Viral Invasion Fitness Across a Continuum from Lysis to Latency† Joshua S

Virus Evolution, 2019, 5(1): vez006

doi: 10.1093/ve/vez006 Research article Downloaded from https://academic.oup.com/ve/article-abstract/5/1/vez006/5476198 by Georgia Institute of Technology user on 23 April 2019 Viral invasion fitness across a continuum from lysis to latency† Joshua S. Weitz,1,2,* Guanlin Li,2 Hayriye Gulbudak,3 Michael H. Cortez,4 and Rachel J. Whitaker5,6

1School of Biological Sciences, 2School of Physics, Georgia Institute of Technology, Atlanta, GA, 30332, USA, 3Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA, 70504, USA, 4Department of Mathematics and Statistics Utah State University, Logan, UT, 84322, USA, 5Department of Microbiology and 6Carl R. Woese Institute for Genomic Biology, University of Illinois Urbana-Champaign, Champaign, IL, 61801, USA

*Corresponding author: E-mail: [email protected]

Abstract The prevailing paradigm in ecological studies of viruses and their microbial hosts is that the reproductive success of viruses depends on the proliferation of the ‘predator’, that is, the virus particle. Yet, viruses are obligate intracellular parasites, and the virus genome—the actual unit of selection—can persist and proliferate from one cell generation to the next without lysis or the production of new virus particles. Here, we propose a theoretical framework to quantify the invasion ﬁtness of viruses using an epidemiological cell-centric metric that focuses on the proliferation of viral genomes inside cells instead of virus particles outside cells. This cell-centric metric enables direct comparison of viral strategies characterized by obligate killing of hosts (e.g. via lysis), persistence of viral genomes inside hosts (e.g. via lysogeny), and strategies along a continuum between these extremes (e.g. via chronic infections). As a result, we can identify environmental drivers, life history traits, and key feedbacks that govern variation in viral propagation in nonlinear population models. For example, we identify threshold conditions given relatively low densities of susceptible cells and relatively high growth rates of infected cells in which lysogenic and other chronic strategies have higher potential viral reproduction than lytic strategies. Altogether, the theoretical framework helps unify the ongoing study of eco-evolutionary drivers of viral strategies in natural environments.

Key words: ecology; evolution; epidemiology; mathematical modeling; microbiology

1. Introduction result, studies of the ecological influence of viruses of microorganisms in natural environments have, for the most part, em- Viral infections begin with the physical interaction between a phasized the impact of the lytic mode of infection. However, the virus particle (the ‘virion’) and the host cell. Infection dynamics spread of viruses through microbial populations need not in- within the cell often culminate in lysis, that is, the active dis- volve the immediate lysis of the infected cell. ruption of the integrity of the cell surface, leading to the death Indeed, many viruses have alternative strategies. Temperate of the host cell and the release of infectious virus particles phage—like phage k—can integrate their genomes with that (Calendar 2005; Dimmock et al. 2007). Virus-induced lysis can be of their bacterial hosts, such that the integrated viral DNA a significant driver of microbial mortality at population scales (the prophage), is replicated along with the infected cell (the (Suttle 2007; Brum and Sullivan 2015; Breitbart et al. 2018). As a lysogen) (Ptashne 2004). Chronic viruses, like the filamentous

†Special Issue Santa Fe Institute Workshop on Integrating Critical Phenomena and Multi-Scale Selection in Virus Evolution

VC The Author(s) 2019. Published by Oxford University Press. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

1 2|Virus Evolution, 2019, Vol. 5, No. 1

phage M13, infect cells and persist episomally (Salivar et al. also extending the scope of applicability to chronic viral strate- 1964; Jaenisch et al. 1970), whereby the genome is replicated gies (where virion production does not require cell lysis). and then packaged into particles which are released extracellu- Further, we show how using a unified metric to characterize larly without necessarily inducing cell washout (Smeal et al. viral invasion dynamics can help predict and explain a contin- 2017a,b). An analogous mode of chronic infection has been ob- uum of infection strategies observed in different environmental served in archaeal virus–host systems (Snyder et al. 2015). contexts. These examples raise a critical question (see Knowles et al.

