Kinetic Monte Carlo Simulations of Statistical-mechanical Models of

Biological Evolution

Per Arne Rikvold and Volkan SevimSchool of Computational Science,

Center for Materials Research and Technology, and Department of Physics,

Florida State University

R.K.P. ZiaCenter for Stochastic Processes in Science and Engineering,

Department of Physics, Virginia TechSupported 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).

DynamicsProbability that an individual of genotype I has F

offspring in generation t before dying is PI({nJ(t)}).

Probability of dying without offspring is (1PI).

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 [1,+1], except MII = 0.

]/)()(/)(exp[1

1)})(({

0tottot NtNtNtnMtnP

JJIJ

JI

Deterministic approximation]1)})[(({)()1( tnFPtntn JIII

)(

2)()( )()})(({)()/(

IKJIKIK OtnPtnL

]/)()(/)(exp[1

1)})(({

0tottot NtNtNtnMtnP

JJIJ

JI

: mutation rate per individual)exp(1

1)(

xxP

Mutations

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

Fixed points for = 0

)})(({)()1( tnFPtntn JIII

Without mutations the equation of motion reduces to

such that the fixed-point populations satisfy

*Jn

*Jn

1)})(({ * tnFP JI

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

IJIJJ

J FNnN10

**tot ~

1)1ln(

M

where is the inverse of the submatrix of MIJ in N-species space.There are also expressions for the individual .

IJ1~ M*Jn

Stability of fixed pointsThe 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:

IJIJ

IJI

IJ

nJ

IIJ FM

N

n

Ftn

tn

I

1*tot

*

~2

1ln1

1)(

)1(~

* M

JKJK

JKJKiJ

i

i

MF

F

tn

tn

11 ~~1exp11

ln)(

)1(ln

MM

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

• = 103

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(F1)]

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

where

S(t) = I [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 > 1000nI [101,1000]nI [11,100]nI [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:

JKJK

JKJKiJ

i

i

MF

F

tn

tn

11 ~~1exp11

)(

)1(

MM

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(F1)]

Species’ lifetime distributions

Stationarity of diversity measures

• 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

Running time and ensemble averages.

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 1/2 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