BACKGROUND
blindness has evolved repeatedly across taxa in caves, creating nearly a thousand cave-dwelling species and many more sub-populations .
surprisingly, many populations of blind individuals experience some level of immigration, which would be expected to prevent the fixation of blindness in a newly established population .
thus, blind cave-dwelling populations of typically sighted species pose an interesting challenge to our understanding of evolutionary biology.
namely, how does significant population differentiation evolve despite homogenizing immigration?
several hypotheses have been put forward to explain the evolution of blindness in cave-dwelling species.
darwin suggested that eyes would be lost by “disuse”  <cit> .
we now consider this hypothesis the “neutral-mutation hypothesis” — random mutations can accumulate in genes or regulatory regions related to sight when, as in caves, there is no purifying selection to eliminate them.
however, the accumulation of mutations causing blindness due to mutation pressure would take a long time to result in fixation of blindness in populations on its own  <cit> .
thus, genetic drift has been proposed to accelerate the evolution of blindness due to mutation pressure .
relaxing selection that maintains the eye, however, also allows for other agents of selection to act on this trait  <cit> .
the “adaptation hypothesis” suggests that there is a cost to an eye; thus, individuals without eyes have greater fitness when eyes are not helpful, resulting in the eventual elimination of seeing individuals.
this cost may either come from the energy required to develop a complex structure or due to the vulnerability of the eye .
alternatively, blindness may evolve not due to direct selection but due to selection for another beneficial trait which results in reduced eye development through pleiotropy  <cit> .
much of the work on the evolution of blindness has focused on cavefishes.
the mexican tetra , which inhabits surface waters and cave systems in mexico, is the most studied species of cave-dwelling fish.
surface and cave forms of this species are distinct, but can hybridize.
the neutral-mutation hypothesis appears to be supported in this cavefish by the observation of a high number of substitutions in putative eye genes .
in addition, no differences have been found in the survival rate for blind and seeing forms between dark and light conditions  <cit> .
the adaptation hypothesis has also been supported by work in this species.
analysis of quantitative trait loci suggests that selection has acted directly to reduce eyes in the cave populations  <cit> .
due to a high metabolic cost of developing and maintaining eye tissue  <cit> , blind fish may have been favored by low resource availability in the dark caves.
however, several lines of evidence do not support blindness evolving for energy conservation  <cit> .
an alternative hypothesis is that selection for improved feeding leads to pleiotropic eye loss without direct selection for blindness.
increased hedgehog signaling affects feeding structures, allowing for better foraging, but also causes the degeneration of eye tissue  <cit> .
while it is clear that direct or indirect selection can lead to blindness despite immigration, the level of selection required to induce blindness in cave populations has not been quantified.
here, we model the effects of migration, selection, and mutation to determine the conditions required for the evolution of blindness.
this model allows us to explore migration-selection-mutation balance.
previous theoretical work has explored this balance generally .
however, understanding the the evolution of cavefish requires application to this system specifically.
for example, in this system, unlike many examples of local adaptation related to the continent-island model, the two populations are nearly fixed for opposite conditions of a trait.
here, we address cavefish evolution specifically by allowing for new mutations and multiple loci potentially related to blindness.
we also focus on population parameters specific to cavefish.
we find that the amount of selection required to oppose the force of immigration is high, but consistent with previous work on metabolic costs in novel environments and selection in other species.
interestingly, drift only impacts blindness in the cave population in a limited range of combinations of selection, dominance, and migration.
methods and
RESULTS
assumptions
we consider two populations: surface-dwelling and cave-dwelling.
we are interested in determining when the cave population will evolve blindness, i.e.
become mostly comprised of blind individuals, as has occurred in numerous natural systems.
we first assume that the surface and cave populations do not experience drift .
additionally, immigration from the surface population into the cave affects the allele frequency in the cave population, but emigration from the cave to the surface does not affect the surface population, as we assume that the surface population is significantly larger than the cave population.
generations are discrete and non-overlapping, and mating is random.
we track a single biallelic locus, where b is the sightedness allele and where b is the blindness allele.
the frequency of b is denoted by q∈  in the surface population and q∈  in the cave population.
on the surface, we assume that blindness is strongly selected against, and q is dictated by mutation-selection balance.
these and all subsequent variables are described in table  <dig>
b
b
q
q
q
′
Δ
q
s
h
m
u
k
n
calculating the frequency of the blindness allele
within the cave, the life cycle is as follows.
embryos become juveniles and experience constant, directional selection with relative fitnesses of w
bb=1+s, w
bb=1+hs, and w
bb= <dig>  where s≥ <dig> and h∈ .
juveniles migrate into and out of the cave such that a fraction m of adults come from the surface and 1−m from the cave, where 0≤m≤ <dig>   adults generate gametes with one-way mutation, where 0≤u≤ <dig> is the probability that a functional b allele becomes a non-functional b allele.
gametes unite randomly to produce embryos.
given this life cycle, we calculate the allele frequency of the daughter generation  via standard equations:
1a qj=q2+qq2+2q+2selection
1b qa=qj+qmimmigration
1c q′=qa+umutation
analysis of the change in allele frequency
the change in allele frequency in one generation is Δ
q=q
′−q.
the first derivative of the dynamics is informative about the behavior of the model under the influence of the different parameters.
selection and mutation are directional forces, and increasing s or u increases Δ
q for 0≤q≤ <dig> .
increasing h causes selection to be more effective at low q, as rare b alleles are exposed to selection, but less effective at high q, as rare b alleles are sheltered from selection; increasing h increases Δ
q if 0<q<1+1+s− <dig> and decreases it if 1+1+s−1<q< <dig> .
migration harmonizes the allele frequency in the cave population towards the surface population allele frequency.
thus increasing m increases Δ
q for low q and decreases Δ
q for high q ≤q, where q
z is a function describing a threshold).
however, increasing q increases Δ
q for 0≤q≤ <dig> .
identifying equilibrium allele frequencies
the model we have developed is an example of migration-selection balance , extended to also include mutation.
an equilibrium exists for this model when Δ
q= <dig>  for small s, there is only one equilibrium, and it is near  <dig>  for large s, there is only one equilibrium, and it is near  <dig>  three equilibria will only exist for moderate levels of selection .
if s=m=u= <dig>  all forces of evolution are eliminated and Δ
q= <dig> for 0≤q≤ <dig>  a lower bound for any valid equilibrium is mq+um+u .
an upper bound for any equilibrium is 1−m .
furthermore, it is important to note that
<dig> q≤mq+um+u⇒q≤q^
fig.
<dig> as selection increases, the evolutionary dynamics of the cave population changes.
when s is low , there is only one equilibrium: near  <dig>  as s increases  the local maximum  increases and crosses the x-axis, producing three equilibria.
when s gets high enough , the local minimum  also crosses the x-axis, resulting in one equilibrium again.
the location of the equilibria are marked using vertical lines at the bottom of the chart.
for all curves m= <dig> , h= <dig>  u=10− <dig>  and q= <dig> .
the figure on the right is an enlarged view of a small part of the figure on the left
indicating that the equilibrium frequency in the cave population will be greater than or equal to the allele frequency in the surface population.
intuitively, this result is obvious as positive selection and one-way mutation only add to the frequency of the blindness allele in the cave.
assuming s> <dig>  the solution to Δ
q= <dig> are the roots of the following cubic polynomial
<dig> g=aq3+bq2+cq+d= <dig>
where
a=−sb=s−h))c=−+u)+sh)d=qm+u
there are three possible roots of this equation, corresponding to three possible equilibria.
depending on the parameter values, eq.
<dig> may have three real roots or one real root and two imaginary roots.
while the values of the roots of this polynomial can be expressed analytically, these equations are too complex to be helpful for understanding the system.
for simplicity, we will let q^ represent any possible equilibrium, and q^a≤q^b≤q^c, stand for the roots of eq.
<dig>
protected polymorphism
rather than tackling the equilibria directly, we first demonstrate that the cave population has a protected polymorphism.
a protected polymorphism exists if the allele frequency moves away from both fixation and extinction, i.e.