2016; Weitz et al. 2017; Knowles and Rohwer 2017): are temper- Downloaded from https://academic.oup.com/ve/article-abstract/5/1/vez006/5476198 by Georgia Institute of Technology user on 23 April 2019 ate or chronic modes prevalent or rare in nature? 2. Materials and methods More than a decade ago, studies of marine, hydrothermal, 2.1 Parameters and soil environments suggested that lysogeny could be more prevalent than assumed based on culture-based analysis of vi- The three model variants we analyze include a set of common rus–microbe interactions (Williamson et al. 2002, 2007, 2008; Paul parameters as well as parameters unique to particular models. 0 2008). This evidence has been augmented by recent studies iden- The parameters include b and b (maximal cellular growth rates, 1 1 tifying viral dark matter—including integrated and extrachromo- h ), K (carrying capacity, ml ), / (adsorption rate, ml/hr), d and 0 1 somal viral sequences—in microbial genomes (Roux et al. 2015; d (cellular death rates, h ), b (burst size), m (virion decay rate, 1 1 Luo et al. 2017; Kim and Bae 2018). Yet, despite increasing evi- h ), and g (lysis rate, h ). For the implicit latent model, param- dence of the relevance of persistent infections in situ the ecologi- eters include p (scaling factor for lysis, 0 < p < 1) and q (scaling cal study of phage has not integrated a common metric to factor for latency, 0 < q < 1), and for the explicit latent model, compare the context-dependent fitness of lytic, temperate, and parameters include 0 < ql < 1 (probability of integration) and c 1 other chronic viral strategies. A landmark theoretical study pro- (rate of induction, h ). The chronic model includes the parame- 1 vides a setting off point for investigating potential benefits of ter a (virion production/budding rate, h ). Additional context non-lytic strategies (Stewart and Levin 1984). Using a combina- on viral life history traits, including constraints and estimation tion of simulations and local stability analysis, this study pro- methods, is described in Weitz (2015). posed that temperate phage could persist over the long term if prophage integration directly enhanced host fitness or enhanced 2.2 Nonlinear dynamics resistance to infections by other lytic phage (‘superinfection immunity’). The same study predicted that oscillations in host 2.2.1 Nonlinear dynamics, obligately lytic strategies population abundances could provide an ecological ‘niche’ for The coupled system of nonlinear ordinary differential equations temperate phage. In essence, if bacterial densities were too low (ODEs) for the obligately lytic system is: to support the spread of lytic phage, then temperate phage al- ready integrated into lysogens could persist until ‘conditions be- zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{logistic growth infectionzffl}|ffl{ cellz}|{ washout dS come favorable for the bacteria to proliferate’ (Stewart and Levin ¼ bSð1 N=KÞ /SV dS dt 1984). Yet this finding does not exclude the possibility that lytic cell washout infectionzffl}|ffl{ z}|{lysis z}|{ dI ; (1) strategies could out-compete temperate strategies—even if lysis ¼ /SV gI d0I at low densities leads to population collapse. dt z}|{lysis infectionzffl}|ffl{ viralz}|{ decay More recently, efforts to understand why viruses should be dV ¼ bgI /SV mV temperate have drawn upon the mathematical theory of bet dt hedging (Maslov and Sneppen 2015). According to this application of bet hedging theory, the temperate strategy enables vi- where S, I, and V denote the densities of susceptible cells, ruses to expand rapidly during stable periods for hosts (via infected cells, and virus particles, respectively (see Beretta and lysis) and mitigate risks of population collapse, particularly dur- Kuang 1998; Weitz 2015). ing unfavorable periods for hosts (via lysogeny). Such estimates of long-term growth rates are limited in their applicability as 2.2.2 Nonlinear dynamics, latent strategies, implicit infections they do not include explicit dynamics of infected cells nor sub- The nonlinear ODEs for the latent system modeled using im- sequent virus-microbe feedback. Moreover, a focus on long-term plicit infections are: estimates of growth does not address whether or not lysis is the advantageous strategy for a virus at a given moment in time. As zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{logistic growth infectionzffl}|ffl{ cellz}|{ washout dS noted by Maslov and Sneppen (2015), ecological models that in- ¼ bSð1 N=KÞ /SV dS dt corporate explicit mechanisms underlying virus–host interac- lysogen growth zfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflffl{ infectionzffl}|ffl{ z}|{lysis cellz}|{ washout tions are required to understand the viability of viral strategies. dL ; (2) ¼ qb0Lð1 N=KÞ þ /SV pgL d0L In this article, we draw upon the foundations of mathemati- dt zffl}|ffl{lysis infectionzffl}|ffl{ viralz}|{ decay cal epidemiology to quantify viral fitness measured in terms of dV ¼ bpgL /SV mV the proliferation of infected cells instead of virus particles. In do- dt ing so, we propose to adapt the basic reproduction number, R0, “arguably the most important quantity in infectious disease epi- where S, L,andV denote densities of susceptible cells, lysogens demiology” (Diekmann et al. 2012) to analyze broad classes of (latent cells), and virus particles, respectively, and the total virus–microbe dynamics, including those characterized by lytic, number of cells is N ¼ S þ L. Here, the relative rate of lysogenic latent, and chronic strategies. In particular, our study unifies growth and cellular lysis is controlled by the scaling factors q previous proposed definitions for purely lytic phage (i.e. termed and p.Whenq ¼ 1andp ¼ 0, then all infections are strictly la- the ‘phage reproductive number’; Jones and Smith 2011; Smith tent and only lead to lysogenic growth. In contrast, when q ¼ 0 and Thieme 2012; Krysiak-Baltyn et al. 2016) and application of and p ¼ 1 then all infections are strictly lytic and only lead to the R0 concept to temperate phage (where virion production cellular lysis. This is a variant of a nutrient-explicit formula- requires cell lysis) (Berngruber et al. 2013; Gandon 2016), while tion considered as part of an analysis of the tradeoffs J. S. Weitz et al. | 3

