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ARTICLE IN PRESS Journal of Theoretical Biology 241 (2006) 928–938 www.elsevier.com/locate/yjtbi Optimality models of phage life history and parallels in disease evolution J.J. Bull The Institute for Cellular and Molecular Biology, Section of Integrative Biology, The University of Texas at Austin, Austin, TX 78712, USA Received 20 October 2005; received in revised form 6 January 2006; accepted 24 January 2006 Available online 17 April 2006 Abstract Optimality models constitute one of the simplest approaches to understanding phenotypic evolution. Yet they have shortcomings that are not easily evaluated in most organisms. Most importantly, the genetic basis of phenotype evolution is almost never understood, and phenotypic selection experiments are rarely possible. Both limitations can be overcome with bacteriophages. However, phages have such elementary life histories that few phenotypes seem appropriate for optimality approaches. Here we develop optimality models of two phage life history traits, lysis time and host range. The lysis time models show that the optimum is less sensitive to differences in host density than suggested by earlier analytical work. Host range evolution is approached from the perspective of whether the virus should avoid particular hosts, and the results match optimal foraging theory: there is an optimal ‘‘diet’’ in which host types are either strictly included or excluded, depending on their infection qualities. Experimental tests of both models are feasible, and phages provide concrete illustrations of many ways that optimality models can guide understanding and explanation. Phage genetic systems already support the perspective that lysis time and host range can evolve readily and evolve without greatly affecting other traits, one of the main tenets of optimality theory. The models can be extended to more general properties of infection, such as the evolution of virulence and tissue tropism. r 2006 Elsevier Ltd. All rights reserved. Keywords: Evolution; Optimality; Bacteriophage; Virulence; Adaptation; Model 0. Introduction ‘‘Natural selection is not evolution.’’ Fisher prefaced his classic book with this caution to remind us that the two phenomena are not the same and to justify developing a theory of natural selection (Fisher, 1930). It is now widely acknowledged that the specific phenotypes favored by natural selection may not evolve, because of genetic constraints, for example. Nonetheless, models assuming natural selection of phenotypes are commonly used to generate predictions and explanations of evolution in the absence of genetic details. An extreme form of this cause–effect inference is the use of optimality models to generate predictions of life history phenotypes (Charnov, 1982, 1993; Krebs and Davies, 1993, 1997; Maynard Smith, 1982; Parker, 1978; Stephens and Krebs, 1986; Williams, 1966). Without knowing mutation rates, distributions of fitness effects, pleiotropy, or even population size, the E-mail address: [email protected]. 0022-5193/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2006.01.027 phenotypic optima predicted by those models are used as the expected trait values of organisms. Use of optimality models in this fashion is not without its harsh critics (Gould and Lewontin, 1979; Lewontin, 1978a, b; Pierce and Ollasen, 1987), but there has been no resolution of the disagreement nor much attempt at resolution. Perhaps the main reason for this stalemate is that the potential problems with the use of optimality models lie in the genetic architecture of the phenotypes being modeled, and there are few phenotypes for which we know the details well enough to evaluate the possible failings of optimality. This paper develops optimality models of two phenotypes important to phage fitness, lysis time and host range. The present work provides important extensions of lysis time evolution and develops a new perspective on host range: selection may favor avoidance of certain hosts, even when infection of those hosts generates viable offspring. The available, detailed understanding of the genetic bases of both phenotypes is reviewed briefly, allowing for ARTICLE IN PRESS J.J. Bull / Journal of Theoretical Biology 241 (2006) 928–938 unusual evaluation of the limitations and value of optimality models. In a more general context, these models of phage evolution can be extrapolated to pathogenic organisms and models of disease evolution. 1. Lysis with one host type Most characterized phages are lytic (Adams, 1959). Lysis is the phage equivalent of ‘‘big bang’’ or semelparous reproduction, a dissolution of the cell that releases all phage progeny produced in the cell up to that point, ending the life of the parent phage. Naively, one might suppose that lysis should be delayed until the phage has produced the maximum possible number of offspring from that cell. This optimum in fact applies in an extreme case, but it ignores the possibility that progeny released early—before offspring number has peaked—may be able to take advantage of an abundance of hosts and produce even more grandchildren than if the phage delayed lysis to gain a few more offspring (Abedon, 1989). Although the basic principles of lysis have been known for at least half a century, the first models of lysis time evolution were published in the last two decades (Abedon, 1989; Wang et al., 1996). Wang et al. provided a simple, graphical model to calculate the optimal lysis time. That model assumed that hosts permanently outnumbered phages and existed at a constant density, so that the phage population was expanding exponentially and indefinitely. In the long-term, therefore, phage population growth rate is b1=g per unit time, where b is burst size (fecundity) and g is generation time. Generation time is faster with shorter lysis times, and shorter lysis times are better for this reason. However, burst size increases with time to lysis, so longer lysis times are better for this reason. These two effects of lysis time impact fitness in opposite ways, and an intermediate lysis time maximizes fitness under appropriate conditions. Solving the Wang et al. optimum should be easy. However, a problem arises in calculating the appropriate generation time, because generation time includes the time from burst until the progeny phage encounter a new host, and this component is a random variable. Use of the average time to infection leads to a bias, perhaps because time to infection follows an exponential distribution, and the exponential is highly asymmetric around its mean. Abedon et al. (2001) simulated an infection process to empirically determine the optimum and avoid this bias; by this method, the bias in the Wang et al. optimum (using the mean time to infection) was serious at low cell densities, although it was minor at high cell densities. It is thus desirable to obtain an analytical expression for this more exact optimum. 