The Yank of Dobzhansky’s Bequest

Abstract

Dobzhansky studied mechanisms of balancing selection using systems of inversions in Drosophila and he soon found that changes in inversion frequencies along generations in experimental populations conformed to the expectation for a simple model of heterosis. However, other more complex modes of selection, like rare male advantage, were later found to affect the maintenance of inversion polymorphisms. Here we show that a more realistic (and complex) model than heterosis—integrating all known fitness component estimates obtained in independent experiments for the ST/CH system of inversions in Drosophila pseudoobscura—not only conforms to but actually also predicts the inversion frequencies. This concludes this line of work and points to other selection mechanisms than heterosis that were also considered by Dobzhansky—frequency- and sex-dependent selection—as potential mechanisms of balancing selection responsible for the maintenance of the inversion polymorphisms in Drosophila.

Appendix

The model of selection we used to obtain our fitness-component-based predicted trajectories has been developed by Álvarez-Castro and Alvarez (2005). Its recurrence equations can be expressed as follows. The adult-to-zygote step can be modeled as

$$ X^{\prime}_{{zy}} = \frac{1}{{\bar w_{{zy}} }}[(f_{1} (X,Z)X + \frac{1}{2}f_{2} (X,Z)Y)(m_{1} (X,Z)X + \frac{1}{2}m_{2} (X,Z)Y)], $$

$$ Y^{\prime}_{{zy}} = \frac{1}{{\bar w_{{zy}} }}\left[ {\left( {f_{1} (X,Z)X + \frac{1}{2}f_{2} (X,Z)Y} \right) \times \left( {\frac{1}{2}m_{2} (X,Z)Y + m_{3} (X,Z)Z} \right) + \left( {\frac{1}{2}f_{2} (X,Z)Y + f_{3} (X,Z)Z} \right) \times \left( {m_{1} (X,Z)X + \frac{1}{2}m_{2} (X,Z)Y} \right)} \right], $$

$$ Z^{\prime}_{{zy}} = \frac{1}{{\bar w_{{zy}} }}\left[ {\left( {\frac{1}{2}f_{2} (X,Z)Y + f_{3} (X,Z)Z} \right) \times \left( {\frac{1}{2}m_{2} (X,Z)Y + m_{3} (X,Z)Z} \right)} \right], $$

(1)

where X, Y and Z, are the frequencies of the three genotypes, ST/ST, ST/CH and CH/CH, the subindex zy means zygote (adult frequencies otherwise), the prime means “at the following generation”, f _i (X,Z) and m _i (X,Z), i = 1,2,3, are the female and male multiplicative frequency-dependent fertilities, and

$$ \bar w_{{zy}} = (f_{1} (X,Z)X + f_{2} (X,Z)Y + f_{3} (X,Z)Z)(m_{1} (X,Z)X + m_{2} (X,Z)Y + m_{3} (X,Z)Z), $$

such that the frequencies $X^{\prime}_{{zy}}$, $Y^{\prime}_{{zy}}$ and $Z^{\prime}_{{zy}}$ add to unity. The adult-to-adult recurrence equations can now be expressed as

$$ X^{\prime} = \frac{{v_{1} (X^{\prime}_{{zy}} ,Z^{\prime}_{{zy}} )}}{{\bar w}}\left[ {\left( {f_{1} (X,Z)X + \frac{1}{2}f_{2} (X,Z)Y} \right)\left( {m_{1} (X,Z)X + \frac{1}{2}m_{2} (X,Z)Y} \right)} \right], $$

$$ Y^{\prime} = \frac{{v_{2} (X^{\prime}_{{zy}} ,Z^{\prime}_{{zy}} )}}{{\bar w}}\left[ {\left( {f_{1} (X,Z)X + \frac{1}{2}f_{2} (X,Z)Y} \right)\left( {\frac{1}{2}m_{2} (X,Z)Y + m_{3} (X,Z)Z} \right) + \left( {\frac{1}{2}f_{2} (X,Z)Y + f_{3} (X,Z)Z} \right)\left( {m_{1} (X,Z)X + \frac{1}{2}m_{2} (X,Z)Y} \right)} \right] $$

$$ Z^{\prime} = \frac{{v_{3} (X^{\prime}_{{zy}} ,Z^{\prime}_{{zy}} )}}{{\bar w}}\left[ {\left( {\frac{1}{2}f_{2} (X,Z)Y + f_{3} (X,Z)Z} \right)\left( {\frac{1}{2}m_{2} (X,Z)Y + m_{3} (X,Z)Z} \right)} \right], $$

(2)

where v _i (X _zy , Z _zy ), i = 1,2,3, are the frequency-dependent viabilities, the zygote frequencies $X^{\prime}_{{zy}}$ and $Z^{\prime}_{{zy}}$ are given by (1), and $\bar w$ is such that X′ + Y′ + Z′ = 1.

The implementation of these equations with overlapping generations consists in recomputing the vector of frequencies at the adult stage, (X, Y, Z), each generation as

$$ (1 - Ov)(X,Y, Z) + Ov(X^{*},Y^{*},Z^{*}), $$

(3)

where the asterisk stands for “at the previous generation” and Ov is the degree of overlapping—expressed as the frequency of individuals from the previous generation that remain in the current one, between zero and one. We considered degrees of overlapping between 0 and 0.25.

Random fluctuations of frequencies could make the populations to on average faster approach the orbit of the stable equilibrium of the genetic system. Thus, we have implemented drift in the model using a binomial distribution (Bürger 2000). In particular, we have further recomputed the vector of frequencies by adding to each frequency a random number drawn from a binomial distribution with index 2N and the frequency divided by 2N as parameter, where N is the number of individuals per generation. The frequencies were normalized back after adding the random numbers. For the population sizes we dealt with—N = 1500 (Dobzhansky and Pavlovsky 1953)—drift had no noticeable effect.

For the time scope of the experiment we reproduce in this article—15 generations (Dobzhansky and Pavlovsky 1953)—it is not necessary to consider long-term phenomena like mutation. We conducted simulations in Mathematica with expressions (1–3), implemented with the fitness components from the literature (Moos 1955; Anderson and Brown 1984; Anderson et al. 1986) in the very same way as described by Álvarez-Castro and Alvarez (2005).