Casting Polymer Nets To Optimize Molecular Codes

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1 Casting Polymer Nets To Optimize Molecular Codes The physical language of molecules Mathematical Biology Forum (PRL 2007, PNAS 2008, Phys Bio 2008, J Theo Bio 2007, E J Lin Alg 2007)

2 Biological information is carried by molecules Self replicating information processing systems

3 Outline The physical language of molecules Molecular code = Mapping symbols to meanings. Codes imply partition of symbol space. Code fitness = Quality + Cost of the partition. Code fitness = free energy of polymer networks. Polymer networks Spin networks. Evolution of networks coding transition where code emerges.

4 Coding theory: Molecular Codes as Maps or Information Channels Molecular code = map relating two sets of molecules. Relation by molecular recognition. Symbols Meanings 64 codons AGA Genetic Code AA trna codon 20 amino acids

5 Diversity of molecular meanings is essential Diversity of amino acids allows proteins to perform a wide variety of functions efficiently.

Semantic challenge of molecular coding is 3 fold Minimize Error Load: Information transfer via molecular recognition in a noisy, crowded milieu:

6 Semantic challenge of molecular coding is 3 fold Minimize Error Load: Information transfer via molecular recognition in a noisy, crowded milieu: Fluctuations, competing lookalikes Minimize Cost: How to construct codes at minimal cost of resources? Maximize Diversity: Enough meanings for efficiency. David Goodsell

7 Molecular code is a noisy communication channel n m meanings n s symbols code n f n m encoded meanings Molecular reading is inherently noisy. Symbol space is a graph: vertices = symbols, edges between symbols confused by misreading. Code = coloring of symbol space. Code emerges when number of encoded meanings n f > 1.

8 Molecular code is a partition of symbol space meanings symbols code meaning islands Coloring partitions symbol space into islands of meanings. Islands separated by self avoiding random walks = polymers net. To be seen: code fitness = Free energy of polymer net. (PNAS 2008)

9 Partition determines fitness of noisy codes Fitness = Error load + Diversity + Cost. Two sources for noise: Noisy mapping (Cost) symbols meanings Misreading = confusion of read symbols (Error load) code

10 Boundary network determines error load Error-load = re ij ij = rij edges boundary i j Boundary network specified by E ij (= 1 if i, j code different meanings). Error load = average chance to cross the boundaries by misreading. Probability of misreading i as j is given by the matrix r ij.

11 Diversity counteracts error load Diversity = # islands = n f H = re w n E ij ij D edges f. Approximated diversity = number of encoded meanings, n f max(diversity) more islands, min(error load) less islands. Quality H E combines interplay of these two conflicting needs. w D measures relative significance error load/diversity.

12 Molecular codes cost chemical specificity Molecular codes are implemented by recognition interactions. Diverse meanings requires specificity = high binding energies E b. Cost ~ average binding site size ~ average binding energy < E b >. Binding probability ~ Boltzmann: P b ~ exp(e b /T). Specificity cost is measured by entropy of mapping S C = <lnp b >. Cost is entropy of ensemble of all possible mappings: all possible partitions and colorings of every partition.

13 Code s fitness combines quality and cost Fitness = Quality + w C x Cost FC = HE wc SC Parameter w C measures significance cost/quality. High w C fuzzy codes, low w C sharp codes. Organism complexity/environment richness low w C. Thermodynamic analogy: Quality H Energy, Cost entropy, w C temperature, Code fitness is free energy, F C : FC/ wc HE/ wc ZC = e = e mappings

14 Code fitness is free energy of polymer network Code fitness = free energy of polymers on the dual graph. Monomers pass along edges E ij = 1and vertices V i = 1. # encoded meanings n f, # vertices n v,# edges n e related by nf = C+ ne nv = C+ Eij Vi edges vertices

15 Fitness is free energy of polymer network From the polymer description Z C e e e networks edges vertices F / w β E / w αv / w C C ij C i C = = α, β are excitation energies of vertex and edge: β = ( r w ) w ln n, α = w + w ln n ij D C m D C m misreading diversity ln(#meanings) Summation is hard (vertices and edges are not independent).

Spin networks enable calculation of fitness Mathematical tool: n = 0 formalism [de Gennes, networks: Zilman & Safran] Assign magnetic spin S ij to each edge.

16 Spin networks enable calculation of fitness Mathematical tool: n = 0 formalism [de Gennes, networks: Zilman & Safran] Assign magnetic spin S ij to each edge. Spin interaction H S and spin free energy, Z S = exp( F S ) = < exp( H S ) >. When n 0: The only non vanishing contributions to F S correspond to a configuration of the polymer network, As a result: F S = F C = code fitness. Spin free energy F S is relatively easy to calculate (mean field). (further details: PNAS 2008)

17 Code emerges at a phase transition Using equivalence to spin networks we trace the evolution of the coding system. Control parameters, w D, w C, r ij, n m, combined into normalized diversity D = w D /w C + ln(n m ), normalized misreading R ij = r ij /w C. Find average mapping as a function of D, R ij. A high dimensional problem (dimension = number of edges). Symmetry reduces the dimension of code space (for regular graphs code space is 1D).