929 1.1. The optimum lysis time Here, an approach to the optimum lysis time is offered that overcomes the problem in calculating generation time. This approach parallels that of Lenski and May (1994) in a somewhat different context. As with Wang et al. (1996), we will calculate the optimum lysis time for a phage cohort in an environment with a constant density of hosts (C), and with a phage density such that multiple infections of the same host are rare. The following differential equation describes the dynamics of phage density, assuming no change in cell density (modified from Levin et al., 1977): P_ ¼ Pðm þ kCÞ þ b edL kCPL , (1) where a superior dot indicates the derivative with respect to time; P is the density of free phage [P(t)]; PL is the density of free phage L time units in the past [P(tL)]; C is the density of uninfected cells; d is intrinsic death rate of cells, whether infected or not; m is the death rate of free phage; k is adsorption rate of phage to cells; b is burst size (number of phage per burst); L is the latent period, the time from infection to lysis (minutes). In contrast to the usual SIR models of infectious disease, this equation describes the free, infectious particles themselves. Infected hosts do not need to be described, because they invariably burst and disappear L minutes after infection, at which point they contribute only to the free-phage pool. Eq. (1) is linear. With constant host density, infections rare, and adsorptions distributed in time, the dynamics will tend to exponential growth (oscillations will decay), so that PðtÞ ¼ Pð0Þ elt . Eq. (1) then becomes (Bull et al., 2006) l elt ¼ ðm þ kCÞ elt þ kbC edL elðtLÞ (2a) or l ¼ m þ kCðb eLðdþlÞ 1Þ. (2b) The long-term, asymptotic population growth rate of the phage is l per unit time. For values of l that deviate from zero, the phage population would either go extinct (lo0) or exceed the host density (l40). The latter would either cause a collapse of host density (Levin et al., 1977) or multiple infections per host (Bull et al., 2006). In either case, the model assumptions no longer apply. Thus, the model is best interpreted as revealing the momentary effect of selection when hosts are rarely encountered by phage. One context in which an excess of hosts might be sustained for long periods of phage growth is a spatial one—the spreading wave of phage across a spatially uniform population of hosts. The model would then apply to the leading edges of the epidemic wave. The conditions assumed by the model can also be achieved experimentally, however. The transcendental equation (2b) is not tractable for further simplification. However, it may be solved for the lysis time optimum by setting the partial derivative of l ARTICLE IN PRESS J.J. Bull / Journal of Theoretical Biology 241 (2006) 928–938 930 with respect to L equal to zero and treating b as a function of L. This approach yields b_ ^ ¼ d þ l, b^ (3) where a hat 4 indicates a value at the optimum (Bull et al., 2006). We denote the quantity d þ l^ as the intrinsic growth rate of the phage, because it describes how fast the phage population is capable of growing, i.e. when cell death rate d is zero. The intrinsic growth rate is specific to cell density C, of course. The intrinsic phage growth rate likewise equals the maximum cell death rate at which the phage can avoid extinction. Wang et al. (1996) extracted published data that indicated an approximately linear accumulation of progeny phage within the cell, starting at the end of the eclipse period and extending well beyond the normal lysis time if lysis was abolished. Therefore, letting b ¼ RðL EÞ, where E is the length of the eclipse period and R is the slope of progeny accumulation after eclipse (Fig. 1), Eq. (3) becomes L^ ¼ E þ 1 d þ l^ . (4) The optimum lysis time is the sum of the eclipse period and the inverse of the intrinsic growth rate. This latter quantity is the time for the phage to increase its numbers by one log (or by a factor e), and represents a type of phage ‘‘generation time,’’ which shortens as the growth rate increases. Although the formula appears simple, l^ is a function of many parameters including lysis time, so it is not trivial to solve. The formula does become tractable if the phage population is at dynamic equilibrium, because l^ ¼ 0, and the optimum is then independent of all but eclipse and intrinsic bacterial death rate. Even when the burst size (slope R) growth rate is not zero, the formula is easily solved numerically. Curiously, this formula applies even when lo0 (provided d þ l40), yet the phage population is declining. Abedon et al. (2001) compared optima obtained by their method with optima obtained from the Wang et al. formula, using the linear accumulation function and the average time to adsorption. As an example, Wang et al. (1996) offered a specific set of values whose optimum was 281.06 min. The corresponding Abedon et al. calculation was 48 min (their Fig. 3). The numerical solution of our Eq. (4) is 47.75 to the nearest fourth-minute (k ¼ 1:04 109 ; E ¼ 10; R ¼ 5; C ¼ 5 105 ). Calculations of optima using (4) for other cell densities largely agree with those of Abedon et al. (their Fig. 3, using the linear progeny function), although at cell densities less than 105, our method gives slightly shorter optimum latent periods (not shown). Thus, our method agrees with the simulation method of Abedon et al. (2001) instead of b1/g, although as Abedon et al. noted, the discrepancy is small at high cell densities. Indeed, the discrepancy even at low cell density is only of quantitative interest. Neither of those previous studies allowed for an intrinsic cell death rate, independent of phage killing, and the cell death rate puts an