Mathematical modelling for variations of inbreeding populations fitness with single and polygenic traits
- Shuhao Sun^{1},
- Fima Klebaner^{1} and
- Tianhai Tian^{1}Email author
Published: 14 March 2017
Abstract
Background
Inbreeding mating has been widely accepted as the key mechanism to enhance homozygosity which normally will decrease the fitness of the population. Although this result has been validated by a large amount of biological data from the natural populations, a mathematical proof of these experimental discoveries is still not complete. A related question is whether we can extend the well-established result regarding the mean fitness from a randomly mating population to inbreeding populations. A confirmative answer may provide insights into the frequent occurrence of self-fertilization populations.
Results
This work presents a theoretic proof of the result that, for a large inbreeding population with directional relative genotype fitness, the mean fitness of population increases monotonically. However, it cannot be extended to the case with over-dominant genotype fitness. In addition, by employing multiplicative intersection hypothesis, we prove that inbreeding mating does decrease the mean fitness of polygenic population in general, but does not decrease the mean fitness with mixed dominant-recessive genotypes. We also prove a novel result that inbreeding depression depends on not only the mating pattern but also genetic structure of population.
Conclusions
For natural inbreeding populations without serious inbreeding depression, our theoretical analysis suggests the majority of its genotypes should be additive or dominant-recessive genotypes. This result gives a reason to explain why many hermaphroditism populations do not show severe inbreeding depression. In addition, the calculated purging rate shows that inbreeding mating purges the deleterious mutants more efficiently than randomly mating does.
Background
In genetic terminology, inbreeding is the breeding of two individuals who are related to each other. Here inbred is defined as the individual that is generated from inbreeding. The closeness of individuals is important in inbreeding, which has substantial influence on the fitness of offspring individuals. Inbreeding depression refers to the reduced survival and fertility of offspring of related individuals. An important mechanism that leads to inbreeding depression is overdomain in which the heterozygote has greater phenotype value and perhaps is more fit than the homozygous state for either of the alleles that it comprises. Therefore there are two major approaches to increase homozygosity but decrease fitness. The first method is to increase homozygosity for partially recessive detrimental mutations; while the second approach is to increase homozygosity for alleles at loci with heterozygote advantage (overdominance). However, it is still under debate which mechanism is more dominat in nature [1–5]. This is an important question in genetics because two differnet mechanisms will lead to different theories regarding the trait values of the crossbred progeny. If the overdominance theory is valid, the mean trait value of crossed lines will return to the equivalence of the outbred population, as heterozygosity will be restored. On the other hand, the partial dominance theory predicts that the mean trait value will exceed that of the outbred population. In this case, in addition to the restore of heterozygosity, crossbred individuals will be purged from their genetic load [6, 7].
Genetic purging is the process for frequency reduction of a deleterious allele. It has been accepted that purging was achieved when inbreeding depression is caused by deleterious recessive alleles. A fitness rebound in inbred populations provides evidence to support the partial dominance mechanism [8, 9]. The effect of genetic purging have been observed in a number of experimental studies [10, 11], but overall, the evidence for purging in plant and animal populations is still limited. In addition, the combined effect of both inbreeding and purging is important for evaluating the evolutionary consequence of inbreeding and for making recommendations in conservation [12]. Although the inbreeding-purging model has been proposed to predict the evolution of the fitness and inbreeding load [13], a number of model parameters should be estimated from observation data. Thus it is high demanded to explore the role of purging in restoring fitness and calculation of purging speed.
The number of phenotypes produced for a given trait depends on how many genes control the trait. There are two types of trait. The single gene trait is controlled by a single gene that has two alleles. On the other hand, polygenic trait is controlled by two or more genes and has 2 or more alleles. Unlike monogenic traits, polygenic traits do not follow patterns of Mendelian inheritance (discrete categories). Instead, their phonotypes vary along a continuous gradient, such as the height of animals and the colour of human skin. Although we have conducted theoretical study for the single trait [14], the study of inbreeding depression for polygenic traits is more challenging and there is limited theoretical result regarding the variation of mean fitness under inbreeding.
In this work we will conduct novel theoretical studies for inbreeding depression in large population with Mendelian and polygenic traits. First, for single locus, we will explore the conditions for the monotonical increase of the mean fitness. We also derive a formula to calculate the purging rate. In addition, for multiple loci, we will find conditions for inbreeding mating reducing the mean fitness of the population. Furthermore, this work will show mechanisms to determine inbreeding depression. As an application, we present additional reasons for the frequent occurrence of self-fertilization populations. The well-known reason is that self-pollination does not need to be visited by animals to produce seed and hence have ecological advantageous under some circumstances [15].
