Repetition and Reversibility in Evolution: Theoretical Population Genetics

Abstract

Repetitiveness and reversibility have long been considered as characteristic features of scientific knowledge. In theoretical population genetics, repetitiveness is illustrated by a number of genetic equilibria realized under specific conditions. Since these equilibria are maintained despite a continual flux of changes in the course of generations (reshuffling of genes, reproduction…), it can legitimately be said that population genetics reveals important properties of invariance through transformation. Time-reversibility is a more controversial subject. Here, the parallel with classical mechanics is much weaker. Time-reversibility is unquestionable in some stochastic models, but at the cost of a special, probabilistic concept of reversibility. But it does not seem to be a property of the most basic deterministic models describing the dynamics of evolutionary change at the level of populations and genes. Furthermore, various meanings of ‘reversibility’ are distinguished. In particular, time-reversibility should not be confused with retrodictability.

Appendices

Appendices

Appendix A.1 provides a formal definition of symmetry by time reversal and of retrodictability . Appendix A.2 applies these definitions to some population genetics models. Appendix A.3 contains the Figs. 13.3, 13.4, 13.5, 13.6 and 13.7.

1.1 Time Reversibility

As discussed in the body of the text, time reversibility corresponds intuitively to a situation where the dynamics follow the same law before and after time reversal . Only deterministic dynamics are considered here.

1.1.1 Continuous Time

In continuous time, a dynamic is typically defined by differential equations . A fairly general definition of differential equations is admitted here:

$$ 0=\mathrm{F}\left({\upalpha}_1,\dots, \mathrm{x},\mathrm{dx}/\mathrm{dt},{\mathrm{d}}^2\mathrm{x}/{\mathrm{d}\mathrm{t}}^2,\dots, {\mathrm{d}}^{\mathrm{n}}\mathrm{x}/{\mathrm{d}\mathrm{t}}^{\mathrm{n}}\right) $$

(13.13)

where the α _i are parameters .

1.1.1.1 Mathematical Notion of Time Reversibility

Time reversal means substituting –t to t in an equation. Time reversibility describes situations where the equation F is still met after time reversal ; that is to say:

$$ 0=\mathrm{F}\left({\upalpha}_1,\dots, \mathrm{x},-1\times \mathrm{dx}/\mathrm{dt},{\left(-1\right)}^2{\mathrm{d}}^2\mathrm{x}/{\mathrm{d}\mathrm{t}}^2,\dots, {\left(-1\right)}^{\mathrm{n}}{\mathrm{d}}^{\mathrm{n}}\mathrm{x}/{\mathrm{d}\mathrm{t}}^{\mathrm{n}}\right) $$

(13.14)

1.1.1.2 Weaker Time Reversibility

We propose a weaker notion of time reversibility , where F is preserved even though the parameters α _i are modified through time reversal . Thus, the criterion becomes:

$$ 0=\mathrm{F}\left({\upalpha}_1^{\hbox{'}},\dots, \mathrm{x},-1\times \mathrm{dx}/\mathrm{dt},{\left(-1\right)}^2{\mathrm{d}}^2\mathrm{x}/{\mathrm{d}\mathrm{t}}^2,\dots, {\left(-1\right)}^{\mathrm{n}}{\mathrm{d}}^{\mathrm{n}}\mathrm{x}/{\mathrm{d}\mathrm{t}}^{\mathrm{n}}\right) $$

(13.15)

A supplementary condition is required for this notion to have a theoretical meaning. The ${\alpha}_i^{\hbox{'}}$ have to be possible values for the α _i . For example if α ₁ is a mass ${\alpha}_1^{\hbox{'}}$ has to be a positive number, but it may be different from α ₁.

1.1.1.3 Retrodictability

The Cauchy-Lipschitz theorem also called Picard–Lindelöf theorem, states that under very general hypotheses on the regularity of F, there is only one trajectory which goes through one initial condition. This theorem is the basis of the ability of the differential equation to define a deterministic process, but it also ensures that we can retrodict the past. Increasing or decreasing the time parameter makes no difference in this theorem. Note, however, that solutions to differential equations may be valid only for a limited time interval, either because a variable becomes infinite or takes a value that has no physical or biological meaning.

As mentioned in the main text, the issue of dynamics sensitive to initial conditions, such as the three-body problem in mechanics have not been discussed here. Let us just mention that in these cases, the impossibility of a perfect measurement of the initial conditions prevents empirical long term predictions and retrodictions because small differences have significant consequences.

