑j=1

(σjMj (x)− τjNj (x)) ln

(n∏i=1

(xici

)γj,i)

=m∑j=1

(σjMj (x)− τjNj (x)) ln(τjNj (x)σjMj (x)

)

The last equality follows from the definition of ΓE and the fact that c is a strong-

equilibrium point.

Lemma 4.4.4. For all x ∈ R>0, (1− x) ln (x) ≤ 0 with equality iff x = 1.

101

Proof. If 0 < x < 1 then 1− x > 0 and ln(x) < 0. If x > 1 then 1− x < 0 and ln(x) > 0.

In either case, the product is strictly negative. If x = 1 then (1− x) ln (x) = 0

Theorem 4.4.5. Let E be a finite, natural event-system of dimension n with positive

strong E-equilibrium point c. Then for all x ∈ Rn>0, ∇gE,c (x) · P E (x) ≤ 0 with equality

iff x is a strong E-equilibrium point.

Proof. Let E = σ1M1 − τ1N1, σ2M2 − τ2N2, . . . , σmMm − τmNm be a finite, natural

event-system of dimension n with positive strong E-equilibrium point c, such that for all

j = 1 to m, σj , τj > 0 and Mj ≺ Nj . Let P = P E and let g = gE,c. By Lemma 4.4.3, for

all x ∈ Rn>0,

∇g (x) · P (x) =m∑j=1

(σjMj (x)− τjNj (x)) ln(τjNj (x)σjMj (x)

)

From Lemma 4.4.4 and the observation that for j = 1, 2, . . . ,m, Mj (x) , Nj (x) > 0 when

x ∈ Rn>0 and by assumption σj , τj > 0, we have,

∇g (x) · P (x) ≤ 0

with equality iff for all j = 1, 2, . . . ,m, σjMj (x) = τjNj (x). This occurs iff x is a strong

E-equilibrium point.

Recall that a function g is a Lyapunov function at a point p for a vector field v iff g

is smooth, positive definite at p and Lvg is negative semi-definite at p [46, p. 131]. For

a finite natural event-system E , P E induces a vector field on Rn. We will show that, if

102

c is a positive strong E-equilibrium point, then gE,c is a Lyapunov function at c for the

vector field induced by P E .

Theorem 4.4.6 (Existence of Lyapunov Function). Let E be a finite, natural event-

system of dimension n with positive strong E-equilibrium point c. Then gE,c is a Lyapunov

function for the vector field induced by P E at c.

Proof. Let g = gE,c. For i = 1, 2, . . . , n:

∂g

∂xi= ln

(xici

)

which are all in C∞ as functions on Rn>0, hence g is in C∞.

∂g

∂xi(c) = ln

(cici

)= 0

establishes that ∇g (c) = 0. For i = 1, 2, . . . , n, for k = 1, 2, . . . , n:

∂2g

∂xk∂xi=δi,kxi

where δi,k is the Kronecker delta function. Hence, for all x ∈ Rn>0, the Hessian of g at

x is positive definite. Therefore, g is strictly convex over Rn>0. Further, g (c) = 0 and

∇g (c) = 0 and g is strictly convex together imply that g is positive definite at c. To

establish g as a Lyapunov function, it remains to show that the directional derivative LP g

of g in the direction of the vector field induced by P = P E is negative semi-definite at c.

This follows from Theorem 4.4.5 since for all x ∈ Rn>0, LP g (x) = ∇g (x) ·P (x) ≤ 0.

103

Henceforth, the function gE,c will be called the Lyapunov function of E at c. The next

theorem shows that finite, natural event-systems satisfy a form of “detailed balance”.

Theorem 4.4.7. If E is a natural, finite event-system of dimension n then all positive

E-equilibrium points are strong E-equilibrium points.

Proof. Let P = P E . Let c ∈ Rn>0 be a positive strong E-equilibrium point. Let x be

a positive E-equilibrium point. That is, P (x) = 0. Hence, ∇gE,c (x) · P E (x) = 0. By

Theorem 4.4.5, x is a strong E-equilibrium point.

The following lemma was proved by Feinberg [30].

Lemma 4.4.8. Let n > 0 be an integer. Let U be a linear subspace of Rn, and let

a = 〈a1, a2, . . . , an〉 and b be elements of Rn>0. There is a unique element

µ = 〈µ1, µ2, · · · , µn〉 ∈ U⊥

such that 〈a1eµ1 , a2eµ2 , . . . , aneµn〉 − b is an element of U .

The next theorem follows from one proved by Horn and Jackson [45]. Our proof is

derived from Feinberg’s [30].

Theorem 4.4.9. Let E be a finite, natural event-system of dimension n. Let H be a pos-

itive conservation class of E. Then H contains exactly one positive strong E-equilibrium

point.

Proof. Let Γ = ΓE . Let c∗ = 〈c∗1, c∗2, . . . , c∗n〉 be a positive strong E-equilibrium point. Let

p ∈ H ∩ Rn>0. For all c ∈ Rn

>0,

104

(1) c is a strong E-equilibrium point

⇔ Γ〈ln(c1), ln(c2), . . . , ln(cn)〉T = Γ〈ln(c∗1), ln(c∗2), · · · , ln(c∗n)〉T . (Lemma 4.4.2)

⇔ Γ⟨

ln(c1

c∗1

), ln(c2

c∗2

), . . . , ln

(cnc∗n

)⟩T= 0

⇔ There exists µ = 〈µ1, µ2, . . . , µn〉 ∈ ker Γ ∩ Rn such that⟨ln(c1

c∗1

), ln(c2

c∗2

), . . . , ln

(cnc∗n

)⟩T= µ.

⇔ There exists µ = 〈µ1, µ2, . . . , µn〉 ∈ ker Γ ∩ Rn such that

ci = c∗i eµi for i = 1, 2, . . . , n.

(2) c ∈ H ∩ Rn ⇔ c− p ∈ (ker Γ)⊥ ∩ Rn. (Definition 4.2.6)

From (1) and (2), c is a positive strong E-equilibrium point in H iff there exists

µ ∈ ker Γ ∩ Rn such that c = 〈c∗1eµ1 , c∗2eµ2 , . . . , c∗neµn〉 and

〈c∗1eµ1 , c∗2eµ2 , . . . , c∗neµn〉 − p ∈ (ker Γ)⊥ ∩ Rn. Applying Lemma 4.4.8 with a = c∗, b = p

and U = (ker Γ)⊥ ∩ Rn, it follows that there exists a unique µ of the desired form.

Hence, there exists a unique positive strong E-equilibrium point in H given by c =

〈c∗1eµ1 , c∗2eµ2 , . . . , c∗neµn〉.

To prove the main theorem of this section (Theorem 4.4.15), we will first establish

several technical lemmas.

Lemma 4.4.10 shows that an event that remains zero at all times along a process can

be ignored.

105

Lemma 4.4.10. Let E be a finite event-system of dimension n, let Ω ⊆ C be non-empty,

open and simply-connected, and let f = 〈f1, f2, . . . , fn〉 be an E-process on Ω. Then either

for all t ∈ Ω, f(t) is a strong E-equilibrium point or there exist a finite event-system E

of dimension n ≤ n, an E-process f = 〈f1, f2, . . . , fn〉 on Ω, and a permutation π on

1, 2, . . . , n such that:

1. If E is physical then E is physical.

2. If E is natural then E is natural.

3. If c = 〈c1, c2, . . . , cn〉 is a positive strong E-equilibrium point, then

c = 〈cπ−1(1), cπ−1(2), . . . , cπ−1(n)〉 is a positive strong E-equilibrium point.

4. For all e ∈ E, there exists t ∈ Ω such that e(f(t)) 6= 0.

5. If E is natural, I ⊆ Ω ∩ R≥0 is connected, 0 ∈ I and f(0) is a non-negative point

then for all t ∈ I ∩ R>0, f(t) is a positive point.

6. For i = 1, 2, . . . , n, if π(i) ≤ n then for all t ∈ Ω, fi(t) = fπ(i)(t).

7. For i = 1, 2, . . . , n, if π(i) > n then for all t1, t2 ∈ Ω, fi(t1) = fi(t2).

Proof. Let m = |E|. Let E1 = e ∈ E | there exists t ∈ Ω, e(f(t)) 6= 0. If E1 = ∅

then for all t ∈ Ω, e(f(t)) = 0, so f(t) is a strong E-equilibrium point and the Lemma

holds. Assume E1 6= ∅ and let m = |E1|. For j = 1, 2, . . . ,m, let σj , τj ∈ R>0 and Mj =∏ni=1X

aj,ii , Nj =

∏ni=1X

bj,ii ∈ M∞ be such that Mj ≺ Nj and σ1M1 − τ1N1, σ2M2 −

τ2N2, . . . , σmMm−τmNm = E1 and σ1M1−τ1N1, σ2M2−τ2N2, . . . , σmMm−τmNm =

E .

106

Let C = i | there exists j ≤ m such that either aj,i 6= 0 or bj,i 6= 0. Let n = |C|.

Let π be a permutation on 1, 2, . . . , n such that π(C) = 1, 2, . . . , n.

For j = 1, 2, . . . , m, let eπ,j = σj∏ni=1X

aj,π−1(i)

i − τj∏ni=1X

bj,π−1(i)

i and let E =

eπ,1, eπ,2, . . . , eπ,m.

It follows that E is a finite event-system of dimension n ≤ n. For i = 1, 2, . . . , n, let

fi = fπ−1(i). Let f = 〈f1, f2, . . . , fn〉.

Let (γj,i)m×n = ΓE . Let (γj,i)m×n = ΓE . It follows that for j = 1, 2, . . . , m, for

i = 1, 2, . . . , n,

γj,i = bj,π−1(i) − aj,π−1(i) = γj,π−1(i). (4.15)

We claim that f is an E-process on Ω. To see this, for k = 1, 2, . . . , n, for all t ∈ Ω :

f ′k(t) = f ′π−1(k)(t) (4.16)

=

m∑j=1

γj,π−1(k)

(σj

n∏i=1

Xaj,ii − τj

n∏i=1

Xbj,ii

) f (t) (4.17)

=

m∑j=1

γj,π−1(k)

(σj

n∏i=1

Xaj,ii − τj

n∏i=1

Xbj,ii

) f (t) (4.18)

=

m∑j=1

γj,π−1(k)

(σj∏i∈C

Xaj,ii − τj

∏i∈C

Xbj,ii

) f (t) (4.19)

=

m∑j=1

γj,π−1(k)

(σj

n∏i=1

Xaj,π−1(i)

π−1(i)− τj

n∏i=1

Xbj,π−1(i)

π−1(i)

) f (t) (4.20)

=m∑j=1

γj,π−1(k)

(σj

n∏i=1

(fπ−1(i)(t))aj,π−1(i) − τj

n∏i=1

(fπ−1(i)(t))bj,π−1(i)

)(4.21)

=m∑j=1

γj,k

(σj

n∏i=1

(fπ−1(i)(t))aj,π−1(i) − τj

n∏i=1

(fπ−1(i)(t))bj,π−1(i)

)(4.22)

107

=m∑j=1

γj,k

(σj

n∏i=1

(fi(t))aj,π−1(i) − τj

n∏i=1

(fi(t))bj,π−1(i)

)(4.23)

=

m∑j=1

γj,keπ,j

f (t) (4.24)

Equation (4.16) follows from the definition of fk. Equation (4.17) follows because f

is an E-process on Ω. Equation (4.18) follows from the definition of E1. Equation (4.19)

follows because, for j ≤ m, if i /∈ C then aj,i = bj,i = 0. Equation (4.20) follows

because π(C) = 1, 2, . . . , n. Equation (4.21) follows from the rules of composition.