Methods
Inbreeding of Mendelian traits
Thus, for random mating (f=0), the mean fitness increases monotonically. Similar observation can be found for the case w _{11}=w _{22} even if f≠1. For example, using a different approach, Ziehe and Roberds [18] has shown that, when caused by symmetric homozygous disadvantage at single locus, inbreeding depression is always less than one-third [16]. Here “symmetric” means w _{11}=w _{22}. Thus for the populations considered in [18], the mean fitness increases monotonically.
Inbreeding depression
Based on this definition the inbreeding coefficient is a function of inbreeding coefficient f, frequency x, degree of dominance h and selection coefficient s, namely δ=δ(x,f,h,s). We first have the following theory for the changes of mean fitness.
Proposition 1
- 1.
the population with overdominance genotype, or with a partial dominant genotype. That is, inbreeding depression occurs.
- 2.
the population with under-dominance genotype, or with a partial recessive genotype. In this case, no inbreeding depression occurs.
Proposition 2
For example, if the degree of dominance satisfies −1<h<0, there would not be any internal equilibrium for a population of self-fertilization plants with f=0.5.
Multiplicative interactions for polygenic traits
Results
Increase of mean fitness for Mendelian traits
We have described the methods for measuring the fitness of inbreeding population in the previous section. The following theorem gives the condition for the increase of mean fitness.
Theorem 1
Proof
Examples for the decreases of mean fitness
f | x | w _{11} | w _{12} | w _{22} | \(\overline w\) | \(\overline w'\) | Δ w |
---|---|---|---|---|---|---|---|
0.1 | 0.62 | 0.8 | 1 | 0.7 | 0.86802 | 0.868018 | <0 |
0.25 | 0.635 | 0.8 | 1 | 0.7 | 0.85041 | 0.85040 | <0 |
0.25 | 0.66 | 0.8 | 1 | 0.7 | 0.85015 | 0.85013 | <0 |
0.5 | 0.7 | 0.8 | 1 | 0.7 | 0.8225 | 0.8224 | <0 |
0.5 | 0.79 | 0.8 | 1 | 0.7 | 0.82047 | 0.82045 | <0 |
0.5 | 0.9 | 0.7 | 1 | 0.4 | 0.7105 | 0.7101 | <0 |
The biological meaning of AMF is that AMF>0 holds if and only if the population does not become extinct. Thus Theorem 1 in fact shows that AMF for directional inbreeding population is always positive because in this case Δ w(x)>0 for each x≠0. The following theorem gives the value of AMP without any additional condition.
Theorem 2
The AMF of any inbreeding population is positive.
Proof
The proof is given in Appendix section. □
Figure 2 b and 2 d schematically shows this point, where the genotype fitness is (1,1.4,0.8) and inbreeding coefficient f=0.25. From Fig. 2 b, when the population goes to the 8th generation, the frequency x becomes 0.6338 and the mean fitness is 1.0831. Then the mean fitness starts to decrease (that is, Δ w(x)<0) until x reaches to the equilibrium (x ^{∗}≈0.6667 in this case). On other hand, Fig. 2 d clearly shows that the positive area which is above the x-axis (and below the curve) is much larger than the negative area which is below the x-axis and above the curve. These figures show that AMF>0 intuitively.
Inbreeding depression
For inbreeding depression, we give the theorem which provides insights into inbreeding mating.
Theorem 3
- 1.
If s<1, δ, as a function of x (3), arrives its maximum at \(x^{*}= 1- \frac {1-\sqrt {1-s}}{s}=\frac {\sqrt {1-s}}{1+\sqrt {1-s}}\) for \(h< \frac {1}{2}\) (partial dominant or over-dominant) and \({\lim }_{x\rightarrow 0}\delta ={\lim }_{x\rightarrow 1}\delta =0\);
- 2.
If s=1 and h≠1,δ is decreasing and \({\lim }_{x\rightarrow 0}\delta =\frac {f(1-2h)}{2(1-h)}\);
- 3.
If s=1 and h=1, then \(\frac {w_{s}}{ w_{o}}\) is not continuous at x=0 and \({\lim }_{x\rightarrow 0}\frac {w_{s}}{ w_{o}}= \infty \);
- 4.
If x≠0, then δ, as a function of h, is decreasing and \({\lim }_{h\rightarrow -\infty }\delta = f\);
- 5.
δ, as a function of s, is strictly increasing;
- 6.
δ, as a function of f, is strictly increasing.
