Editor's evaluation: Dynamics of immune memory and learning in bacterial communities

Abstract

Article Figures and data Abstract Editor's evaluation Introduction Results Discussion Methods Appendix 1 Appendix 2 Appendix 3 Data availability References Decision letter Author response Article and author information Metrics Abstract From bacteria to humans, adaptive immune systems provide learned memories of past infections. Despite their vast biological differences, adaptive immunity shares features from microbes to vertebrates such as emergent immune diversity, long-term coexistence of hosts and pathogens, and fitness pressures from evolving pathogens and adapting hosts, yet there is no conceptual model that addresses all of these together. To this end, we propose and solve a simple phenomenological model of CRISPR-based adaptive immunity in microbes. We show that in coexisting phage and bacteria populations, immune diversity in both populations is coupled and emerges spontaneously, that bacteria track phage evolution with a context-dependent lag, and that high levels of diversity are paradoxically linked to low overall CRISPR immunity. We define average immunity, an important summary parameter predicted by our model, and use it to perform synthetic time-shift analyses on available experimental data to reveal different modalities of coevolution. Finally, immune cross-reactivity in our model leads to qualitatively different states of evolutionary dynamics, including an influenza-like traveling wave regime that resembles a similar state in models of vertebrate adaptive immunity. Our results show that CRISPR immunity provides a tractable model, both theoretically and experimentally, to understand general features of adaptive immunity. Editor's evaluation In this important work, the authors develop a theory for the coevolutionary dynamics of bacteria and phages, where the major evolutionary pressure comes from CRISPR-Cas adaptive immunity in bacteria. Through extensive stochastic numerical simulations and analytical calculations, the article presents a compelling analysis of the emergent properties of immune interactions, in the regime of a single proto-spacer and a single spacer. Some of the trends highlighted by the model are recovered from experimental data. The main results concern how diversity in both phage and bacteria population is linked and is shaped by immunity, and should be of broad interest in immunology. https://doi.org/10.7554/eLife.81692.sa0 Decision letter Reviews on Sciety eLife's review process Introduction Adaptive immunity equips organisms to survive changing pathogen attacks across their lifetime. Many diverse organisms from bacteria to humans possess adaptive immune systems, and their presence shapes the survival, diversity, and evolution of both hosts and pathogens. How adaptive immunity changes the landscape of host-pathogen coexistence, how immune diversity emerges and evolves, and how the pressures of evolving pathogens and adaptive immunity are coupled to produce unique evolutionary outcomes: all of these factors are of fundamental importance to understanding the role of adaptive immunity in populations. These questions have naturally been explored in the vertebrate adaptive immune system, which protects humans and other vertebrates from evolving pathogens. In these organisms, a diverse repertoire of T cell and B cell receptors can rapidly recognize and respond to a wide range of threats. Immune specificity is determined by the unique genetic sequence of each cell’s receptor, and individuals may harbour millions to billions of unique sequences distributed across four or more orders of magnitude of abundance (Desponds et al., 2016; Mora and Walczak, 2019; de Greef et al., 2020). Quantitative frameworks to model immune diversity and clone abundance have revealed that simple low-level interactions can give rise to complex outcomes including broad distributions of clone abundance (Desponds et al., 2016; Mora and Walczak, 2019; de Greef et al., 2020; Mayer et al., 2015; Gaimann et al., 2020; Dessalles et al., 2022), long-lived biologically realistic transient states (Yan et al., 2019; Gaimann et al., 2020), and clonal restructuring following immune challenges (Childs et al., 2015; Puelma Touzel et al., 2020; Sachdeva et al., 2020; Molari et al., 2020; Gaimann et al., 2020). Phenomenological models of pathogen coevolution with the immune system have accelerated our understanding of how the fitness landscape generated by the immune system constrains pathogen evolution (Luksza and Lässig, 2014; Marchi et al., 2019; Yan et al., 2019; Schnaack and Nourmohammad, 2021; Chardès et al., 2022), how the adaptive immune system responds to rapid pathogen evolution (Wang et al., 2015; Nourmohammad et al., 2019; Schnaack and Nourmohammad, 2021; Chardès et al., 2022), and what drives pathogen extinction (Yan et al., 2019; Marchi et al., 2019 or the extinction of particular clonal cell lineages