Analysis of Sexual vs Asexual Reproduction Using Four Measurements

The prevalence of sexual reproduction has long been an outstanding problem of evolutionary biology. In accordance with the mathematical approach employed by past researchers, we propose a mathematical framework to address this problem. We define and derive four measurements, diversity measure DVM, diffusion measure DFM, optimality measure OPM, and survivability measure SVM to compare sexual reproduction with asexual reproduction. We show that DVM increases exponentially in sexual reproduction, while only linearly in asexual reproduction. Hence, sexual reproduction allows species more opportunity to adapt. We also show that DFM is bounded in sexual reproduction and OPM is inversely related to DFM. Thus, sexual reproduction leads to smaller DFM and hence a larger OPM. We further show that SVM is a monotonic increasing function of OPM. Hence, sexual reproduction is better by virtue of producing a more homogeneous population.


Introduction
Why most species reproduce sexually is a question investigated extensively, but not yet answered satisfactorily in biology (1)(2)(3)(4). The prevalence of sexual reproduction suggests that there are major benefits provided by this mode of reproduction. The benefits, however, are not well-understood. In many ways, asexual reproduction seems to be a better evolutionary strategy: only one parent is required, and all of the parent's genes are passed on to its progeny (5)(6)(7)(8)(9)(10)(11)(12).
In a sexual population, the males are unable to produce progeny of their own and females only transfer half their genes to progeny, hence the so called problem of the "two-fold cost of males." Furthermore, sexual reproduction must also overcome obstacles that do not exist in asexual reproduction. Sexually reproducing organisms must spend a great deal of time and energy to find and attract mates. Also, copulation in sexually reproducing organisms leaves both organisms vulnerable to predation.
Despite these and other major drawbacks to sexual reproduction, it remains a very prevalent form of reproduction in most higher order organisms. Biologists have put forth numerous hypotheses for why sexual reproduction is so prevalent. The main hypotheses to explain sexual reproduction typically focus on the benefits of the inherent ability of sexual reproduction to recombine and shuffle genetic information (13)(14)(15)(16)(17)(18)(19)(20). Generally, these hypotheses have relied on fusing observational zoological studies with our understanding of genetics to build their foundations; a very understandable approach given the inherent methodological constraints on the field of evolutionary biology Whereas hypotheses in applied physics or chemistry can be directly tested in experiments with controlled variables in real-time, evolutionary biology deals with "evolutionary time" where the timescales necessary for experimentation far exceed the human lifespan, or human civilization itself for that matter. Such controlled experiments are largely unfeasible for the field of evolutionary biology. As such, even Darwin was forced to use the inductive reasoning approach of accumulating large reservoirs of circumstantial evidence in combination with critical thinking to propound his ideas (21). The field of evolutionary biology has largely inherited this inductive reasoning methodology.
With the dearth of ability to conduct concrete controlled experiments to prove evolutionary principles by a deductive approach, we believe that turning back to a mathematical approach, as done by such past luminaries as JBS Haldane (22) or Kimura and Maruyama (23) can be valuable for addressing the problem of sexual reproduction. We believe that reproductive biology generally follows consistent principles that are amenable to mathematical modeling; as such, a mathematical framework can be used to provide further support for existing ideas, as well as illuminate novel ideas that could be overlooked if not for a deductive mathematical approach.
With a mathematical methodology in mind, we introduce several mathematical constructs to analyze population changes via sexual reproduction or asexual reproduction. We first introduce a diversity measure DV M defined as the number of distinct individuals/genotypes in a population divided by the total number of individuals/genotypes in the population. We derive dynamic equations of DV M for both asexual reproduction and sexual reproduction. We show that DV M increases exponentially in sexual reproduction rather than linearly as in asexual reproduction. This exponential increase in sexual reproduction's diversity DV M is consistent with current hypotheses that posit genetic recombinatorial shuffling from sexual reproduction allows for more rapid adaptation.
We further introduce a diffusion measure DF M defined as the average "distance" of individuals in a population. Here, the distance between two individuals is defined as the percentage difference in their genes. Consequently, DF M measures how homogeneous a population is.
We derive dynamic equations of DF M for both asexual reproduction and sexual reproduction. We show that, despite its diversity, sexually reproducing species are more homogeneous (smaller DF M ) than asexually reproducing species. In other words, sexual reproduction increases diversity (by increasing DV M ) while decreases diffusion (by reducing DF M ). This is less intuitive, but true.
We show that a more homogeneous population has evolutionary advantages over a less homogeneous population (smaller DF M is better). This is because the diffusion measure DF M is inversely related to optimality measure OP M , which measures the average distance of individuals in a population to the optimal individual/genotype who is most suitable in a given environment. More precisely, we show that OP M = 1 − DF M . Therefore, a smaller DF M implies a larger OP M . Hence, a more homogeneous population is more suitable for the environment. Since sexual reproduction leads to smaller DF M versus asexual reproduction, it has advantage over asexual reproduction, in addition to the advantage of having a larger DV M .
The last performance measure we define in this paper is the survivability measure SV M which is the percentage of individuals having progeny. We show that the survivability measure increases as the optimality measure increases (and that survivability measure decreases as the diffusion measure increases). Some theories/hypotheses that have been proposed to address the advantages of sexual reproduction over asexual reproduction, such as genetic recombination, Muller's Ratchet, and the Red Queen Hypothesis, can all be addressed by the mathematical framework using the performance measures proposed in this paper.

