Download Evolution of Sex

Models of Sexual Reproduction in a
Changing Environment
A Short Review
Under supervision of Dr. Lilach Hadany, Department of Plant
Sciences, Faculty of Life Sciences, Tel Aviv University
Amram Shay
18/10/2009
Introduction
Sexual reproduction seems to inhabit all living realms to various degrees thus many
species exist that are mandatory sexual, facultative sexual (such as hermaphrodites) or even
parthenogenesis.
The Evolution of Sex has been and still is a subject for
much debate among researchers of evolution. Many openended questions regarding to the evolution of sex deal with the
process (and conditions) under which sex may develop, or the
stability of the sexual process once it has been acquired. In
Fig 1. The Twofold cost of sex. Rate of
reproduction for sexuals (a) and asexuals (b)
other words – what is the evolutionary advantage of sex in comparison with the asexual
reproductive process?
Often it is cited that in order for sexual reproduction to succeed (and therefore rise in
frequency in population of asexuals) it has to overcome a "twofold cost of sex". This term
refers to the fact that asexual reproduction can multiply at a rate of 2n while the sexuals suffer
from a cost of having only half the population producing offsprings each generation
(Fig 1). This means that in a competition between asexual population and a sexual mutant
appearing in the population (with everything else being equal), the asexuals would reproduce
twice as much and therefore should replace that sexual deviant. As it is common in
evolutionary studies to consider the fitness of an organism as proportional to the amount of
offsprings produced, it may be concluded that asexuals would appear to have higher fitness
than their sexual counterparts. To this twofold cost we may add the "hidden" costs of finding
mate, sexual selection (working differently on males and females) and other costs
incorporated in being sexual. So that in order for sexual reproduction to prosper it has to be at
least twice as successful, in terms fitness, to asexual reproduction and yet sexual
reproduction is common and ubiquitous in contrast with the seemingly heavy cost. Thus the
question of sex ensues: What are the advantages of sex that outweigh its cost?
Many arguments have been proposed to facilitate for this alleged advantage of sex,
one of which is the Red-Queen hypothesis (4). This argument suggests that environment
includes many parasite organisms with life expectancy much shorter than its host, so that
under these conditions parasites adapt much faster to resistance managed by the host's
genome. The two species has to continue co-evolving and it has been suggested that sexual
population has an advantage over asexual population in resisting parasites. Some other
arguments that have been suggested are concerned with increased variance of sexuals.
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The subject and theories concerning the evolution of sex are wide and definitely
beyond the scope of this work, thus, I will limit myself to discussing few models of the
advantage of sexual reproduction in a changing environment, one of which is the
aforementioned Red Queen hypothesis suggested by Van Valen, and two others are models
of a more statistical approach concerning with genetic variance in the population.
Evolution of sex may have been influenced by many effects and forces. Some may be
yet undiscovered mechanisms of evolution while others may have to be further investigated
in order to understand their exact weight and importance.
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Why Changing Environment and Sex?
Changed environment would require the organism to adapt to the new conditions.
Many such adaptations are genetic in origin thus it is intuitively understood that an
environmental change would give advantage to
individuals that carry mutations fitting for dealing
(A)
with the new environment. Just as important are
combinations of genes that lead for faster
adaptation such as suggested by Muller (5, 6). That
is, sex and recombination can be important factors
(B)
in specie's ability to respond to new environment by
bringing together positive mutations that occurred
independently, or creating new positive
combination by the process of genetic
recombination (Fig 2.B). In order for asexual population
Figure 2. Evolution of positive mutations in time. Asexuals (A)
and asexuals (B). A, B and C represent positive mutations
occurring in the population.
to adapt, the positive mutations would have to occur in the same clone (or line of clones)
which is less likely. If positive mutations occur in separate clones the different clones would
then compete with one another which typically lead to loss of one line of clones. This effect
is known as clonal interference and is illustrated in Fig 2. A. Clones C and B "lose" to clone
A and it may take a long period of time until the specie re-acquires the lost positive
mutations B and C, if ever. Therefore it seems that a changing environment, at least
intuitively, would favor sexual and not asexual populations.