upper limit on the lysis time optimum in a stable or growing phage population. A cell death rate might obtain for various reasons—washout in a chemostat, predation from non-phage sources, or death from stress or lethal chemicals (e.g. bacteriocins), or infection by other phages that blocked reproduction by the phage of interest. The optimal phenotypes in these and the following models optimize the growth rate of the phage in a dynamically mixed population of hosts. Constant cell densities have been assumed for mathematical convenience as well as to yield results that can be easily interpreted, but constant cell density is not a realistic assumption for a natural phage environment. It was noted above that, if l40, the phage will outgrow the bacterial population and either reduce the cell density or cause multiple infections. Greater realism would be added, therefore, by allowing cell density to change in response to phage density, as in the early models of Levin et al. (1977). It might seem that l ¼ 0 is the ultimate dynamic equilibrium in such a model, but undamped oscillations are predicted to occur under wide ranges of parameter values in single-host populations, and it is not clear how oscillations affect lysis time optima. 1.2. Numerical examples E Lysis time Fig. 1. Tradeoff of burst size and lysis time. The lower axis is the time since infection at which lysis occurs. No progeny are produced prior to E (eclipse), but progeny output rises linearly thereafter. Optima of phages with three different progeny accumulation functions are shown for different cell concentrations (Fig. 2, top). As shown here and was also evident in Abedon et al. (2001), the optimum changes little across a wide range of high cell densities but changes more dramatically at low cell density. At high cell densities, the differences in optima among different types of phages are due mostly to differences in eclipse period—eclipse puts an ARTICLE IN PRESS J.J. Bull / Journal of Theoretical Biology 241 (2006) 928–938 Fig. 2. (Top) Lysis time optimum versus cell concentration, by numerical solution of (4) to find the value of lysis time (L) that maximizes phage growth rate l, subject to the burst size function b ¼ RðL EÞ, where E is the eclipse period and R is the rate at which phage progeny are produced following eclipse. The optimal lysis time functions are shown for three different burst size functions. (Bottom) Intrinsic growth rate (d+l) versus cell concentration for the same lysis optima curves in the top. In both figures, parameters are ðE ¼ 20 min; R ¼ 200 progeny= minÞ for the solid curve ðE ¼ 20; R ¼ 25Þ for the dashed curve, and ðE ¼ 40; R ¼ 10Þ for the dotted curve. k ¼ 2 109 ml/min for all curves. The highest value of R illustrated here (200) leads to unrealistically high burst sizes but is included to reveal the pattern across a broad range of values. absolute lower bound on lysis time. Overall, changes in cell density have the least impact on lysis time optima for those phages with the highest growth rates (compare top and bottom figures). The lower part of Fig. 2 shows intrinsic growth rates ðd þ lÞ of phages at different cell densities. Lysis time optima in the top panel apply to the growth rates in the bottom panel at the same cell density for the corresponding phage. The optima are not affected by the cell death rate (d), provided the sum (d þ l) remains the same. If l is negative, the phage goes extinct, but the optimum still applies as long as d þ l40. The strength of selection for the optimum may be evaluated for different parameter values and host densities (Fig. 3). The strength of selection for the optimum increases with the curvature at the optimum (in each curve of Fig. 3, the optimum is the highest point). Phages with high growth rate experience stronger selection for the optimum; selection for the optimum of the same phage weakens merely as cell density is lowered (compare the top and middle curves in Fig. 3). Selection is in fact so weak at the optimum on the lower two curves that it would be 931 Fig. 3. Three curves show the growth rates of phages with different lysis times. The three curves differ in the progeny function b ¼ kðL EÞ or in the cell density. From top to bottom, the optima are 26, 34.75, and 66 min. Per minute, the relative growth advantage of a phage expressing the optimum lysis time (‘‘opt’’) over a phage expressing an alternative time (‘‘alt’’) is determined as eðlopt lalt Þ . For these three optima, a rare phage genotype expressing the optimum lysis time (starting frequency of 106) in a population where the common type lysis time was 4 min later than optimum would require 45, 390, and 1903 h (from top to bottom) to reach 50% frequency, so it would not be practical to experimentally evolve the lower two populations close to the optimum. Parameters are ðE ¼ 20 min; R ¼ 25 phage= min; C ¼ 108 cells=mlÞ for the top curve ðE ¼ 20; R ¼ 25; C ¼ 106 Þ for the middle curve, and ðE ¼ 40; R ¼ 10; C ¼ 106 Þ for the bottom curve. k ¼ 2 109 for all curves. tedious experimentally to select a phage population better than to within a few minutes of the optimum. 1.3. Joint evolution of lysis time and lysis time tradeoff The model assumes that all parameters are evolutionarily invariant except burst size and lysis time, and these are forced to covary along a single tradeoff function. In adaptations of phages to novel conditions, it may also be that the lysis time function evolves—a shorter eclipse period, which may in turn result in a different (shallower) slope. In this case, the lysis time can evolve along one function or jump to another. Suppose, for example, that one burst size function is B1 ¼ R1 ðL E 1 Þ and the other is B2 ¼ R2 ðL E 2 Þ. If E 1 oE 2 (without loss of generality), but the slopes obey the reciprocal relationship R2 4R1 , then the two functions will intersect at Lint ¼ ðR1 E 1 R2 E 2 Þ=ðR1 R2 Þ, as in Fig. 4. The question is how lysis time will evolve on this discontinuous surface. The lysis time optimum for a single function is ^ At the point of intersection, both L^ ¼ E þ 1=ðd þ lÞ. functions have the same burst size and lysis time, hence ^ Since the two functions have the same value of ðd þ lÞ. have different eclipse times, it is impossible for the intersection to satisfy the optima for both lines. What is observed in numerical trials is