Gene Conversion Facilitates Adaptive Evolution on Rugged Fitness Landscapes

Abstract

Gene conversion is a ubiquitous phenomenon that leads to the exchange of genetic information between homologous DNA regions and maintains coevolving multi-gene families in most prokaryotic and eukaryotic organisms. In this paper, we study its implications for the evolution of a single functional gene with a silenced duplicate, using two different models of evolution on rugged fitness landscapes. Our analytical and numerical results show that, by helping to circumvent valleys of low fitness, gene conversion with a passive duplicate gene can cause a significant speedup of adaptation, which depends nontrivially on the frequency of gene conversion and the structure of the landscape. We find that stochastic effects due to finite population sizes further increase the likelihood of exploiting this evolutionary pathway. A universal feature appearing in both deterministic and stochastic analysis of our models is the existence of an optimal gene conversion rate, which maximizes the speed of adaptation. Our results reveal the potential for duplicate genes to act as a "scratch paper" that frees evolution from being limited to strictly beneficial mutations in strongly selective environments.

Keywords: adaptive evolution; fitness landscape; gene conversion; gene duplication; pseudogenes.

Figure 1

(A) Cartoon of the gene duplication and conversion. (B) Schematic diagram of the minimal 3 $\times$ 3 model. Short bidirectional arrows denote mutations and green arrows denote gene conversions.

Figure 2

Stochastic simulations of the minimal gene conversion model. Stacked areas of different colors represent the number of individuals in each state according to the lookup table in (A). Colors blue, red, and green correspond to the state of the active gene determining the fitness, different shades of the same color indicate the state of the passive gene. Parameters: (A) $N=800,$ $\delta =0.03,$ $\mu ={10}^{-3},$ $\alpha =0.$ (B) $N=800,$ $\delta =1,$ $\mu ={10}^{-3},$ $\alpha ={10}^{-3}.$ (C) $N=100,$ $\delta =0.01,$ $\mu ={10}^{-4},$ $\alpha =0.$ (D) $N=100,$ $\delta =1,$ $\mu ={10}^{-4},$ $\alpha ={10}^{-4}.$ In all panels, $s=\mathrm{0.1.}$

Figure 3

Stochastic tunneling regime of the minimal model. (A) Normalized fixation time $T_2$ along the gene conversion path, only. Colored dashed lines indicate the approximations of Equation (2) for small (red) and large α (blue). The vertical dashed line indicates the optimal gene conversion rate ${\alpha }_m\approx {\left(s{\mu }^2\right)}^{1/3}.$ Parameters are $\mu ={10}^{-6}$ and $s=\mathrm{0.1.}$ (B) Numerical simulations (circles) and theory according to $T={\left(T_1^{-1}+T_2^{-1}\right)}^{-1}$ for the full minimal model for $N={10}^5,$ $\mu ={10}^{-6},$ $\delta =0.01,$ and $s=\mathrm{0.1.}$ The shaded area indicates the gene conversion rates at which the tunneling approximation for the gene conversion path becomes formally invalid. (C) Fixation time reduction $T_2/T_1$ provided by gene conversion as a function of μ for $s=0.1$ and two different values of δ (the ratio is independent of N), assuming the population size is in the stochastic tunneling regime. The solid lines were calculated with the optimal ${\alpha }_m=2^{-4/3}{\left(s{\mu }^2\right)}^{1/3},$ the dashed lines use fixed ${\alpha }_0={\left(16\cdot {10}^{13}\right)}^{-1/3}$ (the optimal α for the parameters in B). (D) Same as in (C), but for varying selective advantage s with fixed mutation rate $\mu ={10}^{-6}.$

Figure 4

Theory and simulations for deterministic system [9] with $s=\mathrm{0.1.}$ (A) Numerical simulations (symbols) compared to the theoretical predictions of $T_1$ and $T_2$ from Equations (10), (11) (lines) for $\mu ={10}^{-3}$ and $\delta =0.9$ and a range of conversion rates α. (B) Same as in (A), but for smaller $\mu ={10}^{-6}$ and $\delta =\mathrm{0.999.}$ (C) Trajectory of (9) for $\mu =0.001,$ $\delta =0.9,$ and $\alpha =4.83\times {10}^{-7}$ (left filled circle in A). The system takes the direct path to reach higher fitness state 2 via state 1, even though the fraction of the population in state 1 is minuscule and the red shades are not visible. (D) Trajectory of (9) for $\mu ={10}^{-3},$ $\delta =0.9,$ and $\alpha =0.03$ (right filled circle in A). The system reaches state 2 through the conversion process $\left(0,2\right)\to \left(2,2\right).$

Figure 5

Markov chain for a evolving population in the absence of the direct path ($\delta =1$) in the monoclonal approximation. Black arrows denote ordinary mutations, and green arrows denote gene conversions.

