Natural selection can favor ratchet robustness over mutational robustness

The vast majority of fitness-affecting mutations are deleterious. How natural populations evolve to cope is a question of fundamental interest. Previous studies have reported the evolution of mutational robustness, that is, natural selection favoring populations with less deleterious mutations. By definition, mutational robustness provides a short-term fitness advantage. However, this overlooks the fact that mutational robustness decreases finite asexual populations’ ability to purge recurrent deleterious mutations. Thus, mutational robustness also results in higher risk of long-term extinction by Muller’s ratchet. Here, we explore the tension between short- and long- term response to deleterious mutations. We first show that populations can resist the ratchet if either the selection coefficient or the ratio of beneficial to deleterious mutations increases as fitness declines. We designate these properties as ratchet robustness, which fundamentally reflects a negative feedback between mutation rate and the tendency to accumulate more mutations. We also find in simulations that populations can evolve ratchet robustness when challenged by deleterious mutations. We conclude that mutational robustness cannot be selected for in the long term, but it can be favored in the short-term, purely because of temporary fitness advantage. We also discuss other potential causes of mutational robustness in nature.


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The vast majority of fitness-affecting mutations are deleterious. How natural populations evolve 34 to cope is a question of fundamental interest. One widespread hypothesis proposes the evolution 35 of mutational robustness, which would mean that deleterious mutations evolve to become less 36 deleterious (Krakauer and Plotkin 2001;Wilke and Adami 2003). Intuitively, it may seem that 37 organisms with higher mutational robustness will have a fitness advantage over those with lower 38 mutational robustness. Taking that intuition one step further, organisms with higher deleterious 39 5 91 The fitness landscape provides a unifying framework for exploring both mutational robustness 92 and ratchet robustness. First define genotype space, where spatially adjacent genotypes are 93 mutationally adjacent, and then project the fitness of each genotype over the genotype space. 94 Since mutational robustness means deleterious mutations have smaller selection coefficients, 95 regions of the landscape with higher mutational robustness are "flatter". By contrast, "steeper" 96 regions of the landscape have higher ratchet robustness, since selection coefficients are larger. 97 These considerations seem to suggest an intrinsic tension between mutational robustness and 98 ratchet robustness. Ratchet robustness can also be realized by increased ratio of beneficial to 99 deleterious mutation rates, represented on the fitness landscape by the increased probability of 100 going "uphill" rather than "downhill" when locally exploring the genotype space via mutations 101 (Goyal et al. 2012). 102

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The goal of this paper is to identify those features of the fitness landscape that are favored by 104 natural selection as a consequence of the fact that most mutations are deleterious. We investigate, 105 in isolation, the effect of selection coefficient, and the influence of ratio of beneficial to 106 deleterious mutation rates. We find that ratchet robustness protects populations from extinction 107 due to deleterious mutations and should therefore be favored in the long term, although in the 108 short term mutational robustness can also be selected for.    (Kimura 1969). 141 142 Simulations in Fig. 1 and Fig. 2  Muller's ratchet with reference to the time variance in fitness. In Fig. S2, we present the variance 147 in fitness across time at equilibrium for the simulations conducted in Fig. 1A in recorded fitness variance across replicated simulations, due to the stochastic nature of the 164 simulation. However, we find that the second source of variation dominates the other two by at 165 least one order of magnitude (not shown). Therefore, we portray only the second type of 166 uncertainty as error bars in

