Threshold for Life

```MAS.S62 FAB2
2.28.12
The Threshold for Life
http://lslwww.epfl.ch/pages/embryonics/thesis/Chapter3.html
Complexities in Biochemistry
Atoms: ~ 10
Complexion: W~310
Complexity x = 15.8
Atoms: ~ 8
Complexion: W~38
Complexity x = 12.7
DNA N-mer
Types of Nucleotide Bases: 4
Complexion: W=4N
Complexity x = 2 N
Complexity Crossover: N>~8
Synthetic Complexities of Various Systems
Complexity (uProcessor/program):
x ~ 1K byte = 8000
Atoms: ~ 20 [C,N,O]
Complexion: W~ 320
x = 32
Nucleotides: ~ 1000
Complexion: W~41000
x = 2000 = 2Kb
DNA Polymerase
Product: C = 4 states
x=2
x[Product / Parts] =~ .00025
Product: C = 4 states
x=2
x[Product / Parts] =~ .0625
Product: 107 Nucleotides
x = 2x107
x[Product / Parts] =104
x >1 Product has sufficient complexity to encode for parts / assembler
Complexity
Application: Why Are There 20 Amino Acids in Biology?
(What is the right balance between Codon code redundancy and diversity?)
N Blocks of Q Types
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:
W
N!
N!

 ni ! ( N Q) !Q
i
x ( N , Q)  N ln( N )  Q * ( N Q) ln( N Q)  N Q 
40
For a given polymer length N
achieves the half max for
complexity such that:
x ( N , Q*)  0.5F ( N , N )
30
Q*
20
10
500
1000
N
1500
2000
Nucleotides: ~ 150
Complexion: W~4150
Complexity x = 300
Product: 7 Blocks
x=7
x[Product / Parts] =.023
The percentage of heptamers with the correct
sequence is estimated to be 70%
T Wang et al. Nature 478, 225-228 (2011) doi:10.1038/nature10500
Information Rich Replication
(Non-Protein Biochemical Systems)
RNA-Catalyzed RNA Polymerization
14 base extension. Effective Error Rate: ~ 1:103
RNA-Catalyzed RNA Polymerization: Accurate and General RNA-Templated Primer Extension
Science 2001 May 18; 292: 1319-1325
Wendy K. Johnston, Peter J. Unrau, Michael S. Lawrence, Margaret E. Glasner, and David P. Bartel
J. Szostak, Nature,409,
Jan. 2001
Selection of an improved RNA polymerase ribozyme
with superior extension and fidelity
HANI S. ZAHER and PETER J. UNRAU
x[Product / Parts] =~ .1
20 NT Extension
Spring Harbor Laboratory Press.
http://www.uncommondescent.com/biology/j
ohn-von-neumann-an-ider-ante-litteram/
http://web.archive.org/web/20070418081628/http://dra
gonfly.tam.cornell.edu/~pesavent/pesavento_self_repro
ducing_machine.pdf
Implementations of Von Neumann’s Universal Constructor
http://en.wikipedia.org/wiki/Von_Neumann_universal_constructor
Self Replication Simulators
http://necsi.edu/postdocs/sayama/sdsr/java/#l
angton
Langton Loops
http://carg2.epfl.ch/Teaching/GDCA/loops-thesis.pdf
http://carg2.epfl.ch/Teaching/GDCA/loops-thesis.pdf
http://carg2.epfl.ch/Teaching/GDCA/loops-thesis.pdf
http://carg2.epfl.ch/Teaching/GDCA/loops-thesis.pdf
Numbe
r of
States
Neighbor
hood
Number
of Cells
(typical)
Langton's loops[3] (1984): The original selfreproducing 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
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 structuredissolving 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
CA
Replication
Period
(Typical)
Thumbnail
Fault-Tolerant Circuits
Threshold Theorem – Von Neumann 1956
p
n
p
MAJ
n
m ( n 1) / 2
m
P
p
p
p
n
MAJ
MAJ

n=3
p
Recursion Level
p
p
MAJ
p
k
P
K=1
P1  3 p 2 (1  p)  3 p 2
K=2
P2  3( p1 ) 2  3(3 p 2 ) 2  33 p 4
K
For circuit to be fault tolerant
p m (1  p) nm
( 2 k 1)
Pk  3
Pk  3
2 k 1
p
p
 PTh  1 / 3
2k
2k
p
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.