Equation (4.22) follows from Equation (4.15). Equation (4.23) follows by the definition

of fi. Equation (4.24) follows by the definition of eπ,j . This establishes the claim.

With E , n, f and π as described, we will now establish (1) through (7).

(1) Follows from the definition of E .

(2) Follows from 3.

(3) Suppose E is natural. Hence, there exists a positive strong E-equilibrium point

〈c1, c2, . . . , cn〉. For j = 1, 2, . . . , m :

eπ,j(cπ−1(1), cπ−1(2), . . . , cπ−1(n)) = σj

n∏i=1

caj,π−1(i)

π−1(i)− τj

n∏i=1

cbj,π−1(i)

π−1(i)

= σj∏i∈C

caj,ii − τj

∏i∈C

cbj,ii

108

(If j ≤ m and i /∈ C then aj,i = bj,i = 0.)

= ej(c1, c2, . . . , cn)

= 0

Hence, c is a positive strong E-equilibrium point.

(4) Suppose j ≤ m. Then for all t ∈ Ω :

eπ,j(f(t)) = σj

n∏i=1

(fi(t))aj,π−1(i) − τj

n∏i=1

(fi(t))bj,π−1(i)

= σj

n∏i=1

(fπ−1(i)(t))aj,π−1(i) − τj

n∏i=1

(fπ−1(i)(t))bj,π−1(i)

= σj∏i∈C

(fi(t))aj,i − τj∏i∈C

(fi(t))bj,i

= σj

n∏i=1

(fi(t))aj,i − τjn∏i=1

(fi(t))bj,i

(Again, if j ≤ m and i /∈ C then aj,i = bj,i = 0.)

=

((σj

n∏i=1

Xaj,ii − τj

n∏i=1

Xbj,ii

) f

)(t)

= ej(f(t))

Since j ≤ m, therefore ej ∈ E1 and there exists t ∈ Ω such that ej(f(t)) 6= 0. Hence,

for all eπ,j ∈ E , there exists t ∈ Ω such that eπ,j(f(t)) 6= 0.

109

(5) Suppose E is natural, I ⊆ Ω ∩ R≥0 is connected, 0 ∈ I and f(0) is a non-negative

point. It follows that f(0) is a non-negative point and, from Theorem 4.3.5, for

all t ∈ I, f(t) is a non-negative point. Suppose, for the sake of contradiction, that

there exist i0 ≤ n and t0 ∈ I∩R>0 such that fi0(t0) = 0. From Theorem 4.3.5 again,

fi0(0) = 0 and for all t ∈ I : fi0(t) = 0. Since I is an interval and 0, t0 ∈ I, I contains

an accumulation point. Hence, since fi0 is analytic on Ω and Ω is connected, for all

t ∈ Ω :

fi0(t) = 0. (4.25)

It follows that for all t ∈ Ω :

0 = f ′i0(t) =m∑j=1

γj,i0eπ,j(f(t)). (4.26)

We claim that for j = 1, 2, . . . , m, for all t ∈ Ω : γj,i0eπ,j(f(t)) ≥ 0.

Case 1: Suppose γj,i0 = 0. Then γj,i0eπ,j(f(t)) = 0 ≥ 0.

Case 2: Suppose γj,i0 > 0. Then bj,π−1(i0) > 0. Hence,

eπ,j(f(t)) = σj

n∏i=1

(fi(t))aj,π−1(i) − τj

n∏i=1

(fi(t))bj,π−1(i) (4.27)

= σj

n∏i=1

(fi(t))aj,π−1(i) (4.28)

≥ 0 (4.29)

110

Equation (4.28) follows since bj,π−1(i0) > 0 and, by Equation (4.25), fi0(t) = 0.

Equation (4.29) follows since f(t) is a non-negative point, by Theorem 4.3.5.

Hence, γj,i0eπ,j(f(t)) ≥ 0.

Case 3: Suppose γj,i0 < 0. Then aj,π−1(i0) > 0. Hence,

eπ,j(f(t)) = σj

n∏i=1

(fi(t))aj,π−1(i) − τj

n∏i=1

(fi(t))bj,π−1(i) (4.30)

= −τjn∏i=1

(fi(t))bj,π−1(i) (4.31)

≤ 0 (4.32)

Equation (4.31) follows since aj,π−1(i0) > 0 and, by Equation (4.25), fi0(t) = 0.

Equation (4.32) follows since f(t) is a non-negative point, by Theorem 4.3.5.

Hence, γj,i0eπ,j(f(t)) ≥ 0.

This completes the proof of the claim.

From Equation (4.26) and the claim, it now follows that for j = 1, 2, . . . , m, for all

t ∈ Ω :

γj,i0eπ,j(f(t)) = 0 (4.33)

Since i0 ≤ n, there exists j0 ≤ m such that either aj0,i0 6= 0 or bj0,i0 6= 0. If

γj0,i0 6= 0 then, from Equation (4.33), eπ,j0(f(t)) = 0. If γj0,i0 = 0 then, since

γj0,i0 = bj0,i0 − aj0,i0 , it follows that aj0,i0 6= 0 and bj0,i0 6= 0. Hence, Xi0 divides

eπ,j0 . From Equation (4.25), it follows that eπ,j0(f(t)) = 0. Hence, irrespective

111

of the value of γj0,i0 , for all t ∈ Ω : eπ,j0(f(t)) = 0. Since eπ,j0 is an element of

E , this leads to a contradiction with Lemma 4.4.10.4. Hence, for all i ≤ n, for all

t ∈ I ∩ R>0 : fi(t) > 0.

(6) Follows from the definition of f .

(7) For i = 1, 2, . . . , n, if π(i) > n then i /∈ C. That is, for j = 1, 2, . . . ,m : γj,i =

bj,i − aj,i = 0 − 0 = 0. Hence, for all t ∈ Ω : f ′i(t) =∑m

j=1 γj,iej(f(t)) = 0. Hence,

since fi is analytic on Ω, and Ω is simply-connected, for all t1, t2 ∈ Ω : fi(t1) = fi(t2).

We have described, for finite, natural event-systems, Lyapunov functions on the posi-

tive orthant. We next extend the definition of these Lyapunov functions to admit values

at non-negative points.

Definition 4.4.3. Let E be a finite, natural event-system of dimension n with positive

strong E-equilibrium point c = 〈c1, c2, . . . , cn〉. For all v ∈ R>0, let gv : R≥0 → R be such

that for all x ∈ R≥0

gv(x) =

x(ln(x)− 1− ln(v)) + v, if x > 0;

v, otherwise.

(4.34)

Then the extended lyapunov function gE,c : Rn≥0 → R is

gE,c(x1, x2, . . . , xn) =n∑i=1

gci(xi) (4.35)

112

The next lemma lists some properties of extended Lyapunov functions.

Lemma 4.4.11. Let E be a finite, natural event-system of dimension n with positive

strong E-equilibrium point c = 〈c1, c2, . . . , cn〉. Then:

1. gE,c is continuous on Rn≥0.

2. For all x1, x2, . . . , xn ∈ R≥0, gE,c(x1, x2, . . . , xn) ≥ 0 with equality iff

〈x1, x2, . . . , xn〉 = c.

3. For all r ∈ R≥0, the set x ∈ Rn≥0 | gE,c(x) ≤ r is bounded.

4. If Ω ⊆ C is open, simply connected and such that 0 ∈ Ω, f = 〈f1, f2, . . . , fn〉 is

an E-process on Ω such that f(0) is a non-negative point, and I ⊆ R≥0 ∩ Ω is an

interval such that 0 ∈ I then (gE,c f) is monotonically non-increasing on I.

Proof. For i = 1, 2, . . . , n, let gci(x) be as defined in Equation (4.34).

1. For i = 1, 2, . . . , n, gci is continuous on R>0 and limx→0+ gci(x) = ci = gci(0), so gci

is continuous on R≥0. Since gE,c is the finite sum of continuous functions on R≥0,

gE,c is continuous on Rn≥0.

2. Let j ∈ 1, 2, . . . , n. Let g = gcj . For all x ∈ R>0, g ′(x) = ln(x

cj

). If 0 < x < cj

then, by substitution, g ′(x) < 0. Similarly, if x > cj then g ′(x) > 0. Hence, g is

monotonically decreasing in (0, cj) and monotonically increasing in (cj ,∞). From

continuity of g in R≥0, it follows that

For all x ∈ R≥0, g(x) ≥ g(cj) = 0 with equality iff x = cj . (4.36)

113

From Equations (4.35) and (4.36), the claim follows.

3. Observe that limx→+∞ g(x) = +∞. It follows that:

For all r ∈ R≥0, the set x ∈ R≥0 | g(x) ≤ r is bounded. (4.37)

If x1, x2, . . . , xn ∈ R≥0 are such that gE,c(x1, x2, . . . , xn) ≤ r, it follows from Equa-

tions (4.35) and (4.36) that for i = 1, 2, . . . , n : gci(xi) ≤ r. The claim now follows

from Equation (4.37).

4. Let Ω ⊆ C be open, simply connected, and such that 0 ∈ Ω; let f = 〈f1, f2, . . . , fn〉

be an E-process on Ω such that f(0) is a non-negative point; and let I ⊆ R≥0 ∩ Ω

be an interval such that 0 ∈ I. By Lemma 4.4.10 there exists n, E , f , and π

satisfying 4.4.10.1–4.4.10.7. Let c = 〈c1, c2, . . . , cn〉 = 〈cπ−1(1), cπ−1(2), . . . , cπ−1(n)〉.

By Lemma 4.4.10.2, c is a positive strong equilibrium point of E . Then for all t ∈ I,

(gE,c f) (t) =n∑i=1

gci (fi(t)) [Equation (4.35).]