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We use individual-based simulations of finite populations of asexual organisms to study the 193 ability of selection and evolving ratios of beneficial to deleterious mutation rates to protect 194 populations from deleterious mutations. We first assume a constant ratio of beneficial to 195 deleterious mutation rates and examine evolutionary behavior as a function of fitness landscape 196 topography. We then allow the ratio of beneficial to deleterious mutation rates itself to evolve 197 while holding other aspects of the fitness landscape constant. 198 199 Effect of fitness landscape topology 200 We simulate evolution on four kinds of fitness landscapes with increasing complexity, while 201 holding the ratio of beneficial to deleterious mutation (ܷ / ܷ ௗ ) equal to 0.01. We define a 202 fitness peak as isotropic if the fitness of any genotype only depends on its number of deleterious 203 mutations, but not the identity of these mutations. We start by examining evolutionary behavior 204 on fitness landscapes with a single peak where every deleterious mutation shares the same effect 205 regardless of the current genome (Eqn. 1: Fig. 1A inset). This is, by definition, an 206 isotropic peak in the absence of epistasis. Next, we allow the effect of deleterious mutations to 207 increase (Eqn. 1: Epistasis means the dependence of mutational fitness effects on the genetic background. Without 245 epistasis ( Fig. 1), there is no evolution of selection coefficients, since each mutation will always 246 have the same effect. Put differently, epistasis determines how selection changes as deleterious 247 mutations accumulate, shaping both local mutational and ratchet robustness experienced by an 248 evolving population. In order to understand the role of epistasis, we consider the two simplest 12 cases: one where selection strength monotonically increases with the number of deleterious 250 mutations, and the other where selection strength monotonically decreases with the number of 251 deleterious mutations. In other words, we consider a single isotropic peak with either only 252 negative epistasis ( Fig. 2A) or only positive epistasis (Fig. 2C), respectively. 253 254 When epistasis is negative, as deleterious mutations accumulate, the local strength of purifying 255 selection increases, and consequently subsequent deleterious mutations become less likely to 256 accumulate. This hints at a negative feedback between the accumulation of deleterious mutations 257 and the tendency to accumulate more, which potentially could halt Muller's rachet (as previously 258 seen in Kondrashov 1994). On the other hand, when epistasis is positive, as deleterious 259 mutations accumulate, local selection weakens, and deleterious mutations can more easily 260 accumulate. This suggests a positive feedback between the aforementioned two forces, which 261 may render populations more vulnerable to the ratchet than those examined above. Importantly, 262 the critical ܷ ௗ will thus vary across the fitness landscape in response to this variation in the 263 strength of selection 264 265 Our simulations are consistent with this intuition (Fig. 2B, Fig. 2D). In the presence of negative 266 epistasis, when ܷ ௗ is lower than the critical ܷ ௗ at the peak, it is lower than the critical ܷ ௗ 267 anywhere on the landscape. Therefore, selection will drive the population back toward the peak 268 regardless of where on the landscape it is initialized (Fig. 2B). Even if ܷ ௗ is higher than the 269 critical ܷ ௗ at the peak, there exists a point on the landscape where selection exactly offsets such 270 ܷ ௗ , because purifying selection increases monotonically from the peak (see also S1 Text). If a 271 population is initialized below this point, selection locally will be strong enough to push 272 13 populations upward until this point is reached. If instead, a population is initialized above this 273 point, mutation will be strong enough to push the population downward to this point, but no 274 further. Therefore, even under high above this point will experience selection stronger than required to offset ܷ ௗ and evolve to the 282 peak, while populations initiated below this point suffer from selection weaker than needed to 283 offset ܷ ௗ , and will succumb to Muller's ratchet (Fig. 2D). Therefore, we refer to this point as 284 "point of no return". Importantly, even if parameter values are such that populations can 285 equilibrate at the peak, stochastic fluctuations will eventually take them across this point of no 286 return, after which they will succumb to the ratchet. Moreover, as We next demonstrate that populations can indeed evolve to occupy regions of the fitness 295 landscape with negative epistasis when challenged by deleterious mutations. To do so, we 296 construct a fitness landscape composed of two mutationally adjacent isotropic peaks featuring 297 opposite signs of epistasis (Fig. 3A). Populations finding themselves below the point of no return 298 on the positive epistasis side will experience selection weaker than needed to offset mutation and 299 their fitness will decline to the valley, similar to populations declining to the bottom of the 300 landscape in Fig. 2D. Moreover, populations above the point of no return will nevertheless 301 experience stochastic fluctuations and will be eventually carried over the point of no return, at 302 which point they will also evolve to the valley. However, upon arrival at the valley, strongly 303 beneficial mutations become available, drawing populations onto the negative epistasis side of 304 the valley, after which they quickly climb to MSDE (Fig. 3B). (Note that both sides of the valley, and consequently this behavior is driven entirely by differences in the 306 local strength of natural selection.) Since the two peaks share identical selection coefficients at 307 the peak, the positive epistasis side has uniformly higher mutational robustness but uniformly 308 lower ratchet robustness. This extends our understanding of the tension between mutational and 309 ratchet robustness and demonstrates that ratchet robustness, instead of mutational robustness, is 310 likely to evolve in response to high deleterious mutation rates.