Remark 1
This result may provide an explanation to the puzzle for the classical strategy of cultivation. When choosing the offspring (i.e. seeds) of a favourable mutation and then inbreeding between the improved offspring, our results suggest that this inbreeding mating greatly increases the mean fitness of the population. This interesting result explains biological observations but is contradict to the belief that inbreeding mating decreases mean fitness. The key point is that in this case the initial frequency of the favourable mutation normally is very low. However, if the first generation of the population still uses the same selection strategy with the favourable mutation, the initial frequency will be very high, because they all already have the favorable mutation. However, the following Theorem 5 shows that the inbreeding mating for the second generation actually reduces the mean fitness.
Purging rate
Multiplicative interactions for polygenic traits
We first give an example to show that the theory established in the previous section may not be able to explain some experimental observations. For example, for an selfing population with inbreeding coefficient f=0.5, nature selection pressure s=0.05 and h=0 (i.e. a completely dominant mutant), the inbreeding depression at a single locus (3) is δ=0.006. However, the observation in natural populations shows that a completely recessive mutant usually will cost around 20−30% inbreeding depression [9, 23, 24]. This difference suggests that the trait may be determined by multi-genes.
In our previous study, we proposed a theorem for the inbreeding mating without proof. The following result is an updated version of Theorem 3.1 in [14].
Proof
Theorem 4
Assume that a polygenic trait is determined by the genotypes whose relative fitness is (1,1−hs,1−s), where the selective advantage coefficient s is fixed and the degree of dominance h is uniformly distributed over [0,1] (i.e., the trait is determined by the dominant-recessive genotypes). Then inbreeding mating does not decrease the mean fitness of population. □
Note that the inbreeding depression δ is an allele-specific property. However Theorem 4 shows a genotype-specific property which is independent of the choice of alleles.
From Proposition 1, we immediately have the following results.
Proposition 3
If a polygenic trait is determined by partial dominant or over-dominant genotypes, inbreeding mating reduces the mean fitness of the population.
Theorem 5
Proof
The proof is given in Appendix section. □
Remark 2
Recent studies have showed that there are many overdominant loci in rice [3, 4, 26]. In addition, overdominance and epistasis might play an important role as the genetic basis of heterosis in Brassica rapa [27]. The following theorem investigates the relationship between the number of genotypes and the mean fitness.
Theorem 6
If the set of genotypes which determine a polygenic trait contains enough over-dominant genotypes, inbreeding mating reduces the mean fitness of the population for any given initial frequency x≠0, that is, w _{ s }≤w _{ o }.
Proof
The proof is given in Appendix section. □
However, in natural populations, there may not be many overdominance genotypes [28]. Another restriction for over dominance (1,1−hs,1−s) is that h may not be very small. A well-known example is Sickle cell anaemia which occurs when a particular pair of genes carry the ’sickle-cell trait’ which has not been eliminated from the human population by selection. The reason is that there is only one pair of genes carrying the sickle-cell trait and individuals (“carrier”) is highly resistant to malaria. On the other hand, a person whose genes do not carry the sickle-cell trait is susceptible to malaria. For example, according to the data from the World Health Organization, the carrier frequency ranges from 10 to 40% across equatorial Africa. The genotype’s fitness is 1,1−h and 0 with −1.35≤h≤−1.05. Thus h (≥−2s,s=1) is not very small.
Discussions
We now discuss other factors for determining inbreeding depression. Suppose that initially we have two homozygous, inbred lines P _{1} and P _{2} which are used as parents. Let \(\bar P_{1}>\bar {P}_{2}\), where \(\bar {P}\) is the mean fitness or fitness-related trait of P. The genotypes of P _{1} and P _{2} for this gene are A ^{+} A ^{+} and A ^{−} A ^{−} while their hybrid is A ^{+} A ^{−}. Let the average phenotype of the two parents be m and the additive and dominance genetic component of means (the averaged phenotypic value) be \(a_{A} \left (=\bar {P}_{1}-m\right)\) and \(d_{A}\left (=\bar {F}_{1}-m\right)\), respectively, where F _{1} is the first generation of hybrid offsprings of P _{1} and P _{2} and \(\bar F_{1}\) is the mean of the trait of F _{1}. Suppose d _{ A }>a _{ A }>0, that is, P _{1} is overdominant to P _{2}. If inbreeding mating, the mean value is \(\left (\bar {P}_{1}+\bar {P}_{2}\right)/2=m\). However, for cross mating, the mean value is \(\bar {F}_{1}\) which is greater than m. Consider a natural population consisting of equal number of P _{1} and P _{2}. Then inbreeding reduces the mean value of fitness or fitness-related, namely this population exhibits inbreeding depression. On the other hand, Theorem 4 shows that a natural population with few over-dominant genotypes does not exhibit severe inbreeding depression. The next result provides a reason to explain this difference.
