Physics of Information Technology MIT – Spring 2006

PART II

Avogadro Scale Engineering‘COMPLEXITY’

jacobson@media.mit.edu

Homework• I] Nanotech Design:

Find an error function for which it is optimal to divide a logic area A into more than one redundant sub-Areas.

• II] Design Life: (a) Design a biological system which self replicates with

error correction (either genome copy redundancy with majority voting or error correcting coding). Assume the copying of each nucleotide is consumptive of one unit of energy. Show the tradeoff between energy consumption and copy fidelity.

(b) Comment on the choice biology has taken (64 -3 nucleotide) codons coding for 20 amino acids. Why has biology chosen this encoding? What metric does it optimize? Could one build a biological system with 256 – 4 bit codons?

Questions: jacobson@media.mit.edu

Area = A

Area = 2*A/2

Probability of correct functionality = p[A] ~ e A (small A)

Scaling Properties of Redundant Logic (to first order)

P1 = p[A] = e A

P

A

P2 = 2p[A/2](1-p[A/2])+p[A/2]2

= eA –(eA)2/4

Conclusion: P1 > P2

Designing Life

Redundancy

Fault Tolerant

Error CorrectingOther Coding (e.g. Parity)

Fault Tolerant

Error Correcting

Designing Life

Gene1 Gene2 Gene3 Gene1 Gene2 Gene3

I] Fault Tolerant Redundancy

http://www.biochem.ucl.ac.uk/bsm/xtal/teach/repl/klenow.html

1. Beese et al. (1993), Science, 260, 352-355.

Replicate Linearly with Proofreading and Error Correction

Fold to 3D Functionality

template dependant 5'-3' primer extension

5'-3' error-correcting exonuclease

3'-5' proofreading exonuclease

Error Rate:1: 106

100 Steps per second

MutS Repair System

http://www.ornl.gov/hgmis/publicat/microbial/image3.html

[Nature Biotechnology 18, 85-90 (January 2000)]

Uniformed Services University of the Health

Deinococcus radiodurans (3.2 Mb, 4-10 Copies of Genome )

D. radiodurans: 1.7 Million Rads (17kGy) – 200 DS breaksE. coli: 25 Thousand Rads – 2 or 3 DS breaks

Approach 1b] Redundant Genomes

Basic Idea:

M strands of N Bases

Result: By carrying out a consensus vote one requires only

To replicate with error below some epsilon such that the global replication error is:

EP

NM ln

Combining Error Correcting Polymerase and Error Correcting Codes One Can Replicate a

Genome of Arbitrary Complexity

M

N

100 200 300 400 500

10

15

20

25

30

M (

# of

Cop

ies

o f G

enom

e)

N (Genome Length)

EPM

Ribosome

mRNA

Amino Acid

II] Coding

4 Base Parity Genetic CodeLet A=0, U,T=1, G=2, C=3 Use 3+1 base codeXYZ Sum(X+Y+Z, mod 4)

Leu: UUA -> UUAG

http://schultz.scripps.edu/Research/UnnaturalAAIncorporation/research.html

Fault Tolerant Translation Codes (Hecht):NTN encodes 5 different nonpolar residues (Met, Leu, Ile, Val and Phe)NAN encodes 6 different polar residues (Lys, His, Glu, Gln, Asp and Asn)

Local Error Correction:Ribozyme: 1:103

Error Correcting Polymerase: 1:108 fidelity

DNA Repair Systems:MutS System

Recombination - retrieval - post replication repair Thymine Dimer bypass.Many others…

Error Correction in Biological Systems

E. Coli Retrieval system - Lewin

Biology Employs Error Correcting Fabrication + Error Correcting Codes

Physics of Information Technology MIT – Spring 2006

4/101] Von Neumann / McCullough/Winograd/Cowan Threshold Theorem and Fault Tolerant Chips2] Simple Proofs in CMOS Scaling and Fault Tolerance3] Fault Tolerant Self Replicating Systems4] Fault Tolerant Codes in Biology

4/241]Introduction of the concept of Fabricational Complexity2]Examples, numbers and mechanisms from native biology: error correcting polymerase and comparison to best current chemical synthesis using protection group (~feedforward) chemistry.3]Examples from our error correcting de novo DNA synthesis (with hopefully a demo from our DNA synth simulator)4]Error correcting chip synthesis5]Saul's self replicating system with and without error correction

Fabricational Complexity

Ffab = ln (W) / [ a3 fab Efab ]

Ffab = ln (M)-1 / [ a3 fab Efab ]

•Total Complexity•Complexity Per Unit Volume•Complexity Per Unit Time*Energy•Complexity Per unit Cost

Fabricational Complexity

n

n

nFAB mpF ln

1

200 400 600 800 1000

10

20

30

40

50

60

70

A

A G

G T C

A T A C G T …

A G T A G C …

p2p3p

Total Complexity Accessible to a Fabrication Process withError p per step and m types of parts:

Complexity Per Unit Cost:For given complexity n*:

