8 intro ds2 - spatial-computing.org · MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. A...

MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org

MGS « encore un Modèle Général de Simula3on »

and Spa&al Compu&ng

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Jean-‐Louis Giavi:o, Antoine Spicher, Olivier Michel

[email protected] [email protected]

[email protected]


Chronology and warnings

• MGS: start September 2000 •  Spa3al Compu3ng

– Dagsthul 2006 –  SC workshop since 2008

•  The tutorial is –  “unconven3onal programming” oriented –  examples and applica3on based: unconven-onal algorithmic, biology, abstract morphogenesis

–  no complexity and few theory

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A topological manifesto Spa3al compu3ng proposes to celebrate

corporeality of data rather than trying to deny it. Simon Greenworld (MIT medialab)

•  The logical approach in computer science

computa3on = deduc3on (the Curry-‐Howard isomorphism)

•  Other paradigms can be fruiZull : topology

computa&on = moving in some space •  Try to perceive space (and 3me) in programs

(rather than logical opera3ons) purposes: technical, heuris-c, didac-cal

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Material

•  The MGS home page (new home page for the end of the year) h:p://mgs.spa3al-‐compu3ng.org

•  MGS download (new home page for the end of the year) h:p://www.spa3al-‐compu3ng.org/mgs

•  This MGS tutorial h:p://www.spa3al-‐compu3ng.org/mgs:tutorial

•  MGS publica3on h:p://mgs.spa3al-‐compu3ng.org/PUBLICATIONS (old) h:p://repmus.ircam.fr/giavi:o/publica3ons h:p://www.lacl.fr/~michel/doku.php?id=research:start h:p://www.lacl.fr/~michel/www/bib h:p://www.lacl.fr/~aspicher/publica3ons.html

•  The Spa3al Compu3ng home page

h:p://www.spa3al-‐compu3ng.org

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A complete schedule

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A.  Mo3va3on & Applica3on domains 1.  (DS)2 and morphogenesis 2.  Interac3on-‐Based Compu3ng 3.  Spa3al Compu3ng

B.  MGS …

C.  Applica3ons …

D.  Implementa3on …

E.  Current Fron3ers …


Lecture I (Tuesday, September 4)

A.  Mo3va3on & Applica3on domains 1.  Modelling Morphogenesis 2.  (DS)2: dynamical systems with a dynamical structure

B.  MGS 1.  Collec3on 2.  Transforma3on 3.  A Func3onal Language

C.  Applica3ons 1.  Algorithmic Examples 2.  Monoidal Collec3ons 3.  GBF : Cellular and Laice Gaz Automata 4.  Delaunay : the growth of a meristem

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Lecture II (Wednesday, September 5)

A.  Mo3va3on & Applica3on domains 1.  Spa3al Compu3ng 2.  Interac3on-‐Based Compu3ng

B.  MGS 1.  Abstract Combinatorial Complex 2.  Formaliza3on

C.  Applica3ons 1.  Remeshing 2.  Growing a « T » 3.  Self-‐Assembly of Musical Spaces 4.  Growing an ontology with the Li:le Riding Hood 5.  Growing an analogy with Paul Ricœur

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Lecture III (Thursday, September 6)

B.  MGS 1.  Formalisa3on 2.  Stochas3c strategies 3.  Discrete Differen3al Operators

C.  Applica3ons 1.  A generic diffusion operator 2.  Four models for the Paris iGEM’07 project

D.  Implementa3on 1.  Incremental HOAS implementa3on of a DSL 2.  Beyond algebraic data type : a Generic Advanced API for Containers 3.  Generic pa:ern matching

E.  Current Fron3ers From Global to Local , Mul3-‐level and par3al processes, Hybrid Modeling, Time

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MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. http://mgs.spatial-computing.org 9

Modeling and Simula&on of Morphogenesis


Environment characterized by its effects on the system

System described by a state (determined by observation)

10

Modelling morphogenesis: the approach of A. Turing


statet-‐1 statet statet+1 N

statet-‐dt statet statet+dt R

Specifying a dynamical system (for simula3on)

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statet

Specifica&on of •  structure of state •  structure of &me •  evolu&on func&on

H H

H*

∫H(t)dt

state

time

evolution


Morphogenesis as a Dynamical System

Modelling a dynamical system •  state, including space (e.g. fields) •  time •  evolution function