1.1.2 Discrete Time

Dynamics defined by recurrence will now be discussed: p(t + 1) = g(p(t)).

1.1.2.1 Mathematical Notion of Time Reversibility

The transitions t to t + 1 and t + 1 to t follow the same law, that is:

$$ p(t+1)=g(p(t))\to p(t)=g(p(t+1)). $$

1.1.2.2 Weak Time Reversibility

By analogy with the continuous case, changes in the values of the parameters of g in this weaker notion of time reversibility are allowed:

$p\left( t+1\right)= g\left( p(t),{\alpha}_i,\dots \right)\to p(t)= g\left( p\left( t+1\right),{\alpha}_i^{\hbox{'}},\dots \right)$.

Note also that the new parameters must have a physical or biological meaning.

Example: p(t + 1) = p(t)/2 → p(t) = 2p(t + 1). In this case α = 1/2 and α ^′ = 2. See the next case for a dynamic that does not meet this criterion.

1.1.3 Retrodictability

Retrodictability corresponds to the fact that the reverse dynamic is deterministic . A system is retrodictable when a function h exists that satisfies:

$$ p\left( t+1\right)= g\left( p(t),\alpha, \dots \right)\to p(t)= h\left( p\left( t+1\right),{\alpha}^{\prime },\dots \right) $$

Function h may be completely different from g (even though they are of course related).

Example: $p\left( t+1\right)= p{(t)}^2\to p\left( t+1\right)=\sqrt{p(t)}$. This dynamic is retrodictable but not reversible, even in the weak sense.

Counter example: p(t + 1) = 10p(t) − ⌊10p(t)⌋, where ⌊x⌋ is the integer part of x (a digit is lost at every step). For example P(0) = 0.97511 .  . ., P(1) = 0.7511 . ., P(2) = 0.511… But with P ' (0) = 0.87511 .  . ., we get P ' (1) = 0.7511…, which is identical to P(1). This ambiguity prevents defining a deterministic reversed dynamic . This deterministic dynamic does not enable retrodiction.

1.2 Application to Models of Population Genetics

In this part, the reversibility of two classical models of selection in population genetics are discussed. In one case (asexual haploid population), the dynamic is reversible in the weak sense. In the second case (diploid , diallelic locus), the dynamic is retrodictable but not reversible.

1.2.1 Asexual Haploid Population: A Weakly Reversible Model

We start with a classical model of selection at a single locus in a haploid population. W _i is the relative fitness of the allele i and p _i is the proportion of allele i in the population. W(t) is the average fitness of the population at time t.

$$ {\mathrm{p}}_{\mathrm{i}}\left(\mathrm{t}+1\right)={\mathrm{p}}_{\mathrm{i}}\left(\mathrm{t}\right)\frac{{\mathrm{W}}_{\mathrm{i}}}{\mathrm{W}\left(\mathrm{t}\right)} $$

(13.16)

To assess the properties of this dynamic with respect to time reversal , let us write p(t) as a function of p(t + 1), and let us compare the result with Eq. (13.16).

$$ {\mathrm{p}}_{\mathrm{i}}\left(\mathrm{t}\right)={\mathrm{p}}_{\mathrm{i}}\left(\mathrm{t}+1\right)\frac{\mathrm{W}\left(\mathrm{t}\right)}{{\mathrm{W}}_{\mathrm{i}}} $$

(13.17)

Equation (13.17) can also be written:

$$ {\mathrm{p}}_{\mathrm{i}}\left(\mathrm{t}\right)={\mathrm{p}}_{\mathrm{i}}\left(\mathrm{t}+1\right)\frac{1/{\mathrm{W}}_{\mathrm{i}}}{1/\mathrm{W}\left(\mathrm{t}\right)} $$

(13.18)

Here 1/W _i seems to play the same role as W _i in Eq. (13.16), therefore we can write ${W}_i^{\hbox{'}}=1/{W}_i$. However, we have to verify whether 1/W(t) can be interpreted as the mean fitness in the reversed dynamic W^′(t + 1), where the new fitness coefficients are the ${W}_i^{\hbox{'}}$.