Δ
q> <dig> when q= <dig> and Δ
q< <dig> when q= <dig>  for q= <dig>  Δ
q=q m+u and q= <dig> will be an equilibrium only if qm= <dig> and u=0; otherwise Δ
q> <dig> at q= <dig>  for q= <dig> Δ
q=−m and q= <dig> will be an equilibrium if m= <dig>  q= <dig>  or u=1; otherwise Δ
q< <dig>  thus a protected polymorphism always exists except at the edge cases qm=u= <dig> m= <dig> u= <dig>  and q= <dig>  in biological terms, the cave population will be polymorphic despite directional selection for b if there is some immigration from the surface population and the surface population is polymorphic.
validity of equilibria
an equilibrium is only valid in our model if it is real and between ; otherwise, it is not biologically interpretable in this system.
because there is a protected polymorphism, there will be either  <dig> valid, stable equilibrium, or  <dig> valid equilibria in a stable-unstable-stable configuration, depending on the parameter values.
while we have not exhaustively determined the parameter ranges under which there will be only one valid equilibrium, we have determined that if h≥1/ <dig> or if h<1/ <dig> and sh>m+u1+u−m, there will be only one valid equilibrium .
we can also estimate the amount of selection required such that g= <dig> :
<dig> sq=m−uqq−qq+m−hm−−u+q
this equation is not valid for all m∈ .
if the migration rate is low, m<u, no level of selection will make q an equilibrium, as all equilibria will be greater than q. similarly, if the migration rate is high,
m>h+qh+q  no level of selection will make q an equilibrium, as all equilibria will be less than q.
dynamics and the evolution of blindness
the dynamics of the evolution of the cave population depend on the parameter values and the starting allele frequency, q
<dig>  — our model is likely well behaved, e.g.
no limit cycles or chaotic behavior, even though we provide no formal proof of this.
— if there is one equilibrium, then the frequency of b will evolve monotonically towards it, i.e.
qt→q^ as t→∞.
if there are three equilibria, then the frequency of b will evolve monotonically to q^a if q0<q^b and to q^c if q0>q^b.
when the cave population is founded, its initial allele frequency will likely match the equilibrium frequency on the surface .
because q<q^ , the allele frequency in the cave population will increase due to selection until it reaches the lowest equilibrium, i.e.
q
∞= inf{q:0≤q≤ <dig> and Δ
q=0}.
whether blindness evolves in the cave population depends on whether q
∞≥q
∗, where q
∗ is a researcher-chosen threshold for determining that the cave population is a “blind” population.
for example, q
∗= <dig>  would specify that the blindness allele is the majority allele, and q
∗= <dig>  would determine that the blindness allele is approximately fixed.
we can also focus on phenotypes, and let a=q
2+2qh measure the average blind phenotype in the cave population; then
a∞≥a∗⇒q∞≥a∗+h2−h1−2h
we define s
∗ as the minimum level of selection required for cave population to become blind, given the other parameters, i.e.
s∗=inf{s:s>0andq∞≥q∗≫q}
for simplicity, we will only consider values of q
∗ much higher than the surface allele frequency.
if there is one equilibrium, s∗=sq∗; however, if there are three equilibria, q
t will evolve to the lower equilibrium and q
∞≈q≠q
∗ .
thus selection needs to be strong enough such that there is only one equilibrium; therefore,
s∗≈infs:s>0ands≥sq∗andΔ< <dig>  where Δ is the discriminant of eq.
<dig>  figures  <dig>   <dig>  and  <dig> plot analytical solutions for s
∗ based on different thresholds.
when m≫u, the ratio s
∗/m is roughly constant such that if q
∞≥q
∗ then
<dig> s∗m≥maxq∗−qq∗q∗+h,1−6qh+2q−2q2+hq1−3h−6qh <dig>
fig.
<dig> the level of dominance of the blindness allele  affects the level of selection  required to produce blind populations.
each line represents how strong selection must be relative to migration  for blindness to evolve in the cave population for a given level of dominance , where s
∗ is the minimum level of selection required for the cave population to become blind.
regions above the curves produce populations that are blind and regions below do not.
each panel contains a different condition for defining whether the cave population is blind.
a for the blind allele to become the majority allele requires stronger selection when blindness is recessive  compared to when the allele for blindness is dominant.
b for the blind phenotype to become the majority phenotype requires stronger selection when blindness is recessive compared to when the allele for blindness is dominant.
c for the blind allele to become fixed requires stronger selection when the allele for blindness is dominant compared to when it is recessive.
d for the blind phenotype to become fixed requires stronger selection when blindness is recessive.
the curves were calculated analytically with u=10− <dig> and q= <dig>
fig.
<dig> the frequency of the blindness allele  in the surface population affects the level of selection  required to produce blind populations.
the format of this figure follows fig.
2: a shows when the blind allele becomes the majority allele; b shows when the blind phenotype becomes the majority phenotype; c shows when the blind allele becomes fixed; d shows when the blind phenotype becomes fixed.
as q increases, the amount of selection required to evolve blindness in the cave population decreases.
a surface population with a high q can be considered “pre-adapted” to the cave.
the curves were calculated analytically with u=10− <dig> and h= <dig>
fig.
<dig> the mutation rate of the blindness allele  affects the level of selection  required to produce blind populations.
the format of this figure follows fig.
<dig>  as u increases, the amount of selection required to evolve blindness in the cave population decreases for small m, and blindness will evolve regardless of selection.
the curves were calculated analytically with q= <dig>  and h= <dig>
see appendix for derivation.
the neutral-mutation hypothesis
if blindness evolves neutrally in the cave population , the equilibrium allele frequency will be governed by mutation-migration balance, i.e.
q^=mq+um+u.
similar to s
∗, we can define a critical value m
∗, such that if m<m
∗, the cave population will evolve blindness.
m∗=sup{m:q∞≥q∗≫q}=u
clearly, if u= <dig>  the cave population will not evolve blindness without the influence of selection .
however, a completely isolated cave  will evolve blindness if there is mutation .
as the migration rate increases, the equilibrium allele frequency decreases such that if m>m
∗, the cave population will not evolve blindness.
similarly, increasing the mutation rate increases m
∗, allowing blindness to evolve for higher immigration rates and demonstrating the importance of mutation to the evolution of blindness when selection is weak.
recessive blindness
if blindness is recessive , we can evaluate the dynamics of three equilibria in more detail.
first, we will simplify our model by assuming that u≪ <dig> such that 1−u≈ <dig> and
<dig> Δq∝sq21−q−m+qm+u−qm+u
weak-selection approximation
if selection is weak, then an equilibrium exists near q=q.
we use a second-order taylor series at q= <dig> to determine the upper bound on s for the presence of three equilibria .
the second-order series allows us to determine the lower two equilibrium points, although this approximation is inaccurate as q increases.
this approximation gives us
<dig> Δq≈sq2−q+mq+u
after assuming that 1−q≈ <dig>  this equation has two roots, which are the lowest two of three total equilibria,
q^a,1=m+u−2−4s2sq^b,1=m+u+2−4s2s
these two roots exist only if
<dig> 0<2−4s⇒s< <dig>
which provides us with an estimate of the upper bound on s for the presence of three equilibria.
the derivative of eq.
<dig> is dΔqdqq=2sq−, and an equilibrium will be stable if −2<dΔqdqq^< <dig>  from this, it can be easily shown that q^a, <dig> is stable and that q^b, <dig> is unstable.
strong-selection approximation
in order to determine the lower bound on s for the presence of three equilibria, we assume that selection is strong enough such that u/s≈ <dig> and q/s≈ <dig>  therefore,
<dig> Δq∝−qq2−1−mq+m/s
and the equilibria can be described as
q^a,2=0q^b,2=121−m−1−m2−4msq^c,2=121−m+1−m2−4ms
the latter two equilibria will exist only if
s>4m1−m <dig>  which provides us an estimate of the lower bound for the presence of three equilibria.
the derivative of eq.
<dig> is dΔqdqq=−3q2+2q−m/s, and it can be easily shown that q^b, <dig> is unstable and q^c, <dig> is stable.
validity of approximations
by substituting q^a, <dig> and q^b, <dig> back into eq.
<dig>  we obtain Δq=−sq^2q^−qm.
thus, Δ
q≤ <dig>  which indicates that q^a, <dig> overestimates q^a and that q^b, <dig> underestimates q^b.
by substituting q^b, <dig> and q^c, <dig> back into eq.
<dig>  we find that Δq=qm+u.
thus Δ
q≥ <dig>  which indicates that q^b, <dig> overestimates q^b and that q^c, <dig> underestimates q^c.
however, the error in our approximations is slight .
fig.