underlying lysis and lysogeny for marine viruses (Record et al. any population type that has an infectious viral genome, that is, 2016). Of note, this model does not include the absorption of vi- both infected cells and virus particles. rus particles to latent cells nor the degradation of prophage (i.e. ‘curing’). 2.3.1 NGM: obligately lytic interactions We linearize the dynamics of Eq. (1) around the virus-free equi- 2.2.3 Nonlinear dynamics, latent strategies, explicit infections librium, (S; 0; 0Þ where S ¼ Kð1 d=bÞ,andfocusonthe The nonlinear ODEs for latent strategies as modeled using ex- infected subsystem denoted by XðtÞ¼½IðtÞ; VðtÞ>. The linear- plicit infections are: ized infected subsystem dynamics can be written as Downloaded from https://academic.oup.com/ve/article-abstract/5/1/vez006/5476198 by Georgia Institute of Technology user on 23 April 2019 X_ ¼ðT þ RÞX. The decomposition of dynamics separates trans- logistic growth zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{ mission events in T (i.e. corresponding to epidemiological infectionzffl}|ffl{ cellz}|{ washout dS N births) from transition events in R (i.e. corresponding to ¼ bS 1 /SV dS dt K changes in the state of viral genomes, including loss of infec- zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{infection z}|{lysis inductionz}|{ cellz}|{ washout tions). This decomposition also helps in the biological interpre- dI 0 ¼ð1 qlÞ/SV gI þ cL d I tation of the resulting R0 (see Diekmann et al. 2010). Here, the dt ; (3) lysogen growth matrix T can be written as: zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{ cell washout integrationzfflfflffl}|fflfflffl{ inductionz}|{ z}|{ dL N / 0 c 0 ¼ ql SV þ b L 1 L d L 0 /S dt K T ¼ ; (5) z}|{burst infectionzffl}|ffl{ viralz}|{ decay 00 dV ¼ bgI /SV mV dt where the elements denote the birth of individuals of a particular infected class from another class. In this case, there is only where S, I, L, and V denote the densities of susceptible cells, lyti- one kind of birth: an increase in infected cells (I) arising from cally infected cells, lysogens, and virus particles, respectively, horizontally transmitted infections due to virus particles (V). and the total number of cells is N ¼ S þ I þ L. In this model, ql Similarly, the transition matrix is: and ð1 ql) are the probabilities that a new infection enters a lysogenic pathway or a lytic pathway, respectively. When ql ¼ 1 g d0 0 all infections are strictly latent and when ql ¼ 0 all infections are R ¼ : (6) bg /S m strictly lytic. This is a variant of the temperate phage model studied by (Berngruber et al. 2013). Given our focus on invasion Via the NGM theory, the basic reproduction number R fitness, throughout the present analysis we assume c ¼ 0, that is 0 corresponds to the largest eigenvalue of the matrix TR1.The there is no induction of lysogens. i, j matrix elements of R1 correspond to the expected duration in state i of a viral genome that begins in state j.Forthis 2.2.4 Nonlinear dynamics, chronic strategies model, The nonlinear ODEs for the chronic strategies are: 2 3 1 logistic growth infection cell washout 6 0 7 zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{ zffl}|ffl{ z}|{ 6 0 7 dS 1 6 g þd 7 ¼ bSð1 N=KÞ /SV dS R ¼ b g : (7) dt 4 1 5 0 zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{logistic growth infectionzffl}|ffl{ cellz}|{ washout /S þ m g þ d /S þ m dC ; (4) ¼ b0Cð1 N=KÞ þ /SV d0C dt virionz}|{ production infectionzffl}|ffl{ viralz}|{ decay As a consequence, the basic reproduction number is: dV ¼ aC /SV mV dt b/S g R ¼ : (8) 0 /S þ m g þ d0 where S, C, and V denote densities of susceptible cells, chronically infected cells, and virus particles, respectively, and the total number of cells is N ¼ S þ C (see Gulbudak and Weitz 2019). Although the nonlinear population model of Eq. (4) can be 2.3.2 NGM: latent strategies, implicit remapped to the latency model (Eq. 2), this system of equations The linearized infected subsystem dynamics of Eq. (2) where > _ represents distinct mechanistic processes, including establish- XðtÞ¼½LðtÞ; VðtÞ can be written as X ¼ðT þ RÞX where ment of a chronically infected cell and release of virions from qb0ð1 S=KÞ /S chronically infected cells without lysis at a per-capita rate a.Of T ¼ (9) note, this model assumes that infected mother and daughter cells 00 both retain a copy of the viral genome. Partial fidelity of the vertical viral transmission process could be represented via a birth- denotes transmission events and dependent transfer of population from the C to the S states. pg d0 0 R ¼ (10) bpg /S m 2.3 Next-generation matrix approach

