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Levels of selection in positive-strand virus dynamics D. C. KRAKAUER* & N. L. KOMAROVA *Santa Fe Institute, Santa Fe, NM, USA Institute for Advanced Study, Princeton, NJ, USA Keywords: Abstract coevolution; levels of selection; RNA virus; robustness; translation kinetics virulence. Conflicting selection pressures occurring over the life cycle of an organism constitute serious challenges to the robustness of replication. Viruses present a credible model system for analysing problems that arise through evolutionary conflicts of interest. We present a multi-level selection model for the life cycle of positive-strand RNA viruses. The model combines within-cell replication kinetics and protein synthesis, and between-cell population dynamics of virion production and transmission. We show how these two levels of within-host selection interact to produce tradeoffs in the life history strategy of a virus without consideration of host mortality. We find that viruses evolve towards intermediate rather than maximum encapsidation rates. This can be interpreted as selection for intermediate virulence through cellular persistence. We characterize a theoretical persistence threshold arising from the trade-off between genome replication and genetic translation within the cell. We present counter-intuitive relationships whereby increasing genome decay rates and rates of encapsidation lead to increases in the abundance of virusencoded proteins. Data from poliovirus suggest that viruses might be unable to resolve the vertical conflicts of interests among different levels of selection. Introduction Levels of selection problems for a virus Viruses are infectious, obligate intracellular parasites. Within a target host cell, the viral genome is replicated and directs the synthesis of essential virion components by making use of host cellular pathways. Progeny virions are constructed from newly synthesized virion components, and constitute vehicles for the transmission of the viral genome into further target cells. The replication of a virus genome depends upon successful completion of a within-cell phase and a between-cell phase. Those genes encoding for proteins and thereby strategies effective within the cell are often distinct from those influencing the biology of cellular transmission, and with higher levels, including the infection of new hosts. In many cases, including those we consider in this paper, these strategies can come into Correspondence: David C. Krakauer, Santa Fe Institute, Hyde Park Road, Santa Fe, NM 87501, USA. e-mail: [email protected] 64 direct conflict. Strategies effective at ensuring effective replication within the cell lead to reduced infection of susceptible cells. This constitutes a canonical levels of selection problem (Williams, 1992; Keller, 1999) in which replication rates at distinct levels of organization become negatively correlated. The question that arises is, how viruses if at all resolve this conflict and how it relates to the design of the virus life cycle. The virus life cycle Viruses are translational parasites. All RNA of viral origin must be translated by host-cell protein synthesis pathways. Either this is achieved directly as in the case of positive-strand viruses, where the genome can serve immediately as mRNA template, or some additional means of synthesizing mRNAs from the negative-strand template must be employed. The virus typically piggybacks on normal cellular processes and as these are finely regulated and highly compartmentalized, the virus must tune its own replication mechanisms and parameters to exploit those of the host. The essential steps in the J. EVOL. BIOL. 16 (2003) 64–73 ª 2003 BLACKWELL PUBLISHING LTD Virus genome evolution replication cycle are the formation of mRNA (in DNA viruses and negative strand RNA viruses), the synthesis of viral proteins, the replication of viral genomes, the encapsidation of viral genomes by viral proteins and the infection of further susceptible cells. RNA synthesis (replication) is catalysed by virusencoded RNA-dependent RNA polymerase. In some cases, accessory viral proteins or host-derived factors are also required to direct the RNA template or prime the polymerase. The new RNA is synthesized by templatedirected stepwise incorporation of ribonucleotides into the 3¢-OH end (3¢ to 5¢ direction) of the growing negative-strand RNA chain, which extends in the 5¢ to 3¢ direction. Genome synthesis occurs mainly in the cytoplasm, although some viruses synthesize new RNA when bound to cellular membranes. The synthesis of viral proteins during translation proceeds in the 5¢ to 3¢ direction in which the resulting protein is synthesized from the amino to the carboxy terminus. Unlike replication, which is largely under the control of virus-encoded proteins, translation depends almost entirely on host factors. An important implication of these observations is that genome replication and genome expression are mutually exclusive. With replication moving along the positive strand in the 3¢ to 5¢ direction and translation moving 5¢ to 3¢, ribosomes must be cleared from the viral RNA before serving as a template for negative-strand RNA synthesis. Replication