=∑

i:π(i)≤n

gci (fi(t)) +∑

i:π(i)>n

gci (fi(t))

=n∑i=1

gcπ−1(i)

(fπ−1(i)(t)

)+

∑i:π(i)>n

gci (fi(t))

=n∑i=1

gci

(fi(t)

)+

∑i:π(i)>n

gci (fi(t)) [Lemma 4.4.10.6.]

=(gE,c f

)(t) +

∑i:π(i)>n

gci (fi(t)) [Equation (4.35).]

=(gE,c f

)(t) + constant [Lemma 4.4.10.7.]

114

By Definition 4.4.3, for all x ∈ Rn>0, gE,c(x) = gE,c(x). By Lemma 4.4.10.5, for all

t ∈ I ∩ R>0, f(t) ∈ Rn>0. So for all t ∈ I ∩ R>0,

(gE,c f

)(t) =

(gE,c f

)(t).

Then, for all t ∈ I ∩ R>0,

(gE,c f

)′(t) =

(gE,c f

)′(t)

= ∇gE,c(f(t)

)· f ′(t) [Chain rule.]

= ∇gE,c(f(t)

)· P E

(f(t)

)[Definition 4.2.3.]

≤ 0 [Theorem 4.4.5.]

Therefore(gE,c f

)is non-increasing on I ∩ R>0.

By Definition 4.2.3, f is continuous on I; by Theorem 4.3.5, f(I) ⊆ Rn≥0; and

by Lemma 4.4.11.1, gE,c is continuous on Rn≥0; so

(gE,c f

)is continuous on I.

Therefore(gE,c f

)is non-increasing on I. Thus (gE,c f) is a constant plus

a monotonically non-increasing function on I, so (gE,c f) is monotonically non-

increasing on I.

The next lemma makes use of properties of the extended Lyapunov function to show

that E-processes starting at non-negative points are uniformly bounded in forward real

time.

Lemma 4.4.12. Let E be a finite, natural event-system of dimension n. Let α ∈ Rn≥0.

There exists k ∈ R≥0 such that for all Ω ⊆ C open and simply connected and such that

0 ∈ Ω, for all E-processes f = 〈f1, f2, . . . , fn〉 on Ω such that f(0) = α, for all intervals

I ⊆ Ω∩R≥0 such that 0 ∈ I, for all t ∈ I, for i = 1, 2, . . . , n: fi(t) ∈ R and 0 ≤ fi(t) < k.

115

Proof. Since E is natural, let c ∈ Rn>0 be a positive strong E-equilibrium point. Let

g = gE,c.

Let ` = g(α). Let S = x ∈ Rn≥0 | g(x) ≤ `. By Lemma 4.4.11.3, S is bounded.

Hence, let k be such that for all x ∈ S : |x|∞ < k.

Let Ω ⊆ C be open, simply connected, and such that 0 ∈ Ω; let f = 〈f1, f2, . . . , fn〉

be an E-process on Ω such that f(0) = α; and let I ⊆ R≥0 ∩ Ω be an interval such that

0 ∈ I.

From Theorem 4.3.5, for all t ∈ I, for i = 1, 2, . . . , n : fi(t) ∈ R and fi(t) ≥ 0.

Consider the function:

g f |I : I → R

From Lemma 4.4.11.4, for all t ∈ I, g f |I is monotonically non-increasing on I. That

is, for all t ∈ I,

g(f(t)) ≤ ` (4.38)

It follows from Equation 4.38 and the definition of S that f(I) ⊆ S. By the definition

of k, it follows that for all t ∈ I, for i = 1, 2, . . . , n, fi(t) < k.

The next lemma shows that, because E-processes starting at non-negative points are

uniformly bounded in real time, they can be continued forever along forward real time.

Lemma 4.4.13 (Existence and uniqueness of E-process.). Let E be a finite, natural event-

system of dimension n. Let α ∈ Rn≥0. There exist a simply-connected open set Ω ⊆ C,

an E-process f = 〈f1, f2, . . . , fn〉 on Ω and k ∈ R≥0 such that:

1. R≥0 ⊆ Ω.

116

2. f(0) = α.

3. For all t ∈ R≥0, for i = 1, 2, . . . , n : fi(t) ∈ R and 0 ≤ fi(t) < k.

4. For all simply-connected open sets Ω ⊆ C, for all E-processes f on Ω, for all

intervals I ⊆ Ω ∩ R≥0, if 0 ∈ I and f(0) = α, then for all t ∈ I, f(t) = f(t).

Proof. Claim: There exists k ∈ R≥0 such that for all intervals I ⊆ R≥0 with 0 ∈ I, for all

real-E-processes h = 〈h1, h2, . . . , hn〉 on I with h(0) = α, for all t ∈ I, for i = 1, 2, . . . , n:

0 ≤ hi(t) ≤ k.

To see this, let I ⊆ R≥0 be an interval such that 0 ∈ I. Let h = 〈h1, h2, . . . , hn〉 be a

real-E-process on I such that h(0) = α.

From Lemma 4.3.4, there exist an open, simply-connected Ω ⊆ C and an E-process

f = 〈f1, f2, . . . , fn〉 on Ω such that:

1. I ⊂ Ω

2. For all t ∈ I : f(t) = h(t).

From Lemma 4.4.12, there exists k ∈ R≥0 such that for all t ∈ I, for i = 1, 2, . . . , n:

fi(t) ∈ R and 0 ≤ fi(t) < k. That is, for all t ∈ I, for i = 1, 2, . . . , n : 0 ≤ hi(t) < k. This

proves the claim.

Therefore, by [43, p. 397, Corollary], there exists k ∈ R≥0, there is a real-E-process

h = 〈h1, h2, . . . , hn〉 on R≥0 such that h(0) = α and for all t ∈ R≥0,for i = 1, 2, . . . , n :

0 ≤ hi(t) < k.. By Lemma 4.3.4, there exist an open, simply-connected Ω ⊆ C and an

E-process f on Ω such that R≥0 ⊆ Ω and for all t ∈ R≥0, f(t) = h(t). Therefore, for

all t ∈ R≥0, for i = 1, 2, . . . , n : fi(t) ∈ R and 0 ≤ fi(t) < k. Hence, Parts (1,2,3) are

established. Part(4) follows from Lemma 4.3.2.

117

The next lemma shows that the ω-limit points of E-processes that start at non-negative

points satisfy detailed balance.

Lemma 4.4.14. Let E be a finite, natural event-system of dimension n, let Ω ⊆ C be

open and simply-connected, let f be an E-process on Ω, and let q ∈ Cn. If R≥0 ⊆ Ω and

f(0) is a non-negative point and q is an ω-limit point of f , then q ∈ Rn≥0 and is a strong

E-equilibrium point.

Proof. Suppose R≥0 ⊆ Ω, f(0) is a non-negative point, S is the set of ω-limit points

of f , and q ∈ S. By Lemma 4.3.11, q ∈ Rn≥0. By Lemma 4.4.13, there exist an open,

simply-connected Ωq ⊆ C such that R≥0 ⊆ Ωq and an E-process h = 〈h1, h2, . . . , hn〉 on

Ωq such that h (0) = q.

Let c be a positive strong E-equilibrium point. By Lemma 4.4.11.2, gE,c (f (t))

is bounded below and, by Lemma 4.4.11.4, is monotonically non-increasing on R≥0.

Therefore limt→∞ gE,c (f (t)) exists. Since gE,c is continuous, for all α ∈ S, gE,c (α) =

limt→∞ gE,c (f (t)). By Lemma 4.3.11, for all t ∈ R≥0, h(t) ∈ S. Hence, gE,c (h (t)) is

constant on R≥0.

By Lemma 4.4.10, either q is a strong E-equilibrium or there exists a finite event-

system E of dimension n ≤ n, an E-process h = 〈h1, h2, . . . , hn〉 on Ωq, and a permutation

π on 1, 2, . . . , n satisfying 1–7 of Lemma 4.4.10.

Assume q is not a strong E-equilibrium point. By Lemma 4.4.10.6, for i = 1, 2, . . . , n,

for all t ∈ Ωq, hi (t) = hπ−1(i) (t). Let c = 〈c1, c2, . . . , cn〉 = 〈cπ−1(1), cπ−1(2), . . . , cπ−1(n)〉.

By Lemma 4.4.10.3, c is an E-strong equilibrium point.

118

For all v ∈ R>0, let gv be as defined in Equation 4.34 in Definition 4.4.3. Then for all

t ∈ R≥0,

gE,c (h (t))− gE,c(h (t)

)=

n∑i=1

gci (hi (t))−n∑j=1

gcj

(hj (t)

)

=n∑i=1

gci (hi (t))−n∑j=1

gcπ−1(j)

(hπ−1(j) (t)

)=

n∑i=1

gcπ−1(i)

(hπ−1(i) (t)

)−

n∑j=1

gcπ−1(j)

(hπ−1(j) (t)

)=

n∑i=n+1

gcπ−1(i)

(hπ−1(i) (t)

)

But, by Lemma 4.4.10.7, if π (i) > n then hi (t) is constant. Hence, gcπ−1(i)

(hπ−1(i) (t)

)is constant for i = n + 1, n + 2, . . . , n, so gE,c (h (t)) − gE,c

(h (t)

)is constant. Since

gE,c (h (t)) and gE,c (h (t))−gE,c(h (t)

)are both constant, gE,c

(h (t)

)must be constant.

By Lemma 4.4.10.5, for all t ∈ R>0, h (t) is a positive point, so by Definitions 4.4.2

and 4.4.3, gE,c(h (t)

)= gE,c

(h (t)

). Since gE,c

(h (t)

)is constant, d

dtgE,c

(h (t)

)=

∇gE,c(h (t)

)· P E

(h (t)

)= 0. Then, by Theorem 4.4.5 and by continuity, h (0) must

be a strong E-equilibrium point, so for all e ∈ E , for all t ∈ Ωq, e(h(t)) = 0, which

contradicts Lemma 4.4.10.4. Therefore q is a strong E-equilibrium point.

The next theorem consolidates our results concerning natural event-systems. It also

establishes that positive strong equilibrium points are locally attractive relative to their

conservation classes. Together with the existence of a Lyapunov function, this implies that

positive strong equilibrium points are asymptotically stable relative to their conservation

classes [46, Theorem 5.57].

119

Theorem 4.4.15. Let E be a finite, natural event-system of dimension n. Let H be a

positive conservation class of E. Then:

1. For all x ∈ H ∩ Rn≥0, there exist k ∈ R≥0, an open, simply-connected Ω ⊆ C and

an E-process f = 〈f1, f2, . . . , fn〉 on Ω such that:

(a) R≥0 ⊆ Ω.

(b) f(0) = x.

(c) For all t ∈ R≥0, f(t) ∈ H ∩ Rn≥0.

