The Physical Language of Molecules

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1 The Physical Language of Molecules How do molecular codes emerge and evolve? International Workshop on Bio Soft Matter Tokyo, 2008

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

3 Outline Molecular codes = information channels or maps. Fitness of codes = Quality + Cost. Smooth molecular codes emerge at phase transitions. Topology of errors governs emergent code. Evolutionary dynamics of codes.

Challenge of molecular coding Quality: Information transfer via molecular recognition in a noisy, crowded milieu. Recognizer and target fluctuate.

4 Challenge of molecular coding Quality: Information transfer via molecular recognition in a noisy, crowded milieu. Recognizer and target fluctuate. Many competing lookalikes. Weak recognition interactions ~ k B T. Cost: How to construct the molecular codes at minimal cost of resources? David Goodsell

5 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

6 Amino acids are the building blocks of proteins Amino acid = backbone + specific side group. Diversity of amino acids allows proteins to perform a wide variety of functions efficiently.

7 The Genetic Code is highly ordered meaning = polarity Smooth. Degenerate (20 out of 64) meanings symbols symbols Yet, diverse. Generic properties?

8 Q: How do smooth codes emerge and evolve? A: Molecular codes are smooth (1) to withstand noise (2) at a minimal cost.

9 Molecular code is a channel with a quality measure m 1 Encoder ε s 1 r Misreading Distortion c m 2 Decoder d s 2 Quality: D= c = Tr ( ε rdc ) The quality D is the average distortion c of a typical meaning. ε and d determine the quality for a given misreading r. r defines the topology of symbol space.

10 Smooth codes minimize error load meanings Errors (noise) confuse similar, neighboring symbols. Smooth codes neighboring symbols are also similar in meaning. minimal impact of errors. Reading r ~ Laplacian operator in symbol space, Δ s. Quality D ~ elastic energy of symbol space with meanings metric.

11 Molecular codes cost chemical specificity To encode/decode diverse meanings, molecular readers require specificity = high binding energies E b. Cost I ~ average binding energy < E b >. Binding probability ~ Boltzmann: ε ~ exp(e b /T). I ε ε = ln. ε encoder m 1 encoder s 1 Specificity cost is measured by mutual information I.

12 Code s fitness combines quality and cost Quality = Error load + Diversity D = Tr( ε rdc ) + Cost = Chemical Specificity I = ε ln ε Fitness = Quality + Cost/Gain 1 H = D+κ I Gain κ increases with complexity of organism and richness of environment.

13 Survival of the fittest code Population of organisms that compete and evolve according to code fitness, H. Population dynamics: Max(quality) Min (cost) + Mutation Selection Random drift

14 A code is born when gain increases I = Cost Low gain: Cost too high no specificity no correlation no code. Code emerges when channel starts to convey info between symbols and meanings(i 0). Instability of H Continuous 2 nd order phase transition. D = Quality phase transition (PRL 2007, JTB 2007)

15 Codes appear as smooth modes in symbol space codes Instability of H (~free energy) phase transition Code = Smoothest non uniform correlation pattern. no code code Code is smooth 2 nd mode of symbol Laplacian (Courant) = minimal surface tension of meaning islands. Misreading r is the graph Laplacian r ~ Δ s. s 2 s 1 r s 3 s 6 s 5 s 4 s 7

16 Optimal coding is a topological coloring problem Each color denotes a meaning (for example an amino acid). Coloring partitions the symbol space. The code is optimal when every color or meaning has one compact contiguous island of words. Partition described by statistical mechanics of polymer networks. (PNAS 2008)

17 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

18 Topology of a much simpler code CB AA BA CA AB BB CB AC BC CC AC CC CA BC BA BB A B C X A B C AA AB Two letter symbols with 3 bases is embedded on a torus. Euler s characteristic: χ = Vertices Edges + Faces. Genus (# holes): γ = 1 χ/2. Faces are quadrilaterals: Vertices = Faces =9 ; Edges= 18.

19 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

20 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 )

Compact organization reduces impact of errors.

21 Topology determines the optimal coloring Each meaning has single compact domains with one maximum and one minimum (Courant). Compact organization reduces impact of errors. Embedding in R N 1 is tight or convex The code graph contains complete graph K N # meanings = N = coloring(γ) (Banchoff 1965, Colin de Verdiére 1987, TT 2007)

22 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 )

23 Probable recognition errors define the binding sequence space Coloring number estimate: v = 4 L (L=6) e ~ 4 L (3/2)L f ~ 4 L (3/4)L > γ ~ 4 L (3/8)L The coloring # chr(γ) ~ 300

24 Optimal coding is a topological coloring problem optimizing number of meanings AGG AAG CAG TGA AGA CAA CCA TAA TTA AAA ACA ACT 1 coloring( γ ) = ( γ ) ATA AAT ATC AAC GAA GAT A GAC Topology of error Laplacian r governs coding transition. Smoothness limits number of meanings due to tightness of map. The limit is the coloring number, determined by topology (γ). Genetic code γ = coloring number = amino acids. (JTB 2007, ELA 2007)

Reaction diffusion dynamics reach steady state codes Ψ = 2 H Ψ+ μ Ψ

25 Population dynamics: mutations, genetic drift Mutations smear the population in code space. Reaction diffusion dynamics reach steady state codes Ψ = 2 H Ψ+ μ Ψ code t ( μ mutation rate, ψ population density) 1/2 ( ) Ψ exp μ ε ε Other effects : Genetic drift = reproduction fluctuations.

Fitness Jean Pierre = Quality Eckmann + Cost.

26 Thanks Summary Albert Libchaber Molecular Elisha Moses codes = maps or information channels with fitness. Fitness Jean Pierre = Quality Eckmann + Cost. Guy Sella Smooth Roy Bar Ziv codes emerge at phase transitions. Uri Alon Topology of errors governs emergent code. Shalev Itzkovitz Guy Shinar

27 Population dynamics: genetic drift Genetic drift = reproduction fluctuations = Noise. The population migrates between many possible optima. At steady state P(H) ~ exp( H/T) [Sella and Hirsh] with evolutionary temperature ~ 1/(population size). Effective free energy (Potts like, or polymer net) Shifting the critical gain F = H( ε ) + T ε lnε αi αi αi α, i 1/ κ + 1/ N = λ λ 2 c r c (PRL 2007, JTB 2007)

28 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

29 Emergent code is a smooth mode of the error Laplacian on symbol graph Every mode corresponds to a meaning number of modes = number of meanings. Misreading r is the graph Laplacian r ~ Δ s. s 6 Courant s theorem for Δ s : single maximum for each mode s 2 s 1 r s 5 single contiguous domain for each meaning. s 3 Smoothness s 7 s 4

30 ** Statistical mechanics of code evolution Fitness H = D + I/κ Quality + Cost/Gain ~ Free energy Gain κ ~ inverse temperature. Fittest code takes over. Given r, c, κ: min {e,d} H fittest code (e*, d*). Order parameter e ms = s' exp( κ Ems ) exp( κ E ) ms' δe = deviation from randomness.

31 ** Code emerges at a 2 nd order transition Transition at critical gain κ c. Critical temperature depends on r and c: 1/κ c ~ λ r2 λ c. Code is the smooth mode e ms of H that corresponds to 2 nd e.v. of Δ. Three pathways to transition: increase gain. increase accuracy. increase diversity.

32 The transcription network is smooth Transcription factors that bind to similar DNA sequences tend to have similar meanings Meaning is measured by the GO annotation or co regulation Overlapping TFs in Yeast. Vertices are TFs. edges connect TFs with overlapping spheres.

Casting Polymer Nets To Optimize Molecular Codes

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) Biological information