that lysis time optima can exist anywhere along the uppermost function except in a zone around the intersection. As conditions are changed to change the lysis time optimum toward the intersection, a point is reached at which the optimum leaps across Lint onto the other function (the zone of instability around Lint ARTICLE IN PRESS J.J. Bull / Journal of Theoretical Biology 241 (2006) 928–938 932 (slope R2) burst size possible optima (slope R1) E1 E2 Lysis time Fig. 4. Relationship between burst size and lysis time when the phage can evolve on either of two linear tradeoff functions (see Fig. 1 for a single tradeoff). The set of feasible optima are given by the shaded gray zones along the two lines. There is a zone around the intersection of the two lines which is never optimal, because points on one or the other tradeoff yield superior fitness (growth rate). depends on the parameters). Thus, the evolution of lysis time is discontinuous and, at this discontinuity, exhibits a more dramatic shift in lysis time than would occur on a single tradeoff function. 1.4. Genetics of lysis Lysis is not a passive deterioration of the cell. It is an active, rapid dissolution of the cell wall, which enlists the osmotic pressure of the cell to cause a complete liberation of the cell components into the surrounding media. Not all phages lyse their host, however. The filamentous phages (e.g. M13) are assembled at and secreted through the cell membranes without killing (Model and Russel, 1988). Temperate phages (the most-studied example being phage l) can infect and adopt a quiescent state of benign coexistence as a prophage, but whenever they switch from their lysogenic state to produce phage progeny, they exit the cell lytically (Johnson et al., 1981). Two classes of lysis mechanisms have been described (Bernhardt et al., 2002; Young, 1992). The simplest one uses a single phage gene to block cell wall synthesis. In rapidly dividing cells, lysis occurs at cell division (the septum cannot form), so lysis time is only partly controlled by the phage. These mechanisms are found in phages with the smallest genomes, including the RNA phages (e.g. Qb) and the isometric phages of the microviridae (e.g. fX174). The other and most widespread class of lysis mechanism has two components, a lysin (muralytic enzyme) that degrades the cell wall and a transmembrane protein called a holin that permeabilizes the host inner membrane to allow the lysin access to the cell wall (which lies outside the inner membrane). Lysins vary in which chemical bonds of the cell wall they attack, so they have multiple origins. However, the timing of lysis is attributed to the holin, because it is not until the holin creates pores in the inner membrane that lysis can occur; a variety of genetic manipulations of phages indicate that the holin indeed controls lysis time. Holins act in a threshold manner, so that no lysin escapes through the inner membrane until the threshold density of membrane-bound holin is reached, but then there is widespread, rapid escape of lysin (Grundling et al., 2001). In principle, a two-component system times lysis via the onset and level of expression of the holin but also on the holin’s threshold setpoint, the molecular basis of which is not yet understood (Wang et al., 2000; Young, 2002). That the holin threshold is tunable is suggested by extensive sitedirected mutagenesis experiments of holin genes in phages T4 and lambda (Chang et al., 1995; Grundling et al., 2000; Ramanculov and Young, 2001a, b). Not uncommonly, the system is not this simple. Anti-holins are often found as well, non-essential genes (often in-frame with the holin) whose only observed phenotypic effect is a modification of lysis time. Extensive work on these genetic systems indicates that lysis time can evolve with little impact on other phage fitness components, so the assumptions of the models regarding phage genetics appear to be met (i.e. there is little pleiotropy accompanying changes in lysis time). A description of the genes involved in lysis provides a useful understanding of lysis time as pertains to the model here, but it should be realized that this understanding and the model address only a small part of lysis time expression in a natural environment. For an obligately lytic phage, as assumed in our model, there is the complicated issue of lysis time plasticity—the spectrum of lysis times in different hosts and under different physiological conditions. It was noted early that lysis time scales with bacterial growth rate (Ellis and Delbruck, 1939; over a six-fold change in bacterial doubling time effected by temperature), and much of the observed lysis time plasticity in phages is apparently optimal from the evolutionary perspective (Abedon et al., 2001). Our model ignores plasticity by virtue of treating all cells the same. At an even higher level of complexity, lysis inhibition (known only in the T-even phages) is a special type of plasticity in which lysis is delayed if the cell is co-infected by many phages before the normal time of lysis (Abedon, 1990; Adams, 1959). Again, our model ignores lysis inhibition because the rate of infection per cell is assumed to be low. Temperate phage face two lysis timing decisions: (i) how quickly to lyse following an infection adopting the lytic cycle, and (ii) how quickly to lyse after a lysogen is induced into the lytic pathway. The control of lysis genes in the temperate phage lambda, which need to be fully suppressed until induction, has been extensively studied (e.g. Nickels et al., 2002; Roberts et al., 1998). These various complexities lie outside the simple model developed thus far, but they are clearly relevant to lysis time evolution in a natural context and are relevant to interpreting lysis functions in natural phages. ARTICLE IN PRESS J.J. Bull / Journal of Theoretical Biology 241 (2006) 928–938 933 1.5. Implications for evolution of virulence 2. Evolution in populations of heterogeneous hosts Abedon et al. (2003) suggested that phage lysis time was analogous to the virulence of an infection in an animal host: shortening phage latent period increases virulence. This parallel may seem puzzling at first, because a phage infection requires cell/host death for transmission, whereas evolution of virulence theories have been based on parasites that require a