This work will review several models dealing with changing environment and its
relevancy to the evolution of sex
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Red Queen Hypothesis and Host-Parasite Interactions
Host-parasite interaction consists of host, whose life expectancy is long in
comparison with a parasite, whereas the parasite evolves to take advantage of host's
resources, thus lowering the host's fitness. While host evolves to minimize parasite damage,
the parasite evolves to optimize, and not maximize, host damage. This is an important
difference, since a parasite too virulent (e.g., too damaging to host) would soon lower its own
fitness as killing the host would also destroy the parasite before it might have the chance to
transmit and infect other hosts. Almost every biological level suffers from some sort of
parasite and some suggests that sexual reproduction and recombination may confer higher
resistance to parasite interactions (1, 7).
One model for this kind of interaction is the Red Queen hypothesis that got its name
from Lewis Carroll's "Through the Looking Glass" in which Alice and the Red Queen must
run as fast as they can just to keep at the same place. In analogy to host-parasite interactions
this theory suggests that host and parasite co-evolve in response to each other in a sort of
"evolutionary arms race" just to keep in the same place. The presence of parasite with short
life term could be considered as one kind of changing environment while the host is
relatively constant, which argues for faster evolution of the parasite. The model I will discuss
was suggested by Hamilton et al (1, 7).
Model and Assumptions
Simulations were made to include population of 200 individuals that could be either
sexual hermaphrodites or female-parthenogenetic while mode of reproduction was changed
by a single locus. Fecundity and chances of becoming a parent are equally distributed.
However, if number of offsprings is set to be x then it would take one parthenogenetic female
to produce x offsprings or two sexual parents to produce the same number. This incorporated
the twofold cost of sex into the model.
Each individual has a chance of dying each year set by d=1/h=1/14 and reproduction
goes with a juvenile period set by j = 13, so that an individual gets its fair chance at
reproduction at age of 14. Mean age of parenthood of host was G = j + h = 27. This allows
for overlapping generations to be taken into consideration in the model. Also, juvenile period
of parasites was set to be low and death-rate high (in comparison with host).
The only kind of selective force applied was that of parasitic interaction when hosts
may be infected by n species of parasites ( 2 ≤ n ≤ 12 ) each specie has population of 200, and
hosts have k loci for resistance for each specie. Similarly, each parasite specie has k "attack"
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loci that determine its success in the infected host. Both parasites and hosts are haploid (e.g.,
carrying one copy of each chromosome), so that in total count host would have (nk+1) loci,
and each parasite has k loci. Each locus may have one of two alleles: 0 or 1, and the number
of loci, k, was either 1 or 2.
The manner by which interaction occurs is that each year the host gets one random
parasite of each specie, which is n parasites total. Then parasite and host genomes are
checked for match which constitutes the parasites fitness, Sp. Host with loci 0011 and parasite
with 1011 have 3 matching loci, thus Sp=3. Thus the host's fitness may be derived by: (iv)
S = nk − ∑ S p and the hosts with lowest fitness are killed.
Mutation rates m were set to be either m=0.01 or m=0.0001 both of which are
considerably higher in orders of magnitude than what's expected on natural population (8).
Among sexuals, recombination rate, r, was set in a range of values 0 ≤ r ≤ 0.5 .
Results and Discussion
Results of simulations of this model show that sexual reproduction reach high ratio in
the population in all cases but for very low recombination and very high mutation rates, a
setting that almost eliminates the advantage of sex for the host (Figure 3, taken from ref. 1).
