Alternating lysis and lysogeny is a winning strategy in bacteriophages due to Parrondo’s Paradox

Temperate bacteriophages lyse or lysogenize the host cells depending on various parameters of infection, a key one being the host population density. However, the effect of different propensities of phages for lysis and lysogeny on phage fitness is an open problem. We explore a nonlinear dynamic evolution model of competition between two phages, one of which is disadvantaged in both the lytic and lysogenic phases. We show that the disadvantaged phage can win the competition by alternating between the lytic and lysogenic phases, each of which individually is a “loser”. This counter-intuitive result recapitulates Parrondo’s paradox in game theory, whereby individually losing strategies can combine to produce a winning outcome. The results suggest that evolution of phages optimizes the ratio between the lysis and lysogeny propensities rather than the phage reproduction rate in any individual phase. These findings are expected to broadly apply to the evolution of host-parasite interactions.


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Introduction 33 Bacteriophages outnumber all other reproducing biological entities in the biosphere 34 combined, with an estimated instantaneous total of about 10 31 virions across all biomes 35 (Cobián Güemes et al., 2016). Bacteriophages reached this hyper-astronomical 36 abundance using two basic strategies of infection that are traditionally classified as lytic 37 and temperate. Lytic phages enter the host cells and immediately take over the cellular 38 machinery to produce progeny virions, followed by a programmed burst of the cell 39 (lysis) (Young, 2014) which releases progeny virions into the environment for 40 subsequent rounds of infection. In contrast, temperate phages 'decide' to follow the lytic 41 or lysogenic strategy at the onset of infection. In the lysogenic strategy, the phage DNA 42 stably integrates into the host genome becoming a prophage that is inherited by the 43 progeny during cell division and thus propagates with the host without lysis of the host 44 cells. However, upon sensing an appropriate signal, such as DNA damage, a prophage 45 'decides' to end lysogeny and reproduce through the lytic pathway (Oppenheim et al.,46 2005; Ptashne, 2011). Given that an estimated 10 23 infections of bacteria by 47 bacteriophages occur every second (Hendrix, 2003), with profound effects on the global 48 ecology as well as human health (Cobián Güemes et al., 2016;Liu et al., 2016;49 Manrique et al., 2017), the evolutionary processes that shape phage replication 50 strategies of fundamental biological interest and importance. 51 The ability of temperate phages to decide between lysis or lysogeny has drawn 52 considerable attention of theorists aimed at quantifying the conditions where one 53 strategy prevails over the other or, in other words, deciphering the rules of phage lysis 54 vs lysogeny decisions. Temperate viruses that choose lysogeny are effectively 55 4 constrained by cellular binary fission, whereas lytic replication can produce bursts of 56 hundreds and thousands of progeny virions from a single cell. A foundational theoretical 57 study asked the question simply: why be temperate (Stewart and Levin, 1984)? The 58 potential benefits of a non-lytic strategy are realized when host cell densities are too low 59 to support lytic growth that would otherwise collapse one or both populations, and 60 furthermore, the frequency of encounters of the phage particles released upon host lysis 61 with uninfected host cells is low. In essence, lysogeny is advantageous at hard times 62 (Stewart and Levin, 1984). Several recent formal model analyses agree that lysogeny is 63 favored at low host cell density (Maslov and Sneppen, 2015;Wahl et al., 2019;Weitz et 64 al., 2019). However, lysogeny also appears to be the dominant behavior at high host cell 65 abundance (Knowles et al., 2016). The mechanisms driving viruses towards lysogeny at 66 both low and high host cell abundances are not well understood, but differential cellular 67 growth rates and viral adsorption rates appear to contribute (Luque and Silveira, 2020). 68 Collectively, these studies underscore the importance of density-dependent dynamics for 69 infection outcomes. 70 The paradigm for the decisions temperate phages make on the lytic or lysogenic 71 pathway upon infection is Escherichia phage Lambda. Seminal work with Lambda has 72 demonstrated that lysogeny is favored when multiple Lambda virions coinfect the same 73 cell (Kourilsky, 1973). The standard interpretation is that coinfection is a proxy 74 measurement for host cell density, driving Lambda towards lysogeny at low density 75 (Herskowitz and Hagen, 1980). The genetic circuitry underlying Lambda's "lysogenic 76 response" has been meticulously dissected over decades of research (Casjens and 77 Hendrix, 2015;Oppenheim et al., 2005;Ptashne, 2004) and continues to divulge new 78 5 mechanistic determinants (Lee et al., 2018;St-Pierre and Endy, 2008;Trinh et al., 2017;79 Zeng et al., 2010). Directed evolution of Lambda yields mutants with different 80 thresholds for switching from lysogeny to lysis (induction) and such heterogeneity has 81 been observed in numerous Lambda-like phages (Deng et al., 2014;Murphy et al., 2008;82 Refardt and Rainey, 2010). Moreover, the vast genomic diversity of phages implies a 83 commensurately diverse repertoire of lysis-lysogeny circuits, and experiments with 84 phages unrelated to Lambda have revealed a variety of ways evolution constructed these 85 genetic switches (Broussard et al., 2013;Erez et al., 2017;Ravin et al., 1999 P . This counter-intuitive result recapitulates a phenomenon that is known as Parrondo's 97 paradox in game theory (Harmer and Abbott, 1999). Thus, alternating between lysis and 98 lysogeny appears to be intrinsically beneficial for a phage.

