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Kinetic Monte Carlo Simulations of Statistical-mechanical Models of Biological Evolution

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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

- Complicated field with many
unsolved problems.

- Complex, interacting nonequilibrium problems.
- Need for simplified models with universal properties. (Physicist’s approach.)

- 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.

- 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.

- 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).

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.

m: mutation rate

per individual

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

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 .

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:

- 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

- 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

- 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)]
where

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

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

Diversity,

D(t)

Ntot(t),

normalized

nI > 1000

nIe [101,1000]

nIe [11,100]

nIe [2,10]

nI = 1

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

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

Histogram of entropy changes

Histograms of period durations

PSD of D(t)

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

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

- 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

- Predator/prey models
- Community structure and food webs
- Stability vs connectivity
- Effects of different functional responses, including competition and adaptive foraging