We use the next-generation matrix (NGM) approach to calculate R0 denotes transition events. For this model, in mathematical models of interactions between cells and viruses 2 3 (see 3.1 for a definition and interpretation). We follow the conven- 1 6 0 7 tion of Diekmann et al. (2010) in analyzing the subset of the epide- 6 d0 þ pg 7 R1 ¼ 6 7: miological model including infected subclasses. In the case of 4 bpg 1 1 5 (11) viruses of microbes, we denote those infected subclasses to include pg þ d0 /S þ m /S þ m 4|Virus Evolution, 2019, Vol. 5, No. 1

As a consequence, the basic reproduction number is: As a consequence, the basic reproduction number is:

0 qb ð1 S=KÞ b/S pg b0ð1 S=KÞ /S a R0 ¼ þ : (12) R ¼ þ : (21) d0 þ pg /S þ m pg þ d0 0 d0 /S þ m d0

2.3.3 NGM: latent strategies, explicit The linearized infected subsystem dynamics of Eq. (3) 3. Results Downloaded from https://academic.oup.com/ve/article-abstract/5/1/vez006/5476198 by Georgia Institute of Technology user on 23 April 2019 T where XðtÞ¼½IðtÞ; LðtÞ; VðtÞ can be written as X_ ¼ðT þ RÞX 3.1 Cell-centric metric of viral fitness where We consider the spread of viruses through a microbial popula- 2 3 tion. Foundational work in virus-microbe dynamics conceptual- 00ð1 qlÞ/S 4 0 5 T ¼ 0 b ð1 S =KÞ ql/S (13) ized viruses as ‘predators’ and bacteria as their ‘prey’ (sensu 00 0 Campbell 1961; Levin et al. 1977)—a paradigm that has become well-established with time (e.g. Levin 2010). Rather than apply- denotes transmission events and ing the conventions of predator-prey theory, we analyze models of virus-microbe dynamics in terms of the spread of an infec- 2 3 ðg þ d0Þ 00 tious disease (see Fig. 1). The spread of an infectious disease can 4 0 5 R ¼ 0 d 0 (14) be quantified in terms of the basic reproduction number, R0, bg 0 ðm þ /S Þ that is when R0 > 1 then a pathogen is expected to increase its relative abundance in a population (Diekmann et al. 1990, 2010). denotes transition events; in both cases c ¼ 0. For this model, Here we propose the following definition of the basic reproduc- 2 3 tive number for generalized virus–microbe dynamics: 1 6 007 R : the average number of new infected cells produced by a single (typi- 6 g þ d0 7 0 6 7 cal) infected cell and its progeny virions in an otherwise susceptible 1 6 1 7 R ¼ 6 0 0 7: (15) population 6 d0 7 . 4 g b 1 5 0 g þ d0 /S þ m m þ /S This definition of R0 counts viral reproduction in terms of infected cells, as in the study of eco-evolutionary dynamics of As a consequence, the basic reproduction number is: temperate phage (Berngruber et al. 2013; Gandon 2016), rather than in terms of virus particles. In doing so, this definition () builds upon insights from the virocell paradigm (Forterre 2011, b0 S b/S g R0 ¼ max 1 ; ð1 q Þ : (16) 2013), wherein the ‘real living [viral] organism’ (Forterre 2011)is d0 K l /S þ m g þ d0 an infected cell actively reproducing new virions, that is the ‘virocell’. However, here we depart from a strict virocell defini- In the event of exclusively latent interactions, that is q ¼ 1, l tion, by accounting for transmission via virions and transmis- then the basic reproduction number sion via latent infections, for example where viral genomes are integrated into the genomes of their hosts. b0 S R ¼ 1 (17) As we will show, the cell-centric definition of viral fitness 0 d0 K proposed here facilitates comparison of infections caused by ‘vertical’ transmission (i.e. from mother to daughter cell), is equivalent to that of the NGM calculation for the model of la- those caused by ‘horizontal’ transmission (i.e. from an tent strategies with implicit interactions. infected cell to another susceptible cell in the population), and those caused by a combination of both routes (e.g. as in 2.3.4 NGM: chronic strategies chronic viruses). In doing so, R0 quantifies fitness at the indi- The linearized infected subsystem dynamics of Eq. (4) where vidual scale, that is beginning with the virus infection of a sin- ; > _ R XðtÞ¼½IðtÞ VðtÞ can be written as X ¼ðT þ ÞX where gle microbe, and also represents a threshold condition for viral invasion at the population scale. Note that our definition b0ð1 S=KÞ /S T ¼ (18) of fitness does not account for feedback between viral popula- 00 tiongrowthandtheenvironment,anissuewereturntointhe Discussion. denotes transmission events and d0 0 3.2 Obligately lytic viral strategies: a baseline for R ¼ (19) a /S m comparison Dynamics of obligately lytic viruses and their microbial hosts denotes transition events. For this model, can be represented via a set of nonlinear differential equations 2 3 (see Fig. 1 for this and other model schematics; see Section 2 for 1 parameters and equations). The spread of viruses in an other- 6 0 7 1 6 d0 7 wise susceptible population in Eq. (1) were analyzed using the R ¼ 4 a 1 1 5 (20) NGM approach (see Section 2). Via NGM, we find that obligately d0 /S þ m /S þ m lytic viruses should spread when J. S. Weitz et al. | 5 Downloaded from https://academic.oup.com/ve/article-abstract/5/1/vez006/5476198 by Georgia Institute of Technology user on 23 April 2019