<dig> our recessive-blindness equilibria approximations are accurate.
the approximations developed in this paper  are a good fit for calculated values of selection  that result in equilibrium for a given frequency of the blindness allele  using eq.
<dig>  the dashed lines are our approximate bounds for the existence of three equilibria .
other parameters are m= <dig> , u=10− <dig>  and q= <dig>
dynamics
based on these approximations, the dynamics of the recessive-blindness system can be summarized as follows.
first, there are three possible equilibria: q^a≈q^a, <dig>  q^b∈q^b, <dig> q^b, <dig>  and q^c≈q^c, <dig>  second, there are four possible equilibria configurations:  <dig>  2a, 2b, and 2c.
case  <dig>  24<4m1−m2: only one equilibrium exists, and it is stable.
the population will always evolve towards it.
case  <dig>  4m1−m2<24: depending on the strength of s, this case may have one of three possible configurations:
case 2a, 0≤s<4m1−m2: only one equilibrium exists, q^a, and it is stable.
the population will always evolve towards it.
case 2b, 4m1−m2<s<24: all three equilibria exist; q^a and q^c are stable, while q^b is unstable.
if the population starts below q^b, it will evolve towards q^a.
if it starts above q^b, it will evolve towards q^c.
case 2c, 24<s: only one equilibrium, q^c, exists, and it is stable.
the population will always evolve towards it.
furthermore if q
0=q, the selection-threshold for blindness to be established in the cave population is
<dig> s∗≈maxm−uq∗21−q∗−m, <dig>
where q
∗ is the allele-frequency threshold.
additive blindness and multiple alleles
next we investigate a model where blindness is due to many additive  loci of small effect.
this model is motivated by the identification of  <dig> additive loci corresponding to the difference in eye phenotypes between cave and surface populations of a. mexicanus  <cit> .
first, we will make the following assumptions:  there are k unlinked loci with two alleles ,  in the cave population the fitness of an individual is 1+sx2k, where x is the number of blindness alleles the individual carries, and  m, u, q, and q
<dig> are identical at each locus.
because the forces of evolution are equivalent at every locus, they will evolve identically, and the change in allele frequency due to selection is
<dig> qj=q1+qs+s2k1+qs
see the appendix for a derivation.
next, we will simplify our model by assuming that u≪ <dig> such that 1−2k u≈ <dig> and
<dig> Δq∝−sm)q2+)−2k)q+2k
this has a single, stable, valid equilibrium:
q^=s)−2k+−u)−ms+s)2+8ksm+1)2sm+1)  which decreases a k increases.
furthermore, if m≫u
s∗m≥2kq∗
in summary, the frequency of blindness alleles will increase in the cave population until they reach equilibrium, and they will be majority alleles if s≥s
∗>4k m.
finite-population simulations
constant migration
cavefish live in small populations and strong levels of drift may play a significant role in the evolution of blindness in cave species.
to investigate the impact of drift on our recessive-blindness model, we simulated diploid populations of size n= <dig>  by modifying our life cycle  to include a finite population:
13a qj=q2+qq2+selection
13b qm=qj+qmimmigration
13c qa∼binomial/2ndrift
13d q′=qa+umutation
here the adult population consists of 2n alleles sampled with replacement from the post-immigration gene pool.
for every simulation, u=10− <dig> and q= <dig> .
these values were chosen because they are believed to be reasonable estimates, and because we previously examined the impact of varying q and u .
we further explain the impact of altering these choices in the discussion.
we set q
0=q, varied s from 10− <dig> to  <dig>  and varied m from 10− <dig> to  <dig>
we simulated  <dig> replicates for each combination of parameters; simulations were conducted for  <dig>  or  <dig> , <dig> generations.
for each set of parameters, we recorded the average q
′ frequency across these  <dig> populations at specific time points.
our simulation results for finite populations are qualitatively similar to our analytical results for infinite populations.
for high migration rates, the average allele frequency is similar to the infinite model, except that drift allows some populations that have three equilibria to evolve blindness .
however, at low migration rates , populations have low average frequency of b at  <dig>  generations, unless s> <dig>  as immigration decreased, these populations became dependent on de novo mutations to produce b, which is a slow process.
at  <dig> million generations, which is close to the estimated age of cavefish populations  <cit> , the average allele frequency is a better match to the results from the the infinite-population model ; however, it differs in two respects.
when selection is ineffective , the average allele frequency reflects mutation-migration balance.
and  when migration is low , the average allele frequency shows increased variation.
thus drift is the strongest force affecting the change in allele frequencies in the bottom left of fig.
6
c. for n= <dig> , results are qualitatively similar, but stronger selection is required to overcome the stronger effects of drift present at smaller population sizes, which most often leads to loss of the rare blindness allele.
fig.
<dig> populations evolve blindness in the face of immigration only with the help of strong selection.
a the equilibrium frequency of the blindness allele  for an infinite population, and b–d average frequencies of the allele after t generations in finite populations with either constant or episodic migration.
for each combination of selection  and migration  we conducted  <dig> replicate simulations with fixed values of the mutation rate , frequency of the blindness allele in the surface population , and q
0=q.
colors correspond to the frequency of the blindness allele for a given combination of s and m, where blue is high frequency  and red is low .
the solid white line corresponds to the degree of selection required in the infinite population  to result in q
∞> <dig>  .
the area between the solid and dashed lines corresponds to the region where three equilibria exist.
if 2n s≪ <dig>  drift is stronger than selection, and if 4n m≪ <dig>  drift is stronger than migration.
if m q≪u, mutation is the primary force introducing copies of the blindness alleles to the cave population
episodic migration
because cave and surface populations may be connected intermittently due to flooding, we simulated periods of immigration followed by periods of isolation following a first-order markov process.
the probability of switching between from isolation to immigration or vice versa was 10% in each generation.
results for the intermittently connected simulations were nearly identical to previous simulations, with the exception that at high levels of migration and selection, drift was more effective in increasing average allele frequencies .
multiple loci
to determine the effects of drift with multiple loci, we implemented the following individual-based simulation:
14a qj,i=w¯−1∑a=1nwaqa,iselection
14b qm,i=qj,i+qmimmigration
14c qu,i=qm,i+umutation
14d qa,i′∼binomial/2drift
where q
a,i is the frequency of blindness allele at the i-th locus in the a-th individual, w
a is the fitness of the a-th individual, and w¯ is the average fitness.
note that we use fecundity selection in this simulation to reduce its complexity.
simulations of  <dig> individuals were run for  <dig>  generations with u=10− <dig>  q= <dig> , and a grid of s and m values.
the number of loci was k∈{ <dig> , <dig> ,12}.
for each parameter value,  <dig> simulations were run and several summary statistics were calculated: the average frequency of blindness alleles, the average fitness, the average phenotype, and the average genetic load in the cave population.
our stochastic simulations agree with our deterministic results .
as predicted by the deterministic model, increasing the number of loci increased the amount of selection required to evolve blindness in the cave population.
this result is due to the fact that genes with smaller effect size show more genetic load due to migration of surface individuals into the cave.
even when migration was weak, a smaller effect size decreased the strength of selection relative to drift.
fig.
<dig> many loci of small effect require stronger selection than a a single locus of large effect to evolve blindness in the face of immigration.
a the equilibrium frequencies of the blindness allele for an infinite population with a single locus and b average frequencies of the allele after t generations in finite populations of size n .
c and d the impact of multiple loci  on the evolution of blindness.
colors correspond to the frequency of the blindness allele  for a given combination of selection  and migration , where blue is high frequency  and red is low .
the solid white lines correspond to s <dig> ∗ for k= <dig> or  <dig> and the dashed lines correspond to s <dig> ∗ for k=1
discussion
the evolution of blindness in caves has been hypothesized to result from relaxed selection and mutation pressure and/or positive selection for alleles that result in eye loss.
however, the degree to which these factors interact and the theoretical level of selection required to induce blindness have not been quantified previously.
here we have shown that for blindness to evolve via neutral processes, immigration must be rarer than mutation, i.e.
the cave must be almost completely isolated from the surface.
if the cave is not completely isolated from the surface, immigration of surface fish will make it nearly impossible for blindness to evolve in the cave population without strong selection favoring the trait.
when determining whether blindness evolves neutrally or adaptively, it is also important to consider how old cave populations are.
if a blind cave population is much younger than u
− <dig> generations, then it is likely that selection was influential in driving the rapid evolution of the cave population.