(d) For all t ∈ R≥0, for i = 1, 2, . . . , n, 0 ≤ fi(t) ≤ k.

(e) For all open, simply-connected Ω ⊆ C, for all E-processes f on Ω, if 0 ∈ Ω

and f(0) = x then for all intervals I ⊆ Ω ∩ R≥0 such that 0 ∈ I, for all

t ∈ I : f(t) = f(t).

2. There exists c ∈ H such that:

(a) c is a positive strong E-equilibrium point.

(b) For all d ∈ H, if d is a positive strong E-equilibrium point, then d = c.

(c) There exists U ⊆ H ∩ Rn>0 such that

i. U is open in H ∩ Rn>0.

ii. c ∈ U .

iii. For all x ∈ U , there exist an open, simply-connected Ω ⊆ C and an E-

process f on Ω such that

A. R≥0 ⊆ Ω.

B. f(0) = x.

120

C. f(t) → c as t → ∞ along the positive real line. (i.e. for all ε ∈ R>0,

there exists t0 ∈ R>0 such that for all t ∈ R>t0 : ||f(t)− c||2 < ε.)

Proof.

1. Follows from Lemma 4.4.13 and Theorem 4.2.3.

2.(a) and 2.(b) follow from Theorem 4.4.9.

2.(c) Let c ∈ H be a positive strong-E-equilibrium point as in Theorem 4.4.15.2a. Let

g = gE,c. Let T = H ∩ Rn>0. For all x ∈ H ∩ Rn, for all r ∈ R>0, let

Br(x) =y ∈ H ∩ Rn | ‖x− y‖2 < r

Sr(x) =

y ∈ H ∩ Rn | ‖x− y‖2 = r

Br(x) =

y ∈ H ∩ Rn | ‖x− y‖2 ≤ r

Since Rn>0 is open in Rn, it follows that T is open in H ∩ Rn. Therefore, there exists

δ ∈ R>0 such that B2δ(c) ⊆ T . Let δ ∈ R>0 be such that B2δ(c) ⊆ T . It follows that

Bδ(c) ⊆ T .

Since g is continuous and Sδ(c) is compact, let x0 ∈ Sδ(c) be such that g(x0) =

infx∈Sδ(c) g(x). Let U = Bδ(c) ∩ x ∈ T | g(x) < g(x0). It follows that U is open in T .

Since x0 6= c, and by Lemma 4.4.11.2, g(x0) = gE,c(x0) > 0 = g(c). Hence, c ∈ U .

Let x ∈ U . From Lemma 4.4.13, there exist an open, simply-connected Ω ⊂ C and

an E-process f on Ω such that R≥0 ⊆ Ω and f(0) = x.

We claim that for all t ∈ R≥0, f(t) ∈ Bδ(c). Suppose not. Then there exists t0 ∈ R≥0

such that f(t0) ∈ Sδ(c). From the definition of x0, g(x0) ≤ g(f(t0)). Since f(0) = x ∈ U ,

g(f(0)) < g(x0). Hence, g(f(0)) < g(f(t0)), contradicting Lemma 4.4.11.4.

121

To see that f(t)→ c as t→∞ along the positive real line, suppose not. Then there

exists ε ∈ R>0 such that ε < δ and there exists an increasing sequence of real numbers

ti ∈ R>0i∈Z>0 such that ti → ∞ as i → ∞ and for all i, f(ti) ∈ Bδ(c) \ Bε(c). Since

Bδ(c) \ Bε(c) is compact, there exists a convergent subsequence. By Definition 4.3.5,

the limit of this subsequence is an ω-limit point q of f such that q ∈ Bδ(c) \ Bε(c).

From Lemma 4.4.14, q is a strong-E-equilibrium point. Since q ∈ Bδ(c), q ∈ T . From

Theorem 4.4.9, q = c. Hence, c /∈ Bε(c), a contradiction.

We have established that positive strong equilibrium points are asymptotically stable

relative to their conservation classes. A stronger result would be that if an E-process

starts at a positive point then it asymptotically tends to the positive strong equilibrium

point in its conservation class. Such a result is related to the widely-held notion that,

for systems of chemical reactions, concentrations approach equilibrium. We have been

unable to prove this result. We will now state it as an open problem. This problem has

a long history. It appears to have been first suggested in [45, Lemma 4C], where it was

accompanied by an incorrect proof. The proof was retracted in [44].

Open 4.4.16. Let E be a finite, natural event-system of dimension n. Let H be a positive

conservation class of E. Then

1. For all x ∈ H ∩ Rn≥0, there exist k ∈ R≥0, an open, simply-connected Ω ⊆ C and

an E-process f = 〈f1, f2, . . . , fn〉 on Ω such that:

(a) R≥0 ⊆ Ω.

(b) f(0) = x.

(c) For all t ∈ R≥0, f(t) ∈ H ∩ Rn≥0.

122

(d) For all t ∈ R≥0, for i = 1, 2, . . . , n, 0 ≤ fi(t) < k.

(e) For all open, simply-connected Ω ⊆ C, for all E-processes f on Ω, if 0 ∈ Ω

and f(0) = x then for all intervals I ⊆ Ω ∩ R≥0, if 0 ∈ I then for all t ∈ I :

f(t) = f(t).

2. There exists c ∈ H such that:

(a) c is a positive strong E-equilibrium point.

(b) For all d ∈ H, if d is a positive strong E-equilibrium point, then d = c.

(c) For all x ∈ H ∩ Rn>0, there exist an open, simply-connected Ω ⊆ C and an

E-process f on Ω such that:

i. R≥0 ⊆ Ω.

ii. f(0) = x.

iii. f(t)→ c as t→∞ along the positive real line. (i.e. for all ε ∈ R>0, there

exists t0 ∈ R>0 such that for all t ∈ R>t0 : ||f(t)− c||2 < ε.)

In light of Theorem 4.4.15, Open Problem 4.4.16 is equivalent to the following state-

ment.

Open 4.4.17. Let E be a finite, natural event-system of dimension n. Let x ∈ Rn>0.

Then there exists an open, simply-connected Ω ⊆ C, an E-process f on Ω and a positive

strong E-equilibrium point c such that:

1. R≥0 ⊆ Ω.

2. f(0) = x.

123

3. f(t) → c as t → ∞ along the positive real line. (i.e. for all ε ∈ R>0, there exists

t0 ∈ R>0 such that for all t ∈ R>t0 : ||f(t)− c||2 < ε.)

4.5 Finite Natural Atomic Event-Systems

In this section, we settle Open 4.4.16 in the affirmative for the case of finite, natural,

atomic event-systems. The atomic hypothesis appears to be a natural assumption to

make concerning systems of chemical reactions. Therefore, our result may be considered

a validation of the notion in chemistry that concentrations tend to equilibrium. We will

prove the following theorem:

Theorem 4.5.1. Let E be a finite, natural, atomic event-system of dimension n. Let

α ∈ Rn>0. Then there exists an open, simply-connected Ω ⊆ C, an E-process f on Ω, and

a positive strong E-equilibrium point c such that:

1. R≥0 ⊆ Ω,

2. f(0) = α, and

3. f(t) → c as t → ∞ along the positive real line (i.e. for all ε ∈ R>0, there exists

t0 ∈ R>0 such that for all t ∈ R>t0 : ‖f(t)− c‖2 < ε).

It follows from Theorem 4.4.15 that the point c depends only on the conservation

class of α and not on α itself. That is, two E-processes starting at positive points in the

same conservation class asymptotically converge to the same c.

Implicit in the atomic hypothesis is the idea that atoms are neither created nor de-

stroyed, but rather are conserved by chemical reactions. Our proof uses a formal analogue

124

of this idea. Recall from Definition 4.1.11 that if E is atomic then CE(M) contains a unique

monomial from MAE .

Definition 4.5.1. Let E be a finite, natural, atomic event-system of dimension n. The

atomic decomposition mapDE : MX1,X2,...,Xn → Zn≥0 is the function M 7→ 〈b1, b2, . . . , bn〉

such that Xb11 X

b22 · · ·Xbn

n ∈ CE(M) ∩MAE .

The next lemma lists some properties of the atomic decomposition map. Note that

though the event-graph GE is directed, if M and N are monomials and there exists a

path in GE from M to N then there also exists a path in GE from N to M . Informally,

this is because all events are “reversible”.

Lemma 4.5.2. Let E be a finite, natural, atomic event-system of dimension n and let

M,N ∈MX1,X2,...,Xn. Then:

1. DE(M) = DE(N) if and only if CE(M) = CE(N).

2. DE(MN) = DE(M) +DE(N).

Proof. Let D = DE .

(1) D(M) = D(N) = 〈b1, b2, . . . , bn〉 if and only if Xb11 X

b22 · · ·Xbn

n ∈ CE(M) and

Xb11 X

b22 · · ·Xbn

n ∈ CE(N). Then CE(M) = CE(N).

(2) Let D(M) = 〈b1, b2, . . . , bn〉 and D(N) = 〈c1, c2, . . . , cn〉. Then, in GE there is a

path from M to Xb11 X

b22 · · ·Xbn

n ∈ MAE and a path from N to Xc11 X

c22 · · ·Xcn

n ∈ MAE .

It follows that there is a path from MN to Xb1+c11 Xb2+c2

2 · · ·Xbn+cnn ∈ MAE . Hence

D(MN) = 〈b1 + c1, b2 + c2, . . . , bn + cn〉 = D(M) +D(N).

125

Definition 4.5.2. Let E be a finite, natural, atomic event-system of dimension n. For

all i ∈ 1, 2, . . . , n, for all M ∈MX1,X2,...,Xn, DE,i(M) is the ith component of DE(M).

Definition 4.5.3. Let E be a finite, natural, atomic event-system of dimension n. For

all i ∈ 1, 2, . . . , n the function κE,i : Cn → C is given by

〈z1, z2, . . . , zn〉 7−→n∑j=1

DE,i(Xj)zj .

Lemma 4.5.3. Let E be a finite, natural, atomic event-system of dimension n. Then for

all i ∈ 1, 2, . . . , n, the function κE,i is a conservation law of E.

Proof. Let m = |E|, and for j = 1, 2, . . . ,m, let σj , τj ∈ R>0 and Mj , Nj ∈ M∞ with

Mj ≺ Nj be such that E = σ1M1 − τ1N1, . . . , σmMm − τmNm. For i = 1, 2, . . . , n, let

aj,i, bj,i ∈ Z>0 be such that Mj = Xaj,11 X

aj,22 · · ·Xaj,n

n and Nj = Xbj,11 X

bj,22 · · ·Xbj,n

n . Let

(γj,i)m×n = ΓE .