By definition, we have:

$$ \frac{1}{\mathrm{W}\left(\mathrm{t}\right)}=\frac{\mathrm{N}\left(\mathrm{t}\right)}{\mathrm{N}\left(\mathrm{t}+1\right)}={\mathrm{W}}^{\prime}\left(\mathrm{t}+1\right) $$

(13.19)

It is also possible to do this computation on the basis of the analytical expression of $\overline{W}(t)$ which leads to the same result. Then we can conclude that:

$$ {\mathrm{p}}_{\mathrm{i}}\left(\mathrm{t}\right)={\mathrm{p}}_{\mathrm{i}}\left(\mathrm{t}+1\right)\frac{{\mathrm{W}}_{\mathrm{i}}^{\prime }}{{\mathrm{W}}^{\prime}\left(\mathrm{t}+1\right)} $$

(13.20)

This equation has exactly the same form as Eq. (13.16). Therefore the dynamic is weakly reversible.

1.2.2 Sexual Diploid Population : A Retrodictable But Irreversible Model

A population with two alleles at a single locus is now considered. The relative fitnesses are W ₁₁ ≠ 0 and W ₂₂ ≠ 0 for homozygotes and W ₁₂ for heterozygotes, and W ₁₁ > W ₁₂ > W ₂₂. W(t) is the average fitness of the population. p and q are the proportions of the first and second allele in the population. A classical model for this situation is then:

$$ \mathrm{p}\left(\mathrm{t}+1\right)=\frac{{\mathrm{W}}_{11}{\mathrm{p}}^2\left(\mathrm{t}\right)+{\mathrm{W}}_{12}\mathrm{p}\left(\mathrm{t}\right)\mathrm{q}\left(\mathrm{t}\right)}{\mathrm{W}\left(\mathrm{t}\right)} $$

(13.21)

We use q(t) = 1 − p(t) and W(t) = W ₁₁ p ²(t) + 2W ₁₂ p(t)q(t) + W ₂₂ q ²(t).

This leads to a quadratic equation, where p(t) is the variable :

$$ \begin{array}{l}0=\left({\mathrm{W}}_{11}-{\mathrm{W}}_{12}-\mathrm{p}\left(\mathrm{t}+1\right)\left({\mathrm{W}}_{11}-{2\mathrm{W}}_{12}+{\mathrm{W}}_{22}\right)\right){\mathrm{p}}^2\left(\mathrm{t}\right)\\ {}\kern1.76em +\left({\mathrm{W}}_{12}-\mathrm{p}\left(\mathrm{t}+1\right)2\left({\mathrm{W}}_{12}-{\mathrm{W}}_{22}\right)\right)\mathrm{p}\left(\mathrm{t}\right)-\mathrm{p}\left(\mathrm{t}+1\right){\mathrm{W}}_{22}\end{array} $$

(13.22)

The reduced discriminant determines the solutions of such an equation. If the discriminant is positive, there are two mathematical solutions and no solutions if it is negative.

$$ {\Delta}^{\prime }=\left({\mathrm{W}}_{22}{\mathrm{W}}_{11}-{\mathrm{W}}_{12}^2\right)\left(\mathrm{p}\left(\mathrm{t}+1\right)-{\mathrm{p}}^2\left(\mathrm{t}+1\right)\right)+{\mathrm{W}}_{12}^2/4 $$

(13.23)

0 ≤ p(t + 1) ≤ 1, thus (t + 1) − p ²(t + 1) > 0.

If $\gamma ={W}_{22}{W}_{11}-{W}_{12}^2>0$, then clearly Δ^′ is positive. In the opposite case, it is simple to verify that the smallest possible value for Δ^′ is given by p(t + 1) = 1/2, which leads to Δ^′ = W ₂₂ W ₁₁/4 > 0. Therefore, Δ^′ > 0 always holds and the quadratic equation has two mathematical solutions . After computation we get:

$$ \begin{array}{ll}\hfill & \kern-1.3em \mathrm{p}\left(\mathrm{t}\right)\\ {}& \kern-1.8em =\frac{\left({\mathrm{W}}_{12}-{\mathrm{W}}_{22}\right)\mathrm{p}\left(\mathrm{t}+1\right)-{\mathrm{W}}_{12}/2\pm \sqrt{\left({\mathrm{W}}_{22}{\mathrm{W}}_{11}-{\mathrm{W}}_{12}^2\right)\left(\mathrm{p}\left(\mathrm{t}+1\right)-{\mathrm{p}}^2\left(\mathrm{t}+1\right)\right)+{\mathrm{W}}_{12}^2/4}}{\mathrm{p}\left(\mathrm{t}+1\right)\left({\mathrm{W}}_{12}-{\mathrm{W}}_{22}\right)+\left({\mathrm{W}}_{11}-{\mathrm{W}}_{12}\right)\left(1-\mathrm{p}\left(\mathrm{t}+1\right)\right)}\hfill \end{array} $$

