The social evolution of sleep: sex differences, intragenomic conflicts and clinical pathologies

Sleep appears to be essential for most animals, including humans. Accordingly, individuals who sacrifice sleep are expected to incur costs and so should only be evolutionarily favoured to do this when these costs are offset by other benefits. For instance, a social group might benefit from having some level of wakefulness during the sleeping period if this guards against possible threats. Alternatively, individuals might sacrifice sleep in order to gain an advantage over mate competitors. Here, we perform a theoretical analysis of the social evolutionary pressures that drive investment into sleep versus wakefulness. Specifically, we: investigate how relatedness between social partners may modulate sleeping strategies, depending upon whether sleep sacrifice is selfish or altruistic; determine the conditions under which the sexes are favoured to adopt different sleeping strategies; identify the potential for intragenomic conflict between maternal-origin versus paternal-origin genes regarding an individual's sleeping behaviour; translate this conflict into novel and readily testable predictions concerning patterns of gene expression; and explore the concomitant effects of different kinds of mutations, epimutations, and uniparental disomies in relation to sleep disorders and other clinical pathologies. Our aim is to provide a theoretical framework for future empirical data and stimulate further research on this neglected topic.


Basic model of the social evolution of sleep and inclusive fitness predictions
Natural selection favours any gene that is associated with greater individual fitness (Fisher 1930;Price 1970). Assuming vanishingly little genetic variation, this condition may be expressed using the mathematics of differential calculus: dW/dg > 0, where g is the genic value of a gene picked at random from the population and W is the relative fitness of the individual carrying this gene (Taylor 1996). We consider three scenarios (defined by the set A = {U,M,P}), concerning whether the gene's action is independent of its parent of origin (in which case the gene can be considered ignorant of its parent of origin; A = U), whether the gene's action is conditional upon it being of maternal origin (A = M), or whether the gene's action is conditional upon it being of paternal origin (A = P). We assume separate sexes (defined by the set i = {m,f}), such that a given carrier of the gene may be female (i = f) or male (i = m). Accordingly, the appropriate measure of relative fitness is a classreproductive-value-weighted average taken across females and males, i.e. W = ½Wf + ½Wm, where Wf is the relative fitness of the female carrying the gene and Wm is the relative fitness of the male carrying the gene (Taylor 1996;Taylor & Frank 1996). The relative fitness of a female may be written as Wf (xf, yf, zf), where xf is the level of sleep of the focal female, yf is the average level of sleep of the females in the focal patch, and zf is the average level of sleep of the females in the population, with values ranging from 0 (no sleep) to 1 (sleep throughout the whole sleeping period). Similarly, the relative fitness of a male may be written as Wm (xm, ym, zm), where xm is the level of sleep of the focal male, ym is the average level of sleep of the males in the focal patch, and zm is the average level of sleep of the males in the population, with values again ranging from 0 (no sleep) to 1 (sleep throughout the whole sleeping period).
Following the approach of Taylor & Frank (1996) for a class-structured population, we may write dW/dgi|A = ½ (dWf/dgf|A) + ½ (dWm/dgm|A) = ½ ((∂Wf/∂xf)(dxf/dGf)(dGf/dgf|A) + (∂Wf/∂yf)(dyf/dGf')(dGf'/dgf|A) + (∂Wf/∂ym)(dym/dGm')(dGm'/dgf|A)) + ½ ((∂Wm/∂xm)(dxm/dGm)(dGm/dgm|A) + (∂Wm/∂ym)(dym/dGm')(dGm'/dgm|A) + (∂Wm/∂yf)(dyf/dGf')(dGf'/dgm|A)), where: Gf is the focal female's breeding value, Gf' is the average breeding value of the females in the focal patch; dxf/dGf = dyf/dGf' = γf is the mapping between genotype and phenotype in the females; dGf/dgf|A = pf|A is the consanguinity of the genic actor A in the focal female to the female herself; dGf'/dgf|A = pff|A is the consanguinity of the genic actor A in the focal female with a randomly-chosen female on her patch; dGm'/dgf = pfm|A is the consanguinity of the genic actor A in the focal female with a randomly-chosen male on her patch; Gm is the focal male's breeding value; Gm' is the average breeding value of the males in the focal patch; dxm/dGm = dym/dGm' = γm is the mapping between genotype and phenotype in the males; dGm/dgm = pm|A is the consanguinity of the genic actor A in the focal male to the male himself; dGf'/dgm = pmf|A is the consanguinity of the genic actor A in the focal male with a randomly-chose female on his patch; and dGm'/dgm = pmm|A is the consanguinity of the genic actor A in the focal male with a randomly-chosen male on his patch. The consanguinity between a genic actor A to its carrier is the same no matter the class of the genic actor A or the sex that we are considering and, therefore, pf|A = pm|A = p. We divide all terms of the right-side of the equation by p to get the kin-selection coefficient of relatedness (see below ;Bulmer 1994).
If sleep cannot evolve independently in females and males, then γf = 1 and γm = 1. In this scenario, all derivatives are evaluated at xf = xm = yf = ym = zf = zm = z. Accordingly, natural selection favours an increase in the level of sleep in females and males if: where: Cf(z) = ∂Wf/∂xf; Bff(z) = ∂Wf/∂yf; Bfm(z) = ∂Wf/∂ym; Cm(z) = ∂Wm/∂xm; Bmm(z) = ∂Wm/∂ym; Bmf(z) = ∂Wm/∂yf; rff|A = pff|A/p; rfm|A = pfm|A/p; rmm|A = pmm|A/p; and rmf|A = pmf|A/p. If sleep can evolve independently in females and males, then γf = 1 and γm = 0 when analysing sleep in females and γf = 0 and γm = 1 when analysing sleep in males. In this scenario, all derivatives are evaluated at xf = yf = zf and at xm = ym = zm. Accordingly, natural selection favours an increase in the level of sleep in females if: and an increase in the level of sleep in males if:

Inclusive fitness predictions when there are no sex differences and genes do not know their origin
We now assume that there are no sex differences and that genes are ignorant to their origin. Therefore, we can simplify the inequality (A1) to C(z) + B(z)rU > 0, where rU is the average relatedness between a genic actor ignorant to its origin in the focal individual and a random individual in her patch, C(z) is how sleep of the focal individual impacts her own fitness, and B(z) is how the sleep of the focal individual's social partners impacts the fitness of the focal individual. We define a function J(z*, rU) = C(z*) + B(z*)rU, where z* represents a sleep optimum (formally, a convergence stable strategy ;Christiansen 1991;Taylor 1996). Being a sleep optimum means that the population is at its sleep equilibrium and, therefore, J(z*, rU) = 0. To be an evolutionary stable equilibrium, it also needs to be convergent stable and the condition ∂J/∂z* < 0 needs to be met. Making those assumptions, and using the chain rule of derivation, we get dJ/drU = (∂J/∂rU) + (∂J/∂z*)(dz*/drU) = 0 and, rearranging, dz*/drU = -(∂J/∂rU)/(∂J/∂z*). Defining a function S that returns the sign (positive, negative, or zero), we obtain S(dz*/drU) = S(∂J/∂rU) = S(B(z*)) (Pen 2000;Farrell et al. 2015). Consequently, if social partners' sleep improves the individual's fitness (B > 0), then higher relatedness is associated with a higher sleep optimum (dz*/drU > 0); if social partners' sleep decreases the individual's fitness (B < 0), then higher relatedness is associated with a lower sleep optimum (dz*/drU < 0); if social partners' sleep does not affect the individual's fitness (B = 0), then higher relatedness is not associated with a higher or lower sleep optimum (dz*/drU = 0).