Figure 6

Simulations of the full stochastic model (Figure 1B) with $s=\mathrm{0.1.}$ Fixation times T obtained as averages from 5000 independent simulations unless stated otherwise. Symbols represent numerical simulations. (A) Fixation times for $\mu ={10}^{-5}$ and $\delta =1$ as a function of α for different N. Lines represent the monoclonal approximation, (16). (B) Fixation times for $\mu ={10}^{-5}$ and $N=50$ compared to the monoclonal approximation for the gene conversion path (16) (black line), for different values of δ. The colored lines are analytical predictions combining the monoclonal approximation for the direct and the conversion path according to $T={\left(T_1^{-1}+T_2^{-1}\right)}^{-1}.$ (C) Fixation times for $\delta =1$ as a function of N for two different combinations of α and μ, compared to the monoclonal approximation (16) (solid line), the tunneling approximation (dashed-dotted lines) and the deterministic approximation (11) (horizontal dashed lines). Averages for larger N were obtained from progressively fewer numerical simulations. (D) Fixation times for $\mu ={10}^{-3},$ large population size $N={10}^6$ and two different δ from 50 simulations (circles) compared to deterministic simulations (squares) and Equations (10) (color lines) and (11) (black line).

Figure 7

Summary of different approximations. (A) Regimes of validity for the different approximations of the gene conversion path in the minimal model in terms of the population size N and the conversion rate α for fixed $\mu ={10}^{-6}$ and $s=\mathrm{0.1.}$ The vertical dashed line indicates $\alpha =2^{-4/3}{\left(s{\mu }^2\right)}^{1/3}.$ (B) Speed-up $T_2/T_1$ provided by gene conversion across different population sizes for fixed $\mu ={10}^{-6},$ $s=0.1$ and $\delta =0.1$ (the individual fixation times of the two paths can be found in Figure S3 in File S1). Results for each regime are plotted where the population size requirements for both the direct and the gene conversion path are fulfilled and both paths are in the same regime, except for the dashed red line labeled “mixed,” where the direct path is in the tunneling regime and the gene conversion path is in the monoclonal regime (cf. Figure S3 in File S1). Calculations are shown for a fixed gene conversion rate ${\alpha }_0={\left(16\cdot {10}^{13}\right)}^{-1/3}$ (the optimum in the tunneling regime; blue lines) and the optimal α for each regime and population size (red lines).

Figure 8

Fitness evolution in a rugged landscape. (A) Example trajectory of fitness of a single monoclonal population in landscape with $\mathcal{N}=20,$ $\mathcal{K}=19,$ ${\mathcal{N}}_0=10$ and $N={10}^5$ and $\alpha /\mu =10.$ Thick black line indicates actual fitness of the population, thin green line shows the potential fitness of the passive region a simulation with gene conversion, and red dashed line indicates the actual fitness in a simulation without gene conversion. Open circles denote the potential fitness immediately after mutations in nonessential bits of the passive region, solid circles denote the potential fitness ($=0$) after mutations in essential bits, and diamonds denote potential fitness after gene conversion from active to passive region. (B) Ensemble-averaged fitness over time for select conversion rates α (solid lines) and control simulations without gene conversion (dashed lines). Other parameters are identical to those in (A). Fitness values were averaged over 1000 populations and 20 landscapes. (C) Initial average fitness trajectories for a range of different α. Colored sections indicate the intervals which were used to detect the initial rate of fitness increase r. The color corresponds to the value of α, cf. (D). (D) Initial rate of fitness increase r extracted from the data in (C). Dashed lines indicate the approximations of Equations (18) (blue), (19) (yellow), and (20) (red).