Theorem 7
If a natural inbreeding population does not show a high level of inbreeding depression (i.e., δ≤0.2), then most fitness or fitness related loci (QTL) of it exhibits additive and dominant-recessive genotypes.
We conclude that inbreeding depression is determined by not only the inbreeding coefficient but also the genetical structure of the population. This result is consistent with the observation that, in many mixed-mating plant populations, selfing (≈0.2), outcrossing (≈0.4) and the interm1diate (≈0.4) reaches at an equilibrium [29].
Let us consider an example. Goodwillie and Knight [2] measured inbreeding depression in three populations, namely Lake Hennessey (LH), Wantrup Researve (WR) and Ida Clayton Road (IC) of Leptosiphon, whose mean outcrossing rate are 0.06,0.37 and 0.69, respectively. Significant inbreeding depression was observed for the proportion of fertilized ovules that developed into seeds only occurred in the more outcrossing WR and IC populations. Theorem 7 predicts that most genes in population LH are additive and dominant-recessive and both LH and WR have many over-dominant genes.
Another novel contribution of this work is the study for inbreeing depression with polygenic traits. Our theoretical results have shown that inbreeding depression also depends on the structure of the genotypes. The theoretical results support the belief that inbreeding depression is caused more like by over-dominant mechanism rather than dominance-recessive mechanism. An application of these results is to explain why Caenorhabditis remanei has demonstrated to suffer severely from inbreeding depression, while its hermaphroditic relative C. elegans has not [22, 30]. In addition, Theorem 5 shows that inbreeding depression is also determined by the initial frequency, which, for cultivation of improved varieties with a favorable mutation, explains why inbreeding mating does not reduce the mean fitness of the first generation, but does reduce the mean fitness of the second generation. These results may have potential application for cultivation of improved varieties.
Conclusions
In this work we developed mathematical approaches to investigate the the variations of inbreeding population fitness under various conditions. For inbreeding population with a single locus, we proved that the mean fitness increases monotonically for directional selection, which extends the existing result for the random mating substantially [17]. Our results showed that, if most genotypes are additive, inbreeding mating (selfing) does not produce inbreeding depression. In addition, we defined the averaged mean fitness (AMF). Using this concept, we have successfully shown monotonical increase of the averaged mean fitness for any inbreeding population. These results can be applied to study the Mendelian trait or unlinked polygenic trait, and provide the reason to explain why inbreeding mating does not have serious impact on the mean fitness of population. For inbreeding depression with single trait, we have presented an explicit formula to calculate purging rate. Computing results suggest that inbreeding mating does purge deleterious mutants more efficiently than randomly mating does.
Appendix: Theoretic Proofs
Theorem 2
The averaged mean fitness (AMF) of any inbreeding population is positive, that is,
Proof
Also note that P(f)_{ t=1} is a decreasing function of f and [P(f)_{ t=1}]_{ f=0.8}>0 for s≥0.89.
A similar argument can be applied to the case of under-dominance populations. Thus the proof of Theorem 2 is completed.
Theorem 3
Proof
since it is a decreasing function of s.
Let \(\bar y\) be the smaller root of (6−3f)y ^{2}−(8−3f)y+2. Then we have \(\bar y=\frac {2}{6-3f}\) and (6−3f)y ^{2}−(8−3fh ^{2})y+2 is a decreasing function for \(y\le \bar y\).
Theorem 4
If the set of genotypes which determine a polygenic trait contains enough over-dominant genotypes, then inbreeding mating reduces the mean fitness of the population for any given initial frequency x≠0, that is, w _{ s }≤w _{ o }.
Proof
Declarations
Acknowledgments
The authors thank the two anonymous reviewers for their insightful comments and constructive suggestions, which are very helpful to improve the manuscript substantially.
Funding
T.T. is supported by the Australian Research Council (ARC) Discovery Projects (DP120104460) which supports the publication cost of this paper. F.K. is supported by the Australian Research Council (ARC) Discovery Projects (DP120102728).
Availability of data and materials
Not Applicable.
Authors’ contributions
SS and TT conceived and conducted the research. SS carried out the theoretical proof. SS, FK and TT interpreted the results and wrote the paper. All authors edited and approved the final version of the manuscript.
Competing interests
The authors declare that they have no competing interests.
Consent for publication
Not applicable.
Ethics approval and consent to participate
Not applicable.
About this supplement
This article has been published as part of BMC Genomics Volume 18 Supplement 2, 2017. Selected articles from the 15th Asia Pacific Bioinformatics Conference (APBC 2017): genomics. The full contents of the supplement are available online http://bmcgenomics.biomedcentral.com/articles/supplements/volume-18-supplement-2.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
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