Inclusive fitness predictions when there are no sex differences and genes know their origin
We now assume that there are no sex differences and that genes do know their origin. Therefore, we can simplify the inequality (A1) to C(z) + B(z)rA > 0, where rA is the average relatedness between a genic actor A in the focal individual and a random individual in her patch. Following the approach from section 1.1, we get S(dz*/drA) = S(∂J/∂rA) = S(B(z*)) (Pen 2000;Farrell et al. 2015). Therefore, 1) if the sleep of social partners improves an individual's fitness (B > 0), then the sleep optimum is higher for maternal-origin genes than it is for paternal-origin genes (zM* > zP*, where zM* represents a sleep optimum for the maternal-origin genes and zP* represents a sleep optimum for the paternal-origin genes) when relatedness is higher for the former than for the latter (rM > rP, where rM represents the average relatedness between a maternal-origin gene in the focal individual and a random individual in her patch and rP represents the average relatedness between a paternal-origin gene in the focal individual and a random individual in her patch), and the sleep optimum is lower for maternal-origin genes than it is for paternal-origin genes (zM* < zP*) when relatedness is lower for the former than for the latter (rM < rP); 2) if the sleep of social partners decreases an individual's fitness (B < 0), then the sleep optimum is lower for maternal-origin genes than it is for paternal-origin genes (zM* < zP*) when relatedness is higher for the former than for the latter (rM > rP), and the sleep optimum is higher for maternal-origin genes than it is for paternal-origin genes (zM* > zP*) when relatedness is lower for the former than for the latter (rM < rP); and 3) if the sleep of social partners does not affect an individual's fitness (B = 0), then the sleep optimum for maternal-origin genes is equal to that for paternal-origin genes (zM* ≈ zP*) and relatedness does not shape the sleep optimum.

Inclusive fitness predictions when there are sex differences and genes do not know their origin
We now assume that females and males may have different relatedness values to their social partners, with the costs and benefits associated with a given sleeping schedule being the same. We also assume that genes do not know their origin. Therefore, we can simplify the inequalities (A2) and (A3) to Cf(zf) + Bff(zf)rff|U + Bmf(zf)rff|U > 0 and Cm(zm) + Bmm(zm)rmm|U + Bfm(zm)rfm|U > 0, respectively. For simplicity, we assume that Cf = Cm = C and that Bff = Bfm = Bmm = Bmf = B, meaning that the inequalities become C(zf) + B(zf)rf|U > 0 and C(zm) + B(zm)rm|U > 0, and rf|U is the average relatedness for the females in their patch and rm|U is the average relatedness for the males in their patch for a genic actor ignorant to its origin. Following the same strategy as in section 1.1, an evolutionary stable equilibrium also needs to be convergent stable and the conditions ∂J/∂zf* < 0 and ∂J/∂zm* < 0 need to be met (where zf* represents a sleep optimum for the females and zm* represents a sleep optimum for the males). Using the chain rule of derivation, we get dJ/drf|U = (∂J/∂rf|U) + (∂J/∂zf*)(dzf*/drf|U) = 0, which rearranges to dzf*/drf|U = -(∂J/∂rf|U)/(∂J/∂zf*), and dJ/drm|U = (∂J/∂rm|U) + (∂J/∂zm*)(dzm*/drm|U) = 0, which rearranges to dzm*/drm|U = -(∂J/∂rm|U)/(∂J/∂zm*). Defining a function S that returns the sign (positive, negative, or zero), we obtain S(dzf*/drf|U) (Pen 2000;Farrell et al. 2015) and the same conclusions as in section 1.1 applies.
Therefore, 1) if the sleep of social partners improves an individual's fitness (B > 0), then the sleep optimum is higher for females than it is for males (zf* > zm*) when relatedness is higher for the former than for the latter (rf|U > rm|U), and the sleep optimum is lower for females than it is for males (zf* < zm*) when relatedness is lower for the former than for the latter (rf|U < rm|U); 2) if the sleep of social partners decreases an individual's fitness (B < 0), then the sleep optimum is lower for females than it if for males (zf* < zm*) when relatedness is higher for the former than for the latter (rf|U > rm|U), and the sleep optimum is higher for females than it is for males (zf* > zm*) when relatedness is lower for the former than for the latter (rf|U < rm|U); and 3) if the sleep of social partners does not affect an individual's fitness (B = 0), then the sleep optimum for females is equal to that for males (zf* ≈ zm*) and relatedness does not shape the sleep optimum.