Thus the model demonstrates conditions in which sexual reproduction could rise in frequency
and stay stable, yet it is important to note that Hamilton's results show asexual-sexual
population coexistence rather than sex totally taking over the population which is in contrast
to many species in nature such as obligatory sexuals. Suggestions for future research should
be also redirected to facultative species; it would have been interesting to see the results if, in
addition to having one locus for being asexuals\sexual, individuals could be facultative and
have reproduction method base on their fitness. This may provide insights regarding parasite
interactions, for example, in
specie under parasitic-stress
should a successful individual
choose to reproduce sexually or
asexually? On one hand, sexually
reproducing with a lower fitness
individual may have the
offspring acquire other negative
traits, and on the other hand it
may serve to better fight-off
Fig 3. Percent success of the allele for sexuality in the model. Each point is averaged over 10 runs with a
population of 200. Total loci per host ranges from 2 to 12 in a and c and from 4 to 14 in b and d.
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parasites. In order to explore this option it would be best to include also non-parasite
interaction loci into the model, which would also introduce possible deleterious mutations.
While the Hamilton model is claimed to have chosen death rate and juvenile
properties to imitate those of humanoids, other properties of the system were not consistent
with known parameters, for example, mutation rate. Since parasites were set to be all
asexuals, this was explained as to provide them with the ability to evolve quickly enough to
meet with host's defense. This appears to be valid assumption since the higher mutation rate
cannot cause higher rate of deleterious alleles to appear because only attack-related loci were
simulated in this model. It should be noted that such interference may offset the weight of
other factors in the system, such as recombination rate. Recombination may be weakened as
an evolutionary force since gene "shuffling" is provided by unrealistic mutation rates. These
high rates have no other effects so it appears that recombination and mutations are somewhat
interchangeable in this model and while this is also true to some extent in natural population,
this model lacks mutation-rate's natural "backfire", e.g., deleterious mutations.
I also feel that a better application of this model could have been made if higher
polymorphism per loci was simulated (currently alleles are only 0 or 1). This can be justified
both biologically speaking since a locus is likely to have more than 2 alleles, and logically,
by considering the parasite's response to this model's mutation rate. At the current settings
parasite with a mismatching locus has only one direction of evolving, which is necessarily to
a matching locus, while the more accepted approach would be that mutations are more often
disadvantageous.
Moreover, polymorphism would serve hosts by allowing each individual to be more
unique, thus interfering with parasites' matching loci, as well as allowing mismatching loci in
parasite to mutate in a non one-directional manner as was previously pointed out. This dual
effect of polymorphism may serve as more realistic and also may lower the advantage of
sexual reproduction.
While this model is certainly qualitatively interesting and points to possible settings
in which sex could be advantageous, it is lacking since only very few evolutionary factors
have been taken into account. Also, only narrow range of parameters was explored. This, I
may only speculate, is due to computational constraints, and it would still be very interesting
to analyze results of this model under a wider range of parameters (more loci, polymorphism
and bigger populations).
The next level of research would be to ascertain what is the weight of parasite-host
interaction in the complete scheme, thus simulations should be made to include more loci
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(not just parasite-host related ones) and an effort to include realistic mutation rates. This
would also introduce deleterious mutations into the model. Doing so may enable weighing
parasite-host interactions against other evolutionary forces to determine their contribution to
the evolution of sex.
Advantage of Sexual Reproduction in a Changing Environment
The Hamilton model previously introduced is an explicit model, taking into account
actual number of loci, mutations and genes. It was concerned of one particular aspect in
which an organism's environment may change which is parasite-host interactions. But
models may be constructed to incorporate environmental changes in an implicit manner.
Such work was done by Crow (9) and presents a model for sexual and asexual populations
under directional selection, that is, when selection causes the population to evolve in a
manner that moves the distribution curve of a certain trait in the same direction.
The environment of living creatures is constantly changing. The rate of change is a
subject of a different, and very much interesting, academic discussion. The main issue
presented by Crow was how much genetic variance is conserved when comparing sexual and
asexual populations. Genetic variance being important since it may confer better ability to
correspond with selection, that is, more chances of having an allele in the population that is
advantageous under selection. Key point in his approach is that variation is important in
population's ability to respond to changing environment. If there are not enough different
selectable loci available, adaptation to new environment would most likely be hard, or may
even lead to extinction. Unlike the approach of Hamilton (1, 7) who used explicit loci, Crow
argues that the fitness function is composed of many fitness-comprising factors and, since a
large number of these factors exist, they can be treated so that their combined effect on
fitness is normally distributed, so this model does not deal explicitly with one biological
aspect of evolution.
