Complex multilocus population dynamics under mixed reproduction systems Zeev M. Frenkel, Valery M. Kirzhner, and Abraham B. Korol Institute of Evolution and Department of Evolutionary and Environmental Biology, University of Haifa, Mt. Carmel, Haifa 31905, Israel E-mail:
[email protected] Abstract: Here we analyze the effect of deviation from pure sexual reproduction (panmixia) on complex dynamics of multilocus genetic systems subjected to cyclical selection. A rich spectrum of complex dynamic patterns is displayed by this model. It appears that complex dynamics can be very robust to variation of mating parameters and recombination rate. We demonstrate that sex and recombination do not usually simplify the population dynamics of multilocus genetic systems, despite previous generalizations based on single-locus models with selection caused by antagonistic species interaction. Keywords: complex population dynamics, stability of complex dynamics, evolution of reproduction type
1. Introduction Mendel lows by themselves do not produce complex trajectories of corresponding genetic systems (Lyuich, 1992), while this can be possible upon selection. The first (and probably the last) example of complex dynamics for discrete evolutionary operator with constant selection was found by Hastings (1981) using computer simulations, although his cycle is, seemingly unstable. Clearly, if selection is frequency-dependent, a wide diversity of complex limiting behavior (CLB) can be easily demonstrated, from cycles to strange attractors. We have shown earlier that complex dynamics can also be generated by another mechanism, cyclical selection (Kirzhner et al., 1994). It appeared that the ability to display complex dynamics depends also on the reproduction system of the population, pure sexual (panmixia), selffertilization, asexual (clonal), and mixed (various proportions of these, Kirzhner et al., 2004). Temporally varying selection is considered one of the potential mechanisms of evolution of sex and recombination (Otto & Lenormand, 2002). Population dynamics of genetic systems under temporary varying selection can contain complex limiting behavior like auto-oscillations with a long period ("T-cycles" and "supercycles") and chaos-like phenomena. One of the generalizations derived from a study of single-locus ecological-genetic models was that sex reduces the likelihood of complex dynamics and chaos (Ruxton, 1995; Doebeli, 1995; Flatt et al., 2001). According to these authors, the disadvantages of complex dynamics is that population does not stay in optimal state (or does not undergo optimal cyclical fluctuations), exhibits more
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fluctuations, and, therefore has higher probability to loose useful alleles or allele combinations. These disadvantages can lead to invasion of CLB populations allele simplifying system behavior (Doebeli, 1995). In contrast to single-locus models dealing with only meiotic segregation and syngamy components of sex (considered by Ruxton, 1995; Doebeli, 1995, Flatt et al., 2001), we considered multilocus model with genetic recombination, an immanent component of sex. We considered several situations where recombination rates, proportion of asexual progeny, fraction of offspring resulting from selfing were fixed in population or controlled by polymorphic modifier loci. Stabilizing selection with cyclically varying optimum was used as the simplest model of multilocus cyclical selection. In this case, selection coefficients in the evolutionary operator are not depending on population state (i.e., this is not a frequency- or density-dependent selection). Selection of this type can arise under cyclical fluctuation of external ecological factors, temperature, humidity, and others. Also, it may appear in some epidemiological models of infectious diseases (Olsen & Schaffer, 1990). In this paper we analyze the effect of deviation from pure sexual reproduction (panmixia) on complexity of system dynamics. We demonstrate that complex dynamics can be very robust to variation of mating parameters and recombination rate. Our main result is that sex and recombination do not usually simplify the population dynamics of multilocus genetic systems, despite previous generalizations based on single-locus models. 2. The model 2.1 Diploid stabilizing selection for trait with cyclically varying optimum We consider behavior of the multilocus genetic system under stabilizing with periodically changing optimum (Charlesworth, 1993). Let genetic system consist of L the di-allelic loci. Denote by An and an two possible allele at locus n (n=1,..,L). Trait value u is defined for every diploid genotype (i, j) consisting of haplotypes i and j as u(i, j)=!n un(i, j)
(1)
where un(i, j) is the effect of the n-th locus of the genotype (i, j) on the selected trait equal to 2dn, dn(1+hn) or zero for genotypes AnAn, Anan and anan respectively. Here dn is an additive trait effect of allele An over allele an; hn is a dominance trait effect between alleles An and an (-1!hn!1). Fitness coefficients for genotypes (i, j) are defined as wi,j=F(u(i, j)-u0), where u0 is the trait optimum selected for at current generation. We used Gaussian fitness function F(y)=exp{-y2/2"2} that is widespread in population genetics. The above system was studied numerically, under different types of cyclical selection regimes (u0=u0(t)=u0(t+T)), conditioned by an ordered set ((u0(!),#(!)))!=1,2,…,p, where u0(!) is the selected optimum at the !th state, "(!) is the longitude of the !th state, and T= "(1)+"(2)+…+"(p) is the period length.