18 Code emerges at a phase transition Coding n f >1 Free energy = fitness non coding n f = 1 R = r/w C Average spin s D = w D /w C + ln(n m ) Non coding state s = 0, n f = 1. Code conveys no information. Coding state n f > 1 emerges. Code transmits log 2 n f bits/symbol. Abrupt jump (1 st order transition) non coding coding. Emergent network is dense and highly connected.

19 Dense coding network emerges At the transition many meanings appear together. Meanings per symbol n f /n s > 0. R = r/w C Pathways towards transition: Increasing n m, Increasing diversity w D, Decreasing misreading r. D = w D /w C + ln(n m ) edges vertices n f /n s

20 Survival of the fittest code Population of organisms that compete and evolve according to code fitness, F C. Population dynamics: Min(error load) Min (cost) Max(diversity) + Mutation Selection Random drift

21 Evolution in code fitness landscape The population evolves in the space of all possible codes. Dimension of code fitness landscape = # misreading edges = # spins. Population of organisms compete by the code fitness (density Ψ(s)). Evolutionary dynamics effects: Local trapping, mutations, genetic drift. codes code no code s s 1 s 2 r s 6 s 5 s 3 s 4 s 7 (PRL 2007, JTB 2007, PNAS 2008)

22 Trapping in metastable non coding states ψ population density Free energy= fitness non coding R = r/w C Coding D = w D /w C + ln(n m ) Non coding state may remain locally stable. Matastability exists as long as curvature is positive. Metastable network is dilute.

Reaction diffusion dynamics codes Ψ () s = F s Ψ s + Ψ s t 2 C ()

23 Mutations smear population in code space Mutations drive organisms to diffuse to suboptimal codes. Reaction diffusion dynamics codes Ψ () s = F s Ψ s + Ψ s t 2 C () () μ () ( μ mutation rate, ψ population density) Reach Gaussian steady state: 1/2 2 ( μ s s0 ) Ψ exp ( ) s

24 Genetic drift: diffusion between optimal codes Genetic drift = reproduction fluctuations = Noise. The population migrates between many possible optima. At steady state P(F) ~ exp( F/T) [Sella and Hirsh] with temperature ~ 1/(population size). codes Small populations are hotter. Populations effects add external noise to internal coding noise. s (PRL 2007, JTB 2007)

Jean Pierre Eckmann Evolutionary dynamics: Code emerges at a transition.

25 Summary The physical language of molecules Thanks Molecular codes map symbols to meanings by recognition. Albert Code fitness Libchaber = error load + cost + diversity of mapping. Elisha Fitness Moses = free energy of polymer nets (= spin nets). Jean Pierre Eckmann Evolutionary dynamics: Code emerges at a transition. Guy Sella Roy Simple Bar Ziv physical description of evolving molecular information channels. Uri Alon A potential tool to study noisy biological codes (Transcription Shalev Itzkovitz regulation network, genetic code, operons ) Guy Shinar (PNAS 2008, PRL 2007, Phys Bio 2008, J Theo Bio 2007, E J Lin Alg 2007)

26 Additional slides

27 The probable errors define the graph and the topology of the genetic code Symbol (codon) Graph = codon vertices + one letter difference edges ( Hamming = 1 ) AGG AAG K 4 X K 4 X K 4 CAG A T G C X A T G C A X T G C TGA AGA CAA CCA TTA TAA AAA ACA ACT ATA AAT ATC AAC GAC GAA GAT A

28 The surface of the code graph is holey K 4 X K 4 X K 4 A T G C X A T G C A X T G C TGA AGA AGG AAG CAG CAA CCA Holey graph: γ = 41 (lower limit is γ = 25) TTA TAA AAA ACA ACT ATA ATC AAC GAC GAA GAT AAT A K

29 Coloring number is the upper limit for the number of smooth islands What is the minimal number of colors required for a map so that no two adjacent countries have the same color? Coloring number is a topological invariant and 1 ( ) a function of the genus, chr γ = ( + + γ ) # of meanings = chr( γ )

30 Other molecular codes: Transcription regulatory network: Controls gene expression via binding proteins to DNA. Mapping between proteins and DNA is. Number of proteins is limited by the coloring number. Logic design of operons: Logic gates made of binding proteins are smooth. (Itzkovitz, Shinar, Alon, TT, PNAS 2006, BMC 2007 )

31 A sketch for an experiment : 2X2 coding system i 2 binding sites (symbols). α β 2 transcription factors (meanings) after duplication. i j Control gain by environment A(t). Coding transition when 2 nd factor α β becomes advantageous. A(t) A(t) i coding i j transition (Phys Bio 2008) α β α β

The Physical Language of Molecules

The Physical Language of Molecules How do molecular codes emerge and evolve? International Workshop on Bio Soft Matter Tokyo, 2008 Biological information is carried by molecules Self replicating information