Model and Assumptions
Asexual and sexual populations are reproductively isolated in this model. The model
assumes normally distributed fitness function. x is the fitness potential, and ω(x) is the fitness
of individual with fitness potential x. Fitness was dealt using two types of selection: hard
truncation selection – all individuals with fitness potential (x < z) have fitness of 0, while all
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individuals with (x > z) have fitness of 1. Soft truncation selection allows for more "smooth"
fitness function:
0
ω ( x ) = ( x + d − c ) / 2d
1
x<c−d
c−d < x <c+d
x >c+d
I will mainly discuss the results of hard
truncation though results of soft truncation
do not seem to differ by much. It is
Fig. 4. Truncation selection. The fitness, ω(x), is determined by whether
the fitness potential, x, is to the right or left of the truncation point, z.
assumed that all fitness determining factors
are fully hereditable and so, directional selection is expected to lower variance in the
 1
population. Density function of x is Gaussian f ( x ) = 
 2πV
 − ( x − m )2 

exp


 where m, V
2
V



are mean and variance of f(x). The mean fitness is p=p(x) = ϖ . p can also be related as the
∞
proportion of the population saved. Explicitly it can be written p = ∫ ω ( x ) f ( x )dx .
−∞
In the case of hard truncation selection it can be shown that ms=f(z)/p=c where ms is
the mean of f(x) of the selected group, z is the value of x at the truncation point and c is a
constant (Fig. 4, taken from ref. 9 may serve to
illustrate the general behavior of the various functions). Similarly, variance is
Vs = [1 − c(c − z )] ⋅ V0 where V0 is the initial variance in the population.
Since all traits are assumed to be completely heridetable, repeated truncation
selections in asexual population over n generation would be the same as if all selection
occurred in one generation. That is to say that if applying directional selection that saves 75%
of the population each generation for three generations would be equivalent to applying
directional selection for one generation that will save (0.75)3. This enables easily predicting
the the relation of the genetic variance in the nth generation with that of the initial variance
using normal distribution.
In sexual population the variance is partially coserved by recombination so that when
t is generation number Vt +1 =
)
when Vt +1 = Vt ≡ V =
1
{[2 − c ⋅ (c − z ) ⋅ Vt + V0 ]} . Equilibrium is reached
2
V0
.
1 + c(c − z )
Inherent in this model is that the populations remain normally distributed with each
generation and after each truncation (applies to both sexuals and asexuals).
8
Even before properly analyzing the outcome of these equations a basic difference
between sexual and asexual poplations may be seen: Sexuals populations will reach
equilibrium value, thus conserving many more alleles that may be important to adaptation if
direction of selection suddenly changed, while the asexual poplation continues to advance
towards lower variance. It should be mentioned that according to this model, theoretically the
asexual population could have continued to advance to the right end of the distribution, thus
lowering variance even further (Fig. 4), but these ends may not be biologically realistic, so
that even asexual population should stop at some equilibrium variance which is lower than
that of sexuals.
Results and Discussion
Results of this model hint that a sexual population will retain higher variance and
mean fitness in sexual populations than in asexual ones. This difference goes up to an order
of magnitude when inspecting equilibrium values in variance (after 7 generations:
Vasexuals=0.092, Vsexuals=0.611; masexuals=2.739, msexuals=4.584). It may also be seen that mean
values also differ considerably.
Crow notes that changing the direction of selection after several generation still seems
to conserve this difference between sexual and asexual populations behavior with respect to
mean and variance: Asexual variance continue to decrease, and sexual variance tend towards
a relatively higher equilibrium value.