Illustrative model
Life cycle -We consider an infinite diploid population divided into patches (Wright 1931) containing nf females and nm males, with every female mating with every male in her patch, and vice versa. During their sleeping period, females and males spent a proportion of this time sleeping -level of sleep -which is necessary for the maintenance of the organism and to cooperate successfully with social partners in their patch. This is counterbalanced by the presence of an external danger, which deleterious effects increase with the level of sleep, and by the probability of gaining mating opportunities, which decreases with the level of sleep. Specifically, a female's fecundity is: where: m is the minimal amount of sleep that individuals require; bf defines how the benefits of sleep increase throughout the night for the females (close to 0 the benefits grow exponentially, close to 1 the benefits grow linearly); a is the probability of an external danger being present in the environment; and cf is the probability of gaining mating opportunities by sacrificing sleep in females. Therefore, ( " − ) ) * defines the maintenance of the focal female's body through sleep, ( ) defines the mating competition between the females in the group. Likewise, a male's fecundity is: where bm defines how the benefits of sleep increase throughout the night for the males (close to 0 the benefits grow exponentially, close to 1 the benefits grow linearly) and cm is the probability of gaining mating opportunities by sacrificing sleep in males. Therefore, ( 9 − ) ) -defines the maintenance of the focal male's body through sleep, and ( 345 -6 -345 -+ -) defines the mating competition between the males in the group. Following mating, each female produces a large number of offspring, with an even sex-ratio, in proportion to her fecundity. Adults then die. Juveniles then form groups -or buds -of large size at random within their patch and each group either disperse to a random patch with probability dB or remain in the focal patch otherwise (Haldane 1932). After budding dispersal, juveniles can still disperse individually, with females dispersing with probability df and males dispersing with probability dm to a random patch or else remaining in their current patch. Following individual dispersal, nf females and nm males survive at random within each patch to adulthood, returning the population to the beginning of the life cycle.
Natural selection -Female relative fitness in this model is given by: where: " = " | 6 B C+ B ; and > " = " | 6 B CD B , + B CD B , + F CD F . Likewise, male relative fitness in this model is given by: where 9 = 9 | 6 F C+ F . We can now use the inequalities derived in section 1 to reach the marginal fitness equations for the evolution of sleep and, consequently, to derive the optimal level of sleep for the different scenarios explored in the main text (see below).
Relatedness -The relatedness between a genic actor A in the focal female with a randomlychosen female in her patch (including the focal female herself) is approximately given by: where: with probability 3 J * the randomly chosen female is the focal female herself, in which case relatedness is 1; and with probability is a different female, in which case they are only related if they are both locals (1 − " ) . and, if so, their relatedness is defined by the relatedness through the genic actor A (rA) in the focal female. The approximation becomes exact in the limit of vanishingly weak selection. For the relatedness between a genic actor A in the focal female with a random male in her patch: and they are only related if they are both locals (1 -df)(1 -dm) and, if so, their relatedness is defined by the relatedness through the genic actor A (rA) in the focal female. For the relatedness between a genic actor A in the focal male and randomly-chosen male in his patch (including the focal male himself): where: with probability . and, if so, their relatedness is defined by the relatedness through the genic actor A (rA) in the focal male. For the relatedness between a genic actor A in the focal male with a random female in his patch: and they are only related if they are both locals (1 -dm)(1 -df) and, if so, their relatedness is defined by the relatedness through the genic actor A (rA) in the focal male. Note that 9"|I = "9|I and, therefore, we use "9|I to represent both throughout the rest of the document.
Relatedness through the genic actor A between two different juveniles born in the same patch is then given by rA = pA'/p, where pA' is the consanguinity through the genic actor A between two individuals born in the same patch and is defined by picking the genic actor A from the focal individual and a random gene from the other individual and calculating the probability that the two are identical by descent (Bulmer 1994).
and the relatedness between two random individuals born in the same patch is then given by rU = pU'/p (Bulmer 1994). Rearranging, we obtain: But we can also separate the consanguinity between two juveniles in their maternal-and paternal-origin components. That is: and by its turn: Relatedness between two random individuals in the same patch through their maternal-origin genes is then given by rM = pM'/p (Bulmer 1994) and through their paternal-origin genes by rP = pP'/p (Bulmer 1994). Rearranging, we obtain: .
We can replace the equations (A14; A18-19) into the equations (A8-11) to obtain the different coefficients of relatedness for genes ignorant of their origin (A = U), for maternalorigin genes (A = M), and for paternal-origin genes (A = P), respectively.

