J. Theor. Biol. 2006 - PÃ gines de la UAB

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252 ARTICLE IN PRESS J.F. Fontanari et al. / Journal of Theoretical Biology 239 (2006) 247–256 linkage of all templates on a single chromosome (Maynard Smith and Szathma´ry, 1993; Santos, 1998). In that sense, the (partial) failure of group selection is welcome, otherwise it would be difficult to explain gene linkage, and hence the origin of chromosomes, if an arbitrary number of unlinked templates could easily be made to coexist within compartments. To verify how our conclusions are affected by the assumption that there are a countable infinity of vesicles, here we carry out a simple Monte-Carlo simulation to study the situation in which the number of vesicle is fixed to a finite value M. Explicitly, we re-interpret Eq. (2) as the relative fitness of vesicles composed of k 1 ; k 2 ; ...; k d template types and then use a transition probability analogous to Eq. (3) to generate the new vesicle population from the previous one. Essentially, the same stochastic procedure used to model the template replication inside each vesicle is used to model vesicle replication. The results for d ¼ 2 degenerate (s 2 =s 1 ¼ 1) templates fenced in M ¼ 10; 20 and 40 vesicles of capacity N ¼ 20 are depicted in Fig. 4 together with the analytical result for M !1. The data represent averages over 100 independent runs. The point to be noted here is that the behaviour pattern observed in the case of a finite number of vesicles does not differ qualitatively from the pattern predicted by the deterministic theory: the sole effect of finite M is to increase the value of the threshold g c so as to make coexistence more difficult. As expected, the agreement between our analytical results and the Monte-Carlo simulations improves as M increases and, in particular, they become practically indistinguishable for M4500. Note, however, that by fixing the number of vesicles we have ruled out the possibility of extinction of the vesicle population, a realistic ingredient that might pose a serious obstacle to template coexistence when g is set too close to 1. Nevertheless, our results apply provided the vesicle population is not extinct, since then it is likely to be very large. 4. The effect of the error tail The consequences of imperfect replication are catastrophic for templates isolated in compartments of finite capacity: even for a vanishing small mutation probability (u 0) and replication rate (s 0 =s l 0; l ¼ 1; ...; d), the error tail will be fixed in the compartment. In fact, though rare, mutations are recurrent and so they will prevent the fixation of the uncorrupted templates. In addition, the fixation of the error trail is very unlikely to occur because of its small replication rate, but once it does the game is over for the uncorrupted templates. A possible way out of this conundrum is to invoke again group selection—the selection at the vesicle level. The pressing question of how strong group selection must be in order to ensure the survival of templates confined in a vesicle is addressed in this section, where we will concentrate on the interplay between the group selection pressure and the mutation probability in the case of d ¼ 1 and 2 templates. Since in the previous section we have shown that introducing competition between templates does not alter qualitatively the behaviour pattern observed in the case of degenerate templates, henceforth we will consider the situation in which all d uncorrupted templates have the same replication rate, which we set to 1 without loss of generality, while templates in the error tail have replication rate 1 t with t 2 ½0; 1Š. First we consider the case of a single uncorrupted template type (d ¼ 1) in a vesicle of capacity N. Error-tail templates are not included in the initial assignment of templates to vesicles. In Fig. 5, we show the steady-state frequencies of vesicles with several template compositions. All these frequencies vanish at a critical value of the 1 0.2 0.8 0.15 20 0 0.6 16 Ω 2 0.4 40 20 10 Y 0.1 12 8 4 0.2 0.05 0 0 0.2 0.4 0.6 0.8 1 g Fig. 4. Fraction of vesicles with two degenerate types of templates in their composition O 2 as a function of the group selection pressure g for M ¼ 10; 20 and 40 vesicles of fixed capacity N ¼ 20. The solid line is the analytical result for M !1. 0 0 0.05 0.1 0.15 0.2 0.25 0.3 u Fig. 5. Steady-state frequencies of vesicles with i ¼ 0; 4; 8; 12; 16 and 20 copies of the template as a function of the mutation probability u for d ¼ 1, N ¼ 20, t ¼ 0:1 and g ¼ 0:5. We note that Yð20Þ !1 for u ! 0 and Yð0Þ !1 for u ! u c 0:26.