effects of genetic drift
cave populations are likely to be small, and intuitively genetic drift should play a role in the evolution of blindness in the cave population.
in most cases, blindness alleles are rare in the surface population, and drift is expected to lead mostly to sighted populations  <cit> .
drift also reduces the strength of natural selection such that 2ns≫ <dig> for adaptive processes to function .
therefore, smaller populations require stronger levels of selection to eliminate migration load and evolve blindness.
interestingly, drift is only essential to the evolution of blindness in the cave population in a limited range of combinations of selection and migration for which we find three equilibria.
effects of dominance
the amount of selection required for blindness to evolve depends on the migration rate and the level of dominance of the blindness allele .
for example, if q= <dig>  and h= <dig>  the amount of selection needs to be about  <dig> times the migration rate for a blind allele to become the major allele.
conversely, if h>1/ <dig>  the amount of selection only needs to be about three times the migration rate.
the situation is reversed when we look at fixation.
if h= <dig>  selection needs to be about  <dig> times the migration rate for the frequency of the blind allele to exceed 99% in the cave population, and if h= <dig>  it needs to be  <dig>  times greater than the migration rate.
if we focus on phenotypes instead, we see that dominant alleles need lower levels of positive selection to impact the population .
effects of multiple loci
increasing the number of unlinked loci underlying the blindness phenotype increases the amount of selection needed to evolve blindness.
intuitively, blindness alleles with small effect reduce migration load less effectively in the cave population.
with all things being equal, a single allele of large effect would be more likely to sweep through the cave population than multiple alleles of small effect.
however, if alleles of large effect tend to be recessive, they would be less likely to sweep than alleles of small effect that are additive.
magnitude of selection
the magnitude of selection coefficients required by our model to produce blindness given modest levels of immigration are comparable to observations in many species.
levels of selection sufficient to produce selective sweeps in wild populations range from  <dig> – <dig>  .
estimated selection coefficients for drug resistance in plasmodium falciparum were  <dig> – <dig> , leading to fixation in 20– <dig> generations  <cit> .
for a major advantageous allele, the average value of s has been estimated as  <dig>  in plants and  <dig>  in animals  <cit> .
the well-studied three-spine stickleback  exhibits similar strong selection in a novel environment.
in experiments isolating armored sticklebacks in freshwater pools, armor was lost within a few generations due to relaxed selection for defense and positive selection for the lower cost of development in unarmored fish  <cit> .
estimates of selection in this species have ranged from  <dig> – <dig>   <cit> .
the selection coefficient of a blindness allele is determined not only by its impact on the visual system, but also by any other pleiotropic effects, such as enhancement to feeding ability  <cit> .
if an allele produces multiple, adaptive phenotypes, its selection coefficient is more likely to be high enough to promote local adaptation and differentiation between cave and surface populations.
understanding astyanax mexicanus
a. mexicanus is the most well studied cave dwelling species.
this species has inhabited caves for approximately 2– <dig> million years  <cit>  or  <dig> million generations .
given this amount of time, neutral processes might explain the evolution of blindness in cave populations if they were completely isolated.
however, the cave populations are not isolated from the surface populations; they receive immigrants at a rate of 10−4– 10− <dig> per generation  <cit> .
while no fitness difference was detected in laboratory experiments  <cit> , there are  <dig> qtls identified for eye-related phenotypes, and they show a signature of selection directly favoring regressive phenotypes in cave populations  <cit> .
this selection may be due to eye development imposing a high metabolic cost, particularly for juveniles  <cit> , such that individuals with regressive eye-phenotypes require less resources.
however, the precise degree of this selection is unknown.
in this species, our additive model predicts that s would need to be about  <dig> times stronger than m for the  <dig> identified, cave-related alleles  <cit>  to become the majority alleles in cave populations.
for selection to be effective in cave populations  s would at minimum need to be between  <dig>  and  <dig> .
if we required allele frequencies to reach 90%, these estimates would be five times higher!
these estimates are high, but within the range of results found previously for selective sweeps .
we recognize that these estimates could be improved if data was available for surface allele frequencies, mutation rates, and dominance values.
these coefficients are high enough that laboratory experiments could have detected a difference between surface and cave forms; however they did not  <cit> .
one possible explanation of this discrepancy is that selection for blindness is due not to survival or reproductive success but to genotype-dependent dispersal  <cit> .
ninety years ago, lankester  <cit>  proposed that blindness evolves in caves because fish with eyes may be attracted to sources of light and preferentially leave caves.
emigration of fish from the caves to the surface is common in a. mexicanus  <cit> .
in our model, emigration of sighted individuals would act like selection, because individuals with sightedness alleles would systematically leave the cave and not contribute to the gene pool.
even a low level of preferential emigration, e.g.
2%, would provide a significant boost to local adaptation and the evolution of blindness in caves.
it is quite possible that genotype-dependent dispersal combined with lower development costs promotes the elimination of sight in caves despite the immigration of sightedness alleles from the surface.
preferential emigration would also explain the fact that in a. mexicanus only the effects of eye qtls were consistently regressive, and no other phenotypes were as consistent  <cit> .
CONCLUSIONS
we conclude that in most cases strong selection is necessary for the evolution of blind populations in caves.
this result is consistent with two different observations of cavefish:  phototactic fish may leave caves, effectively selecting for the maintenance of mostly blind fish, and  the metabolic cost of eyes is very high.
additionally, the model and results presented in this paper are applicable beyond the evolution of cave populations, expanding existing migration-selection balance theory.
we have developed approximations that allow us to understand the interaction of selection, migration, and mutation.
through simulation we have examined the effects of genetic drift relative to the model and determined that in some situations it can enhance the power of selection to drive local adaptation.
appendix
all the proofs below were validated in mathematica   <cit> .
proposition 1
if m> <dig> or u> <dig>  mq+um+u is a possible equilibrium, and there is no equilibrium less than it.
if m=u= <dig>   <dig> is an equilibrium.
proof
case  <dig>  let f=q
′−q represent the change in allele frequency over one generation .
let q~=mq+um+u.
if s= <dig> and m> <dig> , f= <dig>  and therefore q~ is an equilibrium for these parameters.
furthermore, if s≥ <dig>  f>0∀q∈ <dig> q~.
therefore, there is no equilibrium lower than q~.
case  <dig>  let m=u= <dig>  f= <dig>  □
proposition 2
1−m is a possible equilibrium, and there is no equilibrium greater than it.
proof
let q~=1−m and h= <dig>  since lims→∞f= <dig>  q~ is a potential equilibrium.
furthermore, if 0≤h≤ <dig> and s≥ <dig>  f<0∀q∈q~, <dig>  therefore, there is no equilibrium higher than q~.
□
the derivation of a tighter upper bound can be achieved by not assuming h=0; however, we do not report it at this time.
proposition 3
let s> <dig>  let m> <dig> or u> <dig>  if h≥1/ <dig> or if h<1/ <dig> and sh>m+u1+u−m, g  has exactly one root in .
proof
let m> <dig> or u> <dig>  then g<g and g≤0≤g.
by the intermediate value theorem there is at least one root in .
let s> <dig> and we will show that there is exactly one root for several cases.
case  <dig>  let 1/2<h≤ <dig>  then g< <dig> and g> <dig>  by the intermediate value theorem, g has at least one root below  <dig>  between  <dig> and  <dig>  and above  <dig>  since g is cubic, it can have at most  <dig> roots; therefore, there is exactly one root in .
case  <dig>  let h=1/ <dig>  g reduces to a quadratic equation with one root less than  <dig> and exactly one root in .
case  <dig>  let 1/3≤h<1/ <dig>  d2gdq2≤ <dig>  and g is concave in .
thus g has exactly one root in .
case  <dig>  let 0≤h<1/ <dig> and sh>m+u1+u−m.
then dgdq−∞< <dig>  dgdq0≥ <dig>  dgdq1≤ <dig>  and dgdq0>dgdq <dig>  by the intermediate value theorem, there must be a local minimum in  has exactly one root in .
□
derivation of eq.
5
in order to derive eq.
<dig> we first assume that u= <dig>  then
limm→0sq∗m,h,u,qm=q∗−qq∗q∗+h
however, we also need to determine when Δ
q has only one root.
first we approximate Δ
q by a second-order taylor series near q= <dig>
Δq≈s1−hq2+hs−mq+mq
next we find
limm→0infs:Δs,m,h,u,q<0m=1−6qh+2q−2q2+hq1−3h−6qh <dig>  where Δ is the discriminant of the taylor approximation.
equation  <dig> is the maximum of these two values.
derivation of eq.