Figure 1. Schematic of obligately lytic, chronic, and latent strategies, give a population perspective (top) and individual perspective (bottom). Top, The nonlinear dynamics for each model are presented in Section 2. Bottom, The basic reproduction number accounts for complete cycles beginning with infected cells. In the obligately lytic case, only three virions of many infect cells, these are progeny, aka new mothers. In the chronic case, the mother cell divides once and two progeny virions infect new cells. In the latent case, the mother cell divides three times before it is removed. In all of these examples, the total reproduction number is the same, albeit with differing contributions from horizontal and vertical transmission routes.

infected cells, then only a fraction g=ðg þ d0Þ of infected cells will lyse and release virions before being washed out of the system. Finally, these two probabilities must be multiplied by the burst size b, that is the number of new virions released, to yield the average number of new infectious virions produced by a single virion in a susceptible host population. This product is equal to

the basic reproduction number, Rhor (what was previously termed the phage reproductive number Jones and Smith 2011;

Smith and Thieme 2012; Krysiak-Baltyn et al. 2016). When Rhor exceeds 1 then a single virion produces, on average, more than one virion, of which each in turn produces, on average, more than one virion and so on. Figure 2 shows how viral proliferation varies with life history traits (in this case, the burst size) and the ecological context (in this case, the initial cell density). The criti-

cal value R0 ¼ 1 defines the threshold between regimes of viral extinction and proliferation. Second, we can revisit this same calculation beginning with an assumption that there is a single infected cell in an otherwise susceptible population. In that event, the infected cell pro- Figure 2. Virus reproduction as a function of burst size and susceptible cell den- duces b virions a fraction g=ðg þ d0Þ of the time, of which only a sity. The contours denote the log of R , as measured using Eq. (22), given varia- 10 0 tion in burst size, b, and susceptible cell density, S*. Viruses invade when fraction /S =ð/S þ mÞ are adsorbed before they decay. The