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Alternating between losing lysis and lysogeny strategies results in a winning 103 strategy for a phage 104 A mathematical model was designed to measure the outcomes of infection between 105 two competing phages that differ in their rates of adsorption, mortality, burst size (i.e., 106 number of progeny virions per infection), and, principally, their propensities for lysis or 107 lysogeny. One of the key parameters of infection driving viral replication strategies is 108 host cell density (Knowles et al., 2016;Thingstad, 2000;Weitz et al., 2015). Thus, to 109 realistically capture phage lysis-lysogeny switches, the decisions are not represented by 110 a constant but rather by a probability function (

Fig. 1 | Competition between two phages with different properties. 156
The parameter values are from initial condition 1 in Supplementary Table S1 unless stated otherwise. The two winning strategies employed by 1 P yielded contrasting results for the 165 competitor, namely, 2 P was either driven to extinction or persisted stably at a lower 166 relative abundance. Therefore, we examined the range of lysis-lysogeny decisions by 167 1 P that allow 2 P to coexist over a longer timescale. The timescale of the experiment 168 was increased fivefold relative to that in Fig. 1 Fig. 2A). If the probability of 1 P to lysogenize is increased 173 ( 1 0.7  = ), 1 P wins the competition without driving 2 P to extinction (Fig. 2B).
Interestingly, within the time intervals [1200,2000]  P is immediately driven to extinction after the first wave (Fig. 2C). Combinations of model parameters yielding Parrondo's paradox 196 We then ran a comprehensive simulation to determine the range of parameters that 197 support winning outcomes for 1 P , rather than using a single parameter set. The  higher abundance compared to 2 P when the carrying capacity is relatively large, but 213 the opposite is true when the carrying capacity is small (Fig. 3A), indicating that the 214 total number of cells in the environment affects the ability of a disadvantaged temperate 215 phage to outcompete an advantaged phage. Both phages increase in abundance when the 216 host cell mortality rate decreases (Fig. 3A), reflecting the shared 'interests' of integrated 217 phages and their host cells (Argov et al., 2017). Evidently, decreasing the mortality rate there is still a well-defined dividing region in which 1 P transitions from a loser to a 221 winner (Fig. 3D). In this range, K exerts no obvious effect on the competition result 222 ( Fig. 3B-C) because the total number of cells is relatively small and therefore the 223 growth rates of uninfected hosts ( h ) and lysogens ( l ) do not change drastically. Thus, 224 the mortality rates have a greater impact on the outcome of the competition compared to 225 13 the environmental carrying capacity. 226 The probability function () h  that models the phage lysis-lysogeny switch was 227 demonstrated to substantially impact the outcome of the competition ( Fig. 2A-C). Thus, 228 the effect of each of the four parameters of this function was evaluated across a wide 229 range of values ( Fig. 3E-H). The two parameters with the greatest influence on the 230 outcome of the competition in our initial experiments (Figs. 1-2) were  and  , 231 which set the lower and upper bounds of the probability of a phage to enter the lytic 232 pathway, respectively. A phage is driven towards the lytic pathway by increase of  233 and decrease of  (Fig. 4A). Comparison of the  values between the two phages 234 shows that 1 P outcompetes 2 P in a narrow range where the  value of 1 P is 235 approximately half that of 2 P (Fig. 3E), or in a broad range where  , the value of 2 P 236 is greater than ~0.15, (Fig. 3F). 1 P can also win the competition by changing the  237 and  values while competing against a temperate (Fig. 4B) or purely lytic (Fig. 4C) 238 competitor with fixed  and  values. With regard to the other parameters related to 239 the lysis-lysogeny switch, 1 P can dominate over relatively broad parameter spaces.  240 is the switch threshold, so that a large  causes a phage to switch from lysis to 241 lysogeny in the presence of a relatively large number of hosts (Fig. 3G). The parameter 242  is the switching interval, with a small value of  enabling quick transition to 243 lysogeny with the decline of h (Fig. 3H). Collectively, these results indicate that a 244 disadvantaged phage can win the competition by optimizing multiple parameters 245 involved in switching between lysis and lysogeny. 246 We next examined the life history traits that are traditionally analyzed in models of 247 viral fitness, including burst size (  ), infection rate ( () h  ) and induction rate ( ).