11
to calculate the change in allele frequency due to selection, when there are many alleles of small effect, we first assume that each copy of a cave-adaptive allele adds s2k to the fitness of an individual in the cave population.
therefore, an individual who is homozygous for cave-adaptive alleles at all k loci will have a fitness of 1+s.
next we assume that the allele frequency of each locus has the same frequency, q. therefore, on average each individual in the population carries 2k q copies of cave-adaptive alleles, and the average fitness is w¯=1+=1+qs.
focusing on a specific locus, and averaging over the other k− <dig> loci, we can calculate marginal genotype fitnesses:
wbb=1+qsk−1k+skwbb=1+qsk−1k+s2kwbb=1+qsk−1k
and the marginal allele fitness for b is
wb=qwbb+wbb=1+qsk−1k+qsk+s2k=1+qs+s2k
putting this all together:
qj=qwbw¯=q1+qs+s2k1+qs
additional file
additional file  <dig> mathematica notebook verifying the analysis of our deterministic model.

 blindness has evolved repeatedly across taxa in caves, creating nearly a thousand cave-dwelling species and many more sub-populations. surprisingly, many populations of blind individuals experience some level of immigration, which would be expected to prevent the fixation of blindness in a newly established population. thus, blind cave-dwelling populations of typically sighted species pose an interesting challenge to our understanding of evolutionary biology. namely, how does significant population differentiation evolve despite homogenizing immigration? several hypotheses have been put forward to explain the evolution of blindness in cave-dwelling species. darwin suggested that eyes would be lost by “disuse”. we now consider this hypothesis the “neutral-mutation hypothesis” — random mutations can accumulate in genes or regulatory regions related to sight when, as in caves, there is no purifying selection to eliminate them. however, the accumulation of mutations causing blindness due to mutation pressure would take a long time to result in fixation of blindness in populations on its own. thus, genetic drift has been proposed to accelerate the evolution of blindness due to mutation pressure. relaxing selection that maintains the eye, however, also allows for other agents of selection to act on this trait. the “adaptation hypothesis” suggests that there is a cost to an eye; thus, individuals without eyes have greater fitness when eyes are not helpful, resulting in the eventual elimination of seeing individuals. this cost may either come from the energy required to develop a complex structure or due to the vulnerability of the eye. alternatively, blindness may evolve not due to direct selection but due to selection for another beneficial trait which results in reduced eye development through pleiotropy. much of the work on the evolution of blindness has focused on cavefishes. the mexican tetra, which inhabits surface waters and cave systems in mexico, is the most studied species of cave-dwelling fish. surface and cave forms of this species are distinct, but can hybridize. the neutral-mutation hypothesis appears to be supported in this cavefish by the observation of a high number of substitutions in putative eye genes. in addition, no differences have been found in the survival rate for blind and seeing forms between dark and light conditions. the adaptation hypothesis has also been supported by work in this species. analysis of quantitative trait loci suggests that selection has acted directly to reduce eyes in the cave populations. due to a high metabolic cost of developing and maintaining eye tissue, blind fish may have been favored by low resource availability in the dark caves. however, several lines of evidence do not support blindness evolving for energy conservation. an alternative hypothesis is that selection for improved feeding leads to pleiotropic eye loss without direct selection for blindness. increased hedgehog signaling affects feeding structures, allowing for better foraging, but also causes the degeneration of eye tissue. while it is clear that direct or indirect selection can lead to blindness despite immigration, the level of selection required to induce blindness in cave populations has not been quantified. here, we model the effects of migration, selection, and mutation to determine the conditions required for the evolution of blindness. this model allows us to explore migration-selection-mutation balance. previous theoretical work has explored this balance generally. however, understanding the the evolution of cavefish requires application to this system specifically. for example, in this system, unlike many examples of local adaptation related to the continent-island model, the two populations are nearly fixed for opposite conditions of a trait. here, we address cavefish evolution specifically by allowing for new mutations and multiple loci potentially related to blindness. we also focus on population parameters specific to cavefish. we find that the amount of selection required to oppose the force of immigration is high, but consistent with previous work on metabolic costs in novel environments and selection in other species. interestingly, drift only impacts blindness in the cave population in a limited range of combinations of selection, dominance, and migration. methods and  assumptions we consider two populations: surface-dwelling and cave-dwelling. we are interested in determining when the cave population will evolve blindness, i.e. become mostly comprised of blind individuals, as has occurred in numerous natural systems. we first assume that the surface and cave populations do not experience drift. additionally, immigration from the surface population into the cave affects the allele frequency in the cave population, but emigration from the cave to the surface does not affect the surface population, as we assume that the surface population is significantly larger than the cave population. generations are discrete and non-overlapping, and mating is random. we track a single biallelic locus, where b is the sightedness allele and where b is the blindness allele. the frequency of b is denoted by q∈ in the surface population and q∈ in the cave population. on the surface, we assume that blindness is strongly selected against, and q is dictated by mutation-selection balance. these and all subsequent variables are described in table b b q q q ′ Δ q s h m u k n calculating the frequency of the blindness allele within the cave, the life cycle is as follows. embryos become juveniles and experience constant, directional selection with relative fitnesses of w bb=1+s, w bb=1+hs, and w bb= where s≥ and h∈. juveniles migrate into and out of the cave such that a fraction m of adults come from the surface and 1−m from the cave, where 0≤m≤ adults generate gametes with one-way mutation, where 0≤u≤ is the probability that a functional b allele becomes a non-functional b allele. gametes unite randomly to produce embryos. given this life cycle, we calculate the allele frequency of the daughter generation via standard equations: 1a qj=q2+qq2+2q+2selection 1b qa=qj+qmimmigration 1c q′=qa+umutation analysis of the change in allele frequency the change in allele frequency in one generation is Δ q=q ′−q. the first derivative of the dynamics is informative about the behavior of the model under the influence of the different parameters. selection and mutation are directional forces, and increasing s or u increases Δ q for 0≤q≤. increasing h causes selection to be more effective at low q, as rare b alleles are exposed to selection, but less effective at high q, as rare b alleles are sheltered from selection; increasing h increases Δ q if 0<q<1+1+s− and decreases it if 1+1+s−1<q<. migration harmonizes the allele frequency in the cave population towards the surface population allele frequency. thus increasing m increases Δ q for low q and decreases Δ q for high q ≤q, where q z is a function describing a threshold). however, increasing q increases Δ q for 0≤q≤. identifying equilibrium allele frequencies the model we have developed is an example of migration-selection balance, extended to also include mutation. an equilibrium exists for this model when Δ q= for small s, there is only one equilibrium, and it is near for large s, there is only one equilibrium, and it is near three equilibria will only exist for moderate levels of selection. if s=m=u= all forces of evolution are eliminated and Δ q= for 0≤q≤ a lower bound for any valid equilibrium is mq+um+u. an upper bound for any equilibrium is 1−m. furthermore, it is important to note that q≤mq+um+u⇒q≤q^ fig. as selection increases, the evolutionary dynamics of the cave population changes. when s is low, there is only one equilibrium: near as s increases the local maximum increases and crosses the x-axis, producing three equilibria. when s gets high enough, the local minimum also crosses the x-axis, resulting in one equilibrium again. the location of the equilibria are marked using vertical lines at the bottom of the chart. for all curves m=, h= u=10− and q=. the figure on the right is an enlarged view of a small part of the figure on the left indicating that the equilibrium frequency in the cave population will be greater than or equal to the allele frequency in the surface population. intuitively, this result is obvious as positive selection and one-way mutation only add to the frequency of the blindness allele in the cave. assuming s> the solution to Δ q= are the roots of the following cubic polynomial g=aq3+bq2+cq+d= where a=−sb=s−h))c=−+u)+sh)d=qm+u there are three possible roots of this equation, corresponding to three possible equilibria. depending on the parameter values, eq. may have three real roots or one real root and two imaginary roots. while the values of the roots of this polynomial can be expressed analytically, these equations are too complex to be helpful for understanding the system. for simplicity, we will let q^ represent any possible equilibrium, and q^a≤q^b≤q^c, stand for the roots of eq. protected polymorphism rather than tackling the equilibria directly, we first demonstrate that the cave population has a protected