Cmpf nFAB /ln

*

Where C is cost per step

Cmpf nFAB /ln

*

Fabricational Complexity

Non Error Correcting:

Triply Error Correcting:

Cmpppfn

FAB 3/ln)1(3*332

3

A G T C

A G T C

A G T C

A G T C

50 100 150 200 250 300

20

40

60

80

100

120

140

P = 0.9

n

FAB

FAB

f

f 3

p

0.86 0.88 0.92 0.94 0.96 0.98

500

1000

1500

2000

2500

3000

n = 300

50 100 150 200

0.05

0.1

0.15

0.2

0.25

0.3

FAB

FAB

f

f 3

n

P = 0.85

1] Quantum Phase Space 2] Error Correcting Fabrication 3] Fault Tolerant Hardware Architectures 4] Fault Tolerant Software or Codes

Resources which increase the complexity of a system exponentially with a linear addition of

resources

Resources for Exponential Scaling

…Can we use this map as a guide towards future

directions in fabrication?

Genome (Natural)

Gene Chip (Chemical Parallel Synthesis)

Semi-conductor Chip

High Speed Offset Web TFT DVD-6

Liquid Embossing

Design Rule Smallest Dimension (microns) 0.0003 0.0003 0.1 10 2 0.25 0.2Number of Types of Elements 4 4 8 6 8 2 4Area of SOA Artifact (Sq. Microns) NA 7.E+08 7.E+10 2.E+12 1.E+12 1.E+10 8.E+09Volume of SOA Artifact (Cubic Microns) 6.E+01 5.E+06 7.E+09 2.E+12 1.E+11 7.E+12 8.E+08Number of Elements in SOA Artifact 3.E+09 7.E+04 7.E+12 2.E+10 3.E+11 2.E+11 2.E+11Volume Per Element(Cubic Microns) 2.E-08 8.E+01 1.E-03 1.E+02 4.E-01 4.E+01 4.E-03Fabrication Time(seconds) 4.E+03 2.E+04 9.E+04 1.E-01 7.E+02 3 6.E+01Time Per Element (Seconds) 1.E-06 3.E+02 1.E-08 7.E-12 2.E-09 2.E-11 3.E-10Fabrication Cost for SOA Artifact($) 1.E-07 1.E+02 1.E+02 1.E-01 2.E+03 3.E-02 2.E-01Cost Per Element 3.E-17 2.E-03 2.E-11 6.E-12 6.E-09 2.E-13 1.E-12Complexity 4.E+09 9.E+04 2.E+13 4.E+10 6.E+11 1.E+11 3.E+11Complexity Per Unit Volume of SOA(um 3̂) 7.E+07 2.E-02 2.E+03 2.E-02 5.E+00 2.E-02 3.E+02Complexity Per Unit Time 1.E+06 6.E+00 2.E+08 3.E+11 9.E+08 4.E+10 5.E+09Complexity Per Unit Cost 4.E+16 9.E+02 1.E+11 3.E+11 3.E+08 4.E+12 1.E+12Cost Per Area NA 2.E-07 2.E-09 6.E-14 2.E-09 3.E-12 3.E-11

Fabricational Complexity

http://www.biochem.ucl.ac.uk/bsm/xtal/teach/repl/klenow.html

1. Beese et al. (1993), Science, 260, 352-355.

Replicate Linearly with Proofreading and Error Correction

Fold to 3D Functionality

template dependant 5'-3' primer extension

5'-3' error-correcting exonuclease

3'-5' proofreading exonuclease

Error Rate:1: 106

100 Steps per second

Caruthers Synthesis

DNA Synthesis

http://www.med.upenn.edu/naf/services/catalog99.pdf

Error Rate:1: 102

300 SecondsPer step

Molecular Machine (Jacobson) Group – MIT - May, 2005

Avogadro Scale Engineering

Gene LevelError Removal

Error Rate 1:104Nucleic Acids Research 2004 32(20):e162

In Vitro Error Correction Yields >10x Reduction in Errors

Nucleic Acids Research 2004 32(20):e162

Error Reduction: GFP Gene synthesis

Nucleic Acids Research 2004 32(20):e162

Autonomous self replicating machines from random building blocks

http://alumni.media.mit.edu/~saul/PhD/videos/xtalcontrol1web.mov

1] Consider biological cells which are able to copy their genome using appropriate pieces of molecular machinery (e.g. polymerase).  Assume that the total probability of correctly copying each nucleotide is p=.999 per nucleotide.  Calculate the Total Fabrication Complexity accessible to this system assuming that there are 4 types of nucleotides (i.e. A,G,C,T).  Now assume that we have created a new type of cell which has a genome possessing six different types of nucleotides (i.e. A,G,C,T,X,Y).  If we assume that we wish to keep the total Fabricational Complexity the same what must the probability per nucleotide addition, p, now be? 

2] Consider now the fabricational complexity per unit cost f.  Calculate the threshold probability p for which it is advantageous to use a redundant error correction scheme (such as trible redundancy) and majority voting than no error correction.  Into which regime does biology fall?

HOMEWORK – DUE 5/1/06