Title: Kinetic Monte Carlo Simulations of Statistical-mechanical Models of Biological Evolution
1Kinetic 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
2Biological Evolution and Statistical Physics
- Complicated field with many
- unsolved problems.
- Complex, interacting nonequilibrium problems.
- Need for simplified models with universal
properties. (Physicists approach.)
3Modes 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.
4Models 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.
5Individual-based Coevolution Model
- Binary, haploid genome of length L gives
- 2L different potential genotypes. 01100101
- 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).
6Dynamics
- 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 MIJ quenched, random e -1,1, except MII
0.
7Deterministic approximation
m mutation rate per individual
8Mutations
- Each individual offspring undergoes mutation to
a different genotype with probability m/L per
gene and individual.
9Single or noninteracting species, m 0 Logistic
map
x(t)
x(t1)
Fixed point nI(t1) nI(t)
10Fixed points for m 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 .
11Stability of fixed points
- 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 invaders invasion fitness
12Monte Carlo algorithm3 layers of nested loops
- Loop over generations t
- Loop over genotypes I with nI gt 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 m
13Simulation parameters
- 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 ltlt the total number of
potential genotypes, 2L
14Main 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) expS(t) - where
- S(t) - SI nI(t)/Ntot(t) ln nI(t)/Ntot(t)
- is the information-theoretical entropy
(Shannon-Wiener index).
15Simulation Results
Diversity, D(t)
Ntot(t), normalized
nI gt 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.
16Stability of Quasi-steady States (QSS)
- Multiplication rate of small-population mutant i
in presence of fixed point of N resident species,
J, K
17Active and Quiet Periods
Histogram of entropy changes
Histograms of period durations
18Power Spectral Densities(squared norm of Fourier
transform)
PSD of D(t)
PSD of Ntot(t)/N0 ln(F-1)
19Species lifetime distributions
20Stationarity of diversity measures
Running time and ensemble averages.
- Total species richness, N(t)
- No. of species with nI gt 1
- Shannon-Wiener D(t)
- Mean Hamming distance between genotypes
- Total population Ntot(t)/N0ln3
- Standard deviation of Hamming distance
21Approximate phase-space argument for constant
diversity
- Community of N genotypes can be chosen in
ways - Influenced by N(N-1)/2 different pairs of MIJ and
MJI - Let probability that pair is conducive to forming
stable community be q - Community can be formed in ways
- ? Most probable
- for 2LgtgtN with L13 and q 1/4
22Summary 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., arXivq-bio.PE/0403042
23Current 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