polymorphism. a protected polymorphism exists if the allele frequency moves away from both fixation and extinction, i.e. Δ q> when q= and Δ q< when q= for q= Δ q=q m+u and q= will be an equilibrium only if qm= and u=0; otherwise Δ q> at q= for q= Δ q=−m and q= will be an equilibrium if m= q= or u=1; otherwise Δ q< thus a protected polymorphism always exists except at the edge cases qm=u= m= u= and q= in biological terms, the cave population will be polymorphic despite directional selection for b if there is some immigration from the surface population and the surface population is polymorphic. validity of equilibria an equilibrium is only valid in our model if it is real and between; otherwise, it is not biologically interpretable in this system. because there is a protected polymorphism, there will be either valid, stable equilibrium, or valid equilibria in a stable-unstable-stable configuration, depending on the parameter values. while we have not exhaustively determined the parameter ranges under which there will be only one valid equilibrium, we have determined that if h≥1/ or if h<1/ and sh>m+u1+u−m, there will be only one valid equilibrium. we can also estimate the amount of selection required such that g=: sq=m−uqq−qq+m−hm−−u+q this equation is not valid for all m∈. if the migration rate is low, m<u, no level of selection will make q an equilibrium, as all equilibria will be greater than q. similarly, if the migration rate is high, m>h+qh+q no level of selection will make q an equilibrium, as all equilibria will be less than q. dynamics and the evolution of blindness the dynamics of the evolution of the cave population depend on the parameter values and the starting allele frequency, q — our model is likely well behaved, e.g. no limit cycles or chaotic behavior, even though we provide no formal proof of this. — if there is one equilibrium, then the frequency of b will evolve monotonically towards it, i.e. qt→q^ as t→∞. if there are three equilibria, then the frequency of b will evolve monotonically to q^a if q0<q^b and to q^c if q0>q^b. when the cave population is founded, its initial allele frequency will likely match the equilibrium frequency on the surface. because q<q^, the allele frequency in the cave population will increase due to selection until it reaches the lowest equilibrium, i.e. q ∞= inf{q:0≤q≤ and Δ q=0}. whether blindness evolves in the cave population depends on whether q ∞≥q ∗, where q ∗ is a researcher-chosen threshold for determining that the cave population is a “blind” population. for example, q ∗= would specify that the blindness allele is the majority allele, and q ∗= would determine that the blindness allele is approximately fixed. we can also focus on phenotypes, and let a=q 2+2qh measure the average blind phenotype in the cave population; then a∞≥a∗⇒q∞≥a∗+h2−h1−2h we define s ∗ as the minimum level of selection required for cave population to become blind, given the other parameters, i.e. s∗=inf{s:s>0andq∞≥q∗≫q} for simplicity, we will only consider values of q ∗ much higher than the surface allele frequency. if there is one equilibrium, s∗=sq∗; however, if there are three equilibria, q t will evolve to the lower equilibrium and q ∞≈q≠q ∗. thus selection needs to be strong enough such that there is only one equilibrium; therefore, s∗≈infs:s>0ands≥sq∗andΔ< where Δ is the discriminant of eq. figures and plot analytical solutions for s ∗ based on different thresholds. when m≫u, the ratio s ∗/m is roughly constant such that if q ∞≥q ∗ then s∗m≥maxq∗−qq∗q∗+h,1−6qh+2q−2q2+hq1−3h−6qh fig. the level of dominance of the blindness allele affects the level of selection required to produce blind populations. each line represents how strong selection must be relative to migration for blindness to evolve in the cave population for a given level of dominance, where s ∗ is the minimum level of selection required for the cave population to become blind. regions above the curves produce populations that are blind and regions below do not. each panel contains a different condition for defining whether the cave population is blind. a for the blind allele to become the majority allele requires stronger selection when blindness is recessive compared to when the allele for blindness is dominant. b for the blind phenotype to become the majority phenotype requires stronger selection when blindness is recessive compared to when the allele for blindness is dominant. c for the blind allele to become fixed requires stronger selection when the allele for blindness is dominant compared to when it is recessive. d for the blind phenotype to become fixed requires stronger selection when blindness is recessive. the curves were calculated analytically with u=10− and q= fig. the frequency of the blindness allele in the surface population affects the level of selection required to produce blind populations. the format of this figure follows fig. 2: a shows when the blind allele becomes the majority allele; b shows when the blind phenotype becomes the majority phenotype; c shows when the blind allele becomes fixed; d shows when the blind phenotype becomes fixed. as q increases, the amount of selection required to evolve blindness in the cave population decreases. a surface population with a high q can be considered “pre-adapted” to the cave. the curves were calculated analytically with u=10− and h= fig. the mutation rate of the blindness allele affects the level of selection required to produce blind populations. the format of this figure follows fig. as u increases, the amount of selection required to evolve blindness in the cave population decreases for small m, and blindness will evolve regardless of selection. the curves were calculated analytically with q= and h= see appendix for derivation. the neutral-mutation hypothesis if blindness evolves neutrally in the cave population, the equilibrium allele frequency will be governed by mutation-migration balance, i.e. q^=mq+um+u. similar to s ∗, we can define a critical value m ∗, such that if m<m ∗, the cave population will evolve blindness. m∗=sup{m:q∞≥q∗≫q}=u clearly, if u= the cave population will not evolve blindness without the influence of selection. however, a completely isolated cave will evolve blindness if there is mutation. as the migration rate increases, the equilibrium allele frequency decreases such that if m>m ∗, the cave population will not evolve blindness. similarly, increasing the mutation rate increases m ∗, allowing blindness to evolve for higher immigration rates and demonstrating the importance of mutation to the evolution of blindness when selection is weak. recessive blindness if blindness is recessive, we can evaluate the dynamics of three equilibria in more detail. first, we will simplify our model by assuming that u≪ such that 1−u≈ and Δq∝sq21−q−m+qm+u−qm+u weak-selection approximation if selection is weak, then an equilibrium exists near q=q. we use a second-order taylor series at q= to determine the upper bound on s for the presence of three equilibria. the second-order series allows us to determine the lower two equilibrium points, although this approximation is inaccurate as q increases. this approximation gives us Δq≈sq2−q+mq+u after assuming that 1−q≈ this equation has two roots, which are the lowest two of three total equilibria, q^a,1=m+u−2−4s2sq^b,1=m+u+2−4s2s these two roots exist only if 0<2−4s⇒s< which provides us with an estimate of the upper bound on s for the presence of three equilibria. the derivative of eq. is dΔqdqq=2sq−, and an equilibrium will be stable if −2<dΔqdqq^< from this, it can be easily shown that q^a, is stable and that q^b, is unstable. strong-selection approximation in order to determine the lower bound on s for the presence of three equilibria, we assume that selection is strong enough such that u/s≈ and q/s≈ therefore, Δq∝−qq2−1−mq+m/s and the equilibria can be described as q^a,2=0q^b,2=121−m−1−m2−4msq^c,2=121−m+1−m2−4ms the latter two equilibria will exist only if s>4m1−m which provides us an estimate of the lower bound for the presence of three equilibria. the derivative of eq. is dΔqdqq=−3q2+2q−m/s, and it can be easily shown that q^b, is unstable and q^c, is stable. validity of approximations by substituting q^a, and q^b, back into eq. we obtain Δq=−sq^2q^−qm. thus, Δ q≤ which indicates that q^a, overestimates q^a and that q^b, underestimates q^b. by substituting q^b, and q^c, back into eq. we find that Δq=qm+u. thus Δ q≥ which indicates that q^b, overestimates q^b and that q^c, underestimates q^c. however, the error in our approximations is slight. fig. our recessive-blindness equilibria approximations are accurate. the approximations developed in this paper are a good fit for calculated values of selection that result in equilibrium for a given frequency of the blindness allele using eq. the dashed lines are our approximate bounds for the existence of three equilibria. other parameters are m=, u=10− and q= dynamics based on these approximations, the dynamics of the recessive-blindness system can be summarized as follows. first, there are three possible equilibria: q^a≈q^a, q^b∈q^b, q^b, and q^c≈q^c, second, there are four possible equilibria configurations: 2a, 2b, and 2c. case 24<4m1−m2: only one equilibrium exists, and it is stable. the population will always evolve towards it. case 4m1−m2<24: depending on the strength of s, this case may have one of three possible configurations: case 2a, 0≤s<4m1−m2: only one equilibrium exists, q^a, and it is stable. the population will always evolve towards it. case 2b, 4m1−m2<s<24: all three equilibria exist; q^a and q^c are stable, while q^b is unstable. if the population starts below q^b, it will evolve towards q^a. if it starts above q^b, it will evolve towards q^c. case 2c, 24<s: only one equilibrium, q^c, exists, and it is stable. the population will always evolve towards it. furthermore if q 0=q, the selection-threshold for blindness to be established in the cave population is s∗≈maxm−uq∗21−q∗−m, where q ∗ is the allele-frequency threshold. additive blindness and multiple alleles next we investigate a model where blindness is due to many additive loci of small effect. this model is motivated