This suggests that while sexual populations may respond to abrupt changes in the
environment more slowly, they are more likely to perserve genetic variance which may be
important in the event another environment change (such as sudden changes in direction of
selection). Thus it seems that changing environment would give advantage to the surviving
asexuals in the short term: If any asexuals survive the change and carry the adaptive genes,
they would quickly now reproduce, faster than their sexual counterparts. The change in
environment would also encompass a rather large loss in variation in the long term, so that in
case of a successive rapid environment change the asexual population may be found at a
sheer disadvantage.
Crow suggests that clear advantage to sexual reproduction may be found when
inspecting models that incorporate explicit loci and mutations such as Hamilton (1, 7). Also
noted is the better ability of sexual reproduction to deal with effects such as deleterious
mutations and mutational load, both these costs are much lowered when specie is sexually
reproducing.
9
In my opinion, conclusion derived from this model are quite limited most notably due
to lack of resolution. One may feel that the statistical approach, of making the entire fitness
function normally distributed, might have been carried out to the point in which it yeilds little
biologically-related data. Moreover, this approach requires the directional selection to work
in the same direction of all selectable loci, and no serious discussion was made to justify why
this should be common occurrence in natural populations.
The author also models his equation to regard fully hereditable traits and it was
claimed that the overall fitness function may move in a directional manner if every loci
changed even by a little, which serves somewhat to validate the assumption of hereditability
since it is reasnoble to persume that all traits are at least a little hereditable. Even so, more
work should be required before such assumption should be accepted, especially given that the
author applies directional selection again and again (up to 7 or 8 generations). In this process
many of the less inheritable traits would become completely uninheritable very quickly
(meaning, the lack of variation due to directional selection would make them fixed in the
population, and thus, uninheritable). Therefore, I would advise a more cautious approach in
regarding the results of this model in more than just a few generations, afterwhich a serious
consideration should be given to the effects of mutations and drift.
I feel, that while the implicit approach to modeling may very well be advantegous to
the understanding of the effects of changing enviroment on the evolution of sex, the model
suggested here by Crow is lacking in actual biological factors. Better argument could be
made if biological effects were statistically incroporated into the model such as mutations,
different levels of hereditability, and different rates of envioronmental change, all of which
are not very well represented by this model. A related work done by Waxman and Peck (10),
will be given in the next section of this work where a more complete discussion of the
various suggestions for improvement will be given.
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Sex and Adaptation in a Changing Environment
The work done by Waxman and Peck (10) is a model that incorporates various
biological and environemntal components of evolution such as various change rates of
environment, genetic variace, hereditability, changing optimum values of traits, and more.
This work is intended to show the effect of environmental change on the hereditability of
traits in a natural occruing population, and consider that effect on both sexual and asexual
populations using different number of loci under selection L , rates of change (α) and rates of
mutations (µ).
Model and Assumptions
Sexual and asexual populations, both of which are assumed to be of infinite size, and
sexual population consists of obligatory sexual hermaphrodites under random mating. All
individuals are diploids. Individual's measurement of a trait is denoted by z and the optimal
value by zopt. When zopt does not change in time it leads to stabilizing selection, but zopt can
also be made to change in time. Death rate of each individual increases with distance of z
(z − z )
2
from zopt according to D = 1 +
opt
2V
where V >0 and is inversely proportional to
strength of selection. Simulation usually settled to mean values of D that will be denoted as
D and, as with each parameter, subscript S is used for sexual population and A for asexuals.
All females produce offsprings at rate B and offsprings mature instantly (e.g., unlike
juvenile period such as in the Hamilton model, offsprings are instantly considered as having
chance at producing offsprings). B is cosidered to compensate for deaths at equilibrium, and
thus is set to be B = D .
Each individual has phenotipic value z which is composed by genotipic value, G, and
environmental component, ε so that ε is normally distributed with mean value of zero and
standard deviation Ve. Thus phenotypic value becomes: z = G + ε. The value of G is
determined by number of loci, L which freely recombine, and since all are diploids, in every
individual there are 2L such loci. Each locus i has phenotypic effect xi so that G = ∑i =1 xi .