appears to be regulated according to the natural clearance rate of translating ribosomes (Gamarnik & Andino, 1998; Barton et al., 1999). A kinetic exploration of the intracellular implications of this trade-off is provided by Eigen et al. (1991). Virion production requires that the protein and nucleic acid components produced during replication and synthesis are assembled and sorted within the cell for eventual export. Once at the assembly site, most viruses are capable of self-assembly and packaging of the virion components and genome. The release of the virions from the infected cells varies from non-cytolytic budding, whereby enveloped viruses leave the cell intact during release, to lysis where the cell is destroyed spilling the virions into the extracellular spaces. Once released and mature, these virions travel within the host initiating new infections in susceptible cells for which they express a tropism. We have chosen to focus on the single-stranded, positive-sense RNA viruses, as these viruses possess the simplest of designs. This is because the single RNA genome performs at least three functions: (i) translation into proteins, (ii) replication and (iii) encapsidation by virus-encoded protein for egress from the cell. There is relatively little in the way of regulation, and hence evolutionary tradeoffs can be derived directly from replication kinetics. In order to provide a more concrete understanding of the RNA virus life cycle, we summarize below the life cycle of the poliovirus. J. EVOL. BIOL. 16 (2003) 64–73 ª 2003 BLACKWELL PUBLISHING LTD 65 Poliovirus: a case study The poliovirus is a positive, single-stranded RNA virus (Koch & Koch, 1985). The genome comprises around 7500 nucleotides and encodes between 5 and 10 averagesized proteins. The genome is directly infectious and contains a single major initiation site for translation. The entire coding sequence is translated into a 2200 amino acid polyprotein post-translationally modified into individual protein products. Approximately one-third of the genome is used to encode a capsid protein required as protective packaging for the genome during transmission. The capsid is an icosahedral lattice structure of 60 asymmetrical structural protein units. The life cycle of the virus can be broken down into a number of sequential steps. The first step adsorbtion requires specific host-cell receptor binding. This is followed by entry of the virus genome and structural proteins into the host cell. During uncoating the genome is released from the structural proteins. The virus genome is then translated by host factors, leading to the synthesis of virus proteins detectable at around 1.5 h post-adsorbtion. This interval represents the eclipse period. The total amount of virus RNA produced then increases for 2.5 h, rising to an asymptotic level maintained for about 1 h before declining around the time of cell death. During the phase of maximum genome copy number, up to 6000 full genomic RNAs are synthesized per minute. This value represents the number of particles produced per minute at equilibrium densities. The progeny plus-strand RNA molecules have one of three fates: (1) function as templates for replication (through a negative-strand intermediate), (2) serve as mRNA for translation or (3) associate with capsid proteins in the creation of new progeny virions. The dynamics of replication and translation give rise to a temporal sequence of each of these fates. Early in infection, when virus protein level is low there is a rapid increase in RNA genome copies (fate 1). As the copy number increases a large fraction becomes engaged with ribosomes in protein translation, depleting the store of replicating sequences. This results in a linear phase of genome replication. As translation proceeds, increasing numbers of structural proteins become available, including capsid proteins, and lead to the encapsidation of RNA progeny strands (fate 3). From the population of liberated virions, about 50% penetrate susceptible cells, whereas only 1% leads to a complete virus replication cycle and the establishment of productive infection. Objectives This paper is organized as follows. In the section on ‘The intracellular level of selection’, we describe the intracellular viral dynamics and find two equilibria, one corresponding to a productively infected cell and the other to extinction of virus. We further show that the rate of 66 D. C. KRAKAUER AND N. L. KOMAROVA encapsidation must be bounded from above in order for the virus to persist within the cell. In the section on ‘The extracellular level of selection’, we present the intercellular dynamics and find that increasing the rate of encapsidation is evolutionarily advantageous at this higher level. The optimal encapsidation rate, derived from considering both levels, can be found analytically in terms of the parameters of the system. In the ‘Discussion’ section, our results in relation to empirical studies, where available, are discussed along with the horizontal conflicts of interest among distantly related virus lineages and vertical conflicts of interest among levels of organization. The intracellular level of selection The following model aims to make explicit the dynamical