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) place the evolving population on a region of the landscape exhibiting negative epistasis 319 (Fig. 4A). Concretely, among all the mutational paths leaving the peak, a proportion ‫‬ of them 320 show positive epistasis, while the remaining fraction

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show negative epistasis. We assume 321 that all first mutations share identical fitness effects, so that there is no immediate fitness 322 advantage to lineages entering the region with negative or positive epistasis. However, for all 323 genotypes with greater than one deleterious mutation, fitness is necessarily higher in regions of 324 the landscape with positive epistasis than in regions with negative epistasis (Eqn. 1). In other 325 words, regions of the landscape with positive epistasis have both higher fitness and higher 326 mutational robustness (but lower ratchet robustness) compared with ones with negative epistasis. 327 Note that the only mutational path between regions is through the peak. We  Fig. 4B, Fig. S3B negative epistasis region is still sustained despite lower fitness, thanks to net mutational inflow 342 from the subpopulation at the peak (Table S1). In essence, the subpopulation on the negative 343 epistasis region is at mutation-selection balance: constantly being purified by selection but being 344 regenerated by mutation from the peak. 345 346 However, after ܷ ௗ increases enough that the peak can no longer be sustained (here, the proportion of the population at the peak become negligible (Fig. 4B, Fig. S3C). As a result, 348 the subpopulation residing on the negative epistasis region of the fitness landscape is 349 mutationally disconnected from the peak and is quickly wiped out by selection. subpopulations on the positive epistasis region of the landscape will be favored due to their 360 short-term fitness advantage (Fig. 4C). Such an advantage is amplified by higher deleterious mutations occur, this suggests that another negative feedback may exist between the 387 accumulation of deleterious mutations and the tendency to accumulate more (Goyal et al. 2012). 388 This is reminiscent of the one due to negative epistasis seen above ( Fig. 2A). To gain more 389 insight into this effect, we compare equilibrium fitness in populations evolved when ܷ / ܷ ௗ 390 is allowed to change with that seen in populations evolved under fixed ܷ / ܷ ௗ (Fig. 5). As mechanisms responsible for such negative feedback may be varied, realized via negative 402 epistasis (Fig. 2 & 3), or increasing ܷ / ܷ ௗ (Fig. 5). We note that many surveys of biological 403 The key distinction is that mutational robustness requires tolerating heritable perturbations, 474 which inevitably alters the "starting point" of future generations. Such heritable decay is intrinsic 475 to Muller's ratchet. By contrast, selection for environmental robustness entails non-heritable 476 environmental perturbation. Consequently, the short-term advantage of environmental robustness 477 22 is not offset by any long-term cost, accounting for the absence of an "environmental ratchet". In 478 summary, while mutational robustness may be widespread in nature, we suggest one alternative 479 interpretation for its evolution: namely as a correlated consequence of selection for 480 environmental robustness (de Visser et al. 2003). We address another possibility next. 481 482 Long-term and short-term fate of mutational and ratchet robustness 483 Our findings also illustrate that mutational robustness can be selected for in the short term, 484 despite the danger it imposes in the long term (Fig. 4). This suggests another mechanism for the 485 existence of mutational robustness in natural populations. How populations overcome this 486 shortsightedness remains an open question. One possibility described in a previous study 487 (O'Fallon et al. 2007) is that subdivision can protect populations from myopic selection for 488 mutational robustness. Because selection is more effective at purging deleterious mutations in 489 demes dominated by ratchet robust individuals, in that study net dispersal rates were higher from 490 those demes, and the population in total was thus enriched for such individuals in spite of the 491 short-term disadvantage. We predict that any population structure capable of hindering rapid 492 fixation of mutational robustness will help natural selection favor ratchet robustness. However, a 493