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Kinetic Monte Carlo Simulations of Statistical-mechanical Models of Biological Evolution Per Arne Rikvold and Volkan Sevim School of Computational Science, Center for Materials Research and Technology, and Department of Physics, Florida State University R.K.P. Zia Center for Stochastic Processes in Science and Engineering, Department of Physics, Virginia Tech Supported by FSU (SCS and MARTECH), VT, and NSF Biological Evolution and Statistical Physics • Complicated field with many unsolved problems. • Complex, interacting nonequilibrium problems. • Need for simplified models with universal properties. (Physicist’s approach.) Modes of Evolution • Does evolution proceed uniformly or in fits and starts? • Scarcity of intermediate forms (“missing links”) in the fossil record may suggest fits and starts. • Fit-and-start evolution termed punctuated equilibria by Eldredge and Gould. • Punctuated equilibria dynamics resemble nucleation and growth in phase transformations and stick-slip motion in friction and earthquakes. Models of Coevolution • Among physicists, the best-known coevolution model is probably the Bak-Sneppen model. • The BS model acts directly on interacting species, which mutate into other species. • But: in nature selection and mutation act directly on individuals. Individual-based Coevolution Model • Binary, haploid genome of length L gives 2L different potential genotypes. 01100…101 • Considering this genome as coarse-grained, we consider each different bit string a “species.” • Asexual reproduction in discrete, nonoverlapping generations. • Simplified version of model introduced by Hall, Christensen, et al., Phys. Rev. E 66, 011904 (2002); J. Theor. Biol. 216, 73 (2002). Dynamics Probability that an individual of genotype I has F offspring in generation t before dying is PI({nJ(t)}). Probability of dying without offspring is (1-PI). 1 PI ({nJ (t )}) 1 exp[ - M IJ nJ (t ) / N tot (t ) N tot (t ) / N 0 ] J N0: Verhulst factor limits total population Ntot(t). MIJ : Effect of genotype J on birth probability of I. MIJ and MJI both positive: symbiosis or mutualism. MIJ and MJI both negative: competition. MIJ and MJI opposite sign: predator/prey relationship. Here: MIJ quenched, random e [-1,+1], except MII = 0. Deterministic approximation nI (t 1) nI (t ) FPI ({nJ (t )})[1 - ] ( / L) nK ( I ) (t ) PK ( I ) ({nJ (t )}) O( 2 ) K (I ) P( x) 1 : mutation rate 1 exp( x) per individual 1 PI ({nJ (t )}) 1 exp[ - M IJ nJ (t ) / N tot (t ) N tot (t ) / N 0 ] J Mutations Each individual offspring undergoes mutation to a different genotype with probability /L per gene and individual. Fixed points for = 0 Without mutations the equation of motion reduces to nI (t 1) nI (t ) FPI ({nJ (t )}) such that the fixed-point populations n* J satisfy FPI ({n* (t )}) 1 J This yields the total population for an N-species fixed point: n* N 0 ln( F - 1) 1 n * J * N ~ -1 M tot J J IJ IJ ~ -1 where M is the inverse of the submatrix of MIJ in N-species space. IJ * There are also expressions for the individual n J . Stability of fixed points The internal stability of the fixed point is determined by the eigenvalues of the community matrix nI (t 1) 1 n * M IJ - ln F - 1 - ~ 2 IJ - IJ 1 - I ~ -1 nJ (t ) n* I FN * tot M IJ IJ The stability against an invading mutant i is given by the invader’s invasion fitness: ni (t 1) F n (t ) ln ln i 1 F - 1 exp1 - M iJ M ~ -1 JK ~ -1 M JK JK JK Monte Carlo algorithm: 3 layers of nested loops 1. Loop over generations t 2. Loop over genotypes I with nI > 0 in t 3a. Loop over individuals in I, producing F offspring with probability PI({nJ(t)}), or killing individual with probability 1-PI 3b. Loop over offspring to mutate with probability Simulation parameters • N0 = 2000 • F=4 • L = 13 213 = 8192 potential genotypes • = 10-3 This choice ensures that both Ntot and the number of populated species are << the total number of potential genotypes, 2L Main quantities measured • Normalized total population, Ntot(t)/[N0 ln(F-1)] • Diversity, D(t), gives the number of heavily populated species. Obtained as D(t) = exp[S(t)] where S(t) = - SI [nI(t)/Ntot(t)] ln [nI(t)/Ntot(t)] is the information-theoretical entropy (Shannon-Wiener index). Simulation Results Diversity, D(t) Ntot(t), normalized nI > 1000 nI e [101,1000] nI e [11,100] nI e [2,10] nI = 1 Quasi-steady states (QSS) punctuated by active periods. Self-similarity. Stability of Quasi-steady States (QSS) Multiplication rate of small-population mutant i in presence of fixed point of N resident species, J, K: ni (t 1) F ni (t ) 1 F - 1 exp1 - M iJ M ~ -1 JK ~ -1 M JK JK JK Active and Quiet Periods Histogram of entropy changes Histograms of period durations Power Spectral Densities (squared norm of Fourier transform) PSD of D(t) PSD of Ntot(t)/[N0 ln(F-1)] Species’ lifetime distributions Stationarity of diversity measures Running time and ensemble averages. • Total species richness, N(t) • No. of species with nI > 1 • Shannon-Wiener D(t) • Mean Hamming distance between genotypes • Total population Ntot(t)/N0ln3 • Standard deviation of Hamming distance Summary of completed work • Simple model for evolution of haploid, asexual organisms • Based on birth/death process of individual organisms • Shows punctuated equilibria of quasi-steady states (QSS) of a few populated species, separated by active periods • Self-similarity and 1/t2 distribution of QSS lifetimes leads to 1/f-like flicker noise P.A.R. and R.K.P.Z., Phys. Rev. E 68, 031913 (2003); J. Phys. A 37, 5135 (2004) V.S. and P.A.R., arXiv:q-bio.PE/0403042 Current work and future plans • Predator/prey models • Community structure and food webs • Stability vs connectivity • Effects of different functional responses, including competition and adaptive foraging

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posted: | 9/1/2012 |

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