Research ArticleGENETICS

Stochastic tunneling across fitness valleys can give rise to a logarithmic long-term fitness trajectory

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Science Advances  31 Jul 2019:
Vol. 5, no. 7, eaav3842
DOI: 10.1126/sciadv.aav3842
  • Fig. 1 The model.

    (A) The fitness of a cell is a function of the state of its genome α, with each site having two possible states αi = ±1. The contributions to fitness include both the independent contributions from each of the L sites in the genome (represented by the fields h) and the pairwise interactions between sites that are captured by the symmetric matrix J. (B) The model has the feature of frustration, which can be seen by considering a triplet of sites. In this case, the 4th site is coupled positively to both the 7th and 8th sites (J4,7, J4,8 > 0) and would therefore provide a higher fitness contribution if it has the same sign as both of them. However, this would not satisfy the negative coupling between the 7th and 8th sites (J7,8 < 0). (C) Adaptation dynamics can be considered as a walk in a multidimensional genotypic space. Here, each node represents a genotypic state, with its color indicating its fitness value. Nodes corresponding to fitness peaks are enlarged and have a red border. States connected by blue lines are 1-hamming distance apart. Black arrows trace out an example of a possible adaptation trajectory to a fitness peak. The population can escape from a fitness peak via stochastic tunneling if there are states 2-hamming distance away that are higher in fitness. The red arrow shows an example of such a possible escape path. (D) The fitness landscape consists of multiple local maxima. To increase in fitness, a population away from a fitness maximum can gain a single beneficial mutation, which fixes with probability pf (black arrows). However, once the system is at a fitness maximum (red circles), it will need to acquire multiple mutations (purple dashed arrows) to transit to a state with higher fitness (red arrows). The probability of a successful double mutant emerging from a deleterious single mutant is given by pd. (E) Relative probabilities of the next successful mutation event being a single beneficial mutation (red circles), a single deleterious mutation (blue cross), or a double mutant (green diamonds). When the rank (number of available beneficial mutations) is positive, fixation of beneficial mutations dominates. At a fitness peak (rank = 0), the probability of stochastic tunneling via double mutants dominates. Each data point represents an average over 100 states on a quenched landscape (parameters: L = 200, N = 107, μ = 10−8). (F) For a given L, the relative probability of a double mutant being the next event at a fitness peak goes to 1 above some L-dependent N. Each data point represents an average over 100 randomly drawn fitness peaks, and error bars represent the interquartile range (other parameters: Nμ = 0.1, ρ = 0.05, k = 0.9, Δ = 0.05).

  • Fig. 2 Relaxation to a fitness maximum does not generate a logarithmic fitness trajectory.

    (A and B) Without epistasis (J = 0) (blue curves), the fitness reaches the global maximum quickly and follows a power law F=FmaxFmax1(1+bt)γ with γ = 2. With maximum epistasis (h = 0) (red curves), the fitness trajectory is slower with a power law exponent γ ≈ 0.9 but is still not as slow as a log trajectory if we do not allow escape from local fitness maxima. Here and in all other figures, time t is measured in units of number of mutational attempts (other parameters: L = 200, initial rank = 100, ρ = 0.05, Δ = 0.003).

  • Fig. 3 Logarithmic fitness trajectory emerges from hopping between MSs.

