MAS.S62 FAB2

2.28.12

The Threshold for Life

http://lslwww.epfl.ch/pages/embryonics/thesis/Chapter3.html

Complexities in Biochemistry

Atoms: ~ 10Complexion: W~310 Complexity x = 15.8

Atoms: ~ 8Complexion: W~38

Complexity x = 12.7

DNA N-mer

Types of Nucleotide Bases: 4Complexion: W=4N

Complexity x = 2 N

Complexity Crossover: N>~8

Atoms: ~ 20 [C,N,O]Complexion: W~ 320 x = 32

Product: C = 4 statesx = 2

x[Product / Parts] =~ .0625

Complexity (uProcessor/program):x ~ 1K byte = 8000

Product: C = 4 statesx = 2

x[Product / Parts] =~ .00025

DNA Polymerase

Nucleotides: ~ 1000Complexion: W~41000 x = 2000 = 2Kb

Product: 107 Nucleotidesx = 2x107

x[Product / Parts] =104

x >1 Product has sufficient complexity to encode for parts / assembler

Synthetic Complexities of Various Systems

ComplexityApplication: Why Are There 20 Amino Acids in Biology?(What is the right balance between Codon code redundancy and diversity?)

Qi

iQNN

nNW

!)(!

!!

500 1000 1500 2000

10

20

30

40

N

*Q

Question: Given N monomeric building blocks of Q different types, what is the optimal number of different types of building blocks Q which maximizes the complexity of the ensemble of all possible constructs?

The complexion for the total number of different ways to arrange N blocks of Q different types (where each type has the same number) is given by:

And the complexity is:

N Blocks of Q Types

QNQNQNQNNQN )ln()(*)ln(),( x

For a given polymer length N we can ask which Q* achieves the half max for complexity such that:

),(5.0*),( NNFQN x

.

T Wang et al. Nature 478, 225-228 (2011) doi:10.1038/nature10500

Nucleotides: ~ 150Complexion: W~4150 Complexity x = 300

Product: 7 Blocksx = 7

x[Product / Parts] =.023 The percentage of heptamers with the correct sequence is estimated to be 70%

Information Rich Replication (Non-Protein Biochemical Systems)

RNA-Catalyzed RNA Polymerization: Accurate and General RNA-Templated Primer ExtensionScience 2001 May 18; 292: 1319-1325Wendy K. Johnston, Peter J. Unrau, Michael S. Lawrence, Margaret E. Glasner, and David P. Bartel

RNA-Catalyzed RNA Polymerization

14 base extension. Effective Error Rate: ~ 1:103

J. Szostak, Nature,409, Jan. 2001

RNA (2007), 13:1017–1026. Published by Cold Spring Harbor Laboratory Press.

Selection of an improved RNA polymerase ribozymewith superior extension and fidelityHANI S. ZAHER and PETER J. UNRAU

20 NT Extension x[Product / Parts] =~ .1

http://www.uncommondescent.com/biology/john-von-neumann-an-ider-ante-litteram/

http://web.archive.org/web/20070418081628/http://dragonfly.tam.cornell.edu/~pesavent/pesavento_self_reproducing_machine.pdfhttp://en.wikipedia.org/wiki/File:320_jump_read_arm.gif

http://en.wikipedia.org/wiki/Von_Neumann_universal_constructor

Implementations of Von Neumann’s Universal Constructor

http://necsi.edu/postdocs/sayama/sdsr/java/#langton

Self Replication Simulators

Langton Loops

http://carg2.epfl.ch/Teaching/GDCA/loops-thesis.pdf

http:

//ca

rg2.

epfl.

ch/T

each

ing/

GDCA

/loop

s-th

esis.

pdf

http://carg2.epfl.ch/Teaching/GDCA/loops-thesis.pdf

http://carg2.epfl.ch/Teaching/GDCA/loops-thesis.pdf

CANumbe

r of States

Neighborhood

Number of Cells (typical)

Replication Period

(Typical)Thumbnail

Langton's loops[3] (1984): The original self-reproducing loop. 8 von Neumann 86 151

Byl's loop[4] (1989): By removing the inner sheath, Byl reduced the size of the loop. 6 von Neumann 12 25

Chou-Reggia loop[5] (1993): A further reduction of the loop by removing all sheaths. 8 von Neumann 5 15

Tempesti loop[6] (1995): Tempesti added construction capabilities to his loop, allowing patterns to be written inside the loop after reproduction.

10 Moore 148 304

Perrier loop[7] (1996): Perrier added a program stack and an extensible data tape to Langton's loop, allowing it to compute anything computable.

64 von Neumann 158 235

SDSR loop[8] (1998): With an extra structure-dissolving state added to Langton's loops, the SDSR loop has a limited lifetime and dissolves at the end of its life cycle.