Evolution of sleep cannot evolve independently in females and males
Sentinel model -Here we explore the case where individuals can sacrifice sleep to increase the vigilance in their group but not their mating opportunities (cf = cm = 0). We assume that the benefits that females and males get throughout their sleep is similar (bf = bm = b), that sleep cannot evolve independently in females and males (γf = 1; γm = 1), and diploidy. We now can obtain the derivatives of the left side of the inequality (A1) and obtain the marginal fitness equation for the present model: We can now use the equation (A21) with the values of the main text to get the Figure 1a, with the assumption that genes are ignorant to their origin (A = U). We can also obtain Figure  S1a with that same equation, following the same assumption, but now using different values (see below). Similarly, we can obtain the values of sleep favoured by the maternal-origin genes (A = M) and by the paternal-origin genes (A = P) as shown in Figure 3a and Figure  S3a.
Reproductive model -Here we explore the case where individuals can sacrifice sleep to gain additional mating opportunities but not to increase the vigilance in their group (a = 0). We assume that females and males benefit from this strategy to the same degree (cf = cm = c). We also assume that the benefits that females and males get throughout their sleep is similar (bf = bm = b), that sleep cannot evolve independently in females and males (γf = 1; γm = 1), and diploidy. Using the same approach as above, we get the following marginal fitness equation: We can now use the equation (A23) with the values of the main text to get the Figure 1b, with the assumptions that genes are ignorant to their origin (A = U). We can also obtain Figure S1b with that same equation, following the same assumption, but now using different values (see below). As before, we can also obtain the values of sleep favoured by the maternal-origin genes (A = M) and by the paternal-origin genes (A = P) as shown in Figure  3b and Figure S3b.