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2.2 Panmixia mixed with vegetative reproduction (apomixes) or selfing We consider a population genetics model with a mixed breeding system that includes panmixia, selfing, and asexual reproduction (e.g., apomixis or vegetative propagation). Let each individual with genotype composed of haplotyes i and j produce part ppanm,ij of the progeny by random mating resulting in sex-derived zygotes, part pselfing,ij by pure self-fertilization, and the remainder pasex,ij being a result of precise copying of the maternal zygote; thus ppanm,ij + pselfing,ij + pasex,ij =1 (Marshall & Weir, 1979). If these parameters are equal for all genotypes, then indices ij can be omitted. Denote by zij (zji = zij; zij"0; !i,jzi,j=1) the frequency of zygotes of the (i, j) type; correspondingly, zpanm,ij, zselfing,ij and zasex,ij denote offspring frequency (for individuals of the (i, j)-type) which is produced by random mating, by selfing, and asexually, respectively. Therefore, zij = zpanm,ij + zselfing,ij + zasex,ij = zij ppanm,ij + zij pselfing,ij + zij pasex,ij .
(2)
Transition to the next generation can be described as follows: xi(t+1) is a frequency of haplotype i among haploids produced by generation t.
$ j ,k Pjk %l z panm, jk (t ) , $l $ j ,k Pjk %l z panm, jk (t )
(3)
* z apomixis, ij (t ) ' y ij (t ' 1) ' $ Pkl %i Pkl % j z selfing , kl (t )( , Wmean (t ) +, k ,l )
(4)
xi (t ' 1) &
z ij (t ' 1) &
wij
where Pij#k is the probability of getting a gamete k from zygote (i,j), wij is the fitness of zygote zij(t), and Wmean(t) is the mean fitness of population in generation t, yij(t+1) = xi(t+1) xj(t+1) !k,l zpanm,kl(t) is offspring frequencies obtained via panmixia. 3. Results The dynamic patterns are illustrated by attractors. We distinct three CLB types: (1) Limiting cycle with integer period length dividable by period length of the selection fluctuations (T-cycle). Phase portrait for attractor of this dynamics consists of several points. (2) Limiting behavior characterized by a non-integer period length (supercycle). Phase portrait for attractor of this dynamics can contain continuous curves. (3) Dynamics characterized by strong dependence of the trajectory on even small variations of the initial state (chaotic-like dynamics). Then, like in “stochastic situations”, it is impossible to predict the trajectory of certain deterministic systems if the initial state is not determined exactly. We characterize the level of chaos in the system dynamics by the value of Lyapunov exponent $ (Sandri, 1996). If $ is positive, then phase portrait for attractor of this dynamics can contain filled continuous shapes.
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3.1 Robustness of CLB with respect to variations in the breeding system To demonstrate that CLB can be robust to variations in the breeding system, we consider a system with pure panmixia and mixtures of panmixia with selfing and/or apomixis (Fig. 1). In the case of full panmixia (ppanmixia =1) we obtain a stable supercycle (see Fig. 1a). Changing the breeding system from pure panmixia to a mixture of panmixia and vegetative reproduction (decreasing ppanmixia and increasing pasex), or panmixia and selfing (decreasing ppanmixia and increasing pselfing), had no effect on the type of CLB for a range of parameters: 0 % pasex %0.17 (Fig. 1b) and 0 % pselfing % 0.26 (Fig. 1c). Moreover, CLB was preserved even in the case of simultaneous deviation from full panmixia by a combination of asexual reproduction and selfing (Fig. 1d).