Nourmohammad et al., 2019; Sachdeva et al., 2020). These models have also explored trade-offs such as between immune receptor specificity and cross-reactivity (Mayer et al., 2015; Nourmohammad et al., 2016), between the specificity of host-pathogen discrimination and sensitivity to pathogens (Childs et al., 2015; Downie et al., 2021; Metcalf et al., 2017), between the speed of an immune response and the efficiency of that response (Schnaack and Nourmohammad, 2021), or between metabolic resource use and immune coverage (Chardès et al., 2022). All of these models have shown rich dynamics and qualitatively different states of diversity and evolution arising from simple rules. However, experiments in vertebrates are difficult: vertebrate immunity depends on a complex interplay of many cell types and experiments are time-consuming because of long generation times (Altan-Bonnet et al., 2020). Adaptive immunity in microbes is realized through the CRISPR system, conceptually related to the vertebrate adaptive immune system. The CRISPR system is functionally simple, yet it is incredibly powerful, as indicated by its widespread presence in many diverse bacteria and archaea Koonin and Makarova, 2019 and its experimentally demonstrated ability to provide strong immunity against phages (Paez-Espino et al., 2013; Paez-Espino et al., 2015; van Houte et al., 2016; Bondy-Denomy et al., 2013). Attacking phages expose their DNA to bacteria, and bacteria with a CRISPR immune system acquire small segments of phage DNA, called spacers. They store spacers in their genome and use them to recognize and destroy matching phage sequences in future infections: spacers are transcribed into RNA and guide DNA-cleaving CRISPR-associated proteins to recognize and cut re-infecting phages. Spacers provide a highly specific immune memory of infecting phages, preventing recognized phages from reproducing. In turn, phages can acquire mutations in the protospacer regions of their genome that are targeted by spacers. These features of the CRISPR immune system mean that (a) phage genetic evolution occurs by selection for escape mutants, and (b) the network of CRISPR immune interactions between bacteria and phages can be inferred by sequencing the genomes of co-living bacteria and phages. Spacer acquisition and phage mutation are rare random events, and many such events must be observed in order to understand their impact on populations. Bacteria and phages have short life cycles and can reach large population size, making it possible to build a statistical picture of the impacts of adaptive immunity. The kinetics and interactions of phages and bacteria with CRISPR systems have been the subject of numerous experiments (van Houte et al., 2016; Common et al., 2019; Common et al., 2020; Chabas et al., 2021; Dimitriu et al., 2022; Guillemet et al., 2021). Some themes have emerged from experimental studies of CRISPR immunity: (a) high spacer diversity relative to phage diversity increases the likelihood of phage extinction (van Houte et al., 2016; Common et al., 2020; Guillemet et al., 2021), (b) bacteria become more immune to phages over time (Laanto et al., 2017; Morley et al., 2017; Common et al., 2019; Pyenson and Marraffini, 2020), and (c) phages readily gain mutations (Weinberger et al., 2012a; Paez-Espino et al., 2013; Levin et al., 2013; Pyenson et al., 2017; Watson et al., 2019; Pyenson and Marraffini, 2020; Guillemet et al., 2021; Guerrero et al., 2021a) and sometimes genome rearrangements (Paez-Espino et al., 2015) to escape CRISPR targeting. Explorations of CRISPR immunity in natural environments have also documented ongoing spacer acquisition and phage escape mutations (Weinberger et al., 2012a; Guerrero et al., 2021a). Likewise, previous theoretical work has addressed the impact of parameters such as spacer acquisition rate and phage mutation rate on spacer diversity (Childs et al., 2012; Han et al., 2013; Han and Deem, 2017) and population survival and extinction (Weinberger et al., 2012b), how costs of CRISPR immunity impact bacteria-phage coexistence (Skanata and Kussell, 2021) and the maintenance of CRISPR immunity (Levin, 2010; Weinberger et al., 2012b; Westra et al., 2015; Gurney et al., 2019), how spacer diversity impacts population outcomes (He and Deem, 2010; Weinberger et al., 2012a; Childs et al., 2012; Haerter and Sneppen, 2012; Han et al., 2013; Childs et al., 2014; Bradde et al., 2017; Han and Deem, 2017), and how stochasticity and initial conditions impact population survival (Bradde et al., 2019; Chabas et al., 2018). Notably, foundational work by Childs et al., 2014; Childs et al., 2012 and Weinberger et al., 2012a; Weinberger et al., 2012b found through simulations that spacer diversity readily emerges in a population of CRISPR-competent bacteria interacting with mutating phages. However, the majority of both experiments and theory are based on observations and models of transient phenomena and short-term dynamics, while