live host for transmission. Evolution of virulence theories may thus be most appropriately applied to the non-lethal, filamentous phages that do not lyse their hosts. However, the virulence analogy can also be construed to extend in principle to lytic phages, because progeny production requires a live host, and by halting phage production, the time of lysis determines how much transmission can occur. Indeed, our qualitative results are the same as those of Lenski and May (1994), which addressed the virulence of an infection that transmitted from live hosts continuously. Whether or not one accepts lysis time evolution as a close analogy to virulence evolution, the mathematical structure of both problems is similar. The simple, tractability of phage systems may thus help in exploring some of the issues in the evolution of virulence. The chief architect and advocate of evolution of virulence theories was Ewald (1994), and one of the main evolutionary determinants of virulence in his verbal models was the availability (density) of uninfected hosts: high virulence had little cost if the parasite was transmitted to many new hosts before it killed its previous host. Yet as Lipsitch et al. (1995) pointed out, host density is not a factor in virulence evolution when the parasite reaches dynamic equilibrium, because at equilibrium each infection gives birth to only one new infection during its lifespan, on average, regardless of host density. Result (4) explicitly shows that growth per se of the parasite population selects ‘‘higher virulence.’’ Thus, as is well known from life history theory (Lewontin, 1965; Mertz, 1971), it is only in a growing population that the geometric advantage of early reproduction is favored at the expense of total fecundity. For infections, this translates into a benefit of early transmission at the cost of high host death rate. This point was made previously for virulence evolution (Bull, 1994; Lenski and May, 1994), although the formula here is a more explicit codification of this principle than has been developed before. The numerical illustrations of (4) highlight a subtle difficulty with qualitative applications of evolution of virulence theories. Substantial changes in host density (e.g. 100-fold) have a notable impact on the lysis time optimum for some progeny-lysis time functions but not others, and the impact also depends on which end of the host-density spectrum is considered. Without knowing a variety of quantitative details, effects of changing host density may be too small to detect. Yet those details are almost never known in natural settings. Simple extensions of the preceding model may also be applied to phenotype evolution in populations of mixed hosts. Host quality and host strain no doubt vary in the natural environments phages experience, and in view of the recent discoveries that bacteria in lab cultures exhibit a range of physiologies (physiological tolerance: Wiuff et al., 2005), the phenomenon is also potentially relevant even in the most controlled, homogeneous conditions. Furthermore, when multiple host types are present, lysis time is no longer the only phage phenotype of interest: host range and other forms of specialization can evolve. We first derive the relevant equation for phage growth in a population with multiple host types (two). Assuming fixed densities of two hosts, C1 and C2, and that parameter values are specific to each type of infection (subscripted accordingly), the two-host equation corresponding to (1) is P_ ¼ Pm þ k1 C 1 ðb1 edL1 PL1 PÞ þ k2 C 2 ðb2 edL2 PL2 PÞ. ð5Þ lt Again assuming that PðtÞ ¼ Pð0Þ e , the solution takes the form l ¼ m þ k1 C 1 ðb1 eL1 ðdþlÞ 1Þ þ k2 C 2 ðb2 eL2 ðdþlÞ 1Þ. ð6Þ This equation will be used to address various questions of interest. 2.1. Lysis time With multiple host types, selection on lysis time offers interesting challenges, and intuition is difficult to trust. There is also the messy issue of whether the phage can evolve lysis time on one host independently of the other type—whether lysis time plasticity can be achieved. At one extreme, if the phage can evolve lysis time on one host type fully independently of lysis time on other host types, the single-host results (3) and (4) describe the optimal phage response to each host in the mixed population (eclipse time in the formula is specific to the host type being infected). This result may seem counterintuitive because different host types may have vastly different abundances and thus, by this result, seem to select very different lysis times. But this result is not that lysis time is optimized according to each host independently of the abundance of other hosts. The effects of different hosts on phage growth are united through l, which depends on all infections, not just the type over which lysis time is being optimized. Thus, the growth rate of the phage population affects the optimum equally for rare as well as for common host types. At the other extreme, the phage may be forced to adopt the same lysis time for all hosts. In this case, the result ARTICLE IN PRESS 934 analogous to (3) is P _ 1 j bj k j C j , ¼ P ^ bi ki C i d þ l^ J.J. Bull / Journal of Theoretical Biology 241 (2006) 928–938 (7) i where i, j ¼ 1, 2. This result is similar to (3) except that the b_i and bi functions are averaged across the different hosts, weighted by the force of the infection from that host ðki C i Þ. If the slope of the progeny accumulation function and/or eclipse period varies among host types, then this result has no easy simplification, although a numerical solution should be attainable when the parameters are specified. 