2L
Mutations are incorporated to the model by mutation rate, µ and this value was
usually set as 10-5 per locus which constitutes realistic mutation rate (8). The effect of
mutation on xi is distributed around parental value with m standard deviation of mutant
effect. Distribution of mutant effect in the range y + dy > xi > y given by:
11
(
)
f y−x =
*
(
 − y − x*
exp
2
 2m 2
2πm

1
)
2

 where x* is the parental value of xi. Values of m used


were either 0.1 or 0.2 thus avoiding too extreme mutations.
Hereditability, h2, was considered as dependant of the genetic variance VG
by h 2 =
VG
. Lastly, environmental change was incorporated by moving the optimum
(1 + VG )
value zopt in time: zopt = αt where α is the rate of environmental change. Thus another
important property of the system can be defined: the difference between mean and optimum
phenotype, ξ.
Results and Discussion
Extensive numerical work has been made with this model, using various parameters
for L, µ, m, V and α. All results suggest that a small amount of environmental change may
contribute to manufacturing a lot of variance in the population (example for this analysis in
table 1, taken from ref. 10).
Table 1: Results from the numerical studies for the various quantities reported after they have reached their long-term stationary values.
The columns marked VG,S , h 2 , ξS, and DS refer to sexual populations, and give, respectively, the genetic variance, heritability, the difference
s
between the value of the optimum phenotype and the value of the population mean phenotype, and the average death rate. Same is said for
asexual population, marked with subscript of A instead S. α=0 refers to an unchanging environment.
Parameters of the model (other than rate of environment change, α) were set as follows: L=10, µ=10-5, m=0.2 and V=20. A question mark
appears in cases that were too extreme for calculation without very large amounts of computer time.
It is commonly held that changing environment causes increase in the death rate since
previously adaptive genes become maladapted and thus lower fitness. Upon examining table
1 it may be seen that according to analysis variation tends to increase rapidly with α and also,
that sexual population pays less, in terms of death rate D in maintaining such high variance
(consider a relatively mild rate of change, such as α=0.0664, 2 DS ≅ D A ). That is, asexual
population suffers a cost of adapting to the new environment which about twice as much as
the sexual population. Moreover, the difference between mean and optimum phenotype is
much smaller in sexual populations which shows asexuals are usually farther away from
optimum phenotype. These general trends are common to all shown by Peck and Waxman in
this model results under various parameters (not shown here).
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Changing environment, it seems, conveys an advantage to sexual population in
manner of smaller death rates, so that it may be possible to define α* as the rate of
( )
environmental change for which D A α * = 2 D A (α = 0 ) and α** as the value of α for which
2 DS = D A . Thus it is possible to find the rate of environmental change where death rate of
asexual is twice as high as that of sexuals, which in theory would negate the so called
'twofold cost' of sexual reproduction. Sexuals would live longer due to much smaller death
rate and thus be able to produce twice as much offsprings. For the various parameters used in
this study α** is of order of magnitude ranging between 10-1 – 10-2. These results present an
advantage to sexual reproduction under changing environment in terms of death rate and rate
of adaptation which supposedly may overcome the twofold cost.
Even so, the results must be considered cautiously under the validity of assumptions
made in this model. It is assumed that mutations are equally probable (using m for
distribution of mutant effect) to go either closer or farther from zopt. This may not strictly be
true for a lot of traits in which mutations are highly likely to be deleterious, so that the
theoretical ability of sexuals (and asexuals) to comply with changed environment is also very
much limited by the chance for positive mutation to appear. This, of course, affects of the
hereditability which is a very important a factor in this model and thus requires further
research. Trying to establish experimental evidence of changes in hereditability under
mutations and changing environment might serve to either validate this model, or else adjust
it for a more realistic incorporation of mutant effect.