Diversity, Diffusion, Optimality, and Survivability Measures
In this section, we present the framework of our research. We define diversity measure, diffusion (or inhomogeneity) measure, optimality measure, and survivability measure to describe a species. We also define selective pressure to describe an environment. We show how survivability measure depends on diffusion measure, optimality measure, and selective pressure.
To do this formally and mathematically, let us first introduce the following notations.
N -the number of genes In asexual reproduction, a progeny inherits its genes from one progenitor.
In sexual reproduction, a progeny inherits its genes from two progenitors.  From one generation to the next, the probability of mutation of g i is p.
A mutation causes h% change in g i .
Change of mutation is additive: k mutations cause kh% change in g i .
Q -the total population of a species.
q ∈ Q -an individual/genotype in the species Q.
|g q 1 i − g q 2 i | -distance between g q 1 i and g q 2 i , defined as the percentage difference between g q 1 i and g q 2 i .
If the distance between g w i and g d i exceeds a bound b, that is, |g w i − g d i | > b, then either the mating will not take place or the resultant progeny will not survive.

Diversity measure
Diversity measure describes how diverse a population is. We will argue that a more diverse population has an evolutionary advantage over a less diverse population, as more diversity increases the chance to adapt to the environment.
Diversity measure of a species Q is defined as the number of distinct individuals/genotypes in Q divided by the total number of individuals/genotypes in Q, that is, where |Q| is the number of elements (cardinality) of Q. Clearly, larger DV M (Q) means that the species is more diverse.

Diffusion measure
Distance between q 1 , q 2 ∈ Q is defined as Diffusion measure of a species Q is defined as the average or expected value of DS(q 1 , q 2 ): where E(.) denotes the expectation (mean). Clearly, larger DF M (Q) means that the species is less similar.

Optimality measure
In a given environment, the optimal individual is denoted by o ∈ Q, whose gene is denoted The goal of evolution is for the population to evolve around o, that is, the distance of Q to o reduces as time passes.
The distance of Q from o is defined as the average or expected distance of individuals q ∈ Q to o. One minus this distance is called the optimality measure OP M . Hence, optimality measure of a species Q is one minus the expected value of DS(q, o): The reason to use 1 − E(DS(q, o)) rather than E(DS(q, o)) as optimality measure is that, intuitively, we would like to see that the larger OP M is, the closer Q is to o.
As to be shown in Section V, optimality measure is related to diffusion measure as follows.

Selective pressure
For a given environment, its selective pressure is measured as a threshold SP ∈ [0, 1] such that, if DS(q, o) > 1 − SP , then individual q will not have progeny. The larger SP is, the strong selective pressure is.