2.2. Host range Formula (6) also permits analysis of selection on host use or host range. The evolution of host range in viruses and other organisms has sometimes been considered from the perspective that a broad range is better than a narrow one, except that breadth impairs the ability to infect any one host well (a tradeoff, Agrawal and Lively, 2002; Agrawal, 2000; Bohannan and Lenski, 2000; Frank, 2000). An alternative perspective, adopted here, is to suppose that a phage can potentially infect a wide range of hosts without penalty, but that some hosts are better for phage fitness than others. As an extreme example, it behooves a phage to avoid infecting a cell that will abort its infection or is otherwise not productive, since no progeny will emerge. The same principle may apply to more subtle differences in host quality. The question posed about phage host range is analogous to the diet of a foraging animal (see Discussion). Evolution of host range in this model is strictly evolution of ki, although other forms of phage specialization on host type may be addressed as well (as derived below). When the adsorption rate drops as low as 1011 on a host, that host is effectively avoided, because most collisions between phage and bacterium do not result in infection. As in our model of lysis time, the use of a single parameter for adsorption rate is a great oversimplification of moderately complicated biology. Adsorption is minimally dependent on specific receptors on the bacterial surface, which may be proteins or LPS constituents. The densities of these receptors as well as the overall size of the bacterium (as a target for collisions with phage) may change with bacterial nutrition, altering the adsorption rate physiologically, without any evolution (Adams, 1959). In most phages, the primary determinants of host range are the tail fibers, although chaperones and other tail-associated proteins may be involved as well (e.g. Drexler et al., 1989, 1991; Hashemolhosseini et al., 1994a, b; Letarov et al., 2005). Phage adsorption may respond not only to the presence of specific bacterial receptors, but also to various chemical constituents of the environment (e.g. metal ions, pH: Adams, 1959; Letarov et al., 2005). Phenotypically, adsorption often proceeds in two discernable stages, reversible adsorption and a later, irreversible adsorption (Adams, 1959), and it may be that this temporal partitioning enhances a phage’s ability to discriminate among hosts. In phage T1, the reversible step requires certain cellular receptors, but the irreversible step requires energy provided by the cell (Hancock and Braun, 1976). Evolution of host range can be inferred from the sign of ql=qki , and the main result can be anticipated by inspection of (6): selection favors avoidance of host i if bi eLi ðdþlÞ o1, equivalent to ln bi =Li od þ l. Since bi and Li are properties only of the infection and are independent of bacterial abundance or phage growth rate, multiple types of infection/host can be aligned along one axis, and their positions are fixed regardless of dynamics; host quality in this context is thus measured just by ln bi =Li . The only variable is on the right side of the inequality, l, which changes with host density and with infection rates. The phage is selected to infect both hosts if the inequality ln bi =Li 4d þ l applies to both hosts. If instead d þ l falls in between the ln bi =Li , then the phage is selected to avoid the poorer host. Avoidance of both hosts can be selected only if the phage population is going extinct, and we will not consider that extreme. Further simplification of conditions favoring the avoidance of one host type have not been obtained, but some special cases and qualitative results are evident from (6): (i) If the phage population is at dynamic equilibrium (l ¼ 0), inclusion of host i in the optimal diet requires that each infection of that host produce at least one descendent phage on average, accounting for the possible death of the host before lysis ðbi eLi d X1Þ. (ii) If infection rates of the better host are sufficiently frequent, selection favors avoidance of the poorer host. This result may be argued on the grounds that, when the better host (i) is very common, the phage growth rate will approach ln bi =Li if it infects that host exclusively (at this high-density extreme, essentially no time is spent between lysis of the previous host and infection of a new one). Infection of another host with an inferior ratio must therefore slow growth rate. This result stems partly from the assumption that collisions with one type of host do not delay encounters with the other host type. (iii) If bi eLi ðdþlÞ o1 for a host, selection favors avoidance of that host, and no amount of change in density of that host can reverse the inequality. Decreasing the abundance of the preferred host can shift selection to include the poorer host, however. 2.3. Specializing on one host type when both are infected Even when a phage infects both hosts and is selected to maintain both hosts, selection on phage parameters will typically be stronger on one host type than on the other. This effect could lead to subtle degrees of specialization on one host type. For illustration of this general principle, consider selection for improvements in burst size on the ARTICLE IN PRESS J.J. Bull / Journal of Theoretical Biology 241 (2006) 928–938 different hosts. The relative strength of selection on burst size of host type 1 relative to host type 2 is (found as ðql=qb1 Þ=ðql=qb2 Þ) k1 C 1 eL1 ðdþlÞ . k2 C 2 eL2 ðdþlÞ 935 different physiological states of the same strain. Adaptation to specialize on one host type may lead to even greater selection for specialization, as parameters evolve to become progressively more favorable for the better host. (8) If this ratio exceeds unity, selection is stronger for increasing b1 than b2, and if genetic variation were equal for both burst sizes, selection would tend to increase b1 more than b2, possibly even at the expense of b2. The force of infection kiCi is a major determinant of this selection, but so is lysis time, and lysis time is more important in growing than stable phage populations. Qualitatively similar conclusions apply to selection of adsorption rates and lysis times: selection will be weaker on one infection type than on the other; selection for avoidance of a host type is merely part of a continuum of selection on the ki. 2.4. Implications These results offer a special caution to protocols that adapt phages to populations of heterogeneous hosts and select rapid growth. Such a protocol might be used to improve phage therapy, for example, by passaging a phage from one infected mammal to another, to obtain a phage variant that grows faster on bacteria inside the infected mammal. In such a protocol, the phage might well evolve to avoid poor quality host types, yet those host types could in principle be the therapeutically important ones. Inadvertent selection to avoid poor host types can be mitigated by selecting for phage numbers rather