Evolution of sleep can evolve independently in females and males
Sentinel model -We now assume that sleep can evolve independently in females and males.
As before, we assume that individuals can sacrifice sleep to increase the vigilance in their group but not their mating opportunities (cf = cm = 0), that the benefits that females and males get throughout their sleep is similar (bf = bm = b), that genes are ignorant to their origin (A = U), and diploidy. We now focus on females' sleep (γf = 1; γm = 0). We can obtain the derivatives of the left side of the inequality (A2) and, with that, the marginal fitness equation for the females: We now focus on males' sleep (γf = 0; γm = 1). We can obtain the derivatives of the left side of the inequality (A3) and, with that, the marginal fitness equation for the males: Now we can solve the system of equations (A24-25) to get the optimal solutions for both zf* and zm*, similar to what we did above. Those solutions can then be used to get the Figure 2a, using the values of the main text. Note, however, that using those values means that the equation (A24) is always positive and, as such, females are selected to sleep as much as possible. This result needs to be incorporated into the equation (A25) when trying to find the males' optimal level of sleep. When doing so, there are two solutions but only one makes sense given the assumptions of the model. A similar pattern is present when using the values of Figure S2a. However, after a certain value of male dispersal, the equation is no longer always positive, meaning that we can simply use the solutions of the system of equations (A24-25) to represent the optimal level of sleep for both females and males.
Reproductive model -As in the previous section, here we assume that sleep can evolve independently in females and males, but now we assume that males are the only ones that can sacrifice sleep to increase their mating opportunities (cf = 0) and that individuals do not sacrifice sleep to increase the vigilance in their group (a = 0). Once again, we assume that the benefits that females and males get throughout their sleep is similar (bf = bm = b), that genes are ignorant to their origin (A = U), and diploidy. We now focus on females' sleep (γf = 1; γm = 0). We can obtain the derivatives of the left side of the inequality (A2) and, with that, the marginal fitness equation for the females: We now focus on males' sleep (γf = 0; γm = 1). We can obtain the derivatives of the left side of the inequality (A3) and, with that, the marginal fitness equation for the males: Note that equation (A26) is always positive under the assumptions of the model. Therefore, we can simply assume that females are selected to not sacrifice any sleep. Incorporating such assumptions into equation (A27) means that two solutions are possible for zm*. Only one makes sense given the assumptions of the model and, therefore, we can use the values of the main text to get Figure 2b and, using different values (see below), Figure S2b to represent the males' optimal level of sleep.

Energy allocation
Organisms are selected to allocate energy to basic functions -growth, maintenance, reproduction -in a manner that maximizes energy use. During the waking period, organisms are selected to downregulate costly biological activities, such as the ones that allow the maintanance of the tissues. Those same activities are then upregulated during sleep, a period where energy is not being used by the individual in other activities, such as foraging or collecting environmental information.

Adaptive inactivity
Depending on their ecology, organisms will have periods of time where none of their biological requirements can be satisfied due to biotic and abiotic environmental factors. Organisms are selected to maximize the efficiency of their behaviour, therefore being selected to reduce energy use when activity can be costly and not beneficial. Memory consolidation Sleep promotes consolidation of memories acquired during the waking state into a network of long-term memories. Jenkins and Dallenbach 1924;Karni et al 1994;Maquet 2001;Stickgold 2005;Born et al 2006;Diekelmann and Born 2010.

Synaptic homeostasis
Sleep normalizes synaptic activity to normal levels after a waking period where synapses are being triggered by learning and environmental stimulus, therefore restoring neuronal selectivity and the ability to learn new memories. Tononi and Cirelli 2003;2014. Figure S1. How much an individual should sleep depends on the relatedness between the individuals in a group. When individuals sacrifice sleep in order to remain alert to dangers which may befall the group (a), individuals sacrifice more sleep when relatedness is higher, which is the case when male dispersal is lower. When individuals sacrifice sleep in order to gain an advantage over their mate competitors (b), individuals sacrifice more sleep when relatedness is lower, which is the case when male dispersal is higher. The following parameter values were used for both panels: female dispersal rate d  Figure S4. Genomic imprinting of genes responsible for level of sleep and the effects of possible disruptions. Predictions as to which gene is expressed and which gene is silent -maternal-origin gene (M, orange) or paternalorigin genes (P, blue) -when individuals are sacrificing sleep to protect the group against threats (altruism) or to gain an advantage over their mate competitors (selfishness). We consider an example for a gene that promotes sleep (promoter) and an example for a gene that inhibits sleep (inhibitor). In both cases, we assume male-biased dispersal. Note that for simplicity we assume methylation is associated with gene silencing, as is usually the case in mammals (Bird 2002). In cases where methylation is associated with gene activation the outcome for hypo-methylation is expected to be that shown here for hyper-methylation, and vice versa.