### Background

Where do sex differences come from? The mechanistic differences (females give birth, males inseminate females) are no mystery, but why are there so many subtle differences? There is quite a few of them: morphological differences that arise at puberty, different behavioral predispositions, as well as the differences in prevalence, age of onset, and severity of nearly all human diseases. If you ask this question, you may hear mentions of chromosomes, gamete size, hormones, parental investment, etc. All of these have truth to them (in particular, parental investment is a great way to predict how gender roles in a given species will map to human ones), but they also leave something lacking. The missing piece of the puzzle seems to be a causal explanation of what exactly drives so many subtle differences. One might say, “because females choose males to be this way,” yet though this answer is fairly close to the truth, it also misses something crucial—why would female choose males to be one way and not another? Why would this have any impact on human disease?

It turns out that fairly simple mathematics can help answer our question: many of the above-mentioned sex differences should stem from a fundamental asymmetry: females tend to be choosers, while males tend to be courters. In a reproduction-survival trade-off, the courters sacrifice some of their survival advantage to gain a reproductive advantage. One can equally say that the choosers shift some of their genetic load onto the courters. The male sex then evolves ways to suppress the genes that are harmful to it (resulting in female-biased genes), and to upregulate ones that increase its reproductive success (resulting in male-biased genes). Of course, females also evolve sex-specific regulatory mechanisms, but the asymmetry in reproductive success means this should happen more frequently in males. Because of differential expression triggered during development, a sizeable portion of the genome1 is recruited into this game—not just sex chromosomes.

Is this specific to mate choice? No, the reproduction-survival trade-off is more fundamental. It can lead to intrasexual competition as an alternative to pure mate choice—in fact, the former is more likely to arise. Intrasexual competition should cause similar mutation load imbalance to what we get under mate choice. The trade-off that a male peacock makes with his long tail is well known, but among bovids, polygyny stemming from intrasexual competition (as opposed to mate choice) is also linked to shorter male lifespan. There is another consequence of reproductive asymmetry: the courter may evolve ways to suit the chooser. These could range from something as common as fathers participating in parental care—to a more controversial idea I discuss below.

In a previous post, I introduced the two-locus model of runaway sexual selection. Mark Kirkpatrick developed it to show that a gene which harms male survival but increases reproductive success due to preexisting female preference could sweep through a population, while also increasing the frequency of the female preference gene in the process. In this post, I reuse that model (following Seger and Trivers’ paper) to show that, whenever intralocus sexual conflict exists (a gene variant benefits one sex but harms another), the choosy sex will tend to have an upper hand by shifting some of the burden onto the less choosy one. I also demonstrate that under evolution of traits linked with intrasexual competition (“war paint”), the asymmetry in mutation load should be identical. A Mathematica notebook with code to produce the figures is available here.

### Asymmetry and Conflict

#### The Seger-Trivers Paper

Some time in the late 1970s, Robert Trivers had developed an idea that female choice may evolve in a way that benefits daughters. The idea later appeared in a 1985 book Social Evolution (you can download it from its author’s website) and in a paper that Trivers published with his graduate student Jon Seger (reprinted as Chapter 7 in Natural Selection and Social Theory). The paper makes a two-fold claim: 1) that there is unequal pressure from natural selection on individual survival when only one of the sexes is choosy, and 2) that mate choice should evolve in ways that maximize the fitness of the choosy sex.

Although Russell Lande, in a famous but dry 1980 paper, had given a quantitative expression to the imbalance in genetic load between the sexes, essentially making a similar claim to (1), most of the publications that cited Lande’s paper focused on size dimorphism. The claim (2) had gone almost unmentioned. One reason for its obscurity could be that the mechanism proposed in the paper (selective diffusion) seems to be a weak force that requires population structure. Another reason could be the lack of a good animal example. In their paper, the authors suggest that lower food availability on islands in the Caribbean (compared to the mainland) may favor those female Anolis lizards which prefer larger males. But an alternative explanation is plausible: lower predator pressure on the islands could increase the importance of intrasexual competition, which would increase male size. Finally, a good case can be made that the title of the paper was overly technical-sounding.2

But there is a way to think about this hypothesis that caught my interest. One sometimes hears that stag deer or peacocks carry “handicaps” to prove their survival worth. But how can an organism prove its survival worth through something that lowers its survival?3 The question becomes easier to answer when we consider sexual dimorphism: the females do not carry the handicap. In behavioral ecology terms, this means genic capture should favor female fitness—which is consistent with Trivers’ idea of mate choice benefiting daughters.