it is at long timescales that natural microbial communities experience bacteria-phage coexistence. Some notable experiments have measured long-term coexistence (Paez-Espino et al., 2015; Wei et al., 2011), and long-term sequential sequencing data from natural populations is becoming more available (Gómez and Buckling, 2011; Burstein et al., 2016), but appropriate theories to understand steady-state coexistence, sequence evolution and turnover, and immune memory in microbial populations remain rare. Because the processes of growth, death, and immune interaction are inherently random, understanding population establishment and extinction requires a fully stochastic analysis, and theoretical models that explore long-term coexistence have been partially deterministic to date (Weinberger et al., 2012a; Weinberger et al., 2012b; Childs et al., 2012; Levin et al., 2013; Childs et al., 2014; Santos et al., 2014; Weissman et al., 2018; Gurney et al., 2019). These models do not accurately capture rare stochastic events, in particular mutation, establishment, and extinction. Notable fully stochastic simulations of CRISPR immunity, on the other hand, have lacked rigorous analytic results (Han et al., 2013; Han and Deem, 2017). To understand the emergent properties of immune memory and diversity in microbial populations and how phages and bacteria coexist long-term, we developed a simple theoretical model of bacteria and phages interacting with adaptive immunity. We model a population of bacteria with CRISPR immune systems interacting with phages that can mutate to escape CRISPR targeting, building on our previous work that assumed a clonal population of phages with multiple protospacers in each phage (Bonsma-Fisher et al., 2018). We model phages with single protospacers in this work to efficiently track mutations in large populations over long timescales. We stochastically simulate thousands of bacteria-phage populations across a range of population sizes, spacer acquisition rates, spacer effectiveness rates, and phage mutation rates, and derive analytic expressions for the probability of establishment for new phage mutants, the time to extinction for phage and bacterial clones, and the dependence of bacterial spacer diversity on spacer acquisition rate, effectiveness, and phage mutation rate. Our simulations are fully stochastic and run for many thousands of generations to accurately capture the dynamics of establishment and extinction, yet the underlying model is simple enough to solve analytically. We show that even with the simplest assumptions of uniform spacer acquisition and effectiveness, complex dynamics and a wide range of outcomes of diversity and population structure are possible. We recover and reinterpret experimentally observed feaures: (a) we find that high diversity is not beneficial for bacteria when phage and bacterial diversity is strongly coupled, (b) we show that bacterial immunity can either track new phage mutations rapidly or keep a memory for a long time, but not both, and (c) we find emergent diversity resulting from selection for phage mutations that evade CRISPR targeting, linking diversity to the dynamical quantities of establishment and extinction. We compute bacterial average immunity in our simulations and in available experimental data and show that our model predicts qualitative trends that are visible in data. Finally, we show that adding immune cross-reactivity leads to qualitatively different states of evolutionary dynamics: (a) a traveling wave regime that resembles a similar state in models of vertebrate adaptive immunity (Yan et al., 2019; Marchi et al., 2019; Marchi et al., 2021) emerges when high cross-reactivity creates a fitness gradient for phage evolution, and (2) a regime of low turnover protected from new establishment by the reduced fitness of new phage mutants. Results Bacteria and phages dynamically coexist and coevolve We model bacteria and phage interacting and coevolving in a well-mixed system (Figure 1A and ‘Model’). Bacteria divide by consuming nutrients and phages reproduce by creating a burst of B new phages after successfully infecting a bacterium. Bacteria can contain a single CRISPR spacer that confers immunity against phages with a matching protospacer. Phages are labelled with a single protospacer type, a binary sequence of length L=30 that can mutate to a new type during a burst with probability μ⁢L, where μ is the per-base mutation rate per generation. All simulations begin with a single clonal phage population unless otherwise specified. Figure 1 with 7 supplements see all Download asset Open asset Model description. (A) We model bacteria and phages interacting in a well-mixed vessel. We track nutrient concentration, phage population size (nV), and bacteria population size (nB). Bacteria can either have no spacer (nB0) or a spacer of type i (nBi, ∑inBi=nBs), and phages can have a single protospacer of type j (nVj). With rate α, a phage