Kinetic Monte Carlo Simulations of Statistical-mechanical Models of Biological Evolution - PowerPoint PPT Presentation

About This Presentation
Title:

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

Description:

Kinetic Monte Carlo Simulations of Statistical-mechanical Models of ... Self-similarity and 1/t2 distribution of QSS lifetimes leads to 1/f-like flicker noise ... – PowerPoint PPT presentation

Number of Views:90
Avg rating:3.0/5.0
Slides: 22
Provided by: magne7
Category:

less

Transcript and Presenter's Notes

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


1
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

2
Biological Evolution and Statistical Physics
  • Complicated field with many
  • unsolved problems.
  • Complex, interacting nonequilibrium problems.
  • Need for simplified models with universal
    properties. (Physicists approach.)

3
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.

4
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.

5
Individual-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).

6
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).
  • 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.

7
Deterministic approximation
m mutation rate per individual
8
Mutations
  • Each individual offspring undergoes mutation to
    a different genotype with probability m/L per
    gene and individual.

9
Single or noninteracting species, m 0 Logistic
map
x(t)
x(t1)
Fixed point nI(t1) nI(t)
10
Fixed 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 .
11
Stability 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

12
Monte 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

13
Simulation 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

14
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) expS(t)
  • where
  • S(t) - SI nI(t)/Ntot(t) ln nI(t)/Ntot(t)
  • is the information-theoretical entropy
    (Shannon-Wiener index).

15
Simulation 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.
16
Stability of Quasi-steady States (QSS)
  • Multiplication rate of small-population mutant i
    in presence of fixed point of N resident species,
    J, K

17
Active and Quiet Periods
Histogram of entropy changes
Histograms of period durations
18
Power Spectral Densities(squared norm of Fourier
transform)
PSD of D(t)
PSD of Ntot(t)/N0 ln(F-1)
19
Species lifetime distributions
20
Stationarity 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

21
Approximate 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

22
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., arXivq-bio.PE/0403042

23
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
Write a Comment
User Comments (0)
About PowerShow.com