Clonal Interference in Large Population

1
Clonal Interference in Large Population

• Su-Chan Park and Joachim Krug
• Institute for Theoretical Physics
• Cologne University

2
Clonal interference
From Crow and Kimura (1965)
3
Gerrish-Lenski Picture (1998)
Assumptions

• Infinite sites model
• No epistasis
• Neglect of the multiple mutations
• Genetic drift and Clonal interference are
  independent.

4
GL Picture (continued)
Preliminaries (no interference)

• Fixation Probability
  (c2)
• Fixation time (escaping genetic drift)

• N population size
• m advantageous mutation probability

5
GL Picture (continued)
Criterion for the interference

• Average number of surviving mutations per
  generation

If gtgt no clonal
interference
Clonal interference becomes prominent when
6
GL Picture (continued)

• Number of beneficial mutations

No multiple mutations

• Prob. i) having gt s and ii) escaping drift.

Independence assumption
7
GL Picture (continued)

• Expected number of superior mutations

Poisson distribution

• Fixation probability

8
GL Picture (continued)

• Prob. dist. of the fixed mutations.

• The rate of substitution

• Increase of selectivity per fixation

9
GL Picture (continued)

• The rate of adaptation (fitness increase)

• Conclusion

• slow down by the waste of beneficial mutations.
• speed limit (?) cf. Wilke (2004)
• predicting the parameters in experiment

10
GL Picture (continued)

• Rhythmical Substitution Gerrish (2001)

• E(t) number of fixed mutations until t
• V(t) variance of fixed mutations until t
• fixation events are assumed to be renewal

Universal number(!)
cf random Poisson 1
deterministic periodic 0
11
Criticism on the GL Picture

• Neglect of multiple mutations
• Before fixation, new mutation can hit the
  mutants besides the wild-type.
• Independency assumption
• Prob. of escaping drift should depend on the
  present population distribution.
• Fixation and Origination Processes

fixation
t
origination
12
Numerical Study on the Wright-Fisher model
13
Numerical Study on the Wright-Fisher model

• Substitution processes

14
Numerical Study on the Wright-Fisher model

• Fixation vs Origination Processes

J(k) prob. dist. of fixed mutations per fixation
15
Numerical Study on the Wright-Fisher model

• Geometric distribution

Neutral theory also predicts the geometric
distribution of the fixed mutation per fixation
events. G. A. Watterson (1982)
16
Numerical Study on the Wright-Fisher model
17
Numerical Study on the Wright-Fisher model

• Why is q(N) finite as N gets larger?

largest s ,
Fixation process is stochastic even in the
infinite population limit

• Ratio of the variance to the average for the
  origination process becomes 0.

Origination process becomes deterministic in
the infinite population limit
18
Summary

• GL picture gives rather reasonable prediction for
  the fitness increase rate for the moderate
  population size.
• Rhythm is much stronger than predicted by the GL
  picture (deterministic).
• Can the molecular clock of the HIV be understood
  by our numerical observation?
• T. Leitner and J. Albert (1999)