9 von Neumann 86 151

Evoloop[9] (1999): An extension of the SDSR loop, Evoloop is capable of interaction with neighboring loops as well as of evolution..[10]

9 von Neumann 149 363

Fault-Tolerant Circuits

n MAJ

ppp

MAJMAJ

ppp

MAJ

ppp

k

Threshold Theorem – Von Neumann 1956

mnmn

nm

ppmn

P

)1(2/)1(

kk

pP

pppP

pppP

k2)12(

4322212

221

3

3)3(3)(3

3)1(3

Recursion Level PK=1K=2K

n=3

For circuit to be fault tolerant

3/13 212

Th

k

PppP

kk

n MAJ

ppp

MAJMAJ

ppp

MAJ

ppp

k

Threshold Theorem - Winograd and Cowan 1963

A circuit containing N error-free gates can be simulated with probability of failure ε using O(N poly(log(⋅ N/ε))) error-prone gates which fail with probability p, provided p < pth, where pth is a constant threshold independent of N.

Number of gates consumed: k3Find k such that NpP

kk

k /3 212

2lnln3ln

)/ln(2lnln~

pN

k

)/ln(~3 NPolyk Number of Gates ConsumedPer Perfect Gate is

n p

ppp

MAJp

ppp

p

ppp

k

Threshold Theorem – Generalized

mnmn

m

mnmn

nm

ppmn

pppmn

pP

)1()1()1(2/)1(

02/)1(

2/)1( nnk ckpP

For circuit to be fault tolerant P<p

2/)1( /1 nthreshold ckp

Total number of gates: )( knO

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

Total Area = n*(A/n)

Probability of correct functionality = p[A]

Scaling Properties of Majority Logic

P

A

n segments

knkn

nknmajority pp

kn

P

)1(

2/)1(

2/12/)1( ]0['1

nn Ap

nTo Lowest Order in A

Conclusion: For most functions n = 1 is optimal. Larger n is worse.

Definition: Rich Self Replication

[2] Complexity of Final ProductComplexity of Individual

Building Blocks>Example: DNA

Complexity of Oligonucleotide:N ln 4

Complexity of Nucleotide (20 atoms):Assuming atoms are built from C,O,N,P periodic table: 4 ln 20

Therefore: Rich Self Replication Occurs in DNAIf the final product is a machine which can self replicate itself and if N > ~ 9 bases.

[1] Autonomous

+ + +

+ +

Step 1 Step 2 Step 3

+

Parts

Template

Machine

The Self Replication Cycle

p per base p’ per base

RNA (2007), 13:1017–1026. Published by Cold Spring Harbor Laboratory Press.

Selection of an improved RNA polymerase ribozymewith superior extension and fidelityHANI S. ZAHER and PETER J. UNRAU

20 NT Extension x[Product / Parts] =~ .1

Fabricational Complexity

Fabricational Complexity Per Unit Cost

MpF N ln1

A G T C G C A A T

N

Fabricational Complexity for N-mer or M Types = NMlnFabricational Cost for N-mer = NNp

Where is the yield per fabricational step p

Complexity Per Unit CostComplexity Per Unit Time*Energy

…Can we use this map as a guide towards future directions in fabrication?

Semi-conductor Chip

High Speed Offset Web TFT DVD-6

Liquid Embossing

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

Fabricational ComplexityApplication: Identifying New Manufacturing Approach for Semiconductors

MpF N ln1

Fabricational Complexity Per Unit Cost 2 Ply Error Correction

Non Error Correcting:

2Ply Error Correcting:

A G T C

A G T C

A G T C NppNMNF

2222

ln

20 40 60 80 100

0.6

0.8

1.2

12 FF

p=0.99

Threshold for LifeWhat is the Threshold for Self Replicating Systems?

Measurement Theory

+ + +

+ +

Step 1 Step 2 Step 3

+

Parts

Template

Machine

Replication Cycle

http://en.wikipedia.org/wiki/File:Stem-loop.svg

Error Correcting Exonuclease

(Ruler)

DNA

Number of NucleotidesProb

abili

ty o

f Sel

f Rep

licati

on

NN

N

N

kT

qp

qQp

qQ

kq

q

N

Bond/E-

-1 P :Yield Total

11 :Yield StepPer

:open bonds N ally that Probabilit

3E e :Where

:open is bond single ay that ProbabilitBond

Watson Crick .18 nm

How Well Can N Molecules Measure Distance?

/sandwalk.blogspot.com/2007/12/dna-denaturation-and-renaturation-and.html

200 400 600 800 1000 1200 1400

0.2

0.4

0.6

0.8

1.0

J. Jacobson 2/28/12

Assignment Option #1Design a Rich Self Replicator

• Propose a workable self replicating system with enough detail that it could be built.

• The Descriptional Complexity of the Final Product must exceed the The Descriptional Complexity of the Building Blocks (Feedstock)

• Detail a mechanism for error correction sufficient that errors don’t accumulate from generation to generation.

Assignment Option #2Design an Exponential Scaling

Manufacturing Process•Design a manufacturing process such that on each iteration (e.g. each turn of a crank) the number of widgets produced grows geometrically.

•Detail a mechanism for error correction such that later generations don’t have more errors than earlier ones.

•Human intervention is allowed.

•Proposal should be based on simple processes (e.g. printing).