Figure 1. Robustness of CLB with respect to variation in the breeding system. Three-locus system was considered with r12 = r23= 0.01; d1=d2=d3=1; h1=0.23, h2= -0.1, h3=0.23; p=2; "1= "2=1; u0(1)=5.7, u0(2)=0.6; #2 = 2.5. CLB is displayed under full panmixia and under large deviations from panmixia (e.g., for all pasex %0.17 or pselfing %0.26 this system displays a stable supercycle, but with further increasing of pasex or pselfing only one polymorphic steady state, i.e. single limiting cycle, characterizes the system). The supercycles also proved resistant to a rather large range of simultaneous deviation of pasex and pselfing from zero. A stable supercycle was observed in the case of: (a) full panmixia ppanmixia=1; (b) mixed panmixia and selfing, ppanmixia=0.75 and pselfing =0.25; (c) mixed panmixia and apomixis, ppanmixia=0.83 and pasex=0.17; and (d) mixture of panmixia with selfing and apomixis, ppanmixia=0.8, pselfing =0.1, pasex=0.1. Here and in the next phase portraits, the phase points represent pairs of system characteristics for time moments corresponding to ends of selection periods (i.e., generation numbers dividable by period length of selection).
3.2 Robustness of CLB with respect to variation in recombination rates CLB can be displayed by multilocus systems with weak, intermediate, or tight linkage between selective loci. To ascertain that such important characteristic of sex as recombination does not necessarily simplify the population dynamics, we analyzed various systems at different recombination rates. For example, a 4-locus selected system with d1=0.02, d2=0.03, d3=0.16, d4=0.34; h1= h2= h3 = h4=0; p = 2; "1= "2=1; u0(1)=0, u0(2)=1.1; #2 =0.015; ppanmixia =1 displays a stable polymorphic single limiting cycle for small recombination rates r12=r23=r34=r!0.045 (Fig. 2a). Within this range, for recombination rate r . [0.02-0.045] dumping oscillation around this single limiting cycle were observed. For r=0.046 stable supercycle characterizes limiting behavior of the system. This CLB is observed for all recombination rates r from 0.046 up to
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0.5 (free recombination). With increasing recombination the range of changes of allele frequencies along the supercycle became wider (Fig. 2b,c,d).
Figure 2. Robustness of CLB with respect to variation in recombination rates. Four-locus system was considered with d1=0.02, d2=0.03, d3=0.16, d4=0.34; h1= h2= h3 = h4=0; p = 2; "1= "2=1; u0(1)=0, u0(1) =1.1; #2 =0.015; ppanmixia =1 for different levels of recombination rate r=r12=r23=r34. (a) r=0.045, (b) r=0.05, (c) r=0.1 and (d) r=0.5. This system displays a stable supercycle for all recombination rates r from interval [0.046, 0.5]. Higher recombination rate causes here wider changes of allele frequencies (b,c,d); recombination rates down to 0.045 result in simplification of system limiting behavior to single polymorphic limiting cycle (a).
3.3 Complexity of limiting behavior as the number of points in attractor In spite of the expectations from some previous studies where the dynamic complexity resulted from antagonistic interactions (Ruxton, 1995; Doebeli, 1995; Flatt et al., 2001), no such directed effect of sex and recombination was found in our study. In particular, examples in Figs. 2 and 3a allow concluding that increasing either recombination rate or proportion of sexually derived progeny may increase rather than decrease the area bounded by attractor (i.e. amplitude) of the system dynamics. These examples show that the previous studies cannot be considered as a basis for too far going generalizations about the effect of sex on CLB properties. To illustrate that more sex and recombination may increase rather than decrease the number of points in the attractor, we present bifurcation diagrams with recombination rate, proportion of sex, and proportion of selfing as bifurcation parameters (Fig.3). These examples are characterized by small recombination rates. It seems to be more complicated to find examples of robust T-cycles (with respect to small changing in system parameters and initial state) with intermediate period length in cases of high recombination rate. Nevertheless, we found examples with supercycles with high recombination and even free recombination (see Fig. 2). It is ought to note that set of points in every trajectory is enumerable (because we employed a model with non-overlapping generations). However, it is difficult to test if topological closure (set of limiting points) of phase portrait of supercyclic trajectory is enumerable.