    (A) The average fitness trajectory from an initial MS (green circle) was found by constructing a Markov chain that includes all possible beneficial single mutants that a non-MS (blue circles) can go to and all possible beneficial double mutants that an MS (red circles) can go to. Here, we show a subset of the whole tree, with the thickness of the arrows proportional to −1/ log(λ), where λ is the transition rate. (B) The transition rates out of an MS (red histogram) are typically much slower than those out of a non-MS (blue histogram). (C) Average fitness trajectory increases logarithmically with time. The green points are data obtained from analyzing the Markov chain, while the black line is the fit to the logarithmic function F = 1 + a log(1 + bt). The red, blue, and purple dotted lines are examples of individual trajectories, with the crosses corresponding to fixation events. (D) Average trapping time 〈τ〉 scales approximately linearly with time, showing that as the population evolves, it enters MSs that are harder and harder to get out off. (E) The average τ (blue circles) of an MS seems to increase exponentially with its fitness. The black line is a fit to a straight line. (F) The average number of escape paths from an MS np seems to decrease exponentially with its fitness. The black line is a fit to the exponential function. In both (E) and (F), the averages are taken over states in a single Markov chain, with the weight of each state proportional to the probability of encountering that state, and the error bars represent 95% confidence intervals (parameters: L = 200, μ = 10−8, β = 0.9, Δ = 0.05, ρ = 0.05).

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/7/eaav3842/DC1

    Section SA. Choice of null model for the combined effect of multiple independent mutations

    Section SB. Range of validity for population size N

    Section SC. Structural features of fitness landscapes

    Section SD. Relaxation to local fitness maximum

    Section SE. Hopping between MSs

    Section SF. Batch culture

    Fig. S1. Properties of fitness landscape as a function of L.

    Fig. S2. Properties of fitness peaks as a function of L when additional weak interactions

    between all sites are included.

    Fig. S3. Figure showing how properties of MSs vary with fitness for ρ = 0.01 (black diamonds, Foffset = −3.4), ρ = 0.025 (red circles, Foffset = −3.8), ρ = 0.05 (green triangles, Foffset = −4.2), ρ = 0.075 (blue squares, Foffset = −4.4), and ρ = 0.1 (purple crosses, Foffset = −4.5).

    Fig. S4. The number of connecting MSs ns, which is the number of MSs that the system can transit to next from the current MS, correlates with the number of double mutant escape paths out of a state np.

    Fig. S5. Relaxation toward a single local fitness maximum slows down with increasing degree of epistasis.

    Fig. S6. Changing the distribution of fixation probabilities does not significantly change the functional form of the fitness trajectory.

    Fig. S7. Other distributions for the nonzero elements of the interaction matrix give similar form for the fitness trajectory.

    Fig. S8. Logarithmic fitness trajectories are also observed for different values of ρ.

    Fig. S9. The decay of the two-time correlation function depends on both the time difference Δt and the initial time of the measurement tw, implying that the system ages.

    Reference (45)

  • Supplementary Materials

    This PDF file includes:

    • Section SA. Choice of null model for the combined effect of multiple independent mutations
    • Section SB. Range of validity for population size N
    • Section SC. Structural features of fitness landscapes
    • Section SD. Relaxation to local fitness maximum
    • Section SE. Hopping between MSs
    • Section SF. Batch culture
    • Fig. S1. Properties of fitness landscape as a function of L.
    • Fig. S2. Properties of fitness peaks as a function of L when additional weak interactions
    • between all sites are included.
    • Fig. S3. Figure showing how properties of MSs vary with fitness for ρ = 0.01 (black diamonds, Foffset = −3.4), ρ = 0.025 (red circles, Foffset = −3.8), ρ = 0.05 (green triangles, Foffset = −4.2), ρ = 0.075 (blue squares, Foffset = −4.4), and ρ = 0.1 (purple crosses, Foffset = −4.5).
    • Fig. S4. The number of connecting MSs ns, which is the number of MSs that the system can transit to next from the current MS, correlates with the number of double mutant escape paths out of a state np.
    • Fig. S5. Relaxation toward a single local fitness maximum slows down with increasing degree of epistasis.
    • Fig. S6. Changing the distribution of fixation probabilities does not significantly change the functional form of the fitness trajectory.
    • Fig. S7. Other distributions for the nonzero elements of the interaction matrix give similar form for the fitness trajectory.
    • Fig. S8. Logarithmic fitness trajectories are also observed for different values of ρ.
    • Fig. S9. The decay of the two-time correlation function depends on both the time difference Δt and the initial time of the measurement tw, implying that the system ages.
    • Reference (45)

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