Survivability measure
In a given environment, survivability measure of a species Q is defined as the percentage of individuals having progeny, that is, Survivability measure depends on selective pressure, optimality measure, and diffusion measure as shown in the following theorem, whose proof is in Appendix.

Theorem 1
(1) As the selective pressure SP increases, the survivability measure SV M (Q) decreases.
(2) As the optimality measure OP M (Q) increases, the survivability measure SV M (Q) increases.
(3) As the diffusion measure DF M (Q) increases, the survivability measure SV M (Q) decreases.
In the next two sections, we derive the diversity and diffusion measures for both asexual reproduction and sexual reproduction. Without loss of generality, we assume that all individuals in Q are of the same generation. Denote the m-th generation of the species as Q m . Since

Asexual Reproduction
For asexual reproduction, each individual in the m-th generation q ∈ Q m is produced by an individual in the m-1-th generation q ∈ Q m−1 , denoted as q → q .

Diversity measure
For asexual reproduction, each mutation produces an individual who is different from its parent. The probability that this individual is distinct in Q is proportional to 1 − DV M (Q m−1 ).
The expected number of mutations in one generation is given by |Q| × N × p. Hence, It can be shown ( is small initially, then 1 − DV M (Q m−1 ) ≈ 1 and the above equation can be approximated as In other words, DV M (Q m ) increases linearly.

Diffusion measure
To calculate DS(q 1 , q 2 ), where q 1 → q 1 and q 2 → q 2 , let us consider their i-th genes, g i . Since mutations are independent with probability (w.p.) p, we have the following where δ is used to denote a mutation. Hence, the possible pairs (g ) and probabilities of their occurrences are as follows. (g A mutation causes h% change. The probability that this mutation increases |g Therefore, It can be shown (Section V) that DF M (Q m ) → 100% as m → ∞. In other words, in asexual reproduction, individuals in a species will become less and less similar as time passes.

Sexual Reproduction
In sexual reproduction, each individual in the m-th generation q ∈ Q m is produced by two individuals in the m-1-th generation q w , q d ∈ Q m−1 , denoted as q w , q d → q .
is small initially, then 1 − DV M (Q m−1 ) ≈ 1 and the above equation can be approximated as In other words, DV M (Q m ) increases exponentially with respect to m. It is well-known that exponential increase is much faster than linear increase.

Diffusion measure
To calculate DS(q 1 , q 2 ), where q 1,f , q 1,d → q 1 and q 2,f , q 2,d → q 2 , let us consider their i-th gene, g i . Since mutations are independent with probability p, we have A mutation causes h% change. The probability that this mutation increases |g The reason for min is that, if |g then the progeny will not survive. Therefore, where τ > 0 is the parameter that determines the softness of the approximation of min.
It can be shown (Section V) that DV M (Q m ) → b as m → ∞. This is very different than asexual reproduction, where DF M (Q m ) → 100% as m → ∞. In other words, while asexual reproduction can make individuals in a species completely dissimilar, to the point that a taxonomist could be forced to concede that the population of individuals are in fact multiple "species," whereas, sexual reproduction maintains a species as a related and coherent group.