p
n
p
MAJ
p
p
p
MAJ
MAJ
p
p
p
MAJ
k
3
Number of gates consumed:
p
k
Find k such that
Pk  3
 ln 2  ln( / N ) 
ln 
ln 3  ln p 

k~
ln 2
Number of Gates Consumed
Per Perfect Gate is
3 ~ Polyln( / N )
k
2k 1
p  / N
2k
Threshold Theorem – Generalized
n
p
( n 1) / 2
n m
n m
nm
P  (1  p) 
p (1  p)  p  p (1  p) nm
m( n 1) / 2 m
m 0 m
n
p
p
p
p
p
p
MAJ
n
k
p
p
( n1) / 2
P  ckp
p
p
p
k
For circuit to be fault tolerant P<p
pthreshold 
( n1) / 2
Total number of gates:
1/ ck
k
O(n )
Scaling Properties of Redundant Logic (to first order)
P
A
Probability of correct functionality = p[A] ~ e A (small A)
Area = A
P1 = p[A] = e A
P2 = 2p[A/2](1-p[A/2])+p[A/2]2
= eA –(eA)2/4
Area = 2*A/2
Conclusion: P1 > P2
Scaling Properties of Majority Logic
n segments
P
Total Area = n*(A/n)
A
Probability of correct functionality = p[A]
n k
    p (1  p) nk
k ( n 1) / 2  k 
n
Pmajority n

1
n
( n 1) / 2
 p'[0] A
n 1 / 2
To Lowest Order in A
Conclusion: For most functions n = 1 is optimal. Larger n is worse.
Definition: Rich Self Replication
[1] Autonomous
[2] Complexity of Final Product
Example: DNA
Complexity of Oligonucleotide:
N ln 4
>
Complexity of Individual
Building Blocks
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 DNA
If the final product is a machine which can self replicate itself and if N
> ~ 9 bases.
The Self Replication Cycle
Parts
+
+
+
Template
+
p per base
+
p’ per base
+
Machine
Step 1
Step 2
Step 3
Selection of an improved RNA polymerase ribozyme
with superior extension and fidelity
HANI S. ZAHER and PETER J. UNRAU
x[Product / Parts] =~ .1
20 NT Extension
Spring Harbor Laboratory Press.
Fabricational Complexity
N
A
G
T
C
G
C
Fabricational Complexity for N-mer or M Types =
Fabricational Cost for N-mer =
Where
p
A
A T
N
ln M
N
 Np
is the yield per fabricational step
Fabricational Complexity Per Unit Cost
F1  p ln M
N
Complexity Per Unit Cost
Complexity Per Unit Time*Energy
Fabricational Complexity
Application: Identifying New Manufacturing Approach for Semiconductors
Semi-conductor
Chip
Design Rule Smallest Dimension
(microns)
Number of Types of Elements
Area of SOA Artifact (Sq. Microns)
Volume of SOA Artifact (Cubic Microns)
Number of Elements in SOA Artifact
Volume Per Element(Cubic Microns)
Fabrication Time(seconds)
Time Per Element (Seconds)
Fabrication Cost for SOA Artifact(\$)
Cost Per Element
Complexity
Complexity Per Unit Volume of SOA(um^3)
Complexity Per Unit Time
Yielded Res. Elements Per \$
Cost Per Area
0.1
8
7.E+10
7.E+09
7.E+12
1.E-03
9.E+04
1.E-08
1.E+02
2.E-11
2.E+13
2.E+03
2.E+08
1.E+11
2.E-09
High Speed Offset
Web
TFT
10
6
2.E+12
2.E+12
2.E+10
1.E+02
1.E-01
7.E-12
1.E-01
6.E-12
4.E+10
2.E-02
3.E+11
3.E+11
6.E-14
DVD-6
2
8
1.E+12
1.E+11
3.E+11
4.E-01
7.E+02
2.E-09
2.E+03
6.E-09
6.E+11
5.E+00
9.E+08
3.E+08
2.E-09
0.25
2
1.E+10
7.E+12
2.E+11
4.E+01
3
2.E-11
3.E-02
2.E-13
1.E+11
2.E-02
4.E+10
4.E+12
3.E-12
Liquid
Embossing
0.2
4
8.E+09
8.E+08
2.E+11
4.E-03
6.E+01
3.E-10
2.E-01
1.E-12
3.E+11
3.E+02
5.E+09
1.E+12
3.E-11
…Can we use this map as a guide towards future
directions in fabrication?
Fabricational Complexity Per Unit Cost
2 Ply Error Correction
Non Error Correcting:
F1  p ln M
N
2Ply Error Correcting:
F2 
N ln M

2N 2 p  p

2 N
p=0.99
1.2
F2 F1
20
0.8
0.6
40
60
80
100
A
G
T
C
A
G
T
C
A
G
T
C
Threshold for Life
What is the Threshold for Self Replicating Systems?
Measurement Theory
Replication Cycle
DNA
Parts
+
+
+
+
+
+
Template
Error Correcting
Exonuclease
(Ruler)
Machine
Step 2
Step 3
How Well Can N Molecules Measure Distance?
Probability that a single bond is open : q
Where : q  e -E Bond / kT E Bond  3k
Probability that all N bonds open : Q  q N
Per Step Yield : p  1  Q  1  q N

Total Yield : P  p  1 - q
N
J. Jacobson 2/28/12

N N
http://en.wikipedia.org/wiki/File:Stem-loop.svg
Probability of Self Replication
Step 1
Watson Crick
.18 nm
1.0
0.8
0.6
0.4
0.2
200
400
600
800
1000
1200
Number of Nucleotides
1400
Assignment Option #1
Design 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 #2
Design 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).
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