consequences of conserved features of positive strand RNA virus life cycles, identifying parameters and states that contribute to intracellular behaviour. This simple model will enable us to construct a bottom-up model of extracellular dynamics in the following sections. A more detailed exploration of the intracellular phase is provided by Eigen et al. (1991). We are not so much interested in the ‘Phage clock’ – describing the apparently co-ordinated series of events during virion production – but the critical persistence thresholds of infection, both intracellular and extracellular. In this section, we describe (1) the steps essential to the persistence or extinction of virus genomes within the cell, (2) identify rate-limiting reactions during virion production, (3) characterize parameter dependencies for virus export, and (4) assess the impact of selection pressures acting at the level of mature, infectious virions on within-cell genome replication and protein synthesis pathways. Throughout we shall adopt the following notation: • m for the number of mRNA strands; • R for the number of ribosomes; • p for the abundance of virus protein; • y for [mp], the mRNA–virus protein complex or virion; • z for [mR], the mRNA–ribosome or translation complex. Intracellular dynamics We shall assume that a constant fraction 0 £ b £ 1 of the total virus-encoded protein synthesized is RNA-dependent RNA polymerase. We can then write down the following reactions as irreversible kinetic relations describing a part of the infectious life cycle of an RNA virus: k1 m þ bp ! y; k2 y ! 2m þ bp; k3 m þ R ! z; ð1Þ ð2Þ ð3Þ k4 z ! m þ R þ p; k5 m þ p ! out: ð4Þ ð5Þ The positive-strand RNA genome serves directly as mRNA – binding with polymerase to form a complex y at a rate k1 The polymerase replicates the genome and is released out of complex where it is ready to participate in further reactions at a rate k2. The genome is also bound to ribosomes to form the complex z within which virus proteins are synthesized at a rate k4. RNA present in the translation complex is not available for replication by ribosomes. The new genomes and proteins are then packaged and egress from the cell at a rate k5. The ki are the reaction rates for each process, and by assuming mass action, we can write these reactions down as the following system of differential equations: m_ ¼ k1 bmp þ 2k2 y k3 mR þ k4 z k5 mp dm m; ð6Þ R_ ¼ k3 mR þ k4 z dR R; ð7Þ p_ ¼ k1 bmp þ k2 by þ k4 z k5 mp dp p; ð8Þ y_ ¼ k1 bmp k2 y dy y; ð9Þ z_ ¼ k3 mR k4 z dz z; ð10Þ where we have additionally included death rates (di for each of the components). We shall assume that dR ¼ dz ¼ 0: ð11Þ We make this assumption because the host-derived factors are constitutively expressed within the infected cell to serve ongoing cellular processes. Note that under this assumption, R_ þ z_ ¼ 0 (this follows from adding equations (7) and (10)). In other words, we have RðtÞ þ zðtÞ ¼ Rð0Þ þ zð0Þ C init : ð12Þ Upon infection, p(0) ¼ y(0) ¼ z(0) ¼ 0. This is because the viral proteins have not yet been synthesized. The initial infectious dose of the virus determines m(0), whereas R(0) is the homeostatically regulated abundance of ribosomes within the cell and is always greater than zero. Once a cell is infected, genome replication and protein production follows. This leads to the establishment of a productively infected cell. At a certain point (in poliovirus 4 h after adsorbtion), the cell can be thought of as reaching a dynamical equilibrium. This equilibrium does not imply that components are no longer being produced or destroyed in some defined sequence, but rather that the time-averaged abundance of each component has reached an approximately constant turnover. Where we have described only five creative kinetic parameters, in reality there are many more. This is because each J. EVOL. BIOL. 16 (2003) 64–73 ª 2003 BLACKWELL PUBLISHING LTD Virus genome evolution reaction that we treat as a single step, comprises a number of steps in nature. For example, the parameter k5 which we term the rate of encapsidation, involves the assembly of the virion, transport within the cell, fusion with the cell membrane and release from the cell surface. Our parameters are best thought of as ratelimiting steps of these events. For this reason, determining the precise value of the parameters from published studies becomes very difficult. We therefore speak of parameters in terms of their relative values, and only discuss those results where absolute values are not required. Viral equilibria System (6)–(10) under condition (11) has two equilibria. One of them is given by m ¼ p ¼ y ¼ z ¼ 0; R ¼ R~ 0; ð13Þ where R~ is an arbitrary constant. In terms of the initial condition of the system, we have R~ ¼ C init (this is a consequence of (12)). Local stability analysis tells us that the solution (13) is always (neutrally) stable. The other equilibrium of the system is given by the following expressions: Re ¼ dm ½dp ðdy þ k2 Þ þ me ðbk1 k2 ð2 bÞ QÞ =ðk3 me QÞ; ð14Þ pe ¼ dm ðdy þ k2 Þ=Q; ye ¼ bdm k1 me =Q; ze ¼ Re k3 me =k4 ; where Q ¼ bk1(k2 ) dy) ) (k2 + dy)k5 and me > 0 is an arbitrary constant. The existence condition can be written as Q ¼ bk1 ðk2 dy Þ ðk2 þ dy Þk5 > 0; ð16Þ dp ðdy þ k2 Þ þ me ðbk1 k2 ð2 bÞ QÞ > 0: ð17Þ If we assume that dy ¼ 0; ð18Þ all the expressions simplify greatly. Instead of (14) and (15), we have Re ¼ dm ½dp þ me ðbk1 ð1 bÞ þ k5 Þ ; k3 me ðbk1 k5 Þ ð19Þ dm ; bk1 k5 ð20Þ bdm k1 me ; k2 ðbk1 k5 Þ ð21Þ pe ¼ ye ¼ ze ¼ Re k3 me =k4 : Solution (19)–(22) exists if k5 < bk1 : ð22Þ J. EVOL. BIOL. 16 (2003) 64–73 ª 2003 BLACKWELL PUBLISHING LTD ð23Þ This condition highlights an essential instability in the positive-strand RNA virus cellular kinetics. It states that productive infection can only be achieved when the rate of encapsidation lies below the rate of polymerase– genome complex formation. The intuitive explanation for this requirement is that when the rate of encapsidation becomes too large, replication is unable to preserve genomes within the cell as they are being removed more quickly than they are being produced. There is a need for fine-tuning of parameters to ensure the continued production of virus. The precise position of the equilibrium depends on the virus-determined initial infectious dose, and the hostdetermined abundance of ribosomes. The value me is related to these initial conditions and is found from Re þ ze ¼ C init ; ð24Þ and equals me ¼ b þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 4ac ; 2a ð25Þ where a ¼ dmk3[bk1(1 ) b) + k5] > 0, b ¼ dm[dpk3 + k4(bk1(1 ) b) + k5)] ) k3k4(k1b ) k5)Cinit and c ¼ dmdpk4 > 0. Let us denote D > Dc ð15Þ 67 dm ½dp k3 þ k4 k1 bð2 bÞ : k4 ½k3 C init þ dm ð26Þ A simple analysis of formula (25) shows that in order for the solution, me, to exist and be positive, we need to satisfy k5 < bk1 Dc : ð27Þ If this condition holds, the system relaxes to the virus equilibrium [(19)–(22)]. If condition (27) is violated, the system relaxes to the first, trivial fixed point (13), where no virus is present. Figure 1 illustrates this point. We simulated the system of equations with the same initial conditions for the case where k5 satisfied condition (27) (plot (a)) and also for the case where k5 violated condition (27), but satisfied condition (23) (plot (b)). In the former case, we have a stable non-trivial virus equilibrium whereas in the latter case the virus disappears from the cell as time progresses. It is important to note that when the system converges to the nonproductive fixed point, virus RNA decays exponentially from the initial infecting dose. There is no transient burst in production that might allow the virus to persist at the population level. This result demonstrates that for reasonable initial conditions, the critical threshold relating encapsidation and genome replication, given by inequality (23), is never reached. This is because the dynamical system 68 D. C. KRAKAUER AND N. L. KOMAROVA (a) 2.0 y 1.5 1.0 R p z m 0.5 0.0 0 100 t 200 300 (b) 2.0 1.5 R 1.0 0.0 0 m,y,z p 0.5 100 200 bution to the production/decay of mRNA from processes (3) and (4). This is hardly surprising because during these processes, the mRNA is conserved. Now, if we look at the first set of brackets in (28) and use equation (9), we can see that the corresponding contribution is non-zero and is equal to bk1pm. This is a manifestation of the fact that during processes (1) and (2), mRNA is produced, and the rate of production equals bk1p. Now, we can balance the three contributions to the change of m: production, virus egress from the cell and death, given by the three terms: ðbk2 p k5 p dm Þm ¼ 0: t 300 Fig. 1 The time-evolution of the variables of the system, m,R,p,y and z, in the two cases: (a) k5 ¼ 1.3 < bk1 ) Dc and (b) k5 ¼ 3, bk1 ) Dc < k5 < bk1 (below). In the former case, the system relaxes to the virus equilibrium (19)–(22), the calculated values for the equilibrium are marked by horizontal dotted lines. In the latter case, the system relaxes to the trivial fixed point (13), where no virus is present. The other parameters are chosen to be k1 ¼ 5, k2 ¼ 0.2, k3 ¼ 5, k4 ¼ 0.2, b ¼ 0.8, dm ¼ 1 and Cinit ¼ 1. imposes an additional accessibility restriction on the parameter values, in effect ensuring that whenever a productive equilibrium is observed, it is certain to lie some distance away from the critical threshold. In the vicinity of the threshold, the virus population evolves towards extinction. Parameter dependence of the virus equilibrium Let us examine the form of the virus equilibrium [(19)– (22)]. In particular, formula (20) strikes us as counterintuitive at first glance. Why should the amount of protein produced grow with the death rate for the mRNA,dm? Surely reducing the half-life of the mRNA would lead to a reduction in its protein product. Furthermore, why does p grow as the rate of its removal from the cell, k5 increases? We might expect that the amount of protein within the cell decline with increasing rates of encapsidation. These counterintuitive results can be appreciated by examining more closely the implications of equilibrium on the dynamical system. Expression (20) can be interpreted in the following intuitive way. Let us recall the equation for the dynamics of m: m_ ¼ ðk1 bmp þ 2k2 yÞ þ ðk3 mR þ k4 zÞ k5 mp dm m: ð28Þ The two terms in brackets on the right-hand side correspond to processes (1) and (2), the second brackets contain the contribution of the processes (3) and (4), the term –k5mp is process (5) and the last term is the decay of mRNA. Let us start from the second set of brackets in equation (28). From (10) it follows that at equilibrium, )k3mR + k4z ¼ 0, which means that there is no contri- ð29Þ Formula (20), in which increasing the decay of mRNA or increasing the rate of encapsidation leads to an increase in the abundance of protein, follows immediately. To ensure equilibrium, an increase in the decay of mRNA or an increase in the rate of encapsidation must be matched by an increase in the replication rate of the virus genome. At equilibrium, these conditions are required to prevent selection of the trivial, zero equilibrium for the virus. Dynamics of a simplified system System of equations (6)–(10) is non-linear, and the general analytical solution cannot be obtained. So far we have only found the equilibrium points of the system. It is however possible to characterize the dynamics by recognizing important properties of virus biology. We consider an approximation to system (6)–(10) under the assumption that some of the processes have a much faster time scale. In particular, complex formation is much faster than either genome replication or protein synthesis. Another assumption we are making is that the system is far from threshold, i.e. k5 bk1. The advantage is that the system now becomes linear, and we obtain analytical expressions for all the variables as a function of time. We assume that k1 ; k3 k2 ; k4 ; k5 : ð30Þ From equations (19)–(22), it follows that R 1/k3 and p 1/k1, so it is convenient to re-scale the variables as R ¼ R¢/k3, p ¼ p¢/k1. Substituting this into system (6)–(10) and neglecting small terms, we can see that R¢ ¼ k4z/m, p¢ ¼ (k4z + k2by)/(bm) and there are only three differential equations left: m_ ¼ ð2 bÞk2 y k4 z dm m; ð31Þ y_ ¼ k4 z þ k2 yðb 1Þ; ð32Þ z_ ¼ 0; ð33Þ with initial conditions m(0), y(0) and z(0). Note that the initial conditions are not the same as the ones in the full system; effectively, they correspond to the values of m, y and z after a short initial stage of the dynamics where quick, of the order of t 1/k1 or 1/k3, readjustment J. EVOL. BIOL. 16 (2003) 64–73 ª 2003 BLACKWELL PUBLISHING LTD Virus genome evolution takes place. In particular, it can be shown that the correct initial condition for z is z(0) ¼ Cinit. System (31)–(33) is linear, and its eigenvalues are 0, )dm and )k2(1)b). The solution of system (31)–(33) can be written as follows: 0 1 0 1 0 k4 1 m 1 dm ð1bÞ B C B C B C @ y A ¼ A1 @ k4 A þ A2 @ 0 Aedm t k2 ð1bÞ z 0 B þ A3 @ 1 ð2bÞk2 dm ð1bÞk2 1 1 0 C ð1bÞk2 t Ae ð34Þ 0 where the constants are expressed in terms of the initial conditions for m, y and z: A1 ¼ zð0Þ; A2 ¼ mð0Þ þ ðk2 þ dm Þk4 zð0Þ dm k2 ð2 bÞyð0Þ ; dm ðdm ð1 bÞk2 Þ A3 ¼ yð0Þ k4 zð0Þ : k2 ð1 bÞ The stable fixed point of this system is given by 1 0 0 1 k4 me dm ð1bÞ C @ ye A ¼ C init B @ k4 A; k2 ð1bÞ ze 1 ð35Þ ð36Þ ð37Þ proceeds at a much faster rate than between cell transmission. Dynamics Let S be the abundance of susceptible cells, I the abundance of infected cells and V the amount of free virus. We have the following standard system: S_ ¼ r cVS d1 S; ð39Þ I_ ¼ cVS d2 I; ð40Þ V_ ¼ dI d3 V ; ð41Þ where r determines the rate at which susceptible cells are generated from source tissues, c, the efficiency of infection when viruses meet susceptible cells, di are decay constants and d ¼ k5m(t)p(t) is the rate at which the virus is liberated from infected cells as described in the previous within-cell model. We assume that the solution m(t)p(t) relaxes very quickly to its equilibrium value, so we can replace it with d ¼ k5 me pe ; ð38Þ and Re ¼ dm(1)b)/k3, pe ¼ dm/(k1b). One sees from the solution that the dominant factor in determining the time required for the system to reach equilibrium is the smaller of the quantities (1 ) b)k2 and dm. In particular, the smaller the rate at which new genomes are replicated, k2 the longer it takes for the system to converge to its equilibrium. The extracellular level of selection The previous results all relate to the within-cell phase of virus infection. However, a virus must leave a cell and infect new susceptible cells to ensure that the life cycle is re-established in further target cells. Parameters critical for between-cell dynamics need not be the same as those within the cell: the biological basis for cell tropism and infectivity (population level parameters) are different from those describing replication and synthesis (cellular parameters). Both levels of replication need to satisfied. In this section, We explore how virus population dynamics feeds-back to influence the selection of parameters determining cellular kinetics. An important assumption is that the time scale over which the population of susceptible cells is infected is greater than the time required for an individual cell to reach its maximum rate of virion production. In other words, within-cell kinetics J. EVOL. BIOL. 16 (2003) 64–73 ª 2003 BLACKWELL PUBLISHING LTD 69 ð42Þ where me and pe are the equilibrium values, and treat d as a constant within the system. In this way, we have combined two time scales of infection, the rapid time scale of within-cell kinetics and the relatively slow time scale of