Rhor > 1 or, equivalently when log 10Rhor > 0. Contours denote combinations of product represents the number of newly infected cells produced ðb; S Þ of equivalent Rhor. Additional parameters that affect viral reproduction by a single infected cell in an otherwise susceptible population. 1 are u ¼ 6:7 10 10 ml/h and m ¼ 1/24 h . The product is the same, but in this alternative approach we have counted proliferation in terms of a viral life cycle that starts and ends inside cells. Although both interpretations—the /S g R ¼ b > 1; (22) virion-centric and the cell-centric—lead to equivalent estimates hor /S þ m g þ d0 of R0, we will use the cell-centric definition to unify subsequent comparisons across a spectrum of viral strategies. where b is the burst size, / is the adsorption rate, m is the decay rate of virus particles, 1=g is the latent period, d0 is the loss 3.3 Latent viral strategies rate of infected cells, and S is the equilibrium density of susceptible cells. This threshold value represents the (exclusively) In this section, we consider the dynamics of latent viral strate- horizontal contributions to the basic reproduction number. gies, such as temperate phage, in which proliferation may be ei- This inequality can be understood in two ways (see Fig. 1). ther horizontal or vertical (but not both simultaneously). First, First, consider a single virion. Virions successfully adsorb we focus on the case where virus genomes exclusively integrate to susceptible hosts at a rate /S. In contrast, virions decay at with host cell genomes which can then be passed on to daugh- aratem. The factor /S=ð/S þ mÞ denotes the probability that ter cells. We use the cell-centric interpretation as before, and a virion is adsorbed before it decays. Adsorption need not lead consider infection dynamics given a single lysogen in an other- to lysis, instead given a lysis rate of g and a loss rate d0 of wise susceptible population (see Eq. 2). Via the NGM analysis 6|Virus Evolution, 2019, Vol. 5, No. 1

detailed in Section 2 (see Section 2.3.2), we find that lysogens proliferate when

b0ð1 S=KÞ R ¼ > 1; (23) ver d0 where b0 denotes the division rate of infected cells, K is the density of susceptible cells in the absence of viruses, and other Downloaded from https://academic.oup.com/ve/article-abstract/5/1/vez006/5476198 by Georgia Institute of Technology user on 23 April 2019 parameters are equivalent to the obligately lytic model. Here the value of R0 is derived from vertical transmission of viral genomes among lysogens. The basic reproduction number can be interpreted mecha- nistically. The term b0ð1 S=KÞ represents the birth rate of lysogens, which decreases with increasing number of cells— whether susceptibles or lysogens. Given that d0 is the death rate of lysogens, the term 1=d0 denotes the average lifespan of an individual lysogen. Therefore, this reproduction number is equal to the average number of newly infectious cells produced in the lifetime of the original infection (see Fig. 1). When Rver exceeds one, then a single lysogen will beget more than one lysogen, on average, and those lysogens will do the same, and Figure 3. Basic reproduction number of temperate viruses as a function of sus- so on. ceptible cell density. The increasing (red) line denotes the horizontal R0 if tem- As is evident, lysogens reproduce more frequently when perate phage infect then always lyse cells. The decreasing (blue) line denotes they are subject to less competition with hosts, that is when S the vertical R0 if temperate viruses always integrate with their hosts. Relevant parameters are b ¼ 50, u ¼ 6:7 1010 ml/h, K ¼ 7:5 107 ml–1, and b0 ¼ 0:32, 0.54 is small relative to K. Given the value of S, the basic reproduc- and 1 h1 as well as d0 ¼ 0:75, 0.44, and 0.24 h1 for the three lysogeny curves 0 0 tion number can be written as Rver ¼ðb =d Þ=ðb=dÞ, where b and from bottom to top respectively. d denote the maximal cellular growth rate and cellular death rate of susceptible cells, respectively. Hence, if lysogens have more advantageous life history traits than do susceptible cells infection, both vertical and horizontal transmission can take then viruses can spread exclusively via vertical transmission. place concurrently. Via a NGM analysis, we find that a small This benefit of lysogeny applies in the immediate term and con- number of chronically infected cells will spread in a population stitutes direct support for how a lysogen that benefits its host when can also benefit the virus. However, if lysogeny comes with a 0 0 cost (i.e. b =d lower than b/d), then vertical transmission alone horizontal vertical zfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{ will not be enough for Rver > 1. Note that Rver is a monotoni- a /S b0ð1 S=KÞ R þ > 1; (24) cally decreasing function of S , such that increased abundan- chron d0 /S þ m d0 ces—all things being equal—diminishes the advantage for vertical transmission. where a is the virus particle production rate and all other This analysis raises the question: does a strictly lytic or parameters are equivalent to those defined in the obligately strictly lysogenic strategy have a higher basic reproduction lytic and latent models. In the horizontal pathway, the chronic number? Recall that the horizontal R0 is an increasing function cell will remain viable for an average duration of 1=d0. In that of susceptible cell density, that is when there are more hosts time, the chronic cell will produce new virions at a rate a,of then the value of horizontal transmission increases. The value which only /S=ð/S þ mÞ will survive to enter another cell. of Rhor and Rver cross at a critical value, Sc (see Fig. 3). For S > Sc, Concurrently, the chronic cell will divide initially at a rate then horizontal transmission is favored and for S < Sc then ver- b0ð1 S=KÞ, which when multiplied by the average cell duration tical transmission is favored. To further examine this finding, of 1=d0 yields the expected number of daughter cells, that is rep- we extended the current model to include an explicit represen- resenting vertical transmission. A chronic virus will spread at tation of lysogeny, that is in which infection can lead to lysis or the population scale due to the combination of transmission via to initiation of lysogeny (see Section 2). In the event that induc- horizontal and vertical components. The spread of chronic vi- tion is rare, we find that R0 converges to the maximum of the ruses depends on both infected cell traits and virion-associated horizontal and vertical fitness, with precisely the same func- traits. As a consequence, it would suggest that chronic viruses tional dependencies on S as in Fig. 3 (see calculation in Section should evolve to improve the sum of horizontal and vertical re- 2.3.3). Hence, whether latency is modeled implicitly or explic- production. Trade-offs likely constrain the evolution of virion itly, we find that there is a critical susceptible cell density above release rates and cell duration. For example, increasing the vi- which horizontal modes of infection should be favored over ver- rion production rate, a, may improve horizontal reproduction, tical modes, when measured in terms of invasion fitness. but if doing so increases cell washout, d0, then the overall Analysis of dynamics for latent viruses that includes feedback change in Rchron may be negative. As a result, it is possible that of induction processes will be the subject of follow-up work. chronic viruses could have the largest reproduction number in an intermediate density regime (see example in Fig. 4). Understanding the pleiotropic effects of changes to chronic vi- 3.4 Chronic viral strategies rus genotypes may provide one route to characterizing the evo- Finally, we consider the dynamics of ‘chronic’ virus strategies, lution of viral strategies in which both horizontal and vertical or what have been termed ‘persister’ or ‘producer’ strains in transmission rates operate concurrently (Goldhill and Turner other contexts (see Fig. 1 and the Section 2). In a chronic 2014). J. S. Weitz et al. | 7