Then for j = 1, 2, . . . ,m:

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σjMj − τjNj ∈ E

⇒ Mj ∈ CE(Nj) [Definition 4.1.9]

⇒ DE(Mj) = DE(Nj) [Lemma 4.5.2]

⇒n∑i=1

aj,iDE(Xi) =n∑i=1

bj,iDE(Xi) [Lemma 4.5.2]

⇒n∑i=1

(bj,i − aj,i)DE(Xi) = 0

⇒n∑i=1

γj,iDE(Xi) = 0 [Definition 4.2.1]

It follows that for all j ∈ 1, 2, . . . ,m, for all k ∈ 1, 2, . . . , n,

n∑i=1

γj,iDE,k(Xi) = 0

Therefore, for all k ∈ 1, 2, . . . , n, ΓE · 〈DE,k(X1), DE,k(X2), . . . , DE,k(Xn)〉T = 0.

Since the vector 〈DE,k(X1), DE,k(X2), . . . , DE,k(Xn)〉T is in the kernel of ΓE , by Theo-

rem 4.2.2, κE,k is a conservation law of E .

Lemma 4.5.4. Let E be a finite, natural event-system of dimension n. Let M,N ∈M∞

and let q ∈ Cn. If M ∈ CE(N) and q is a strong E-equilibrium point and M(q) = 0, then

N(q) = 0.

Proof. Let 〈v0, v1〉 be an edge in GE . Then there exist e ∈ E and σ, τ ∈ R>0 and

T,U, V ∈M∞ such that e = σU − τV and v0 = TU and v1 = TV .

127

Assume v0(q) = 0. Then either T (q) = 0 or U(q) = 0. If T (q) = 0 then v1(q) = 0.

If U(q) = 0 and q is a strong E-equilibrium point, then e(q) = σU(q) − τV (q) = 0, so

V (q) = 0. Therefore v1(q) = 0. The lemma follows by induction.

We are now ready to prove Theorem 4.5.1.

Proof of Theorem 4.5.1. Since α is a positive point, it is in some positive conservation

class H. By Theorem 4.4.15:

1. There exists exactly one positive strong E-equilibrium point c ∈ H.

2. There exist an open and simply-connected Ω ⊆ C and an E-process f on Ω such

that R≥0 ⊂ Ω and f(0) = α.

3. For all t ∈ R≥0, f(t) ∈ H ∩ Rn≥0.

4. There exists k ∈ R≥0 such that for i = 1, 2, . . . , n, for all t ∈ R≥0, fi(t) ∈ R and

0 ≤ fi(t) ≤ k.

Let tjj∈Z>0 be an infinite sequence of non-negative reals such that tj → ∞ as

j →∞. Then f(tj)j∈Z>0 is an infinite sequence contained in a compact subset of Rn,

so it must have a convergent subsequence. Let q = 〈q1, q2, . . . , qn〉 ∈ Cn be the limit

point of a convergent subsequence of f(tj)j∈Z>0 . H and Rn≥0 are both closed in Cn, so

q ∈ H∩Rn≥0. Since E is natural and q is an ω-limit of f , q must be a strong E-equilibrium

point by Lemma 4.4.14.

Assume, for the sake of contradiction, that q /∈ Rn>0. Let i ∈ 1, 2, . . . , n be such that

qi = 0. Let N ∈ CE(Xi)∩MAE . Since E is atomic, a unique such N exists. It follows from

128

the definition of event graph that Xi ∈ CE(N). By Lemma 4.5.4, N(q) = Xi(q) = qi = 0.

It follows thatN 6= 1. Hence, there existsXa ∈ AE such thatXa dividesN andXa(q) = 0.

For all j ∈ 1, 2, . . . , n such that DE,a(Xj) 6= 0, let Mj ∈ CE(Xj) ∩MAE . Then Xa

divides Mj , so Mj(q) = 0. Again by Lemma 4.5.4, Xj(q) = Mj(q) = 0, so qj = 0. It

follows that for all j ∈ 1, 2, . . . , n either DE,a(Xj) = 0 or qj = 0 so

κE,a(q) =n∑j=1

DE,a(Xj)qj = 0.

Since κE,a is a conservation law of E by Lemma 4.5.3 and q is an ω-limit point of f , it

follows that

κE,a(α) = 0. (4.39)

For all j, DE,a(Xj) is nonnegative, and α is a positive point, so for all j ∈ 1, 2, . . . , n,

DE,a(Xj)αj ≥ 0. But DE,a(Xa) = 1 and αa > 0 so κE,a(α) > 0, contradicting equa-

tion (4.39). Therefore q ∈ Rn>0. Since c is the unique positive strong E-equilibrium point

in H, c = q.

Let U ⊆ H ∩ Rn>0 be the open set stated to exist in Theorem 4.4.15.2c. Since c is an

ω-limit point of f , there exists t0 ∈ R>0 such that f(t0) ∈ U . Again by Theorem 4.4.15,

there exist Ω ⊆ C and an E-process f on Ω such that R≥0 ⊆ Ω and f(0) = f(t0) and

f(t) → c as t → ∞. By Lemma 4.3.3, for all t ∈ R≥0, f(t + t0) = f(t). Therefore,

f(t)→ c as t→∞.

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Chapter 5

Zeta Event-Systems

The mathematician Leopold Kronecker once said that “God created the integers, all else

is the work of man.” Is it possible to give formal meaning to the idea that the integers

were created from more primitive things? If so, that might give insight into why they

are as we see them now. In this chapter, we will describe how event-systems provide the

means to view the integers as the outcome of a self-assembly process. The work presented

in this chapter is the result of collaboration between Leonard Adleman and myself, and

the first person plural pronouns in this chapter refer to those authors.

In event-systems, we usually think of the reacting species as chemical compounds.

The systems themselves, however, have no intrinsic ties to chemistry. It is possible to

think of the reactants as other objects, such as physical objects being self-assembled or

even as pure numbers.

In chemistry, the atomic hypothesis states that all substances are composed of a

unique set of atoms. Atomic event-systems capture this idea: every reactant in an atomic

event-system is made up of a unique set of atoms. The atomic hypothesis in chemistry is

analogous to the fundamental theorem of arithmetic in mathematics: all natural numbers

130

are composed of a unique set of primes. One can imagine assembling the natural numbers

from the prime numbers by multiplication.

In this chapter, we will define a class of event-systems called zeta event-systems. We

will use event-systems to model the mathematical operation of multiplication. Intuitively,

these event-systems model chemical systems in which the reactants are natural numbers.

The numbers can react with one another to form their product.

Applying the law of mass action to an event-system (see Chapter 4) gives a set of

differential equations that tracks the time evolution of the concentration of each species.

If the species are identified with natural numbers, the reactions are identified with the

multiplication operator, and in the beginning the only species present are prime numbers,

then we can “watch” the natural numbers being created through time as primes combine

through multiplication.

Thermodynamic properties such as pressure and temperature have meaning in event-

systems, even if a particular system does not model chemical reactions. Pressure in event-

systems can be defined as the sum of the concentrations of all species. Temperature factors

into the specific rates: Other things being equal, at temperature T the specific rates are

proportional to e−1/T . In the zeta event-systems defined in this chapter, the reactions

and reaction rates are chosen carefully so that the system has a strong equilibrium point.

This equilibrium point is special in that, if the event-system is at temperature s, the

pressure at equilibrium is ζ(s), where ζ is the Riemann zeta function. Formal definitions

and proofs will be presented in Section 5.3.

131

The motivation for doing this is to increase our understanding of the Riemann zeta

function. By using event-systems and their associated event-processes, we add a dimen-

sion to the zeta function: time. The hope is that we can watch the numbers self-assemble

through time to create the zeta function, and that this self-assembly process can shed

light on the zeta function itself. In particular, the locations of the zeros of the Riemann

zeta function are extremely important in mathematics. Starting from initial conditions

that have no zeros anywhere in the complex plane, we may be able to “watch” the system

evolve and see how zeros arise. This may lead to insight on their final destinations, ideally

on the line with real part 12 .

5.1 Riemann Zeta Function

Countless books, both technical and popular, have been written about the Riemann zeta

function. This function has a fascinating history and is immensely important in mathe-

matics. Here we will only give a brief outline of its history, properties, and importance.

The Riemann zeta function ζ : C→ C is defined for Re(s) > 1 by

ζ(s) =∞∑n=1

1ns. (5.1)

When s = 1, the sum becomes the harmonic series, which diverges, as proved by

Nicole d’Orseme in the fourteenth century. When s = 2, the sum becomes the Basel

series, which Leonhard Euler proved converges to π2/6 in 1735 [21].

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The Riemann zeta function can be analytically continued to the entire complex plane

(with a simple pole at s = 1) by

ζ(s) =1

1− 21−s

∞∑n=0

12n+1

n∑k=0

(−1)k(n

k

)(k + 1)−s. (5.2)

Euler also proved a deep connection between the Riemann zeta function and the prime

numbers [28]. For all s ∈ R>1,

ζ(s) =∞∑n=1

1ns

=∏

p prime

(1− p−s)−1 (5.3)

It is this connection for which the Riemann zeta function is best-known and most-studied.

Carl Friedrich Gauss (in 1793) and Adrien-Marie Legendre [54] (in 1796) indepen-

dently conjectured that the prime numbers are distributed in a certain way, though

Gauss’s work was not published until 1863 [33]. In its simplest form, the conjecture says

that

limn→∞

π(n)n/ ln(n)

= 1. (5.4)

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where π(n) is the number of primes less than or equal to n. This can be stated more

succinctly as π(n) ∼ n/ ln(n). This conjecture became known as the Prime Number

Theorem. Gauss later improved the estimate to

π(n) ∼ Li(n) where Li(n) =∫ n

2

1ln(x)

dx. (5.5)

In 1859 Berhnard Riemann presented a paper titled “On the Number of Primes Less

Than a Given Quantity” [69]. As the title indicates, his paper was on the distribution

of primes. It was in this paper that Riemann generalized Euler’s result to the complex

numbers and laid out a plan to prove the conjecture of Gauss and Legendre. The function

defined in Equations (5.1) and (5.2) bears Riemann’s name because of the results in this

paper. It was also in this paper that he stated the Riemann Hypothesis: all the non-trivial

zeros of the Riemann zeta function lie on the line Re(s) = 12 .

In 1896 the Prime Number Theorem was proved independently by Jacques Hadamard

and Charles de la Vallee Poussin [40, 20], which established the correctness of the con-

jecture of Gauss and Legendre. Hadamard and Vallee Poissin did not prove the Riemann

Hypotheses; instead they proved the weaker result that there are no zeros of the Riemann

zeta function with Re(s) ≥ 1. To date, no one has proved the Riemann Hypothesis and

it is considered one of the greatest open questions in mathematics. Proving the Riemann

Hypothesis would strengthen the Prime Number Theorem by tightening the bounds on

the error function: how much π(n) and Li(n) differ.