The inaccuracy of mutational modeling may extend even further when considering
population size: Since population was assumed to be infinite it would be certain that positive
mutation would appear and take the population closer to zopt, which might not occur in
natural finite populations. This may be addressed in this model by expanding it to explicitly
contain population sizes and evaluate how the model reacts to changing population sizes and
comparing to available experimental data.
Another possible complication is pleiotropy and epitasis: A gene may affect many
traits and selection may vary in direction toward each trait, or may be involved in a process
whose failure may halt its downstream processes, thus the effect of mutation becomes a more
complex expression.
I would suggest a different approach towards this set of problems: Expanding this
model to contain several "clusters" of loci with different properties so that each individual
may have several loci-types that behave differently. Such suggested clusters may be QTLs
(quantitative trait loci) in which mutations are expected to have less severe effect towards
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fitness, and another may be 'structural genes' that would represent groups of genes in which
mutations are likely to have severe deleterious effects. This kind of expansion can be applied
by using this same model of Waxman and Peck but with different mutation rates, mutational
effect distribution, number of loci, etc, for each cluster of genes. A new z will be
incorporated into the model which can be made to be: z=G1+G2+G3…+ε when each Gi will
be determined according to its own parameters, thus new classes of behavior can be easily
incorporated into this very same model. It would still require changing the manner, in which
mutations are considered to incorporate different distribution types. Experimental results then
could be used to set relative amounts of loci for each cluster of genes; this of course may
differ extensively, depending of specific phylum or class of organisms.
14
Discussion
Several models of changing environment were reviewed each providing additional
angle into the problem of modeling such a complex system. As with all models, there usually
is tradeoff between accuracy and computational power. Some models can be well constructed
while being too specific thus limiting their value such as with the model suggested by
Hamilton (1, 7) which considers only parasite-host interactions and does little to incorporate
any other effects. This is why, while having very interesting results, it is still very difficult to
evaluate the role and weight of this interaction in the complete set of environmental factors.
Other models such as the one suggested by Crow (9) are too general in my opinion in
such way that severely limits its value in understanding the contributing factors to the
evolution of sex. Being too generic it fails to incorporate many biological aspects and thus it
is difficult to attribute the said results to any biological component of the system under
scrutiny. This may also make it harder to relate the results to any experimentally available
data. Having said that, I feel that approaching the modeling of changing environment can still
benefit from works such as that of Crow: Biological systems certainly include many
components and this approach to have them normally distributed may very well be justified.
So perhaps having a series of "independent" (for nothing in biological system can truly be
said to be independent) components of fitness, each normally distributed with its own
properties, will be a better approach as it would be able to correlate theoretical and
experimental data.
In this regard, the work of Waxman and Peck (10) is very well made model. It
consists of some biologically relevant data with various adjustable parameters, which can be
used to investigate many different behaviors of sexual vs. asexual populations. While this
model may not be perfect, and certainly has some disadvantages, I find the approach very
useful and think more should be done to improve upon this model (one such a way was
suggested in previous section discussing their work).
All things considered, it is still unclear how much sexual selection remains stable
though it seems that conservation of variability is an important factor. I would suggest
building a model similar to that of Peck and Waxman but in which the fitness determining
expression would be a series of independent factors (keeping in mind that it should be as
simple as possible and taking into account for the model to remain computationally feasible).
These factors can be terms for parasite-host interactions, inbreeding depression or any other
relevant factors.
15
References
(1) Sexual Reproduction as an Adaptation to Resist Parasites, Hamilton W. D. et. al, Proc. Natl. Acad.
Sci. USA vol 87, pp. 3566-3573, 1990.
(2) Evolutionary Genetics, Maynard Smith, Oxford University Press, 2nd Edition, 1999
(3) Population Genetics A Concise Guide, John H. Gillespie, John Hopkins University Press, 3rd Edition,
1997.
(4) A new evolutionary law. Leigh Van Valen, Evolutionary Theory, 1:1-30, 1973
(5) H. J. Muller, Am. Nat. 66: 118-38, 1932.
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