C : continuous, D: discrete

PDE Coupled ODE

Iteration of functions

Cellular automata …

state C C C D …

time C C D D …

space C D D D …


Modelling morphogenesis: the approach of A. Turing

•  Uniform maDer, con&nuous-‐oriented system descrip&on è SPACE based One choice is to ignore cells completely, e.g., Physiome models 3ssues as con3nua with bulk mechanical proper3es and detailed molecular reac3on networks, which is computa3onally efficient for describing dense 3ssues and non-‐cellular materials like bone, extracellular matrix , fluids, and diffusing chemicals, but not for situa-ons where cells reorganize or migrate.

versus

•  Cell-‐oriented discrete system descrip&on è ENTITY based Mul3-‐cell simula3ons are useful to interpolate between single-‐cell and con3nuum-‐3ssue extremes because cells provide a natural level of abstrac3on for simula3on of 3ssues, organs and organisms. Trea3ng cells phenomenologically reduces the millions of interac3ons of gene products to several behaviors: most cells can move, divide, die, differen3ate, change shape, exert forces, secrete and absorb chemicals and electrical charges, and change their distribu3on of surface proper3es.

(CompuCell3D manual)

Aggregate-‐ vs. En&ty-‐based models


Modeling morphogenesis: paDerning vs. development

4 Spicher, Michel & Giavitto

data) and the content (chemical data) are clearly decoupled, making these modelsusually fall into two categories2:

• formalisms that focus on pattern formation in an initially homogeneous but staticsubstrate, and

• generative formalisms that specify the creation and the evolution of a dynamicshape, irrespective of the processes that may take place within the shape.

Examples of the first category are given by reaction-diffusion systems, activator-inhibitor models, or random Boolean networks. They rely on various model of dy-namical systems (Table 1). Examples of the second category include Lindenmayersystems (L-systems) [36], membrane computing [46], graph grammars [49], self-assembly systems, and others (Table 2).

Table 1 Formalisms that can be used to specify structured dynamical systems according to (a) theunderlying space in which the patterning process takes place, (b) a continuous or discrete repre-sentation of time, and (c) the state variables of the system’s components. “Numerical Solutions”refer to explicit numerical solutions of partial differential equations (PDE) or systems of coupledordinary differential equations (ODE).

C: continuousD: discrete PDE Coupled ODE Numerical

SolutionsCellular

Automata(a) Space C D D D(b) Time C C D D(c) States C C C D

Table 2 Generative formalisms that can be used to specify the evolution of a shape, accordingto the topology connecting the components of the shape. In a multiset, all elements are consideredconnected to each other. In a sequence, elements are ordered linearly: this case includes lists and ex-tends to tree-like structures (lists of lists). Uniform structures involve a regular neighborhood, e.g.,a rectangular lattice where each element has exactly four neighbors. Group-Based Fields (GBF,Section 2.1.4) are a powerful tool relying on mathematical groups to represent such structures.The first four cases describe spatial structures that can be accurately depicted by a graph. Beyondgraphs, nD combinatorial structures are used to define arbitrary connections between componentsof various dimensions. The MGS language (next section) can handle all these types of shapes.

Topology: multiset sequence uniform arbitrary graph nD combinatorialstructures

Formalism: membranesystem

L-systems GBF map L-systems,graph-grammars

MGS

2 This distinction is somewhat contrived. For instance, cellular automata have been devised tostudy self-reproduction of distinct entities, but these entities are represented by specific patterns ina predefined medium.

generative formalism

C : continuous, D: discrete

PDE Coupled ODE Iteration of functions

Cellular automata …

state C C C D …

time C C D D …

space C D D D …

(vs. dynamical systems)


Modeling morphogenesis: pa:erning in a predefined medium

Alexander Gavrilovitch Gurwitsch (1875-‐1954) Morphogene3c Fields


Modeling morphogenesis: pa:erning in a predefined medium

Compatible with •  the notion of morphogenetic field •  cell fate But •  there is evidence for

feedback loops between the shape and the process inhabiting the shape

from E. Haenkel (cited by C. Goodman-‐Strauss): example of a nega3ve curvature surface. Curvature can be controlled while the surface is growing along a ‘front’


The medium/process problem


•  (px, py) •  (vx, vy) •  (px’, py’) •  (vx’, vy’)

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a falling ball

at any -me a state is a posi-on and a speed

A dynamical system (DS)



a falling ball a developing embryo

the structure of the state (chemical and mechanical state of each cell)

is changing in -me A dynamical system with a dynamical structure

(DS)2

•  (px’, py’) •  (vx’, vy’)

at any -me a state is a posi-on and a speed

A dynamical system (DS)


