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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

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Kinetic monte carlo simulations of statistical mechanical models of biological evolution
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
Biological Evolution and Models of Biological EvolutionStatistical Physics

  • Complicated field with many

    unsolved problems.

  • Complex, interacting nonequilibrium problems.

  • Need for simplified models with universal properties. (Physicist’s approach.)

Modes of evolution
Modes of Evolution Models of Biological 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


    stick-slip motion in friction and earthquakes.

Models of coevolution
Models of Coevolution Models of Biological Evolution

  • Among physicists, the best-known coevolution model is probably the Bak-Sneppenmodel.

  • 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
Individual-based Coevolution Model Models of Biological Evolution

  • 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 Models of Biological Evolution

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).

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: MIJquenched, randome [-1,+1], except MII = 0.

Deterministic approximation
Deterministic approximation Models of Biological Evolution

m: mutation rate

per individual

Mutations Models of Biological Evolution

Each individual offspring undergoes mutation to a different genotype with probability m/L per gene and individual.

Fixed points for m 0
Fixed points for Models of Biological Evolutionm = 0

Without mutations the equation of motion reduces to

such that the fixed-point populations satisfy

This yields the total population for an N-species fixed point:

where is the inverse of the submatrix of MIJ in N-species space.

There are also expressions for the individual .

Stability of fixed points
Stability of fixed points Models of Biological Evolution

The internal stability of the fixed point is determined by the eigenvalues of the community matrix

The stability against an invading mutant i is given by the invader’s invasion fitness:

Monte carlo algorithm 3 layers of nested loops
Monte Carlo algorithm: Models of Biological Evolution3 layers of nested loops

  • Loop over generations t

  • Loop over genotypes I with nI > 0in t

    3a. Loop over individuals in I, producing F offspring with probability PI({nJ(t)}), or killingindividual with probability 1-PI

    3b. Loop over offspring to mutate with probability m

Simulation parameters
Simulation parameters Models of Biological Evolution

  • N0 = 2000

  • F = 4

  • L = 13 213 = 8192 potential genotypes

  • m= 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
Main quantities measured Models of Biological Evolution

  • Normalized total population, Ntot(t)/[N0ln(F-1)]

  • Diversity, D(t), gives the number of heavily populated species. Obtained as D(t) = exp[S(t)]


    S(t) = - SI [nI(t)/Ntot(t)] ln [nI(t)/Ntot(t)]

    is the information-theoretical entropy (Shannon-Wiener index).

Simulation results
Simulation Results Models of Biological Evolution





nI > 1000

nIe [101,1000]

nIe [11,100]

nIe [2,10]

nI = 1

Quasi-steady states (QSS) punctuated by active periods. Self-similarity.

Stability of quasi steady states qss
Stability of Quasi-steady States (QSS) Models of Biological Evolution

Multiplication rate of small-population mutant i in presence of fixed point of N resident species, J, K:

Active and quiet periods
Active and Quiet Periods Models of Biological Evolution

Histogram of entropy changes

Histograms of period durations

Power spectral densities squared norm of fourier transform
Power Spectral Densities Models of Biological Evolution(squared norm of Fourier transform)

PSD of D(t)

PSD of Ntot(t)/[N0 ln(F-1)]

Species lifetime distributions
Species’ lifetime distributions Models of Biological Evolution

Stationarity of diversity measures
Stationarity of diversity measures Models of Biological Evolution

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
Summary of completed work Models of Biological Evolution

  • 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
Current work and future plans Models of Biological Evolution

  • Predator/prey models

  • Community structure and food webs

  • Stability vs connectivity

  • Effects of different functional responses, including competition and adaptive foraging