Weitz et al. 2017; Knowles and Rohwer 2017). Our analysis makes it evident that evaluating the benefits of latent or chronic strategies also requires consideration of the intracellular infection status of hosts. For example, in our models, strictly (or par- tially) vertically transmitted viruses may be favored when new susceptible hosts are scarce and when infections benefit host competitive fitness, for example through growth or survival.

This may help to explain the inverse relationship between in- Downloaded from https://academic.oup.com/ve/article-abstract/5/1/vez006/5476198 by Georgia Institute of Technology user on 23 April 2019 ducible lysogen fraction and total microbial abundances in marine environments (Brum et al. 2016). The present approach adapts an epidemiological framework beyond temperate phage (as analyzed by Berngruber et al. 2013; Gandon 2016) to include obligately lytic and chronic strategies. In doing so, we have focused our analysis on short-term invasion scenarios. Comprehensive understanding of viral strategies requires analysis of both short- and long-term dynamics (Day and Gandon 2007). This is particularly relevant given that evolu- Figure 4. Viral strategies with the highest R0 vary with susceptible host density, tionary dynamics need not lead to the maximization of R0 including exclusively vertical (bold blue, left), mixed (bold green, middle), and horizontal (bold red, right) modes of transmission. Relevant parameters are (i) (reviewed by Lion and Metz 2018). For example, evolution is not for obligately lytic viruses (red): b ¼ 100, u ¼ 6:7 1010 ml/h, and m ¼ 0.13 h1; expected to maximize R0 given mixed transmission modes (e.g. (ii) for chronic viruses (green): b0 ¼ 0:68 h1, d0 ¼ 0:63 h1, a ¼ 20 h1, u ¼ horizontal and vertical) (Cortez 2013). Indeed, long-term fates 3:4 1010 ml/h, and m ¼ 0.04 h1; (iii) for temperate viruses, given vertical trans- are influenced by the Malthusian (i.e. exponential) growth rate 0 1 0 1 7 –1 mission (blue) b ¼ 0:54 h , d ¼ 0:44 h , where K ¼ 7:5 10 ml in all three of viruses which we denote as r. The basic reproduction number scenarios given variation in S. R0 and r are related, but they are not equivalent (Wallinga and Lipsitch 2007)—R0 measures the speed of viral proliferation in generations (i.e. at the individual level) whereas r measures the 4. Discussion speed of viral proliferation in time (i.e. at the population level)

