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.
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
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).