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 we can ask which Q* 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 RNA (2007), 13:1017–1026. Published by Cold Spring Harbor Laboratory Press. http://www.uncommondescent.com/biology/j ohn-von-neumann-an-ider-ante-litteram/ http://en.wikipedia.org/wiki/File:320_jump_read_arm.gif 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 (1984): The original selfreproducing loop. 8 von Neumann 86 151 Byl's loop (1989): By removing the inner sheath, Byl reduced the size of the loop. 6 von Neumann 12 25 Chou-Reggia loop (1993): A further reduction of the loop by removing all sheaths. 8 von Neumann 5 15 Tempesti loop (1995): Tempesti added construction capabilities to his loop, allowing patterns to be written inside the loop after reproduction. 10 Moore 148 304 Perrier loop (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 (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 (1999): An extension of the SDSR loop, Evoloop is capable of interaction with neighboring loops as well as of evolution.. 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) nm ( 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 ~ Polyln( / N ) k 2k 1 p / N 2k Threshold Theorem – Generalized n p ( n 1) / 2 n m n m nm P (1 p) p (1 p) p p (1 p) nm m( n 1) / 2 m m 0 m n p p p p p p MAJ n k p p ( n1) / 2 P ckp p p p k For circuit to be fault tolerant P<p pthreshold ( n1) / 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) nk k ( n 1) / 2 k n Pmajority n 1 n ( n 1) / 2 p' 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  Autonomous  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 RNA (2007), 13:1017–1026. Published by Cold 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 /sandwalk.blogspot.com/2007/12/dnadenaturation-and-renaturation-and.html 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).