than phage growth rate, hence by allowing the infected culture to proceed until phage numbers have reached a plateau. This practice allows the non-specialist time to grow on the poor hosts after the specialist has exhausted the good hosts. An interesting consequence of selection for avoidance of the poorer host may operate in a spatial context. If an epidemic wave spreads out through a population of hosts and cell density is sufficiently uniform and high across a large region, selection could favor phages at the leading edge of the epidemic that specialize on the better host (if the better host is sufficiently dense). Because of their superior growth rate, these specialists will spread ahead of ‘‘generalist’’ phage that do not specialize. In turn, the generalist will then encounter a population that consists largely of the poorer host, and if there is any appreciable tradeoff in ability to infect one host type versus the other, the generalist may evolve to specialize on the poor host. If infection of both bacterial types is maintained, specialization can evolve to improve performance on one host type more than on the other (Section 2.2, above). As one example of how this specialization could be accomplished, if the two host types differed in concentration of a factor used by the phage (e.g. a sigma factor), the phage could evolve to grow best in the concentration of that factor found in the better host. Thus, specialization could evolve even when the different host types are merely 3. Discussion The models developed above are optimality models, commonly used and commonly criticized in evolutionary biology. For experimental purposes, the genetic and evolutionary tractability of phages exceeds that of virtually all other ‘‘organisms’’ that have been pursued with an optimality perspective, so it seems important to consider how phages can inform us about optimality approaches. There are at least three different levels of optimality model tests to consider: (A) natural history: a comparison of phenotypes of natural organisms to predicted optima; (B) genetic systems: whether the genetic bases of the phenotypes match expectations from the model; and (C) experimental evolution. We consider each of these in turn. (A) A major goal in biology is to understand natural variation, and optimality models offer a simple and useful step in that direction. In a typical application, an optimum predicted by the model is compared to observed phenotypes of organisms in the appropriate environmental contexts (Charnov, 1982; Parker, 1978). The populations being tested are assumed to have experienced a constant environment long enough that their phenotypes have evolved to equilibrium. Regardless of whether this approach is reasonable for some organisms, at face value, it cannot be easily applied to phages for two reasons. First, the most basic properties of phage natural environments are unknown (host strains, host physiologies, host densities), so that it is not possible to predict an optimum for a wild phage. Second, the models developed here make such unrealistic assumptions about the phage environment (e.g. constant cell density) that they certainly would not apply to any natural phage. If we did know the natural environment of a phage, it might be possible to develop numerical approaches to predict optima. Yet this accomplishment poses a new problem: in ignorance of which phenotype set is available to the organism, it becomes difficult to interpret the optimality of observed phenotypes, except in the simplest of environments. Consider lysis time. The simplest environment would be a single host maintained at constant density and physiology, for which there is a straightforward optimum. Once multiple host types are infected, the optimality prediction depends on how well the phage can evolve lysis time on one host independently of others. If the relevant infection parameters were known for the different host types but the degree of lysis time independence was unknown, one could fit a modified optimality model with a parameter specifying the degree of lysis time independence across host types. But as less and less of the underlying biology of lysis is known, this approach becomes increasingly problematic. Phenotypes that qualitatively or even ARTICLE IN PRESS 936 J.J. Bull / Journal of Theoretical Biology 241 (2006) 928–938 quantitatively match a model may do so for the wrong reason (as shown for lysis: Heineman et al., 2005), and this problem presumably increases as more parameters are required to fit the model. These impediments stand in the way of simple applications of the comparative method. In specific applications, it may be feasible to obtain information that enables progress in understanding evolution in the natural environment, as when the environment is well defined and/or the susceptible host density can be ascertained. Furthermore, experimental systems might be developed to approximate some natural environments (e.g. see Weld et al. (2004) for a unique blend of empirical and mathematical applications to phage therapy in a gut). (B) Optimality models may also shed light on the evolution of genetic systems (Bull, 1983; Bull et al., 2004; Charnov and Bull, 1977). If the typical environment experienced by a phage is heterogeneous for host type and/or is temporally variable in host abundance and host type (as seems plausible), we can infer from the models that lysis time and host range are both often selected for change and/or plasticity. In some cases, the only conceivable response is evolution, but in others it is plasticity. At a minimum, therefore, phages should have evolvable, modular genetic systems for lysis time and host range that allow phenotypic change with few mutations and without major pleiotropic consequences on other traits. The mechanism for achieving lysis time should also be responsive to cell physiology (hence allowing plasticity). As described in Section 1.3 above and elaborated previously, this perspective is supported for lysis time mechanisms in all but the smallest phage genomes (Bull et al., 2004). The ‘‘genetic systems’’ perspective is also compatible with the mechanisms known to affect host ranges of tailed phages. Tailed phages have highly evolvable, modular proteins for attachment (the tail fibers) that have been shown to be important in expanding