I will begin with Trivers’ verbal model from Social Evolution:

Imagine a gene in a female leading her to choose a male whose genes improve her daughters’ chance of surviving by 10%, while decreasing her sons’ survival by a similar amount. At first glance, genes for this kind of female preference appear to be neutral in their evolutionary effects: what is gained in daughters is lost in sons. But since the genes for this kind of female choice will have increased in one generation among females, more females will choose this way, giving a benefit to the males they prefer. (353)

Such a situation is incredibly common: a gene for smaller body size may be beneficial to a female (in helping her avoid predators) but detrimental to a male (who would be less successful in warding off rivals). Trivers’ verbal model suggests that female body size should be, on average, closer to the species’ optimum than male size. Similar logic can be applied not only to body size, but to almost any trait one can think of, including behavior. Note that if the gender roles were reversed (as is seen in a few bird species such as cassowary, phalarope, or jacana), then it is the females who would fall under a survival disadvantage.

This may seem clear enough, even trivial. Interestingly—and perhaps somewhat disconcertingly, the asymmetry in genetic load is blind to what the females actually prefer. It results from greater variance in male reproductive success, not from how females choose (though of course, very choosy females will increase the variance in male reproductive success). In the first two models I discuss, such asymmetry is caused by mate choice. In the last one, it is caused by intrasexual competition linked to “war paint” displays. I show that both cases lead to the same genetic load asymmetry.

I already explained the two-locus model of trait-preference co-evolution in a previous post, so I will not dwell as much on details. As before, we have a preference gene with alleles $P_1$ and $P_2$ (with frequencies $p_1$ and $p_2$) and a trait gene with alleles $T_1$ and $T_2$ (with frequencies $t_1$ and $t_2$). Unlike in the previous model, the trait gene affects survival of both males and females. The viabilities (survival rates) can range from 0 to 1 and are given in the following table:$\def\sp{^}$

The fractions of $T_2$ carriers among males and females that survive to maturity are:

$$\begin{eqnarray} t_2\sp{m} &=& \alpha_2 t_2 ⁄ (\alpha_1 t_1 + \alpha_2 t_2) \tag{1a} \\ t_2\sp{f} &=& \beta_2 t_2 ⁄ (\beta_1 t_1 + \beta_2 t_2), \tag{1b} \end{eqnarray}$$

while the corresponding $T_1$ fractions are simply $t_1\sp{m} = 1 - t_2\sp{m}$ and $t_1\sp{f} = 1 - t_2\sp{f}$. Using a method explained in the previous post, the frequencies of the two trait alleles in the next generation are determined as:

$$\begin{eqnarray} t_1' &=& (U_{11} p_1 + U_{21} p_2 + t_1\sp{f}) ⁄ 2 \tag{2a} \\ t_2' &=& (U_{12} p_1 + U_{22} p_2 + t_2\sp{f}) ⁄ 2, \tag{2b} \end{eqnarray}$$

where $U_{21}$ stands for the fraction of $P_2$ females that mate with $T_1$ males (and so on). Under fixed-preference mate choice, the fractions of $P_1$ and $P_2$ females that mate with $T_1$ and $T_2$ males respectively are:

$$\begin{eqnarray} U_{11} &=& a_1 t_1\sp{m} ⁄ (t_2\sp{m} + a_1 t_1\sp{m}) \tag{3a} \\ U_{22} &=& a_2 t_2\sp{m} ⁄ (t_1\sp{m} + a_2 t_2\sp{m}), \tag{3b} \end{eqnarray}$$

where $a_1$ and $a_2$ denote the degree of female bias towards the respective male types (when $a_2 = 3$, the ratio of $T_2 : T_1$ males that mate with $P_2$ females is 3:1). The remaining fractions are simply $U_{12} = 1 - U_{11}$ and $U_{21} = 1 - U_{22}$.