The interplay between state and form

a developing embryo

the state as well as the structure of the state is changing in 3me (chemical and mechanical state of each cell as well as the arrangement of the cells)

A dynamical system with a dynamical structure (DS)2

Dynamics ON form Dynamics OF form F

orm

State

“Strange loop”


Waddington : epigene3c landscape

Link Li & Mhairi Towler, 2012

Conrad Waddington (1905-‐1975)


Such cycles are not new in biology

Tibor Ganti’s CHEMOTON (1971)

metabolic cycle

template molecules

membranogenic molecule

… a “chemical automaton”… (also called fluid automata)

Varela’s AUTOPOIESIS (1974)

“an autopoietic unity is able to self-generate owing to a reaction network taking place within its

own boundary” (the reaction network makes all components of the unit, including the boundary)

M. Eigen & P. Schuster: hypercycle, W. Fontana & L. Buss: organization …


•  Complex systems •  Whose structure evolves over the 3me

–  The structure constrains the behavior of the system that modifies the structure … –  The phase space cannot be defined a priori –  The evolu3on func3on cannot be specified globally

•  Examples –  Biology

•  Molecular biology •  Developmental biology

–  Physics •  Soz ma:er mechanics,

mul3-‐scale systems •  General rela3vity

–  Urbanism City growth,

traffic control, … –  Computer science

Internet, sensor network, reconfigurable nanobots, knowledge network, …

DS² : Dynamical System with a Dynamical Structure

© David S. Goodsell 1999

© Tecplot

© L. Sanders - EUROSIM

© C. Harrison - Clusterball project 24


What has changed since Turing’s &me

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What has changed since Turing’s &me

P. Prusinkiewicz, c.2003 Diffusion and reac3on in a deformable surface (E. Coen’s expanding canvas metaphor). Spring-‐mass system. No topological change.


Sozware as Science ?

•  Intelligibility The en-re process should be accessible for analysis into a finite, not very large number of stages, each stage being represented as a monotonic func-on of some definite ini-al condi-ons and a single variable such as -me, or distance, etc. (Gurwitsch, 1944) à compress behavior or shape in few rules

•  Simula3on is only a first step: models must enable reasoning à stay close to mathema3cal formalism

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M.-P. C

ani & F. B

ertails

A program is a formal object (and some form of reasoning on it is possible) but a 106 lines of codes is not an explanation !


(…) An invagina3on of a germ layer may be explained on a basis of a pressure difference between the two surfaces (or sides) or by cell movements, and so forth. This can be considered as an 'explana3on' un3l we ask about the origin of pressure differences, or the mechanisms involved in cell movement, etc. However, ques3ons of this kind become trivial when a larger process, rather than its individual components becomes the main problem. Suppose for a moment that each element in the succession A, a, C… can be explained separately, e.g. A as a swelling, a as a chemical reac3on, etc. Interes3ng as they may be, these explana3ons are of subordinate importance when related to the main ques3on: Why indeed is a regular (emphasis added) succession of these obviously quite different processes taking place at all? Most biological problems are of this kind and all of embryogenesis is just such a single problem. Here we require a peculiar or, maybe, original explanatory principle... A process may become accessible to explana3on only insofar as one can succeed in subs3tu3ng [understanding of] a purely phenomenological mul3plicity and diversity of events [for understanding] of a less diverse and less arbitrarily created picture correctly reflec3ng reality. The main aim of such a construc3on would be as follows. The en3re process should be accessible for analysis into a finite, not very large number of stages, each stage being represented as a monotonic func3on of some definite ini3al condi3ons and a single variable such as 3me, or distance, etc. If this cannot be realized, we consider a given set of events as scien3fically inaccessible. On the other hand, even a par3al success of such an enterprise is an obvious step forward."

(Gurwitsch, 1944) cited by Beloussov in “Life of Alexander G. Gurwitsch and his relevant contribu3on to the theory of morphogene3c fields”, Int. J. Dev. Biol. 41, 771-‐779 (1997)

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COMBINING PATTERNING AND DEVELOPMENT

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Pa:erning vs. Growth

•  Dynamics ON form pa:erning, mo3f forma3on in a predefined form – Diffusion, reac3on-‐diffusion, transport (con3nuous models) –  cellular automata (discrete models)

•  Dynamics OF form growth, deforma3on of a shape – Deforma3on of elas3c bodies (con3nuous models) –  Lindenmayer systems (discrete models)