host range (Drexler et al., 1989, 1991; Hashemolhosseini et al., 1994a, b; Tetart et al., 1996, 1998; Werts et al., 1994; Yoichi et al., 2005). Of course, such ad hoc, correlational evidence can be interpreted in multiple ways. In our case, the evolvability and modularity of phage host range mechanisms may have evolved more from selection for host range expansion than from selection for host avoidance. Nonetheless, this use of optimality models should be valuable in generating novel hypotheses that can be subjected to further testing. (C) Phages offer the advantage of enabling short-term selection experiments, and optimality models can be tested directly by this approach. In the simplest design that is relevant to the model here, cells would be grown to a standard density and physiological state before phage are added. By controlling the initial phage density and time between transfers, each passage can avoid high multiplicity while cells are maintained in a relatively constant state, thus closely matching the assumptions underlying our calculation of l. These artificial conditions are clearly simplistic, yet if evolution fails to meet our expectations in these types of studies, it is feasible to explore why. Precedents exist for the experimental evolution of lysis time (Abedon et al., 2003; Heineman et al., 2005; Wang, 2006), but only a couple conditions and phage types have been explored with this approach. Host range is also amenable to experimental selection. In the extreme, a phage can be selected to avoid one host by engineering that host to abort infections. Once phages that avoid a host have been isolated, more subtle tests of the models can be conducted in which mixtures of discriminating and nondiscriminating phages are grown on mixes of host types differing in density and quality. Testing optimality models with direct selection experiments is relatively new, but when used with microbes, it offers some of the most comprehensive and powerful analyses of evolution to be found (Abedon et al., 2003; Dekel and Alon, 2005; Dykhuizen, 1993; Suiter et al., 2003; Wang, 2006). 3.1. Extensions: virulence, foraging, and multicellular hosts Parallels between optimal lysis time and evolution of virulence were described above. Although the evolution of virulence of human pathogens has been an active topic over the last two decades (Dieckmann et al., 2002; Ewald, 1994; Frank, 1996), support for evolution of virulence theories is questionable (Ebert and Bull, 2003). Does the empirical work on phages, which so far supports the optimality models, bolster models for evolution of virulence in human pathogens? I suggest that the support is at best weak simply because there are major biological differences between phage lysis time and virulence of the pathogens of multicellular organisms. Phage lysis time provides an ideal fit to optimality because it matches the assumptions of those models well. The tradeoff between lysis time and burst size in phages corresponds to the virulence-transmission tradeoff in the simplest evolution-of-virulence models, and for phages, this tradeoff is well established and does not appear complicated by other dimensions of the life history. The problem in extrapolating from lysis time to pathogens of higher organisms is precisely that those tradeoffs are more complex in higher organisms. Thus for a human pathogen, a higher parasite load may increase its transmission, all else equal, but the higher load may also elicit a stronger host immune response and thus decrease transmission. Likewise, virulence may manifest only after a significant amount of transmission has occurred (Day, 2003), and virulence may stem from the infection of tissues that are not important to transmission. None of these potential complications applies to phages. Thus, phages offer the best match of biology to the tradeoff between virulence and transmission assumed in many of the evolution of virulence models, and any success phages enjoy in this respect does not extend to more complicated host–parasite systems. The host range model considered here offers some direct parallels to the earliest optimal foraging models (Emlen, 1966; MacArthur and Pianka, 1966). In the original ARTICLE IN PRESS J.J. Bull / Journal of Theoretical Biology 241 (2006) 928–938 foraging models, the function maximized is energy intake, and the classic result is that the optimal diet is a set of ‘‘best’’ food items, ranked according to energy content and handling time. An optimal diet consists of all items whose energy/handling time ratio exceeds a threshold and no items whose ratio is below the threshold. An unintuitive result from that model is that changes in the abundance of items included in the optimal diet can influence the items included in the optimal diet, but the changes in the abundance of food items excluded from the optimal diet does not. In simple terms, an item that is unsuitable because of its energy/handling time ratio remains unsuitable whether it is common or rare. However, increasing the abundance of a preferred item can drive lesser items out of the optimal diet. The phage results here match these optimal foraging results, even though phage growth rate (fitness) is the phage property that is maximized in the model. The host range results for phages may not apply to most viruses of terrestrial organisms. A key assumption in the models is that a virus avoiding host type 1 will not only ultimately contact host type 2 but will experience no delay in infecting host type 2 by its collisions with host type 1. This assumption suits a liquid environment of host encounters, but it likely does not fit many terrestrial environments, in which the virus is unlikely to have more than one opportunity to infect a host. The model may have parallels to viral infections within complex hosts, however. Whenever the tissue initially infected by a virus differs from the tissue disseminating the virus to the next host, the infection can be represented as an epidemic population of virus spreading from one tissue to another within a host and ultimately out of that host. (Analogous models have been developed for persistent infections, which are equilibrium rather than epidemic pathogen populations within the host: Kelly et al., 2003; Orive et al., 2005.) 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