Now that we have a complete model specification, we can do the algebra explained previously, set $a_2=3$, selection coefficient to 0.4 and reproduce Figure 1 from Seger-Trivers paper (the color red means the fitness of $T_2$ allele is higher than that of $T_1$, blue means the opposite):

The evolution is expected to proceed upwards in the red zone and downwards in the blue zone. The thick black line shows stable equilibria where the rate of evolution is zero. The left pane here is the the same as in the figure from the previous post except with its axes flipped. The right pane shows what happens when it is the $T_2$ females (rather than $T_2$ males) who suffer viability penalty, everything else being equal.

A geneticist would be interested in allele fixation (when allele completely takes over the population). In the left pane, the chance of $T_2$ fixation is some non-zero value (assuming a cryptic preference for $T_2$ can arise through an unrelated process), while in the right pane it is close to zero. We can crudely approximate the probability of fixation of each allele by measuring the distances indicated with the arrowhead intervals. Because this measure does not accurately reflect actual probability (which would depend on the prior probability of cryptic preference arising among females), I will refer to it as “chance of fixation,” or $\phi$. We can find closed-form expressions for $\phi_1$ and $\phi_2$ and plot each as a function of selection pressure $s$:

The left pane shows the chance of fixation of $T_1$ allele, and the right pane shows the same for the $T_2$ allele. The blue line is when $T_2$ males are penalized ($\alpha_2 = 1 - s$), and the orange one is when $T_2$ females are penalized ($\beta_2 = 1 - s$). We conclude that penalizing the $T_2$ males (as opposed to females) results in a greater chance of fixation of either allele. Between $s=0.40$ and $s=0.67$ (right pane), there is a distinct discrepancy: penalizing $T_2$ males still allows us to fixate the harmful $T_2$ allele, while penalizing $T_2$ females cannot result in fixation.

#### A Fully Symmetric Model

To make the model fully symmetric, consider the following. Under condition A (blue line), the $T_2$ allele reduces the survival of males but increases that of females by the same amount. Under condition B (orange line), the same allele increases the survival of males but decreases that of females. Furthermore, unlike in the previous models, the $P_1$ females prefer $T_1$ males instead of mating randomly, while the $P_2$ females prefer $T_2$ males to the same degree ($a_1 = a_2 = 3$ under fixed-preference and $c_1 = c_2 = 1$ under best-of-2 models). Here is what the chance of fixation looks like under varying selection pressure:

The left pane shows the chance of fixation of $T_2$ allele under fixed-preference mate choice, the right one under best-of-2 model detailed in the previous post. Unsurprisingly, the asymmetry (seen as the vertical distance between the blue and orange lines) increases with selection pressure. This effect is stronger under best-of-2 mate choice. Now instead of varying selection pressure, we will vary the degree of preference among females towards the respective male types ($a_1 = a_2 = a$ and $c_1 = c_2 = c$). We get:

This is somewhat more surprising. As the degree of preference increases, the asymmetry decreases. A little bit of basic algebra gives us closed-form expressions for the deltas. If we set

$$k = -\frac{(s-2) s\sp{3}}{(s-1)\sp{2}}, \tag{4}$$

which increases monotonically with $s$, then under fixed-preference mate choice, $\Delta\phi_2 = k a /(a\sp{2} - 1)$, and under best-of-2 model, $\Delta\phi_2 = k/(2 c)$. It is obvious that the deltas are always positive when $a>1$ and $c>0$ and that their value declines as preference increases. Using this knowledge, we can construct a seemingly pathological case. Suppose $c_1 = c_2 = 0.4$ under the best-of-2 mate choice, which corresponds to a moderate degree of preference. Then if we set $\alpha_2 = \beta_1 = 0.5$ (left pane) and $\alpha_1 = \beta_2 = 0.5$ (right pane), we get:

The arrows show the approximate directions of evolution (see previous post for details). Although these plots are mirror images of each other, the story they tell is not symmetric. An allele that increases the survival of females by 2× while decreasing the survival of males by 2× can invade together with fairly moderate female preference for that allele from the point of origin (left pane). Meanwhile, the allele that decreases the survival of females by the same amount while increasing the survival of males cannot invade (note the arrow pointing down in the right pane). Furthermore, there is a chance of fixation of the allele that hinders male survival, while there is no chance of fixation of the allele that hinders females even if a cryptic female preference for the associated trait arises beforehand.

#### Evolution of War Paint

The first question that sprang to my mind after reproducing Seger-Trivers results was whether the same holds for other forms of sexual selection. Pure female choice-driven ornaments are probably fairly rare in nature (though the elaborate trains of peacocks and plumage of the birds of paradise are clear cases of it). A much more common type of ornament can be thought of as “war paint.” Fisher described it thusly:

As a propagandist the cock behaves as though he knew that it was advantageous to impress the males as the females of his species, and a sprightly bearing with fine feathers and triumphant song are quite as well adapted for war-propaganda as for courtship. (139)

A classic experiment on stickleback fish showed that the color red (a relatively inexpensive trait) may serve such a role in this species. The logic of the model is as follows: the $T_2$ males are aggressive and also express some visual trait which is preferred by $P_2$ females. The survival cost paid by the $T_2$ males is dependent on the frequency of that allele in the population. When the frequency is low, fights are rare (because $T_1$ males are not aggressive) and the ornament is not expensive, but as $T_2$ frequency approaches 1.0, the survival cost approaches its maximum value (fighting off other aggressive $T_2$ males is energetically expensive). For simplicity, I borrow the notation from the previous post. The frequencies of the two trait alleles after selection among all males in the population are:

$$\begin{eqnarray} t_1\sp{m} &=& t_1 ⁄ M \tag{5a} \\ t_2\sp{m} &=& (1 - s t_2) t_2 ⁄ M, \tag{5b} \end{eqnarray}$$

where $M = t_1 + (1 - s t_2) t_2$ is the average male viability. This is the only change needed to the model. I now plot the equivalent of the first figure from the previous post, with the left pane showing the fixed-preference model and the right pane showing the best-of-2 model (all variables are unchanged):

Unsurprisingly, male displays with a frequency-dependent cost are much easier to evolve than ones with a fixed cost—hence we expect them to be more frequent in nature. Now what about asymmetry in chance of fixation and genetic load? It turns out we get exactly the same results as with the pure mate-choice ornament, that is $\Delta\phi=k a /(a\sp{2}-1)$ under fixed-preference and $\Delta\phi=k/(2 c)$ under best-of-2 models.

### Last Words

The consequences of asymmetric choice may also apply to various seemingly irrational sociocultural phenomena—initiation rituals, body modification, fashion trends, and so on, though obviously these call for more dedicated models than what I present here.4 It is interesting to think how far the principle of genetic load increase due to artificial selection can be extended. Although we do not exchange any genes with dogs (hence sexual selection models are inappropriate here), almost all of the dog breeds we know today have significantly shorter lifespan than their wild ancestor, the wolf.

The idea of mate choice benefiting daughters, in my opinion, does better justice to some aspects of sexual selection than the handicap principle for reasons I explained above. For any biologists reading this, it may be interesting to revisit studies that tried to demonstrate the working of the handicap principle from this alternative perspective.

1. In Drosophila flies, male-biased genes show greater inter-species diversity, which indicates both greater mutational load and faster evolution of those sequences. In D. melanogaster, a half of all genes show sex bias: 4,073 genes are female- and 2,876 are male-biased (same source). A 2015 study on six species of birds found elevated rate of coding sequence evolution in male-biased genes, which correlated with intensity of sexual selection.