134

In 1900 David Hilbert posed a list of 23 open problems at the Paris conference of the

International Congress of Mathematicians. Many of these problems turned out to be very

influential during the 20th century. Among the problems was the Riemann Hypothesis.

In 2000 the Clay Institute posed a list of seven open problems. They offer a one

million dollar prize for a solution to each of the problems. Again, among the problems is

the Riemann Hypothesis.

5.2 Basic Definitions and Notation

The event-systems in this chapter differ in several important ways from the event-systems

in Chapter 4. First, all zeta event-systems presented here are both infinite and infinite-

dimensional. This means that many of the definitions and theorems from Chapter 4

do not apply. Although it is often easy to extend the definitions and theorems from

Chapter 4 to the infinite case, great care must be taken not to introduce subtle problems.

As the goal of this chapter is not to give a complete description of what is known about

infinite event-systems, but to prove the existence and properties of zeta event-systems,

we will only formally extend the definitions and theorems needed for the task at hand.

Second, each zeta event-system is not an individual event-system, but a class of event-

systems that depend on a parameter, the system’s temperature. However, every complex

number can be used as the temperature parameter and the resulting system does turn

out to be an event-system (see Definition 5.2.2).

Third, the quantity of interest in this chapter is a measurement of the state of the

system. Most simply, this will be the pressure of the system (the sum of the concentrations

135

of all of the species). However, in some zeta event-systems, at some temperatures, at some

times, this infinite sum diverges. Later in this chapter, we will extend the definition of

pressure to mitigate this problem.

Finally, using Definition 4.2.3 for event-process leads to systems of infinitely many,

nonlinear, complex-valued, differential equations. These systems of differential equations

are difficult to analyze. We will approach this difficulty in two ways: we will define

“simplified event-processes” that keeps many of the features of the usual event-processes

but are more amenable to analysis, and we will use numerical methods to simulate the

event-processes of subsets of zeta event-systems.

The goal of this chapter is to present a class of event-systems with the property that

the pressure at equilibrium is the Riemann zeta function of the system’s temperature.

How the “pressure” of an event-system is measured and how the “temperature” of an

event-system affects it will be defined more formally in Sections 5.2.1 and 5.2.2, respec-

tively. Informally, however, in this chapter we will present event-systems in which species

Xn has, as its unique strong equilibrium value, n−s, where s is the temperature of the

system. By considering the pressure of the system (sum of the concentrations of all of

the species) at this equilibrium, we recover Equation (5.1), which defines the Reimann

zeta function on a half-plane.

To begin we will need notation to represent an infinite-dimensional event-system pa-

rameterized by temperature. Recall from Notation 4.1.1 that

C∞ =∞⋃n=1

C[X1, X2, · · · , Xn],

136

a monic monomial is a product of the form∏∞i=1X

eii where the ei are non-negative integers

all but finitely many of which are zero, and M∞ is the set of all monic monomials of C∞.

A holomorphic function f : C → C is zero-free iff for all x ∈ C, f(x) 6= 0. Let

F = f : C → C | f is holomorphic and zero-free. It is well-known that F = ef | f is

holomorphic. The set of parameterized binomials is Bp,∞ = σM + τN | σ, τ ∈ F and

M,N are distinct elements of M∞ and M ≺ N.

Definition 5.2.1. A parameterized event-system E is a nonempty, countable subset of

Bp,∞.

Given a parameterized event-system E , it is possible to evaluate the reaction rates at

s ∈ C resulting in a (nonparameterized) event-system E(s).

Definition 5.2.2. Let E be a parameterized event-system and let s ∈ C. Then E(s), or

E evaluated at s, is the event-system σ(s)M + τ(s)N | σ, τ ∈ F and M,N ∈ M∞ and

σM + τN ∈ E.

That E(s) is an event-system follows immediately from the the facts that σ and τ are

holomorphic (hence defined at s) and zero-free (hence nonzero at s).

Event-systems, in general, can be infinite-dimensional. However, Definition 4.1.6 only

defines points as finite-dimensional complex vectors. The following extends the defini-

tion to the countably infinite case, keeping the convention from Chapter 4 that points

are written in a bold symbol and the individual components that make up a point are

subscripted, normal-weight versions of that symbol:

Definition 5.2.3. Let CZ>0 be the set of all functions α : Z>0 → C. Then α ∈ CZ>0 is

a point in CZ>0 and for all n ∈ Z>0, αn = α(n).

137

Given a finite-dimensional event-system E of dimension n, one can evaluate an event

e ∈ E a point α ∈ Cn to get a value e(α) ∈ C. In infinite-dimensional event-systems, an

analogous result holds: Given an infinite-dimensional event-system E , one can evaluate

an event e ∈ E at a point α ∈ CZ>0 to get a value e(α) ∈ C. Recall that even though

E is infinite-dimensional, each event in E is a binomial, each term of which is a nonzero

complex number times a monic monomial in M∞, which can have only finitely many

factors.

If E is a finite-dimensional event-system of dimension n, a point α ∈ Cn may be a

strong equilibrium point of E by Definition 4.1.7. This definition has a straight-forward

extension to infinite-dimensional event-systems.

Definition 5.2.4. Let E be an infinite-dimensional event-system. A point α ∈ CZ>0 is a

strong equilibrium of E iff for all e ∈ E , e(α) = 0.

5.2.1 Pressure

The simplest definition of pressure in an event-system is simply the sum of the concen-

trations of each of the species. Let α ∈ CZ>0 , then

pres(α) =∞∑n=1

αn (5.6)

Note that at some concentrations α, pres(α) can diverge.

If s ∈ C is such that Re(s) > 1 and for all n ∈ Z≥1, αn = n−s, then pres(α) =∑∞n=1 n

−s = ζ(s) by Equation (5.1), which can be analytically continued to the entire

138

complex plane using Equation (5.2). Strictly by analogy, for all s ∈ C, we will define the

“continuation” of pressure to be

pres(α) =1

1− 21−s

∞∑n=0

12n+1

n∑k=0

(−1)k(n

k

)αk+1. (5.7)

5.2.2 Temperature

In chemistry, the specific rates of a reaction depend on the temperature at which the

reaction takes place. The Arrhenius equation describes how the specific rates vary with

the temperature. In the zeta event-systems in this chapter, the specific rates will be given

by the Arrhenius equation. The Arrhenius equation is usually given in the form

σ(T ) = Aσe−Ea,σ/RT and

τ(T ) = Aτe−Ea,τ/RT , (5.8)

where R is the universal gas constant, Aσ and Aτ are constants called the prefactors

of the reaction, and Ea,σ and Ea,τ are constants called the activation energies of the

reaction. Usually Aσ, Aτ , Ea,σ and Ea,τ are determined empirically for each reaction. For

convenience, rather than use the actual temperature T , the parameter s in zeta event-

systems will be the inverse of the temperature T :

s =1T. (5.9)

139

Zeta event-systems were created to model an abstract idea: multiplication. As such,

there is no experimental set-up in which the constants in the Arrhenius equation can be

empirically measured, and we are free to choose them however we wish. In creating zeta

event-systems, we will use two methods for choosing the constants. These are the same

two methods for both the forward rate σ and the backward rate τ , so in what follows,

A = Aσ and Ea = Ea,σ or A = Aτ and Ea = Ea,τ . The first method of choosing the

constants is

A = 1 and

Ea = 0. (5.10)

Using these constants in the Arrhenius equation, it is easy to verify that the specific rate

is the constant 1 at all complex temperatures T and all inverse temperatures s.

The other method we will use to choose the constants in the Arrhenius equation will

depend on the event: specifically the event’s number in some well-defined ordering of the

events. Let n ∈ Z>0 be the event’s number and let R be the universal gas constant. The

second method of choosing the constants is

A = 1 and

Ea =−R

ln(n)where ln(n) is taken to be real. (5.11)

Using these constants in the Arrhenius equation, it is easy to verify that, at temperature

T = 1/s, the specific rate is n−1/T = n−s.

140

Note that in both methods, the specific rates are analytic and zero-free in s = 1/T

(but not in T itself, which is one of the reasons for using s as the parameter rather than T ).

In the zeta event-systems in this chapter, all specific rates will come from the Arrhenius

equation with the prefactor and activation energy given in Equation (5.10) or (5.11), but

both methods may be combined in a single event (i.e., σ(s) given by Equation (5.10)

and τ(s) given by Equation (5.11)). Hence, all zeta event-systems are parameterized

event-systems.

5.2.3 Processes

The stoichiometric matrix is only defined for finite event-systems (Definition 4.2.1), but

it is possible to compute each entry of the stoichiometric matrix in infinite and infinite-

dimensional event-systems:

Definition 5.2.5. Let E = e1, e2, . . . be an infinite-dimensional event-system and let

i, j ∈ Z≥1. Let ej = σM + τN , where M,N ∈M∞ and M ≺ N . Then γj,i is the number

of times Xi divides N minus the number of times Xi divides M .

Similarly, the definition of event-process can be extended to infinite and infinite-

dimensional event-systems:

Definition 5.2.6 (Event-process). Let E = e1, e2, . . . be an infinite-dimensional event-

system and for all j ∈ Z≥1, let ej = σM + τN , where M,N ∈ M∞ and M ≺ N . For all

i ∈ Z≥1, let Pi =∑∞

k=1 γk,iek. Let Ω ⊆ C be a non-empty, simply-connected, open set.

For all i ∈ Z≥1, let fi : Ω → C. Then f = 〈f1, f2, . . .〉 is an E-process on Ω iff for all

i ∈ Z≥1:

141

1. f ′i exists on Ω and

2. f ′i = Pi f on Ω.

Such event-processes are difficult to work with analytically, so in Section 5.5 we will

use numerical methods to analyze them. For some infinite-dimensional event-systems,

however, it is possible to define a “simplified” event-process that is possible to analyze

analytically.

Definition 5.2.7 (Growth event-system). Let E be an event-system. E is a growth event-

system iff

1. for all e ∈ E , there exist j ∈ Z≥1, k ∈ Z≥0 such that k < j, i1, i2, . . . , ik ∈ Z≥0, and

σ, τ ∈ C \ 0 such that e = σXj + τ∏k`=1X

i`` and

2. for all j ∈ Z≥1, e = σXj + τN ∈ E | σ, τ ∈ C \ 0, N ∈M∞ is finite.

Note that in Definition 5.2.7, k is allowed to be 0, so events of the form σXj + τ

are allowed in growth event-systems. Intuitively, in growth event-systems every species

is composed only of smaller species, and only in a finite number of different ways.