cellular infection. An equilibrium of system (39)–(41) is given by S ¼ rd1, I ¼ V ¼ 0. This point is stable if R0 < 1, with R0 ¼ cdr/ (d1d2d3). The value R0 is the familiar basic reproductive ratio. For R0 > 1, the virus is able to invade the population of susceptible cells and we have another stable fixed point: S ¼ d2 d3 =ðcdÞ; ð43Þ I ¼ r=d2 ð1 1=R0 Þ; ð44Þ V ¼ dr=ðd2 d3 Þð1 1=R0 Þ: ð45Þ Increasing the rate of production of susceptible cells or increasing the value of d always leads to an increase in the equilibrium abundance of virus. Now, let us assume that there are two types of viruses, a wild-type V and a rare mutant V*, which are liberated from infected cells at the corresponding rates d and d*.The number of cells infected by each kind of virus is denoted by I and I*. We have S_ ¼ r cðV þ V ÞS d1 S; ð46Þ I_ ¼ cVS d2 I; ð47Þ I_ ¼ cV S d2 I; ð48Þ V_ ¼ dI d3 V ; ð49Þ V_ ¼ d I d3 V : ð50Þ 70 D. C. KRAKAUER AND N. L. KOMAROVA An equilibrium of this system is given by equations (43)– (45), and I* ¼ V* ¼ 0. This solution is stable as long as R0 > 1 and d > d*. This means that the virus for which the quantity k5m0p0 is maximized is stable with respect to invasion of any other strain of virus differing in the value of the population level parameter d. An evolutionary stable encapsidation rate We know that population dynamics favours a steady increase in the rate of egress of virus from infected cells. However, we need to consider the impact of this population-level selection on the stability of the withincell virus dynamics. We can use the results of the section on ‘The intracellular level of selection’ to find the conditions under which the quantity k5m0p0 is maximized. Let us assume that dp ¼ 0: ð51Þ With this simplification, which amounts to assuming that the rate of decay of virus-encoded proteins is very slow, all the expressions become much more concise and it is easier to carry out the analysis. We stress, however, that very similar behaviour is observed for nonzero dp. We find after some calculation that the evolutionarily stable strategy for a virus is to have k5 ¼ bk1 Dopt ; ð52Þ where the optimal value of D is given by Dopt ¼ 1þ ð2 bÞbk1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð1 bÞk3 C init =dm ð53Þ Figure 2 illustrates this result: the quantity bk1)Dopt optimizes the curve k5m0p0 as a function of k5. We also present the result for the case dp > 0, which is qualitatively similar. From formula (53), we can see that increasing the decay rate of virus-encoded proteins reduces the maximum value that k5m0p0 can attain. Increasing the rate of polymerase–genome complex formation (k1), reducing the rate of ribosome–genome complex formation (k3) or increasing the decay rate of the virus genome (dm) all cause the optimum value of the parameter k5 to be reduced. In other words, these all favour a reduction in the rate of encapsidation of genomes by virus-encoded proteins for egress from the infected cell. tion. A trade-off between these two within host levels of selection is established, limiting virus replication rates, and hence limiting virulence in the absence of host mortality. Viral genomic architectures and life cycles are highly diverse. General classifications of viruses are often composites of these two features (Wildly, 1971). The first feature – architecture – relates to the organization of the virus genes and associated proteins, whereas the second feature – life cycle – is a dynamical property defined at the level of populations of infected cells. Evolutionary approaches to biology endeavour to explain the relationship between organization and dynamics. Frequently, even within evolutionary studies, these are treated as functionally independent. This is often for reasons of tractability. Thus, evolutionary models will assign payoffs or fitness values to onedimensional entities, variously described as genes, genomes, phenotypes or populations, without considering their detailed architecture and concentrate on their dynamics and long-term distributions. This is problematic, not least because the bulk of experimental research in biology is largely concerned with organization and tends to neglect dynamics. For example, virologists interested in virulence study virulence genes and their expression and consequences within the cell, whereas theorists tend to treat virulence as synonymous with high replication rates from infected cells. The world of viruses, as a result of their relative simplicity, provide a credible model system in which these two features can be integrated. We can treat viruses 0.12 k 5 m e pe b k1 – ∆ opt dp = 0 0.1 b k1 0.08 0.06 dp > 0 b k 1– ∆ c 0.04 0.02 Discussion We develop a model of virus replication dynamics in which the within-cell kinetics and between-cell population dynamics are both described explicitly. We characterize a bottleneck that arises as a result of incompatibilities between virus genome replication and virus protein synthesis. This bottleneck limits the maximum rate of virion production by infected cells, and opposes selection for maximum rates of cellular infec- 1 2 3 k5 4 Fig. 2 The quantity d ¼ k5m0p0 as a function of k5 for the two cases: (i) dp ¼ 0 and (ii) dp ¼ 0.01. The other parameters are as in