Simulation Results
There are two tasks for simulations. The first task is to verify The second is to compare sexual reproduction with asexual reproduction by means of the diversity, diffusion, and optimality measures.
We use MATLAB, a popular engineering software to perform simulations. For the first task, we calculate DF M and OP M using Equations (2) and (3).
Since DS(q 1 , q 2 ) and DS(q, o) are percentages, we represent q ∈ Q by a number between 0 and 1. We generate K random numbers, denoted, with a slight abuse of notation, as Q = {q 1 , q 2 , ..., q K }, and then calculate where, without loss of generality, we let o = q 1 .
We run the above simulations R times and take the average. For different K and R, the results are shown in Table 1. For the second task, we simulate the derived dynamic equations, which are summarized below, to compare sexual reproduction with asexual reproduction. Since it is a comparison, only the relative values (rather than the absolute values) of the diversity, diffusion, and optimality measures are of importance.
Diversity measure: Diversity measure for asexual reproduction is given in Equation (5). Diversity measure for sexual reproduction is given in Equation (8).
Diffusion measure: Diffusion measure for asexual reproduction is given in Equation (7). Diffusion measure for sexual reproduction is given in Equation (10).
Optimality measure Diffusion measures for both asexual and sexual reproductions are given by Equation (11).   From Figure 1, it is clear that the diversity measure of sexual reproduction is larger than and increases faster than that of asexual reproduction. Larger diversity measure allows a species to evolve faster. Hence, in terms of diversity measure, sexual reproduction has an advantage over asexual reproduction.
On the other hand, from Figure 2, diffusion measure of sexual reproduction is bounded and smaller than that of asexual reproduction. By Theorem 1, smaller diversity measure leads to larger survivability measure. Hence, in terms of diffusion measure, sexual reproduction also has an advantage over asexual reproduction.
Comparing the horizontal axis of Figure 1 with the horizontal axis of Figure 2, we note that the advantage of sexual reproduction over asexual reproduction in terms of diffusion measure takes a longer time to show than the advantage in terms of diversity measure. This probably explains why the effects of diversity measure have been so readily latched onto and investigated by biologists, whereas the potential evolutionary consequences of diffusion measure's effects have been overlooked and not been investigated in any comprehensive manner. To illustrate this benefit of genetic similarity and maintenance of adaptational advantages in a biological context, we take the example of a developed ecosystem that is near its "climax community". Such a system can be considered relatively stable with high biodiversity. The high biodiversity means there is more competition for the same limited resources. In order for all these species to successfully live amongst each other, selective pressures have caused each species to develop specific adaptations that allow it to occupy an exclusive niche. Deviating away from such adaptations, which means deviating away from the species' niche, results in competition with other species that occupy other niches. These other species are highly adapted for their niches; consequently, the "deviant organism" is unlikely to survive the competition.

Diversity vs Diffusion
Sexual reproduction maintains adaptational advantages and minimizes the conversion of precious resources to the production of deviant organisms that are unlikely to survive.
In summary, both DV M and DF M are distinct principles that can provide a more nuanced approach to investigating the advantages of sexual reproduction over asexual reproduction. (1) Sexual reproduction leads to a more diverse population and hence allows more rapid adaption to environments. (2) Sexual reproduction is a species stabilization mechanism that naturally maintains genetic homogeneity and species identity. (3) Asexual reproduction, which does not have this inherent species stabilization mechanism, leads to genetic inhomogeneity and no definitive species identity. (4) Sexual reproduction is beneficial because the maintenance of species identity maintains desired adaptational advantages, which is important when selective pressures are strong. In the next section, we use DV M and DF M to explain some theories/hypotheses that have been proposed to address the advantages of sexual reproduction over asexual reproduction. (2) Muller's Ratchet: Muller's Ratchet is reflected in the optimality measure OP M . As shown in Figure 2, OP M decreases in both sexual reproduction and asexual reproduction as time goes by. This is exactly what Muller's Ratchet anticipates. However, in sexual reproduction, OP M will reach a limit and no longer decreases. In other words, sexual reproduction can restore genotypes to optimal fitness.

Advantages of Sexual Reproduction
(3) Red Queen hypothesis: The Red Queen Hypothesis can also be explained by DV M .
Sexually reproduced parasites and hosts have much larger DV M than asexually reproduced parasites and hosts as shown in Figure 1. Larger DV M allows parasites and hosts to better adapt in the battle with hosts evolving resistances to parasites and parasites evolving ways to get past those resistances.

Proof of Theorem 1
Let us enumerate individuals in Q based on their distance to o, that is, let where K = |Q|, such that Define a distance function ρ : K → [0, 1], where K = {1, 2, ..., K}, as Then ρ is a monotonic increasing function, that is, where ⇒ denotes "implies".
From the definition of survivability, we have where ⇔ denotes "if and only if". Therefore, To prove Part (1) of Theorem 1, consider two selective pressures SP 1 , SP 2 with SP 1 ≤ SP 2 .
We have where SV M i (Q) is the survivability measure under selective pressure SP i , i = 1, 2.
For better simulations, we approximate min by a continuous function as follows.