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Evidently, a large burst size results in a greater number of infectious progeny viruses, so 249 that the phage with the larger burst size will win the competition against th one with the 250 smaller burst size (Fig. 3I)

Fig. 3 | The regions of the parameter spaces yielding Parrondo's paradox in the competition 262
between two phages. 263 The parameter values are from initial condition 1 in Supplementary Table S1 unless stated otherwise.  (Fig. 4A) (Fig. 4B). This effect is crucial for the phage in an adverse environment. 288 The fitness of both phages is greater than the host fitness, and the phage in the lytic 289 phase is fitter than the phage in the lysogenic phase (Fig. 4B). In the competition 290 between the virulent phage and temperate phage (Fig. 4C) 304 We demonstrate here that a phage with inferior life history traits can outcompete a 305 competitor with superior traits over a broad set of conditions by switching between the 306 lysis and lysogenic infection strategies. Neither the lytic nor the lysogenic phase of the 307 disadvantaged phage is competitive on its own, yet a winning outcome is achieved by 308 alternating between the two strategies (Fig. 1). This counterintuitive result is formulated 309 as Parrondo's paradox, whereby alternating between two losing strategies can result in a 310 winning outcome. This effect is manifested at different levels of biological organization 311 (Cheong et al., 2019), such as the alternation between unicellular and multicellular 312 phases in organismal life history (Cheong et al., 2018), between phases of activity and 313 dormancy in predator and prey (Tan et al., 2020) or between nomadic and colonial life 314 styles (Tan and Cheong, 2017). 315 Several studies have examined the conditions in which lysogeny is more favorable 316 than lysis for phages. Vacillations in host cell population density, due to environmental 317 downturns or other factors, could lead to extended periods of low host availability that 318 are too low to support lytic infection (Stewart and Levin, 1984). Lysogenizing host cells 319 under these conditions allays the risk of population collapse (Maslov and Sneppen, 320 2015). As the host cell population recovers, this strategy of leveraging binary cellular 321 fission for vertical transmission becomes inferior to horizontal transmission (that is, 322 lytic reproduction) (Li et al., 2020;Weitz et al., 2019). Consistent with lysogeny being 323 favorable at low host cell availability, our experiments indicate that, when the number 324 of host cells oscillates, temperate phages that favor the lysogenic phase persist, whereas 325 those that prefer lysis as well as purely lytic phages go extinct ( Fig. 1-2).

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Fewer studies have investigated the parameters of infection that dictate the 327 outcome between two phages competing for a host cell. The ubiquity and biological 328 importance of inter-virus competition is perhaps best reflected by the broadly 329 distributed and diverse superinfection exclusion mechanisms that prevent secondary 330 infections of lysogens by other phages (Dedrick et al., 2017;Gentile et al., 2019;331 Mavrich and Hatfull, 2019). Mathematical models predict that temperate phages can 332 invade microbial populations in the presence of a competing lytic virus, so long as they 333 confer a minimal level of superinfection immunity (Li et al., 2020). In one of our 334 experiments, the disadvantaged phage indeed confers superinfection immunity to the 335 host cell during lysogeny and drives the competitor to extinction (Fig. 2). With the 336 decrease in the probability to lysogenize, the disadvantaged phage is instead driven to 337 extinction. Notably, we identified multiple regions of the parameter space where the two 338 phages can coexist, with the disadvantaged phage that alternates between lysis and 339 lysogeny reaching higher abundance (Figs. 3,4). 340 Our work identifies the conditions in which a disadvantaged phage can not only 341 coexist with but outcompete a phage with superior life history traits by cycling between 342 two losing strategies, a clear case of Parrondo's paradox. Parrondo's paradox applies to 343 large regions of the parameter space, with propensities for lysis or lysogeny being the 344 primary parameters of infection that determine this winning outcome (Fig. 2). It should 345 be emphasized, however, that many combinations of parameters favor a pure lytic or 346 lysogenic strategy. 347 On closer examination, the advantage of alternating modes of interaction with the 348 host by a virus does not appear truly paradoxical, being tightly linked with the 19 oscillations of the host population that are well known from the classic, 350 Lotka-Volterra-type prey-predator models (Roughgarden, 1979). Indeed, once the host 351 cell density drops, due to the cell lysis by the virus, the latter switches to lysogeny, but 352 when the host population recovers, switching to lysis becomes advantageous.
where  is the total number of cells, which equals  for each phage parameter were collected from recent analyses of phage lysis-lysogeny decisions 396 (Pleška et al., 2018;Sinha et al., 2017;Wahl et al., 2019). 397 398

Parameter Functions 399
The infection function () h  is modelled by the Holling type II functional response 400 (Rosenzweig and MacArthur, 1963), which is monotonically increasing. This function takes the form 401 where  is the efficiency, and  is the handling time that phages need to infect the host cell.      Table S1).

Supplementary Information
Alternating lysis and lysogeny is a winning strategy in bacteriophages due to Parrondo's