Fig. 1. The vertical lines indicate the important values of k5: the threshold value, bk1, which is never reached (plot b), the existence condition for the virus equilibrium, bk1)Dc (for dp ¼ 0, equation (26)) and the selected value, bk1)Dopt (for dp ¼ 0, equation (53)). J. EVOL. BIOL. 16 (2003) 64–73 ª 2003 BLACKWELL PUBLISHING LTD Virus genome evolution as a microcosm of the problems of evolution, and ‘carve’ up the virus world into groups according to combined organizational and dynamical features. In this paper, we have presented some speculations on one of these groups, the single-stranded, positive-sense, RNA viruses. In our approach, organization is treated using a kinetic model (developmental model) of replication, and this is then related to mutation and selection at the population level through a separation of time scales: the organization emerges quickly, whereas the feedback from the population dynamics occurs more slowly. This multilevel selection approach highlights important constraints, or life history tradeoffs, that arise during the course of evolution. We describe some of these below. (1) We derive a critical persistence threshold representing a balance between replicating virus genomes within the cell, and synthesizing proteins in order to export virions from the cell. If the rate of export is too high, all genomes are removed from the cell before they have an opportunity to replicate. The replication of the positive-sense RNA hepatitis A virus (HAV) in BS-C-1 cells was examined under singlecycle growth conditions using strand-specific probes for detection of viral RNA species (Anderson et al., 1988). The results of this study suggest that encapsidation of positive-strand HAV RNA inhibits transcription at all times during the growth cycle, thereby reducing the pool of replicating RNA and the final yield of infectious HAV. In other words, encapsidation threatens to exhaust the store of RNA genomes within the cell. It seems that one way around this for the virus is to ensure that genomes be replicated before they are encapsidated. This strategy is found for flavivirus RNA genomes (Khromykh et al., 2001). Another solution is to divide the RNA into satellite RNA destined for replication and normal RNA destined for encapsidation. This is the solution favoured by velvet tobacco mottle virus (Hanada & Francki, 1989). It has even been suggested that persistent infection with enteroviruses in the central nervous system could depend on defective transcription, thereby biasing kinetics towards replication and away from protein synthesis and encapsidation. Evidence from coxsackie virus in human skeletal muscle cells does not support this position (Gow et al., 1997). (2) The model tells us that increasing the rate of decay of the virus genome (mRNA), or increasing the export of protein from the cell, both lead to an increase in the total amount of protein within the cell at equilibrium. For most viruses, the half-life of mRNAs varies according to the quantity of protein required within the cell. Long-lived RNAs generally encode structural proteins that are required most of the time, whereas shortlived RNAs are often associated with regulatory proteins that are only required transiently. For any RNA that is required constantly, the decay rate must not exceed the production rate; otherwise, the RNA will disappear. Hence at equilibrium, any increase in the rate of decay J. EVOL. BIOL. 16 (2003) 64–73 ª 2003 BLACKWELL PUBLISHING LTD 71 must be matched by an increase in production. The immediate outcome of this requirement is that increasing the decay rate can in some instances lead to an increase in the abundance of RNA. Herpes simplex virus carries within its virion a host shutoff protein (Vhs) that is delivered into cells prior to virus gene expression. Herpes also encodes the protein within its genome for synthesis following late gene expression. The effect of the Vhs protein is to reduce the stability of mRNA in infected cells (Zelus et al., 1996). The protein does not discriminate between host-derived and virus-derived mRNA and leads to an accelerated decay of both. The standard explanation for Vhs is that the protein reduces competition between virus- and host-derived mRNAs for access to translational machinery, producing a net benefit for the virus. We demonstrate, however, that such a strategy can increase genomic RNA without invoking host factors. An opposite finding is observed in the papilloma viruses, in which transformed cell lines producing benign tumours express unstable RNAs rich in AU nucleotides. In malignant, cervical carcinoma cells, as a consequence of integration into the host genome, the E6 and E7 gene mRNAs lack the AU rich sequences, and are consequently more stable (Jeon & Lambert, 1995). This increased stability is associated with a higher concentration of tumour-specific proteins. These two cases serve to illustrate that RNA decay rates are in themselves insufficient to determine protein abundance. Estimates of proteins require knowledge of the full chemical reaction scheme and decay rates for proteins. (3) The fully analytical solution to our system (under certain assumptions) shows that the rate of synthesis of new viral genomes from the