Definition 5.2.8 (Simplified event-process). Let E be a growth event-system. For all

i, j ∈ Z≥1 let

γj,i =

−1 if there exist σ, τ ∈ C \ 0 and N ∈M∞ such that ej = σXi + τN

0 otherwise.

142

For all i ∈ Z≥1 let Pi =∑∞

j=1 γj,iej . Let Ω ⊆ C be a non-empty, simply-connected,

open set. For all i ∈ Z≥1 let fi : Ω → C. Then f = 〈f1, f2, . . .〉 is a simplified E-process

on Ω iff for all i ∈ Z≥1:

1. f ′i exists on Ω and

2. f ′i = Pi f on Ω.

5.3 Zeta Event-Systems

There are several different ways to formulate event-systems that have the Riemann zeta

function as the pressure of a strong equilibrium point. Below, we will define two different

zeta event-systems. The zeta event-systems below, as well as many other zeta event-

systems, share a common feature: the events are multiplicative. That is to say, all the

events have the form

Xn −k∏i=1

Xmi where n =k∏i=1

mi, or (5.12)

Xn − n−s. (5.13)

Intuitively, events of the form in Equation (5.13) “force” species Xn to have, as its

strong equilibrium value, n−s. Similarly, events of the form in Equation (5.12) force

species Xn to have, as its strong equilibrium value, the product the equilibrium values of

species Xm1 , Xm2 , . . . , Xmk . Taken together, this forces all species Xn to have, as their

strong equilibrium values, n−s.

143

In the first zeta event-system, Eζ,1, only pairwise multiplication of numbers is modeled,

and all pairwise multiplications of numbers greater than two are included. In chemistry,

reactions in which two reactants combine to form one product are called “bimolecular”

reactions, hence the name bimolecular zeta event-system.

Definition 5.3.1 (Bimolecular zeta event-system). For all s ∈ C, let Rs,1 = Xp − p−s |

p ∈ Z>1 and p is prime, let Cs,1 = Xnm−XnXm | n,m ∈ Z>1, and let Zs,1 = X1−1.

Then the zeta event-system Eζ,1 is the infinite-dimensional, parameterized event-system

Rs,1 ∪ Cs,1 ∪ Zs,1.

In Eζ,1 each species can be “built” from smaller species in many different ways. For

example, X12 −X2X6 and X12 −X3X4 are both events in Eζ,1 that build the number 12

from smaller numbers.

In the second zeta event-system, Eζ,2, there is only one event used to build each species

from smaller species: each species can only be built from its prime decomposition.

Definition 5.3.2 (Prime zeta event-system). Let s ∈ C. Let Zs,2 = X1 − 1. For all

p ∈ Z>1, let τp : C→ C be such that for all s ∈ C, τp(s) = p−s. Let Rs,2 = Xp− τp | p ∈

Z>1 prime. Let Cs,2 = Xn −Xp1Xp2 · · ·Xpk | n ∈ Z>1 is composite and p1p2 · · · pk is

the prime factorization of n. Then the zeta event-system Eζ,2 is the infinite-dimensional,

parameterized event-system Rs,2 ∪ Cs,2 ∪ Zs,2.

The following theorem states that zeta event-systems deserve that name: they have

a strong equilibrium point at which the pressure is the Riemann zeta function of the

temperature.

Theorem 5.3.1. For all s ∈ C, the point c = 〈1−s, 2−s, . . . , n−s, . . .〉 is such that

144

1. c is a strong Eζ,1(s)-equilibrium

2. c is a strong Eζ,2(s)-equilibrium

3. if Re(s) > 1 then pres(c) = ζ(s)

4. pres(c) = ζ(s).

Proof. Parts 1 and 2 follow immediately from the definitions of Eζ,1(s) and Eζ,2(s) and

from the fact that for all s ∈ C, for all n,m ∈ Z≥1, n−sm−s = (nm)−s. Parts 3 and 4

follow from Equations (5.1) and (5.2), respectively.

At temperature s, the pressure at a strong Eζ,1-equilibrium or a strong Eζ,2-equilibrium

is the Riemann zeta function’s value at s. But what about the dynamics? Starting from

all-zero initial conditions, does the event-process or simplified event-process asymptoti-

cally approach this equilibrium in forward real time?

Using Definition 5.2.6 of event-processes, this question turns out to be difficult to

answer analytically. In Section 5.5 we will give evidence that, at some temperatures, the

event-process does appear to approach the strong equilibrium, while at other tempera-

tures, the event-process appears chaotic.

However, using Definition 5.2.8 of simplified event-processes makes this question more

amenable to analysis. It turns out that both Eζ,1 and Eζ,2 have similar dynamics for both

the event-processes and simplified event-processes. For the rest of this section, we will

analyze only Eζ,2 but a similar analysis works for Eζ,1 to produce similar results.

Theorem 5.3.2. For all s ∈ C, for all i ∈ Z≥1 there exist fs,i : C → C such that

f s = 〈fs,1, fs,2, . . . 〉 is a simplified Eζ,2(s)-process on C and f s(0) = 〈1, 0, 0, . . . 〉. Further,

145

if Re(s) > 1 then pres(f s(t)) → ζ(s) as t → ∞ in the positive real direction and for all

s ∈ C, pres(f s(t))→ ζ(s) as t→∞ in the positive real direction.

Proof. Let s ∈ C and let E = Eζ,2(s). For all e ∈ E , there exist n, k ∈ Z≥1 and primes

p1, p2, . . . , pk ∈ Z≥1 such that

1. e = Xn − 1 (if n = 1) or

2. e = Xn − n−s (if n is prime) or

3. e = Xn−Xp1Xp2 · · ·Xpk (if n is composite and p1p2 . . . pk is the prime factorization

of n).

Note that for all n ∈ Z≥0 there is exactly one event in E of the form σXn + τN , for

some σ, τ ∈ C and N ∈M∞. Therefore, by Definition 5.2.8, there is exactly one j ∈ Z≥0

such that γj,n is nonzero. Hence,

Pn =

1−Xn if n = 1

n−s −Xn if n is prime

Xp1Xp2 · · ·Xpk −Xn if n = p1p2 · · · pk is the prime factorization of n

We would like to find f s = f = 〈f1, f2, . . . 〉, where for all i ∈ Z≥1, fi : C → C, such

that for all i ∈ Z≥1, f ′i = Pi f on C. To do this, we note that Pi f gives the following

system of ordinary differential equations:

146

For all prime p, and for all composite n such that p1p2 · · · pk is the prime factorization

of n,

f ′1 = 1− f1 (5.14)

f ′p = p−s − fp (5.15)

f ′n = fp1fp2 · · · fpk − fn (5.16)

Equation (5.14) is a one-dimensional, first-order, linear, ordinary differential equation

whose trivial solution

f1(t) = 1 (5.17)

satisfies the initial condition f1(0) = 1.

Let p ∈ Z>1 be prime. Then Equation (5.15) is a one-dimensional, first-order, linear,

ordinary differential equation whose solution

fp(t) = p−s(1− e−t) (5.18)

satisfies the initial condition fp(0) = 0.

Let n ∈ Z>1 be composite, let k and p1, p2, . . . , pk ∈ Z≥1 be such that p1p2 · · · pk is

the prime factorization of n. Then Equation (5.16) is a multi-dimensional, first-order,

147

nonlinear, ordinary differential equation. However, in light of the solutions given in

Equation (5.18), it can be transformed into

f ′n = p−s1 (1− e−t)p−s2 (1− e−t) · · · p−sk (1− e−t)− fn

= n−s(1− e−t)k − fn

= n−sk∑i=0

(k

i

)(−e−t)i − fn

= n−sk∑i=0

(k

i

)(−1)ie−it − fn (5.19)

Now, Equation (5.19) is a one-dimensional, first-order, linear, ordinary differential

equation. Let

g(t) = n−sk∑i=0

(k

i

)(−1)ie−it (5.20)

Then f ′n(t) = g(t)−fn(t), or equivalently, f ′n(t)+fn(t) = g(t). This has the well-known

solution

fn(t) = e−t(∫ t

1eτg(τ)dτ + κ

)(5.21)

where κ is a constant of integration. Using the initial condition fn(0) = 0,

κ = −∫ 0

1eτg(τ)dτ

=∫ 1

0eτg(τ)dτ (5.22)

148

Then

fn(t) = e−t(∫ t

1eτg(τ)dτ +

∫ 1

0eτg(τ)dτ

)= e−t

(∫ t

0eτg(τ)dτ

)= e−t

(∫ t

0eτn−s

k∑i=0

(k

i

)(−1)ie−iτdτ

)

= n−se−tk∑i=0

(k

i

)(−1)i

∫ t

0eτe−iτdτ

= n−se−tk∑i=0

(k

i

)(−1)i

∫ t

0eτ(1−i)dτ

= n−se−t

((k

0

)∫ t

0eτdτ −

(k

1

)∫ t

01 dτ +

k∑i=2

(k

i

)(−1)i

∫ t

0eτ(1−i)dτ

)

= n−se−t

((et − 1)− kt+

k∑i=2

(k

i

)(−1)i

i− 1(1− et(1−i))

)

= n−se−t

((et − 1)− kt+

k∑i=2

(k

i

)(−1)i

i− 1−

k∑i=2

(k

i

)(−1)i

i−1 et(1−i)

)

Let Hk be the kth harmonic number. Then, using the identity∑k

i=2

(ki

) (−1)i

i−1 = k(Hk−1),

fn(t) = n−se−t

((et − 1)− kt+ k(Hk − 1)−

k∑i=2

(k

i

)(−1)i

i− 1et(1−i)

)

= n−s − n−s(e−t + e−tk(1 + t−Hk) +

k∑i=2

(k

i

)(−1)i

i− 1e−it

)

= n−s − n−se−t(

1 + k(1 + t−Hk)−k−1∑i=1

(k

i+ 1

)(−1)i

ie−it

)

149

As t → ∞ in the positive real direction, fn(t) → n−s. So if Re(s) > 1 then

pres(f(t))→∑∞

n=1 n−s = ζ(s) by Equation (5.1). For all s ∈ C,

pres(f(t)) =1

1− 21−s

∞∑n=0

12n+1

n∑k=0

(−1)k(n

k

)fk+1(t)

→ 11− 21−s

∞∑n=0

12n+1

n∑k=0

(−1)k(n

k

)(k + 1)−s

= ζ(s)

by Equation (5.2).

The above theorem suggest the following definition and corollary.

Definition 5.3.3. Let s ∈ C. For all n ∈ Z≥1, let fs,n : C→ C be such that for all t ∈ C

1. if n = 1 then fs,1(t) = 1

2. if n is prime then fs,n(t) = p−s(1− e−t)

3. if n is composite, k ∈ Z>1 and p1, p2, . . . , pk are such that p1p2 · · · pk is the prime

factorization of n, and Hk is the kth harmonic number, then

fs,n = n−s − n−se−t(

1 + k(1 + t−Hk)−k−1∑i=1

(k

i+ 1

)(−1)i

ie−it

)

Then f s = 〈fs,1, fs,2, . . .〉.