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Figure 3. Bifurcations of limiting attractor, with r, pselfing, and pasex as bifurcation parameters, for a system displaying T-cycle. Graphs illustrate projection of attractors on axis of allele A1 frequencies for different values of bifurcation parameter (BP): (a) BP is rate of sex (panmixia) vs. asex (apomixix). (b) BP is the rate of selfing in the case of pure sexual reproduction. (c) BP is recombination rate in the case of full panmixia. Presented examples express that it is not probably true, that more sex and recombination simplifies the system limiting behavior (at least in the sense of number of points in attractor). Parameters of the systems: (a) d1=d2=d3=1; h1=0.9, h2=0.45, h3=0.9; p=4; "1="3=4, "2="4=2; u0(1)=5.7, u0(3)=1.5, u0(2)=u0(4)=3.0; #2=0.30; r=r12=r23= r34=0.01. Rate of sex (panmixia vs. apomixis) changing from 0.3 to 1.0. (b) The same system like in (a) with recombination rates r=r12=r23=r34=0.007; proportion of progeny reproducing by random mating (panmixia vs. selfing) changing from 0.7 to 1.0. (c) d1=d2=d3 =1; h1=0.9, h2=0.5, h3=0.9; p=4; "1="3=3, "2="4=1; u0(1)=5.6, u0(3)=1.3, u0(2)= u0(4) =3.0; #2=0.21; ppanmixia=1. r=r12=r23=r34 changing from 0.004 to 0.012.
3.4 Complexity of limiting behavior measured as divergence of the system trajectories caused by small variations of the initial state Chaotic dynamics of deterministic systems is considered as an extreme manifestation of complexity of non-linear dynamics. The degree of chaos can be estimated with calculation of Lyapunov exponent value $. The example presented in Figure 4 shows that chaos can be characteristic of the system behavior in a wide range of proportion of psex=1-pasex. Moreover, in contrast to the statements that sex simplifies population behavior, complex dynamics was observed only for relatively high values of psex. Thus, for psex . [0.49, 0.876] we found a T-cycle with period length equal to two periods of selection changing; for psex . [0.877, 0.9156] – T-cycle with period length equal to four periods, and for psex>0.9157 T-cycles, supercycles and chaotic-like dynamics are characteristic of system behavior. This system displays chaotic behavior with very high positive $ ($&0.05) even in the case of pure panmixia (Figs. 4a). With increasing psex, complexity may change non-monotonically. In the presented example, starting from psex=0.96 we observed a trend of increasing complexity ($) with increasing level of sex. Negative values of $ in Fig. 4 correspond to stable T-cycles (see Fig. 4b). Three attractors of the presented system with different level of psex are shown in Fig. 4c, d and e. 4. Discussion Our results show that complex dynamics can be very robust to variation of mating parameters and recombination rate. In particular we found that more sex and recombination do not usually simplify the population dynamics of genetic systems, despite previous generalization based on results with single-
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locus models. In general, depending on system parameters, more sex and recombination can either simplify or complicate the system dynamics.
Figure 6. Example of changing of level of chaos with increasing of rate of sex (panmixia vs. asex). Parameters of the model: d1 = d2 = d3 =1; h1 = 0.9, h2 = 0.45, h3 = 0.9; p = 4; "1 = "3 = 3, "2 = "4 = 3; u0(1)= 5.5, u0(3)= 1.2, u0(2)= u0(4)= 3.1; #2 = 0.20; r=r12=r23=r34=0.005. Rate of sex (panmixia vs. apomixis) changing from 0.93 to 1.0. (a) Lyapunov exponent $ value; (b) Bifurcations of limiting attractors in projection on P1 = frequency of allele A1; (c, d and e) Examples of nice attractor form changing with changing of psex=1-pasex: (c) ppanmixia=0.95; (d) ppanmixia=0.996; (e) ppanmixia=1.0.
The foregoing statement that sex and recombination do not necessarily simplify the system dynamics is also relevant for systems with antagonistic selection. Numerous examples of CLB in such systems were described in literature (Hamilton et al., 1990; Korol et al., 1994; Gavrilets & Hastings, 1995; Kirzhner et al., 1999). Kirzhner (unpublished) found that chaotic-like behavior and T-cycles can alternate with increasing of recombination rate in models of host-parasite interactions (not shown). Ruxton (1995) demonstrated that in some cases of single-locus competition, increased rate of panmixia (vs. selfing) may also complicate the system dynamics. Using single-locus host-parasite models, Flatt et al. (2001) found that increasing of proportion of sexual reproduction can complicate the dynamics from T-cycles to chaotic-like. It can also result in doubling of Tcycle period. Nevertheless, these examples were considered by their authors as small exceptions from a general tendency in wide range of parameters changes. Our interpretation is that the predominance of this pattern was mostly due to their use of single-locus models reflecting only meiotic segregation and syngamy, and neglecting recombination as a major sex component. Acknowledgements: Israeli Ministry of Absorption, Israeli Ministry of Science and the University of Haifa Graduate School supported this study.
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