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Moran process and Wright-Fisher process favor low variability

  • * Corresponding author: Jan Rychtář

    * Corresponding author: Jan Rychtář 
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  • We study evolutionary dynamics in finite populations. We assume the individuals are one of two competing genotypes, $ A $ or $ B $. The genotypes have the same average fitness but different variances and/or third central moments. We focus on two frequency-independent stochastic processes: (1) Wright-Fisher process and (2) Moran process. Both processes have two absorbing states corresponding to homogeneous populations of all $ A $ or all $ B $. Despite the fact that types $ A $ and $ B $ have the same average fitness, both stochastic dynamics differ from a random drift. In both processes, the selection favors $ A $ replacing $ B $ and opposes $ B $ replacing $ A $ if the fitness variance for $ A $ is smaller than the fitness variance for $ B $. In the case the variances are equal, the selection favors $ A $ replacing $ B $ and opposes $ B $ replacing $ A $ if the third central moment of $ A $ is larger than the third central moment of $ B $. We show that these results extend to structured populations and other dynamics where the selection acts at birth. We also demonstrate that the selection favors a larger variance in fitness if the selection acts at death.

    Mathematics Subject Classification: Primary: 92D15, 92D25, 60J80.

    Citation:

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  • Figure 1.  Fixation probabilities for the Wright-Fisher process (left) and the Moran process (right). Darker areas correspond to higher fixation probabilities; $ 0.1 $ corresponds to a random drift. For $ N = 10 $, we considered genotypes $ G_i; i = 1, \ldots, 11 $ that have a fitness $ i+1 $ with probability $ 1/i $ and $ 1 $ with probability $ 1-1/i $. The average fitness of $ G_i $ is $ 2 $, the variance is $ i-1 $. For every pair of $ i,j $, we run $ 10^5 $ simulations starting with a single $ G_i $ individual among $ G_j $ individuals. Note that the individuals' fitness is sometimes more than double the expected value

    Figure 2.  Left: A graphical representation of the genotypes $ G_p $ for $ p\in\{0.1, 0.2, \ldots, 0.9\} $. A fitness of $ G_p $ is $ \mu-x_p $ with probability $ p $ and $ \mu+y_p $ with probability $ 1-p $ where $ \mu = 4 $, $ x_p = (1-p)y_p/p $ and $ y_p = \sqrt{((1-p)^2/p + (1-p))^{-1}} $. The average fitness of $ G_i $ is $ \mu $ represented by the horizontal dotted line, the variance is $ 1 $. The genotypes are color coded by $ p $. The center of a disc corresponds to the fitness, the area of a disc corresponds to the probability of attaining such a fitness. The thick black curve is the third central moment $ s_p $ of the genotype $ G_p $ with values of the right $ y $ axis. Right: Fixation probabilities for the Moran process. The darker the color, the larger the fixation probability; $ 0.2 $ corresponds to a random drift. For $ N = 5 $ and every pair of $ p_i,p_j $, we run $ 10^7 $ simulations starting with a single $ G_{p_i} $ individual among $ G_{p_j} $ individuals

    Figure 3.  Fixation probabilities for the Db-Moran process. At every step, an individual is selected to die with the probability inversely proportional to their fitness and is then replaced by a copy of a randomly selected remaining individuals. For $ N = 10 $, we considered genotypes $ G_i; i = 1, \ldots, 11 $ as in Figure 1; a fitness of $ G_i $ is $ i+1 $ with probability $ 1/i $ and $ 1 $ with probability $ 1-1/i $. The average fitness of $ G_i $ is $ 2 $, the variance is $ i-1 $. For every pair of $ i,j $, we run $ 10^5 $ simulations starting with a single $ G_i $ individual among $ G_j $ individuals

    Figure 4.  Errors in approximation of $ 1/y $ by Taylor polynomials $ P_n(y) $ at $ y_0 $. The approximation by a higher degree polynomial is better only on $ (0,2y_0) $ and worse on $ (2y_0,\infty) $

    Figure 5.  The mean square errors (MSE) of the estimate of $ \mathbb{E}[\pi_A'|\pi_A]-\pi_A $ by $ E_2 $ (left) and by $ E_3 $ (right). The darker the color, the larger the error. For $ N = 10 $, we considered genotypes $ G_i; i = 1, \ldots, 21 $; a fitness of genotype $ G_i $ is $ i+1 $ with probability $ 1/i $ and $ 1 $ with probability $ 1-1/i $. The average fitness is $ 2 $, the variance is $ i-1 $. For every $ \pi_A\in\{1/N, \ldots, (N-1)/N\} $, we run $ 10^6 $ simulations to estimate $ \mathbb{E}[\pi_A'|\pi_A]-\pi_A $ numerically by the average different $ \widehat{E} $. We then calculated the MSE as $ \frac{1}{N-1}\sum_{\pi_A\in\left \{\frac1N, \ldots, \frac{N-1}{N}\right\}} \left(\widehat{E}-E_2\right)^2 $

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