We have proposed a unified theoretical framework to measure (Park et al. 2019). To illustrate this point, note that Rver in Eq. the spread of viruses within microbial populations when utiliz- (23) remains unchanged if lysogen birth and death rates (b0 and ing strategies spanning a continuum from lysis to latent to d0) are multiplied by the same constant factor, whereas the chronic. By defining viral reproduction in terms of infected cells, lysogen’s population growth rate would change because it is we are able to directly compare the spread of obligately lytic vi- equal to the difference of b0 d0 and not their ratios. ruses, latent viruses, and chronic viruses in the context of non- More generally, the threshold condition R0 > 1 indicates linear population models (see Fig. 1). The invasibility of a newly whether the population growth rate r is positive, but does not introduced virus is measured in terms of the basic reproduction predict changes in fitness given viral-host feedback. For exam- number, specifically adapted to the life cycle of viral infections ple, virus proliferation depletes susceptible hosts, thereby de- of microbial hosts. creasing the ‘effective’ viral fitness of the obligately lytic At its core, the theoretical framework re-envisions life his- pathway—an outcome concordant with prior findings from tory theory for viruses that infect microorganisms. In our calcu- mathematical models and eco-evolutionary experiments in- lations, a focal virus genome inside a cell can be thought of as a volving temperate phage k and E. coli (Berngruber et al. 2013). In ‘mother virus’. These mother viruses may lyse cells and pro- the work of Berngruber et al. (2013), obligately virulent phage in- duce ‘juvenile’ offspring, that is virus particles. When a virion creased in relative number in environments initially dominated successfully infects a susceptible host this new infection by susceptible cells, whereas temperate phage strains exhibited becomes, once again, a mother virus. This is an example of hori- higher population level growth when susceptible cells were sub- zontal transmission. For latent and chronic viruses, the viral ge- sequently depleted via lysis. Extrapolating to chronic strategies, nome inside an infected cell may be passed on to both cells we expect that viral production may shift from horizontal to upon division. This division is equivalent to direct reproduction vertical as viruses proliferate through a microbial population. A of a mother virus, bypassing the juvenile state. This is an exam- full analysis requires consideration of the invasion of subse- ple of vertical transmission. Combinations of these two scenar- quent mutant viruses in the environment set by resident ios emerge in applying NGM theory for calculating the basic viruses. reproduction number of viral strategies (see Figs 2–4 and Systematic analysis of the evolution of viral traits spanning Section 2). lysis, latent, and chronic strategies in an ecological context is The critical invasion fitness of a virus strategy, as calculated likely to draw upon a substantial body of work on the evolution in terms of R0, depends on life history traits as well as suscepti- of virulence (Ewald 1994; Lenski and May 1994; Lipsitch et al. ble cell density. Obligately lytic viruses have increasing values 1996; Frank 1996; Turner et al. 1998; Brown et al. 2002; Ebert and of R0 in populations with larger numbers of susceptible hosts. Bull 2003; Alizon et al. 2009). Priority areas include the evolution This trend is consistent with experimental findings that fitness of traits when viral particles can persist for long periods in the of virulent phage kcI857 declines with decreasing susceptible environment (similar to epidemiological models of the ‘curse of cell density (Berngruber et al. 2013). More broadly, we contend the pharaoh’ (Bonhoeffer et al. 1996; Gandon 1998)) and the evo- that the link between strategy and susceptible host density will lution of transmission mode itself (Antonovics et al. 2017). In inform ongoing debates on conditions that favor lysogeny and moving forward, one immediate opportunity is to assess how other persistent infections in marine systems; debates that viruses of microbes evolve virulence levels, or even strategy have focused on the ratio of free virus particles and total micro- types, when co-infecting the same microbial population. For ex- bial densities (Knowles et al. 2016; Thingstad and Bratbak 2016; ample, analysis via the cell-centric approach implies that lytic 8|Virus Evolution, 2019, Vol. 5, No. 1

viruses may reduce niche competition between cells, increasing Beretta, E., and Kuang, Y. (1998) ‘Modeling and Analysis of a the benefits of vertical transmission, and enable invasion by la- Marine Bacteriophage Infection’, Mathematical Biosciences, 149: tent or chronic viruses (Gulbudak and Weitz 2019). This finding 57–76. is consistent with repeated evidence of coinfection in microbial Berngruber, T. W. et al. (2013) ‘Evolution of Virulence in genomes between ssDNA filamentous phage (Inoviridae that Emerging Epidemics’, PLoS Pathogens, 9: e1003209. have a chronic lifestyle) and dsDNA phage (Caudovirales, that Bonhoeffer, S., Lenski, R. E., and Ebert, D. (1996) ‘The Curse of the can transmit via lysis) (Roux et al. 2015). 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