30


The interplay between state and form a developing embryo

the state as well as the structure of the state is changing in 3me

(chemical and mechanical state of each cell as

well as the arrangement of the cells)

A dynamical system with a dynamical structure (DS)2

Dynamics ON form Dynamics OF form F

orm

State

31


(DS)2 versus DS

•  Processes in the form are topologically (geometrically) meaningful e.g. growth rate

•  Topological (geometric) informa3on is meaningful e.g. domain of diffusion, informa3on transfert

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Topological informa3on is meaningful

A basic example [T. Gross, Adapta3ve Networks]: –  Infec3ous disease spreading across a network – Node are either suscep-ble or infected

–  Suscep3ble nodes linked to an infected node become infected with probability α

–  Infected nodes recover with probability β

33


Topological informa3on is meaningful

The disease persists in the network depending: –  of the probability α

–  of the probability β

–  of the mean excess degree d the average number of links that one finds connected to a node that is reached by following a random link: a global topological informa3on

•  d > threshold: every node will be infected once in a while (in a large network)

•  By watching the behavior of a single node (local observa-on) for a sufficiently long period of 3me, we can es3mate if the mean excess degree (global topological informa-on) exceeds the threshold

•  Simple mechanism can put the system at the cri-cal point! (no special reason to be in a generic state)

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The interplay between form and topology is meaningful

•  A suscep3ble node is allowed to cut a link with an infected node with a certain probability decreasing when the delay between its own infec3ons increase

•  Now there is a feedback loop between topologies and states of the network the dynamics of the prevalence of the disease depends on the network topology, but the evolu3on of the network topology also depends on the prevalence of the disease

Simple mechanism can put the system at the cri@cal point! no special reason to be in a generic state

e.g. synchroniza3on of growth rate in flat leaf

(and it can be worst if there is no 3mescale separa3on)

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OK: the coupling is important. What can we do?

1.   “Dynamics ON form” toward DS2 –  parameterize (control) the shape by some quan33es

(e.g. curvature for a manifold or adjacency matrix for a graph)

–  link these quan33es with processes in the shape

•  growth depending on concentra3on

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Differen&al geometry is not enough

•  Encoding a shape into con3nuous parameters is difficult

•  It does not handle topological changes very well

apical elonga3on depends of 3me

•  Ok, some3mes they are some tricks (e.g. level set) but they are very smart tricks

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OK: the coupling is important. What can we do?

1.   “Dynamics ON form” toward DS2 –  parameterize (control) the shape by some quan33es

(e.g. curvature for a manifold or adjacency matrix for a graph)

–  link these quan33es with processes in the shape

•  growth depending on concentra3on

2.   “Dynamics OF form” toward DS2 –  enhance the form by parameters

–  put a dynamic on these parameters example: module in Lindenmayer systems

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Lindenmayer systems

39

The growth of Anabaena Catenula •  two cell types : vegeta3ve and heterocyst •  asymetric division : one is smaller than the other •  polarized cell (lez/right orienta3on)

A a B b

a -> aB A -> bA b -> aB B -> bA a a B a B b A a B b A a B b A a B b A a B b A a B b A a B b A ...


A good example of PaDerning+Development in a declara&ve formalism: Lindenmayer systems with modules

•  The structure of a tree can be coded by a string of parenthe3sed symbols

•  A symbol is an elementary part of the plant

•  The symbol between [ and ] represents a sub-‐tree

•  Addi3onal conven3ons are used to represent main axis, orienta3on, depth, etc.

•  A rule s0 → s1 s2 s3 …

represents the evolu3on of s0 P. P

rusi

nkie

wic

z


Diffusion and reac3on in a linear growing medium M. Hammel and P. Prusinkiewicz (1996)

The following rules state that a differen3ated cell (heterocyst) returns to a vegeta3ve state if the concentra3on of the ac3vator is too low. In addi3on, if the cell is large enough, it con3nues to grow.

e / (D(e) & (e.a < thr) | (e.x >= shorter*gr)) => {type ="C", a=e.a/gr, h=e.h/gr, x=e.x*gr, p=e.p};

The following rule specifies when a cell with a lez polarity divides. Only vegeta3ve cells can divide (hence the predicate C in the rule guard) and it must be large enough. The volume of the two daughter cells remains the same, so there is no varia3on in the concentra3on.

e / (C(e) & (e.x >= lm) & (e.p == L))

=> {type="C", a=e.a, h=e.h, x=e.x*shorter, p=L},

{type="C", a=e.a, h=e.h, x=e.x*longer, p=R};

