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

iQN

N

n

NW

!)(

!

!

!

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(),(

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

),(5.0*),( NNFQN

.

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 Extension

Science 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/

GD

CA/l

oops

-the

sis.

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

ppm

nP

)1(2/)1(

kk

pP

pppP

pppP

k2)12(

4322212

221

3

3)3(3)(3

3)1(3

Recursion Level P

K=1

K=2

K

n=3

For circuit to be fault tolerant

3/1

3 212

Th

k

P

ppPkk

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: k3

Find k such that NpPkk

k /3 212

2ln

ln3ln)/ln(2ln

ln

~

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

ppm

nppp

m

npP

)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

k

nP

)1(

2/)1(

2/1

2/)1(]0['

1 n

nAp

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 3̂) 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 NppN

MNF

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).