by the identification of additive loci corresponding to the difference in eye phenotypes between cave and surface populations of a. mexicanus. first, we will make the following assumptions: there are k unlinked loci with two alleles, in the cave population the fitness of an individual is 1+sx2k, where x is the number of blindness alleles the individual carries, and m, u, q, and q are identical at each locus. because the forces of evolution are equivalent at every locus, they will evolve identically, and the change in allele frequency due to selection is qj=q1+qs+s2k1+qs see the appendix for a derivation. next, we will simplify our model by assuming that u≪ such that 1−2k u≈ and Δq∝−sm)q2+)−2k)q+2k this has a single, stable, valid equilibrium: q^=s)−2k+−u)−ms+s)2+8ksm+1)2sm+1) which decreases a k increases. furthermore, if m≫u s∗m≥2kq∗ in summary, the frequency of blindness alleles will increase in the cave population until they reach equilibrium, and they will be majority alleles if s≥s ∗>4k m. finite-population simulations constant migration cavefish live in small populations and strong levels of drift may play a significant role in the evolution of blindness in cave species. to investigate the impact of drift on our recessive-blindness model, we simulated diploid populations of size n= by modifying our life cycle to include a finite population: 13a qj=q2+qq2+selection 13b qm=qj+qmimmigration 13c qa∼binomial/2ndrift 13d q′=qa+umutation here the adult population consists of 2n alleles sampled with replacement from the post-immigration gene pool. for every simulation, u=10− and q=. these values were chosen because they are believed to be reasonable estimates, and because we previously examined the impact of varying q and u. we further explain the impact of altering these choices in the discussion. we set q 0=q, varied s from 10− to and varied m from 10− to we simulated replicates for each combination of parameters; simulations were conducted for or, generations. for each set of parameters, we recorded the average q ′ frequency across these populations at specific time points. our simulation results for finite populations are qualitatively similar to our analytical results for infinite populations. for high migration rates, the average allele frequency is similar to the infinite model, except that drift allows some populations that have three equilibria to evolve blindness. however, at low migration rates, populations have low average frequency of b at generations, unless s> as immigration decreased, these populations became dependent on de novo mutations to produce b, which is a slow process. at million generations, which is close to the estimated age of cavefish populations, the average allele frequency is a better match to the results from the the infinite-population model; however, it differs in two respects. when selection is ineffective, the average allele frequency reflects mutation-migration balance. and when migration is low, the average allele frequency shows increased variation. thus drift is the strongest force affecting the change in allele frequencies in the bottom left of fig. 6 c. for n=, results are qualitatively similar, but stronger selection is required to overcome the stronger effects of drift present at smaller population sizes, which most often leads to loss of the rare blindness allele. fig. populations evolve blindness in the face of immigration only with the help of strong selection. a the equilibrium frequency of the blindness allele for an infinite population, and b–d average frequencies of the allele after t generations in finite populations with either constant or episodic migration. for each combination of selection and migration we conducted replicate simulations with fixed values of the mutation rate, frequency of the blindness allele in the surface population, and q 0=q. colors correspond to the frequency of the blindness allele for a given combination of s and m, where blue is high frequency and red is low. the solid white line corresponds to the degree of selection required in the infinite population to result in q ∞>. the area between the solid and dashed lines corresponds to the region where three equilibria exist. if 2n s≪ drift is stronger than selection, and if 4n m≪ drift is stronger than migration. if m q≪u, mutation is the primary force introducing copies of the blindness alleles to the cave population episodic migration because cave and surface populations may be connected intermittently due to flooding, we simulated periods of immigration followed by periods of isolation following a first-order markov process. the probability of switching between from isolation to immigration or vice versa was 10% in each generation. results for the intermittently connected simulations were nearly identical to previous simulations, with the exception that at high levels of migration and selection, drift was more effective in increasing average allele frequencies. multiple loci to determine the effects of drift with multiple loci, we implemented the following individual-based simulation: 14a qj,i=w¯−1∑a=1nwaqa,iselection 14b qm,i=qj,i+qmimmigration 14c qu,i=qm,i+umutation 14d qa,i′∼binomial/2drift where q a,i is the frequency of blindness allele at the i-th locus in the a-th individual, w a is the fitness of the a-th individual, and w¯ is the average fitness. note that we use fecundity selection in this simulation to reduce its complexity. simulations of individuals were run for generations with u=10− q=, and a grid of s and m values. the number of loci was k∈{,,12}. for each parameter value, simulations were run and several summary statistics were calculated: the average frequency of blindness alleles, the average fitness, the average phenotype, and the average genetic load in the cave population. our stochastic simulations agree with our deterministic results. as predicted by the deterministic model, increasing the number of loci increased the amount of selection required to evolve blindness in the cave population. this result is due to the fact that genes with smaller effect size show more genetic load due to migration of surface individuals into the cave. even when migration was weak, a smaller effect size decreased the strength of selection relative to drift. fig. many loci of small effect require stronger selection than a a single locus of large effect to evolve blindness in the face of immigration. a the equilibrium frequencies of the blindness allele for an infinite population with a single locus and b average frequencies of the allele after t generations in finite populations of size n. c and d the impact of multiple loci on the evolution of blindness. colors correspond to the frequency of the blindness allele for a given combination of selection and migration, where blue is high frequency and red is low. the solid white lines correspond to s ∗ for k= or and the dashed lines correspond to s ∗ for k=1 discussion the evolution of blindness in caves has been hypothesized to result from relaxed selection and mutation pressure and/or positive selection for alleles that result in eye loss. however, the degree to which these factors interact and the theoretical level of selection required to induce blindness have not been quantified previously. here we have shown that for blindness to evolve via neutral processes, immigration must be rarer than mutation, i.e. the cave must be almost completely isolated from the surface. if the cave is not completely isolated from the surface, immigration of surface fish will make it nearly impossible for blindness to evolve in the cave population without strong selection favoring the trait. when determining whether blindness evolves neutrally or adaptively, it is also important to consider how old cave populations are. if a blind cave population is much younger than u − generations, then it is likely that selection was influential in driving the rapid evolution of the cave population. effects of genetic drift cave populations are likely to be small, and intuitively genetic drift should play a role in the evolution of blindness in the cave population. in most cases, blindness alleles are rare in the surface population, and drift is expected to lead mostly to sighted populations. drift also reduces the strength of natural selection such that 2ns≫ for adaptive processes to function. therefore, smaller populations require stronger levels of selection to eliminate migration load and evolve blindness. interestingly, drift is only essential to the evolution of blindness in the cave population in a limited range of combinations of selection and migration for which we find three equilibria. effects of dominance the amount of selection required for blindness to evolve depends on the migration rate and the level of dominance of the blindness allele. for example, if q= and h= the amount of selection needs to be about times the migration rate for a blind allele to become the major allele. conversely, if h>1/ the amount of selection only needs to be about three times the migration rate. the situation is reversed when we look at fixation. if h= selection needs to be about times the migration rate for the frequency of the blind allele to exceed 99% in the cave population, and if h= it needs to be times greater than the migration rate. if we focus on phenotypes instead, we see that dominant alleles need lower levels of positive selection to impact the population. effects of multiple loci increasing the number of unlinked loci underlying the blindness phenotype increases the amount of selection needed to evolve blindness. intuitively, blindness alleles with small effect reduce migration load less effectively in the cave population. with all things being equal, a single allele of large effect would be more likely to sweep through the cave population than multiple alleles of small effect. however, if alleles of large effect tend to be recessive, they would be less likely to sweep than alleles of small effect that are additive. magnitude of selection the magnitude of selection coefficients required by our model to produce blindness given modest levels of immigration are comparable to observations