Linear cell structure

morphogene concentration

time


•  Rewri3ng system –  Used to formalize equa3onnal reasoning –  A genera3ve device (grammar) –  Replace a sub-‐part of an en3ty by an other –  Set of rewri3ng rules α à β

•  α: pa:ern specifying a sub-‐part •  β: expression evalua3ng a new sub-‐part

•  Example: arithme3c expressions simplifica3on

Rewri&ng systems (and abstract transi3on systems)

x

+

0

x

y

+

x x

+

y


A non-‐standard interpreta3on

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e1 + e2 → … e1 can be a cell and e2 a signal e1 and e2 can be interac3ng cell

+ is the possibility of interac-on between en33es (or some other rela3onships)

→  is the passing of 3me, a local evolu3on, a transi3on, the concre3za3on of the interac3on

Examples: if e is a cell and i a biochemical signal

e + i → e’ growth (evolu3on of e on signal i) e + i → e+i’ quorum sensing e + i → e’ + e’’ division e + i → . apoptose


Rewri3ng systems in simula3on

Complex systems Rewri&ng techniques

Modelling Specifica3on State (space) Data structure

hierarchical and tree organiza3ons arbitrary complex organiza-ons

formal trees (or terms) ?

Evolu&on func&on Set of rules interac@ons à evolu3on

local evolu&on laws α : pa:ern à β : expression rewri3ng rules

Simula3on Applica3on Trajectories

Time management Deriva3ons Rule applica3on strategy

discrete, event-‐based, synchronous vs. asynchronous

maximal parallel, sequen3al, determinis3c, stochas3c

ßà


•  local evolu-on rules mandatory when you cannot express a global func3on/rela3on because the domain of the func3on/rela3on is changing in 3me

•  interac3on based approch the l.h.s. of a rule specifies a set of elements in interac-on, the r.h.s. the result of the interac3on

•  the phase space is well defined but not well known a genera3ve process enumerates the elements but membership-‐test can be very hard

•  various kind of -me evolu-on (for the same set of rules)

•  demonstra-on by induc-on on the rules or on the deriva3on (e.g. growth func3on in L system)

Proper3es


How to extend to arbitrary spa3al structure?

•  Anabaena was « easy » because of the linear uniform structure •  How to handle the complex spa3al structure of a cell?

David S. Goodsell


Three challenges and some tools

compilation, engineered emergence

validation, analysis, collective

properties

limits, non-standard analysis

approximation, numerical resolution,

partitionning

DNA

cellular machinery

Cellular machinery

DNA

global local

discrete

software/data hardware/programme

population

individual Molecules, Compartments,

Cells

concentration, time, fields

continuous


Acknowledgements MGS: Antoine Spicher, Olivier Michel, Julien Cohen S&S Bio : Hanna Klaudel, Franck Delaplace, Hugues Berry, Przemek Prusinkiewicz, Annick Lesne… Spa3al Compu3ng: Jacob Beal, Fréderic Gruau, René Doursat…

Examples: Pierre Barbier de Reuille, Christophe Godin, Samuel Bo:ani, the Paris iGEM’07 team…

Some figures are borowed from Olivier Michel, Antoine Spicher, Pierre Barbier de Reuille, Franck Delaplace, Hugues Berry (INRIA), the iGEM Paris 2007 and many others.

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•  Antoine Spicher •  Olivier Michel •  PhD and other students J. Cohen, P. Barbier de Reuille, E. Delsinne, V. Larue, F. Le3erce, B. Calvez, F. Thonerieux, D. Boussié and the others... •  Collabora&ons •  A. Lesne (IHES, stochas3c simula3on) •  P. Prusinkiewicz (Calgary, declara3ve modeling) •  P. Barbier de Reuille (meristeme model) •  C. Godin (CIRAD, biological modeling) •  H. Berry (LRI, stochas3c simula3on) •  G. Malcolm (Liverpool, rewri3ng) •  J.-‐P. Banâtre (IRISA, programming) •  F. Delaplace (IBISC, synthe3c biology) •  P. Di:rich (Jena, chemical organiza3on) •  E. Goubault (CEA, topological formaliza3on) •  F. Gruau (U. PXI, language and hardware) •  P. Liehnard (Poi3er, CAD, Gmap and quasi-‐manifold)

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Thanks http://mgs.spatial-computing.org

8 intro ds2 - spatial-computing.org · MGS @ UCNC’2012. J.-L. Giavitto, A. Spicher. A...

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