ARTICLE IN PRESS J.F. Fontanari et al. / Journal of Theoretical Biology 239 (2006) 247–256 253 mutation probability u c 0:26, except for Yð0Þ that tends to 1, indicating that the error tail has become fixed in all vesicles. This figure endorses the portrait of package models in equilibrium as a quasispecies of compartments with internal competition, as demonstrated for the stochastic corrector model (Szathma´ry and Demeter, 1987) and, in addition, demonstrates the existence of an error threshold u c beyond which the uncorrupted template is lost from the population. A better characterization of this threshold is given in Fig. 6 that shows the fraction of vesicles that contain at least one copy of the uncorrupted template O 1 as a function of the mutation probability for several values of the group selection pressure. The point here is to emphasize that u c goes to zero when the group selection pressure is turned off, in agreement with the qualitative reasoning presented in the beginning of the section. The effect of changing the vesicle capacity N, depicted in Fig. 7, is quite complex and somewhat puzzling. For small u, increasing N favours the persistence of the uncorrupted template as expected since the effect of drift is attenuated in this case. However, for large u the situation is reversed, resulting that low capacity vesicles become able to tolerate much higher mutation rates. The reason for that is as follows. According to Eq. (1), for large N the survival probability is significant only for those vesicles with a large number of templates. Vesicles with few templates have survival probability only slightly larger (order of 1=N) than the baseline. Thus, by diminishing the number of uncorrupted templates, large mutation rates will weaken the efficacy of group selection to maintain the templates in large vesicles. We note that in the deterministic regime O 1 becomes the step function O 1 ¼ 1 for uot and O 1 ¼ 0 otherwise. These results are summarized in Fig. 8 where the critical mutation probability is presented against the group selection pressure. For g ¼ 0 we find u c ¼ 0 regardless of the values of t and N; however, this critical value jumps to a finite value for any arbitrarily small value of g for t not Ω 1 1 0.8 0.6 0.4 0.2 500 100 20 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 u Fig. 7. Fraction of vesicles with at least one uncorrupted template O 1 as a function of the mutation probability u for d ¼ 1, g ¼ 0:5, t ¼ 0:1 and N ¼ 2; 10; 20; 100 and 500. u c 1 0.8 0.6 0.4 0.2 0.9 0.5 0 0.2 0.4 0.6 0.8 1 g Fig. 8. Critical value of the mutation probability u c above which the uncorrupted template is lost as a function of the group selection pressure g for d ¼ 1, N ¼ 20 and t ¼ 0:1; 0:5 and 0.9. 10 0.1 2 Ω 1 1 0.8 0.6 0.4 0.2 0.1 0.5 0.8 0.9 0 0 0.2 0.4 0.6 0.8 1 u Fig. 6. Fraction of vesicles with at least one uncorrupted template O 1 as a function of the mutation probability u for d ¼ 1, N ¼ 20, t ¼ 0:1 and g ¼ 0:1; 0:5; 0:8; 0:9 and 1. 1 too small. Although our numerical results cannot guarantee the existence of a real discontinuity at g ¼ 0, at least the extreme sensitivity of the threshold to a non-zero value of g is clearly demonstrated. We turn now to the analysis of the case of two uncorrupted template types (d ¼ 2) confined in vesicles of finite capacity. As before, error-tail templates are left out of the initial assignment of templates to vesicles. Fig. 9 illustrates the typical equilibrium behaviour pattern of the fraction of vesicles of capacity N ¼ 20 containing m ¼ 0; 1 and 2 uncorrupted template types (regardless of their numbers). The group selection pressure, set to g ¼ 0:5, can maintain the coexistence of the two template types in more than 90% of the vesicles in the absence of replication errors, u ¼ 0. Initially, increasing the mutation probability (up to u ¼ 0:05) results in an increase of vesicles carrying only one type of uncorrupted template (m ¼ 1) at the
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