interacts with a bacterium. If the bacterium does not have a matching spacer, the phage kills with probability pV and produces a burst of B new phages, while for bacteria with a matching spacer that probability is reduced to pVs=pV⁢(1-e), 0≤e≤1. Bacteria without spacers that survive an attack have a chance to acquire a spacer with probability η, and bacteria with spacers lose them at rate r. Lower inset: average immunity is the weighted average pairwise immunity between spacer-containing bacteria and phages, given by 1-∑i,jnBi⁢nVj⁢pV⁢(i,j)pV⁢∑i,jnBi⁢nVj. The probability of a phage with protospacer j successfully infecting a bacterium with spacer i is pV⁢(i,j). (B) Three time points in a typical simulation with C0=104, e=0.95, η=10-4, and μ=10-5. Coloured circles represent unique protospacer or spacer sequences; shared sequences are shown with the same colour. The size of each circle is proportional to clone size, and new mutants are shown radially more distant from the centre. (C) Ten individual clone trajectories vs simulation time for phages (top) and bacteria (bottom). The mean clone size is shown with a horizontal dashed line. (D) Total phage, bacteria, and nutrient concentration as a function of phage success probability pV. Markers show an average over five independent simulations for different values of pV with C0=104,η=10-3,e=0.95, and μ=10-7. Solid lines show theoretical predictions for different constant values of effective e. As pV decreases, phages go extinct at a critical value given by A=1, where A=(B⁢pV-1)⁢(1-f)⁢αf⁢g. (E) Total phage and bacteria population size as a function of average bacterial immunity to phages. Colours indicate the fraction of simulations in which phage or bacteria go extinct before a set endpoint. Solid lines show the mean-field prediction. Error bars are the standard deviation across three or more independent simulations. Coexistence occurs across a wide range of parameters but is not guaranteed: below a certain success probability pV0=1B⁢(g⁢f(1-f)⁢α+1), phages are not able to reproduce often enough to overcome their base death rate due to outflow and adsorption and are driven extinct (grey area of Figure 1D; Bonsma-Fisher et al., 2018). In this expression, g is the bacterial growth rate, B is the phage burst size, α is the phage adsorption rate, f=F/(g⁢C0) is a normalized outflow rate, and C0 is the inflow nutrient concentration. This is the same extinction threshold reported by Payne et al., 2018 as the cutoff for achieving herd immunity in a well-mixed bacterial population. To a first approximation, phages must successfully infect every 1/B bacteria they encounter, but if bacteria are growing quickly, then phages must do better to overcome bacterial growth, leading to the extra terms in this expression (see ‘Phage extinction threshold’). We write this extinction threshold as A=(B⁢pV-1)⁢(1-f)⁢αf⁢g. Above the phage extinction threshold (A>1), the phage population size increases with increasing pV but eventually decreases again as bacterial numbers are driven too low to support a large phage population (Bonsma-Fisher et al., 2018). A similar non-monotonicity as a function of the probability of naive bacterial resistance (1-pV) was described in theoretical work by Weinberger et al., 2012b. In our model the position of the peak in phage population size as a function of infection success probability is determined by e, the effectiveness of CRISPR spacers against phage; increasing e pushes the peak to higher pV (Figure 1D). While e is a constant parameter that determines the outcome of pairwise interactions between bacteria and phages, the bacterial population as a whole possesses an average immunity to phages that is a weighted average of all the possible pairwise interactions (Figure 1A inset). It is the overall average immunity that determines population outcomes, which we describe in detail in ‘Pathogen and host diversity must be considered together’. To focus on regimes where bacteria and phages coexist, we select parameters within the deterministic coexistence regime to explore bacteria-phage coevolution. Even in this regime, stochastic extinction will eventually come for one or both populations in simulations (Figure 1E), though the timescale of extinction may be extremely long for large population sizes (Badali and Zilman, 2020). Phages are more susceptible to stochastic extinction than bacteria because of their large burst size B which increases their overall population fluctuations (Appendix 3). The length of coexistence before stochastic extinction depends on population size as well as simulation parameters and initial conditions (see ‘Stochastic population extinction’): phage populations can be rescued from extinction by high mutation rate (Figure 1—figure supplement 3) or high initial protospacer diversity (Figure 1—figure supplement 6), but are more likely to go extinct if spacer effectiveness is high (Figure 1—figure supplement 1). Conversely, bacteria