Corollary 5.3.3. For all s ∈ C, f s is a simplified Eζ,2-process.

Proof. Follows immediately from the proof of Theorem 5.3.2.

150

5.4 Visualizing Event-Processes

To get a better understanding of the Riemann zeta function and zeta event-systems, a

technique called “domain coloring” can be used to visualize complex-valued functions of

a complex variable [29]. In this technique, each point in the complex plane is assigned

a color. In the coloring scheme used in this chapter, hue represents the argument of

a complex number and intensity and brightness represent the magnitude of a complex

number (see Figure 5.1). Complex numbers that are close to zero are nearly white and

complex numbers with large magnitude are nearly black.

More formally, let HSB map [0, 1]3 to colors in the HSB color space (Hue, Saturation,

Brightness). Then for all s ∈ C, the color of s is given by

s 7→ HSB(

arg(s)/(2π),max( |s|+ 0.1, 1),max(0.9/(1 + e.5|s|−4 + .1), 1)).

Complex-valued functions of a complex variable can be visualized using this technique.

Given a function f : C → C, each point s ∈ C is depicted using the color given by

HSB(f(s)).

Figure 5.1a is a representation of the identity function f(s) = s, for s = −10− 10i to

s = 10 + 10i. Note the nearly white spot in the center of the figure, which represents the

single root (i.e., zero) of the identity function.

Figure 5.1b shows the Riemann zeta function ζ(s) for s = −20− 10i to s = 20 + 50i.

Note the nearly white region near the negative real axis. In this region, the value of the

Riemann zeta function is close to zero. Indeed, several of the trivial zeros of the zeta

function are in this region. There is also a vertical line of nearly white spots that have

151

-10 -5 0 5 10

-10

-5

0

5

10

(a) Coloring of the identity function, f(s) = s,for s = −10− 10i to z = 10 + 10i.

-20 -10 0 10 20

-10

0

10

20

30

40

50

(b) Coloring of the Riemann zeta function ζ(s)for s = −20− 10i to s = 20 + 50i.

Figure 5.1: Visualizing complex-valued functions of a complex variable using domaincoloring. The identity function is shown in (a) and the Riemann zeta function is shownin (b). Positive real numbers are shown in varying shades of red, negative real numbersin varying shades of cyan, etc.

real part one-half. These are nontrivial zeros of the Riemann zeta function. There is also

a nearly black spot located at 1 + 0i, indicating the simple pole of the zeta function.

It is possible to use this technique to visualize the simplified Eζ,2(s)-process f s as

defined in Definition 5.3.3. Figures 5.2a to 5.2h show f s(t) for s = −10−5i to s = 10+50i

and for various t.

At t = 0 (as shown in Figure 5.2a) the event-process f s is at its initial conditions:

for all s ∈ C, f s(0) = 〈1, 0, 0, . . .〉. There are color variations throughout the image and

152

-10 -5 0 5 10

0

10

20

30

40

50

(a) t = 0

-10 -5 0 5 10

0

10

20

30

40

50

(b) t = 1.23× 10−7

-10 -5 0 5 10

0

10

20

30

40

50

(c) t = 1.27× 10−4

-10 -5 0 5 10

0

10

20

30

40

50

(d) t = 7.31× 10−3

-10 -5 0 5 10

0

10

20

30

40

50

(e) t = 0.130

-10 -5 0 5 10

0

10

20

30

40

50

(f) t = 1.21

-10 -5 0 5 10

0

10

20

30

40

50

(g) t = 7.49

-10 -5 0 5 10

0

10

20

30

40

50

(h) t = 35.0

Figure 5.2: Images of the simplified Eζ,2(s)-process f s, for s = 10− 5i to s = 10 + 50i.

153

periodic poles at Re(s) = 1. These features arise from using Equation (5.7) to measure

pressure, specifically the factor of 11−21−s .

At t = 35 (as shown in Figure 5.2h) the event-process f s has nearly converged to the

Riemann zeta function, at least in the range of temperatures (s values) shown. Note the

similarity between Figure 5.2h and Figure 5.1b.

In analyzing Figure 5.2 (and similar figures later in this chapter), we will sometimes

refer to features “moving”. By this, we mean that there exist s1, s2 ∈ C and t1, t2 ∈ R≥0

such that a feature (value, zero, pole, etc.) at f s1(t1) is also at f s2(t2). This can usually

easily be seen in the figure as a particular color shifting position from one image to the

next.

In Figure 5.2b, a dark band can be seen on the left side of the image, indicating that

the magnitude of the pressure becomes large for temperatures with large negative real

part. To the right of the dark band is a periodic colored band, indicating a region of

moderate pressure, and to the right of that, a light-colored band. Careful examination of

this light-colored band reveals many zeros. In Figures 5.2c and 5.2d, these bands appear

to move to the right.

These images give evidence that, as time moves from t = 0 in the positive real direc-

tion, the zeros of pres(f s(t)) move in approximately the positive real direction, at least

initially. It appears that there are three distinct types of zeros:

1. Those that move to a complex number s with Re(s) > 0 such that ζ(s) = 0, i.e.,

they move to a non-trivial zero of the Riemann zeta function.

154

2. Those that move to a complex number s such that f s(0) is a pole. We call these

“annihilation zeros”.

3. Those that initially move in approximately the positive real direction, but eventually

move to the negative real axis, to a trivial zero of the Riemann zeta function.

5.5 Numerical Simulations of Event Processes

In Section 5.3, we showed that for all s ∈ C, the zeta event-systems Eζ,1(s) and Eζ,2(s) have

a strong equilibrium point c such that pres(c) = ζ(s), and that Eζ,2(s) has a simplified

event-process that asymptotically converges in forward real time to this strong equilibrium

point. What about the event-processes (as opposed to the simplified event-processes) of

these zeta event-systems?

Since finding an explicit solution to the differential equations given by event-processes

of zeta event-systems is difficult, we have simulated subsets Eζ,1 and Eζ,2 numerically.

Rather than use infinitely many species Xi, the event-systems are truncated at a finite

value k and contain only events in C[X1, X2, X3, . . . , Xk]. Transforming the event-system

this way makes it a finite event-system, but it is still not physical nor natural because of

the complex reaction rates. Therefore the results in Chapter 4 do not guarantee that the

process will approach an equilibrium, stay within a compact set, nor even exist. It appears

from the numerical simulations, however, that under many conditions the event-processes

do approach the desired equilibria.

We have simulated Eζ,1 and Eζ,2, (and other zeta event-systems) for various values of

s and k. Initially, the simulations were written in C++. The Runge-Kutta method (both

155

with and without adaptive stepsize control) and the Bulirsch-Stoer method were used

to integrate the differential equations, with the algorithms as described in [65]. Later

attempts were programmed in Mathematica, using its NDSolve method to numerically

integrate.

The results from one simulation of a ζ event-system are shown in Figure 5.3. In this

simulation, k = 50 and the region of temperature simulated was the same as in Figure 5.2:

s = −10− 5i to s = 10 + 50i.

At t = 0 (as shown in Figure 5.3a), one can see the initial conditions of the event-

process. Figure 5.3a is identical to Figure 5.2a, indicating that the event-process and the

simplified event-process begin have the same state at t = 0.

Figures 5.3b and 5.3c appear similar to Figures 5.2b and 5.2c, at least qualitatively.

The black band appears on the left side of the images, with a band of periodic color to

its right, and a band of nearly white to the right of that. These bands appear to move

to the right, just as they did in Figure 5.2.

However, beginning in Figure 5.3c and becoming more pronounced in subsequent

images, areas of gray color appear in the nearly black regions. The gray pixels repre-

sent temperatures s at which the numerical simulation became unstable and could not

continue.

Beginning in Figure 5.3c and becoming more pronounced in subsequent images, dis-

continuities in color in the critical strip can be seen. Areas where discontinuities appear

eventually become gray: the discontinuities are a early sign that the simulation is be-

coming unstable. The discontinuities and instabilities in the simulation indicate that the

event-process is chaotic at some temperatures.

156

In the regions where the discontinuities and instabilities do not appear (right side

of Figure 5.3h and some places in the critical strip), it appears the system converges to

the equilibrium given in Theorem 5.3.1, albeit more slowly than in the simplified event-

process.

5.6 Discussion

Explorations of zeta event-processes and simplified zeta event-processes have hinted at

tantalizing properties of zeta event-systems. Of particular interest, of course, is the

behavior of the zeros of the event-processes. Numerical studies of these systems show the

following recurring themes.

1. The zeros of zeta-processes move continuously as a function of time.

2. The zeros of zeta-processes begin (at time 0) at the point at (complex) infinity, and,

at least initially, move in the positive real direction through positive real time.

3. There are three types of zeros of zeta-processes:

(a) Those that eventually become trivial zeros of the Riemann zeta function (trivial

zeros).

(b) Those that eventually become nontrivial zeros of the Riemann zeta function

(nontrivial zeros).

(c) Those that eventually “annihilate” poles introduced by the factor of 11−21−s

that appears in pres (annihilation zeros).

157

One might suspect that the three types of zeros are independent of one another.

Perhaps surprisingly, this appears not to be the case. Numerical studies of simplified

zeta-processes suggest the three types of zeros are related through complex time. That is

to say, as t moves through C, a trivial zero may become a nontrivial zero or an annihilation

zero.

The pressure of the simplified Eζ,2-process can be viewed as a complex-valued function

of two complex variables, temperature s and time t. From numerical explorations, it

appears this function is a two-dimensional meromorphic function. Further, it appears

that the zero set of this function forms a one-dimensional complex manifold and can be

viewed as the Riemann surface of a complete analytic function. While the structure of the

branch points of this complete analytic function appears to be quite intricate, the function

may contain substantial information about the structure of the zeros of the Riemann zeta

function. In particular, the value on each branch as t approaches infinity in the positive

real direction appears to converge to the position of one of the trivial, nontrivial, or

annihilation zeros of the Riemann zeta function. Coming to understand this complete

analytic function could aid in the understanding of the Riemann zeta function.

158

(a) t = 0 (b) t = 6.38× 10−8 (c) t = 1.48× 10−2 (d) t = 0.335

(e) t = 4.14 (f) t = 30.8 (g) t = 66.2 (h) t = 100

Figure 5.3: Images of the numerical simulation of a Eζ,2(s)-process, for s = 10 − 5i tos = 10 + 50i. Gray indicates temperatures and times where the numerical simulationbecame unstable and could not continue.

159

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