in many species. levels of selection sufficient to produce selective sweeps in wild populations range from –. estimated selection coefficients for drug resistance in plasmodium falciparum were –, leading to fixation in 20– generations. for a major advantageous allele, the average value of s has been estimated as in plants and in animals. the well-studied three-spine stickleback exhibits similar strong selection in a novel environment. in experiments isolating armored sticklebacks in freshwater pools, armor was lost within a few generations due to relaxed selection for defense and positive selection for the lower cost of development in unarmored fish. estimates of selection in this species have ranged from –. the selection coefficient of a blindness allele is determined not only by its impact on the visual system, but also by any other pleiotropic effects, such as enhancement to feeding ability. if an allele produces multiple, adaptive phenotypes, its selection coefficient is more likely to be high enough to promote local adaptation and differentiation between cave and surface populations. understanding astyanax mexicanus a. mexicanus is the most well studied cave dwelling species. this species has inhabited caves for approximately 2– million years or million generations. given this amount of time, neutral processes might explain the evolution of blindness in cave populations if they were completely isolated. however, the cave populations are not isolated from the surface populations; they receive immigrants at a rate of 10−4– 10− per generation. while no fitness difference was detected in laboratory experiments, there are qtls identified for eye-related phenotypes, and they show a signature of selection directly favoring regressive phenotypes in cave populations. this selection may be due to eye development imposing a high metabolic cost, particularly for juveniles, such that individuals with regressive eye-phenotypes require less resources. however, the precise degree of this selection is unknown. in this species, our additive model predicts that s would need to be about times stronger than m for the identified, cave-related alleles to become the majority alleles in cave populations. for selection to be effective in cave populations s would at minimum need to be between and. if we required allele frequencies to reach 90%, these estimates would be five times higher! these estimates are high, but within the range of results found previously for selective sweeps. we recognize that these estimates could be improved if data was available for surface allele frequencies, mutation rates, and dominance values. these coefficients are high enough that laboratory experiments could have detected a difference between surface and cave forms; however they did not. one possible explanation of this discrepancy is that selection for blindness is due not to survival or reproductive success but to genotype-dependent dispersal. ninety years ago, lankester proposed that blindness evolves in caves because fish with eyes may be attracted to sources of light and preferentially leave caves. emigration of fish from the caves to the surface is common in a. mexicanus. in our model, emigration of sighted individuals would act like selection, because individuals with sightedness alleles would systematically leave the cave and not contribute to the gene pool. even a low level of preferential emigration, e.g. 2%, would provide a significant boost to local adaptation and the evolution of blindness in caves. it is quite possible that genotype-dependent dispersal combined with lower development costs promotes the elimination of sight in caves despite the immigration of sightedness alleles from the surface. preferential emigration would also explain the fact that in a. mexicanus only the effects of eye qtls were consistently regressive, and no other phenotypes were as consistent.  we conclude that in most cases strong selection is necessary for the evolution of blind populations in caves. this result is consistent with two different observations of cavefish: phototactic fish may leave caves, effectively selecting for the maintenance of mostly blind fish, and the metabolic cost of eyes is very high. additionally, the model and results presented in this paper are applicable beyond the evolution of cave populations, expanding existing migration-selection balance theory. we have developed approximations that allow us to understand the interaction of selection, migration, and mutation. through simulation we have examined the effects of genetic drift relative to the model and determined that in some situations it can enhance the power of selection to drive local adaptation. appendix all the proofs below were validated in mathematica. proposition 1 if m> or u> mq+um+u is a possible equilibrium, and there is no equilibrium less than it. if m=u= is an equilibrium. proof case let f=q ′−q represent the change in allele frequency over one generation. let q~=mq+um+u. if s= and m>, f= and therefore q~ is an equilibrium for these parameters. furthermore, if s≥ f>0∀q∈ q~. therefore, there is no equilibrium lower than q~. case let m=u= f= □ proposition 2 1−m is a possible equilibrium, and there is no equilibrium greater than it. proof let q~=1−m and h= since lims→∞f= q~ is a potential equilibrium. furthermore, if 0≤h≤ and s≥ f<0∀q∈q~, therefore, there is no equilibrium higher than q~. □ the derivation of a tighter upper bound can be achieved by not assuming h=0; however, we do not report it at this time. proposition 3 let s> let m> or u> if h≥1/ or if h<1/ and sh>m+u1+u−m, g has exactly one root in. proof let m> or u> then g<g and g≤0≤g. by the intermediate value theorem there is at least one root in. let s> and we will show that there is exactly one root for several cases. case let 1/2<h≤ then g< and g> by the intermediate value theorem, g has at least one root below between and and above since g is cubic, it can have at most roots; therefore, there is exactly one root in. case let h=1/ g reduces to a quadratic equation with one root less than and exactly one root in. case let 1/3≤h<1/ d2gdq2≤ and g is concave in. thus g has exactly one root in. case let 0≤h<1/ and sh>m+u1+u−m. then dgdq−∞< dgdq0≥ dgdq1≤ and dgdq0>dgdq by the intermediate value theorem, there must be a local minimum in has exactly one root in. □ derivation of eq. 5 in order to derive eq. we first assume that u= then limm→0sq∗m,h,u,qm=q∗−qq∗q∗+h however, we also need to determine when Δ q has only one root. first we approximate Δ q by a second-order taylor series near q= Δq≈s1−hq2+hs−mq+mq next we find limm→0infs:Δs,m,h,u,q<0m=1−6qh+2q−2q2+hq1−3h−6qh where Δ is the discriminant of the taylor approximation. equation is the maximum of these two values. derivation of eq. 11 to calculate the change in allele frequency due to selection, when there are many alleles of small effect, we first assume that each copy of a cave-adaptive allele adds s2k to the fitness of an individual in the cave population. therefore, an individual who is homozygous for cave-adaptive alleles at all k loci will have a fitness of 1+s. next we assume that the allele frequency of each locus has the same frequency, q. therefore, on average each individual in the population carries 2k q copies of cave-adaptive alleles, and the average fitness is w¯=1+=1+qs. focusing on a specific locus, and averaging over the other k− loci, we can calculate marginal genotype fitnesses: wbb=1+qsk−1k+skwbb=1+qsk−1k+s2kwbb=1+qsk−1k and the marginal allele fitness for b is wb=qwbb+wbb=1+qsk−1k+qsk+s2k=1+qs+s2k putting this all together: qj=qwbw¯=q1+qs+s2k1+qs additional file additional file mathematica notebook verifying the analysis of our deterministic model. 
we estimate that the selection coefficient for blindness would need to be at least for blindness to evolve in the model cave-organism, astyanax mexicanus. our results indicate that strong selection is required for the evolution of blindness in cave-dwelling organisms, which is consistent with recent work suggesting a high metabolic cost of eye development. this model captures the interaction of three forces: selection favoring alleles causing blindness, immigration of sightedness alleles from a surface population, and mutations creating blindness alleles. we investigated the dynamics of this model and determined selection-strength thresholds that result in blindness evolving in caves despite immigration of sightedness alleles from the surface.
 blindness has evolved repeatedly in cave-dwelling organisms, and many hypotheses have been proposed to explain this observation, including both accumulation of neutral loss-of-function mutations and adaptation to darkness. investigating the loss of sight in cave dwellers presents an opportunity to understand the operation of fundamental evolutionary processes, including drift, selection, mutation, and migration.  here we model the evolution of blindness in caves. this model captures the interaction of three forces: selection favoring alleles causing blindness, immigration of sightedness alleles from a surface population, and mutations creating blindness alleles. we investigated the dynamics of this model and determined selection-strength thresholds that result in blindness evolving in caves despite immigration of sightedness alleles from the surface. we estimate that the selection coefficient for blindness would need to be at least for blindness to evolve in the model cave-organism, astyanax mexicanus.  our results indicate that strong selection is required for the evolution of blindness in cave-dwelling organisms, which is consistent with recent work suggesting a high metabolic cost of eye development. electronic supplementary material the online version of this article contains supplementary material, which is available to authorized users. keywords migration-selection balancepopulation geneticsmodels/simulationsmutationshttp://dx.doi.org/ /100006755school of life sciences, arizona state universitybarrett honors college, arizona state universityhttp://dx.doi.org/ /100000153division of biological infrastructuredbi-1356548cartwright reed a. issue-copyright-statement© the author 2017 