are more likely to go extinct if spacer effectiveness or spacer acquisition rate are low (Figure 1—figure supplement 4). Population survival and persistence in natural populations is impacted by additional factors we do not address in our model, including immigration (Volkov et al., 2003; Chabas et al., 2016, niche partitioning Simek et al., 2010; Weitz et al., 2013; Mills et al., 2013; Badali and Zilman, 2020; Voigt et al., 2021), environmental fluctuations (Abreu et al., 2020; Voigt et al., 2021), and spatial structure (Haerter et al., 2011; Haerter and Sneppen, 2012; Heilmann et al., 2010; Heilmann et al., 2012; Simmons et al., 2018; Skanata and Kussell, 2021). Across a wide range of coexistence parameters, our simulations show continual phage evolution and bacterial CRISPR adaptation in response (Figure 1B). New phage protospacer clones arise from a single founding clone by mutation, and a small fraction of new mutants grow to a large size and become established. Once phage clones become large, bacteria acquire matching spacers and an immune bacterial subpopulation becomes established. The specific protospacer and spacer types present in the population continually change as old types go extinct and new types are created by phage mutation, but the average total diversity and average overlap between bacteria and phage remains constant at steady state (Figure 8). Both bacteria and phage clones stochastically go extinct, completing the life cycle of a clonal population (Figure 1C). Phages drive stable emergent sequence diversity New phage protospacer clones continually arise and go extinct in our simulations, generating turnover in clone identity in the population. Despite constant turnover, however, the total number of clones remains fixed at steady state. We use the mean number of bacterial clones at steady state, designated m, as a measure of system diversity. This choice of diversity measurement is equivalent to the Hartley entropy of the clone size distribution, a special case of the Rényi entropy (Mora and Walczak, 2016; Altan-Bonnet et al., 2020). This definition weights all clones equally regardless of their abundance; such a measurement is not appropriate when clone size distributions are very broad and small clones may be unsampled, but is reasonable when clone size distributions are relatively narrow and all clones are sampled (Mora and Walczak, 2016). In our simulation results, both bacteria and phage populations exhibit relatively narrow clone size distributions across a range of parameters, with the exception of low values of spacer acquisition η (Figure 2A, Figure 2—figure supplement 3, Figure 2—figure supplement 4). Even at low η, however, clone size distributions are approximately exponential, indicating that they are not scale-invariant and that the mean clone size still captures important information about the full clone size distribution. Figure 2 with 4 supplements see all Download asset Open asset Diversity depends sub-linearly on parameters. (A) Bacteria and phage clone size distributions normalized to the measured mean clone size for C0=105, μ=3×10-7, and e=0.95. As η increases, both clone size distributions become more sharply peaked. (B) The mean number of bacterial clones depends only on a combined parameter in the limit of small average immunity (generally coinciding with high C0). (Inset) The mean number of bacterial clones can be predicted by numerically solving Equation 1 for m. The two lowest values of η are shown with lighter shading. Error bars are the standard deviation across three or more independent simulations. What determines clonal diversity? Many factors that correlate with transient diversity have been experimentally identified, such as phage extinction and slower phage evolution at high bacterial spacer diversity (van Houte et al., 2016; Common et al., 2020) and maintenance of a diverse bacterial population when exposed to diverse phages (Paez-Espino et al., 2015; Common et al., 2019; Guillemet et al., 2021; Lopatina et al., 2019), but a conceptual framework to understand emergent diversity has remained elusive. For instance, while initial high spacer diversity puts low-diversity phage populations under intense pressure to the point of driving them extinct (van Houte et al., 2016; Common et al., 2020), is the same true for emergent bacterial diversity after an extended period of coexistence? Is observed high bacterial spacer diversity indicative of successful bacterial escape from phage predation or an indicator of increased phage pressure? In our model, phage and bacterial diversity is tightly coupled: the number of large phage clones is approximately the same as the number of bacterial clones (Figure 22). This is also the case in experimental coevolution data from Paez-Espino et al., 2015: the number of phage protospacer types is on the same order of magnitude as the number of bacterial spacer types across most similarity thresholds (Figure 72). There