(13.24)

Because W ₁₁ > W ₁₂ > W ₂₂, the denominator is always positive, and the relations between the roots and the coefficients show that the two solutions have different signs. The only solution which has biological meaning is therefore the positive solution:

$$ \frac{\mathrm{p}\left(\mathrm{t}\right)=\left({\mathrm{W}}_{12}-{\mathrm{W}}_{22}\right)\mathrm{p}\left(\mathrm{t}+1\right)-{\mathrm{W}}_{12}/2+\sqrt{\upgamma \left(\mathrm{p}\left(\mathrm{t}+1\right)-{\mathrm{p}}^2\left(\mathrm{t}+1\right)\right)+{\mathrm{W}}_{12}^2/4}}{\mathrm{p}\left(\mathrm{t}+1\right)\left({\mathrm{W}}_{12}-{\mathrm{W}}_{22}\right)+\left({\mathrm{W}}_{11}-{\mathrm{W}}_{12}\right)\left(1-\mathrm{p}\left(\mathrm{t}+1\right)\right)} $$

(13.25)

$$ \mathrm{p}\left(\mathrm{t}\right)=\mathrm{h}\left(\mathrm{p}\left(\mathrm{t}+1\right)\right) $$

(13.26)

Consider now time reversal . Equation (13.21) gives p(t + 1) as a function of p(t). This corresponds to function g in the definition of reversibility above (Appendix A.1.2). p(t + 1) = g(p(t)), is a rational function, that is to say a fraction of polynomials. When γ ≠ 0, the square root in the definition of p(t) as a function h of p(t + 1) implies that h is not a rational function. The difference between g and h is then more than just a change of coefficient. In this case, retrodiction is possible but time is not reversible in either the weak or the strong sense.

Interestingly, the case where the square root disappears, γ = 0, corresponds to ${W}_{12}=\sqrt{W_{22}{W}_{11}}$. Thus, in this case, the fitness of heterozygotes is the geometric average of the fitnesses of homozygotes, which can be interpreted as a form of linearity in the effects of the alleles . Reciprocally, when γ ≠ 0, the change of the form of the dynamics by time reversal stems from the non-linearity in the effects of the alleles in heterozygotes .

Fig. 13.5

Effect of selection when W ₁₁ < W ₁₂ = W ₂₂ (After Jacquard 1971). This is the typical case of selection against a recessive gene. Allele 1 is disadvantaged only as homozygote, and is eliminated very slowly when rare. W always increases. Selection against a recessive gene is the most frequent case among the standard situations represented in Figs. 13.3 through 13.7. Figure 13.8 represents the shape of the trajectory of gene frequencies for this case

Fig. 13.6

Effect of selection when W ₁₂ < W ₁₁ < W ₂₂ (After Jacquard 1971). The heterozygote is disadvantaged (underdominance). There are two equilibria. Δp is either negative or positive. W always increases, but moves towards two possible stable equilibria, corresponding to p = 0 and p = 1; $\widehat{p}$ is an unstable equilibrium , corresponding to a minimal value for W. The question is debated whether this situation is realized in nature or not, and what its biological meaning is. But it is probably a very rare case

Fig. 13.7

Effect of selection when W ₁₁ < W ₂₂ = W ₁₂ (After Jacquard 1971). Known as overdominance or ‘advantage of the heterozygote’. There are two unstable equilibria, p = 0 and p = 1, and one stable equilibrium, $\widehat{p}$, corresponding to a maximal value for W, which always increases. Δp is either negative or positive. This is the best-known polymorphic equilibrium, with ample proof of its existence in nature

Fig. 13.8

Selection against a recessive allele for various selective coefficients (From Roughgarden 1979). Compare with Fig. 13.5, which gives the underlying dynamics. As almost all trajectories in models of selection with discrete generations, this trajectory is obtained by computer iteration. s is the ‘selection coefficient’. The greater s, the stronger the selection against the genotype. The ‘coefficient of selection’ is a convenient way of treating the relative selective values. If the selective value of the favoured genotype(s) is 1, the selective value of the unfavoured genotype (here the recessive homozygote) is 1 – s

Fig. 13.9

Selection with heterozygote superiority (From Roughgarden 1979). Compare with Fig. 13.7, which gives the underlying dynamics. In this example, the selection against the homozygote aa is twice as intense as the selection against AA. All trajectories converge to the same value of $\widehat{p}$. If the two homozygotes were equally selected against, the equilibrium would be obtained for $\widehat{p}$ = 0.5