polymerase complex is rate limiting and determines the time to reach equilibrium: the higher the replication rate the more quickly a fixed point is reached. The virus life cycle is made up from a number of consecutive steps. Although the cleavage of polyproteins and the subsequent assembly of capsids can be protracted, they are rarely rate-limiting. The rate-limiting step is most often associated with the production of new protein (k4) (Borovec & Anderson, 1993) or the replication of the virus genome (k2) (Anderson et al., 1988). Reduced rates of protein synthesis by polymerase can be sufficiently important to form the basis of restricted host ranges (Lemm et al., 1990). (4) We showed that evolution favours those viruses that maximize their rate of egress from infected cells. We then used this result to determine the evolutionary optimum value of the virus-determined rate parameter k5 – the rate of virus encapsidation and export from the infected cell. We found that k5 is bounded from below and from above by the intracellular and intercellular selection forces, and that an intermediate optimum exists. This optimum is greater for lower rates of genome 72 D. C. KRAKAUER AND N. L. KOMAROVA decay and lower rates of polymerase binding. Thus, population level selection feeds-back onto parameters influencing within-cell kinetics. (5) Instabilities arising through competing selection pressures at the intracellular and extracellular levels give rise to critical points in virus persistence, limiting virus virulence. There has been much discussion on the evolution of virulence (Frank, 1996; see also the volume by Diekman et al., 2002). In formal treatments of this problem virulence is modelled as increasing rates of virion production (Nowak & May 1994). Competition among virus strains within a single infected host is thought to produce the most productive strain capable of effective transmission between hosts. Thus host mortality and transmission opportunity place a ceiling on the virulence (e.g. Bonhoeffer & Nowak, 1994; Haraguchi & Sasaki, 2000). We have shown that an explicit coupling of the intracellular life cycle with the population dynamics leads to an additional kinetic restriction on the evolution of virulence: viruses that emerge too quickly from within the cell are incapable of persistent infection. This suggests that population selection increasing the encapsidation rate of viral genomes might effectively rid cells of infection. Evolutionary approaches to virulence could benefit from considering in more detail the mechanisms of virus replication within cells, as increasing fitness as measured at the cellular level, can correspond to the reduction of long-term fitness within cells. (6) Competition among the levels of selection organizes the virus life cycle. As a result of the distinct adaptations required for within-cell virus replication and protein production, and between-cell tropism and transmission, virus genomes come under selection for the expression of structurally independent traits. However, these traits are rarely functionally independent, as the previous discussion has illustrated. This suggests that selection for efficient kinetics will often oppose selection for efficient cellular population dynamics. This evolutionary conflict of interest between two levels of selection is played out in the structure of the virus life cycle. The poliovirus example presented in the Introduction suggests that this conflict is not always resolved. Whereas about 50% of polioviruses penetrate susceptible cells, only 1% establishes a productive infection. Traits valuable at the within-cell level do not perform as well as traits required at the level of the population of cells. Such a large reduction in viability might only be sustainable because of the large number of virus particles generated, reflecting the relationship between genomic stability and population size (Krakauer & Plotkin, 2002). A further dimension of the problem not addressed in the model is the consequence of heterogeneity in the genetic structure of the virus population. Thus, genetic conflicts arise both vertically among levels of organization and horizontally among more distantly related virus lineages. The study of Turner & Chao (1999) on the within-cell competition dynamics of the /6 phage shows how within-cell kinetic competition can lead to a mean reduction of virus fitness defined at the level of cellular propagation. As the within-cell phase becomes more intense, the ability to perform at the cellular level is increasingly compromised. This study also suggests how promoting competition at a suitable level of selection might be exploited towards therapeutic advantage, in which cellular pathology is promoted, in order to minimize further infection (Krakauer, 2002). Acknowledgments DCK thanks the John D. and Catherine T. MacArthur foundation, the National Science Foundation, and the US Department of Energy. NLK thanks the Alfred P. Sban Foundation, the Ambrose Monnell Foundation and the National Science Foundation. Thanks also to the anonymous referees for their constructive comments. References Anderson, D.A., Ross, B.C. & Locarnini, S.A. 1988. 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