is evidence that this coupling of diversity may also occur in the wild: a recent longitudinal study of Gordonia bacteria interacting with phage in a wastewater treatment plant identified 14 high-coverage phage genotypes and 11 high-coverage bacterial variants based on CRISPR spacer sequence (Guerrero et al., 2021a). Using the tight correspondence between bacterial diversity and phage diversity in our model, we calculate the overall steady-state diversity by balancing the effective phage clone mutation rate μ¯=1g⁢C0⁢α⁢B⁢pV⁢(1-e-μ⁢L)⁢(1-e⁢νm)⁢nV⁢nB, phage clone establishment probability Pe⁢s⁢t, and the time to extinction for large phage clones Te⁢x⁢t (details in ‘Measuring diversity): (1) m=Pestμ¯Text Equation 1 arises from the simple statement that the number of large clones must be equal to their establishment rate (Pe⁢s⁢t⁢μ¯) multiplied by their average time to extinction (Te⁢x⁢t). This relationship successfully predicts the number of bacterial clones at steady state across a wide range of parameters and a wide range of diversity values (Figure 2B inset, Figure 2—figure supplement 1, and Figure 2—figure supplement 2). At the lowest value of η the prediction tends to overestimate the number of clones – in this regime, low acquisition means that phage clones go extinct because of clonal interference before bacteria are able to acquire spacers (Figure 16, ‘Measuring diversity’). Through approximations (‘Analytic approximations for diversity’), we find that diversity depends on a single combined parameter to the power 1/3 (Figure 2B, Equation 2), and this parameter is proportional to spacer effectiveness e, the probability of bacterial survival followed by spacer acquisition (1-pV)⁢η, and the phage mutation rate μ⁢L. (2) m≈(4eη(1−pV)μL(gC0(1−f))3B2α2pV3r)13 Each of these parameters intuitively increases diversity (e.g., a higher phage mutation rate means that phage diversity increases and bacterial diversity follows suit). What is surprising is that their combined effect on diversity is to a power much less than 1: this 1/3 exponent means that if the mutation rate increased tenfold, the diversity would only increase by about a factor of two (Appendix 2—figure 1). In contrast, a simple neutral model of cell division with mutations gives a linearly proportional increase in diversity for the same increase in mutation rate (Appendix 2—figure 2). To understand where the dependence of diversity on these key parameters come from, we look more closely at each component expression. The effective phage mutation rate μ¯ depends linearly on the parameter μ:μ¯≈gC0(1−f)fμLαPV, while both the probability of establishment and the time to extinction depend inversely on diversity m:pest≈1mceη(1−PV)gC0(1−f)BpVr (Equation 5) and Text≈1m2gC0(1−f)fαBpV (Equation 173, Appendix 3). By comparison with Equation 1, we find that m3 depends approximately linearly on mutation rate, resulting in the weak m∝μ13 dependence on mutation rate. The dependence of diversity on both e and η comes from the probability of phage establishment since μ¯ depends only very weakly on these parameters through its dependence on total population sizes and Te⁢x⁢t depends explicitly on m alone, not e or η. The phage probability of establishment is proportional to e⁢η⁢(1-pV)m (Equation 5), and as before, this gives m3∝e⁢η⁢(1-pV). Bacteria are more successful at high η⁢(1-pV)⁢e, which increases the phage establishment probability. Previous theoretical work has predicted that diversity increases as spacer acquisition rate increases (Childs et al., 2012); here, we provide a quantitative prediction for this dependence. In the following sections, we explore phage establishment in more detail. What determines the fitness and establishment of new mutants? We find that diversity emerges in our model from the balance of phage clone establishment and extinction. However, only some phage mutants escape initial stochastic extinction and survive long enough to become established. What determines the fate of a new phage mutant? In our model, a single phage mutation in a protospacer can completely overcome CRISPR targeting, which means that new phage mutants can infect all bacteria equally well and their initial growth rate s0 is independent of CRISPR: s0≈α⁢nB⁢(B⁢pV-1)-F, where F is the chemostat flow rate (a shared death rate for phages and bacteria). Surprisingly, however, even once bacteria start to acquire matching spacers, the probability of establishment for new phage mutants is still well-described by theory in which CRISPR targeting only influences total average population sizes (Figure 3C); that is, the specific interaction between a phage and its matching clone can be ignored. Intuitively, this is because phage clones must grow to a certain size before bacteria encounter them enough to begin to acquire spacers, and this size turns out to be large enough to avoid stochastic extinction (Figure 15). The probability of phage establishment is 2⁢s0B⁢(s0+δ0), where