Clustered Regularly Interspaced Short Palindromic Repeats (CRISPR), together with associated genes (cas), form the CRISPR–cas adaptive immune system, which can provide resistance to viruses and plasmids in bacteria and archaea. Here, we use mathematical models, population dynamic experiments, and DNA sequence analyses to investigate the host–phage interactions in a model CRISPR–cas system, Streptococcus thermophilus DGCC7710 and its virulent phage 2972. At the molecular level, the bacteriophage-immune mutant bacteria (BIMs) and CRISPR–escape mutant phage (CEMs) obtained in this study are consistent with those anticipated from an iterative model of this adaptive immune system: resistance by the addition of novel spacers and phage evasion of resistance by mutation in matching sequences or flanking motifs. While CRISPR BIMs were readily isolated and CEMs generated at high rates (frequencies in excess of 10−6), our population studies indicate that there is more to the dynamics of phage–host interactions and the establishment of a BIM–CEM arms race than predicted from existing assumptions about phage infection and CRISPR–cas immunity. Among the unanticipated observations are: (i) the invasion of phage into populations of BIMs resistant by the acquisition of one (but not two) spacers, (ii) the survival of sensitive bacteria despite the presence of high densities of phage, and (iii) the maintenance of phage-limited communities due to the failure of even two-spacer BIMs to become established in populations with wild-type bacteria and phage. We attribute (i) to incomplete resistance of single-spacer BIMs. Based on the results of additional modeling and experiments, we postulate that (ii) and (iii) can be attributed to the phage infection-associated production of enzymes or other compounds that induce phenotypic phage resistance in sensitive bacteria and kill resistant BIMs. We present evidence in support of these hypotheses and discuss the implications of these results for the ecology and (co) evolution of bacteria and phage.
The experimental demonstrations, in 2007 and 2008, that the Clustered Regularly Interspaced Short Palindromic Repeats that abound in the genomes of the vast majority of archaea and nearly half of bacteria can serve as part of an adaptive immune system (CRISPR–cas) that protects these prokaryotes against lytic phage [1] and conjugative plasmids [2] raised many intriguing questions and stimulated a great deal of research. Between 2007 and 2012, according to PubMed, the number of articles with the acronym CRISPR in their title and/or body has been doubling every 20 months. Understandably, the majority of these articles are reports or reviews of the genetic, biochemical, and molecular mechanisms by which (i) the CRISPR–cas system acquires the 26–72 base pair sequences (spacers) of DNA from infecting phage, generating Bacteriophage Insensitive Mutants (BIMs), or from invading plasmids, forming Plasmid Interfering Mutants (PIMs); (ii) the small RNAs coded for by these spacers abort subsequent infections by phage and plasmids with DNA sequences corresponding to those in these spacers (protospacers) [3], [4], [5], [6]; (iii) the phage can evade this resistance and generate CRISPR Escape Mutants (CEMs) by modifying the DNA sequences of protospacers or protospacer adjacent motifs (PAM) [3], [7], [8], [9]; (iv) the Cas proteins participate in these processes [1], [8], [9]; and (v) the existence, distribution and genetic diversity of CRISPR–cas systems in the archaea and bacteria came about [10], [11], [12] Some of this research has been devoted to the practical applications of CRISPR and the CRISPR–cas system. Long before the immunological role of CRISPR in these microorganisms was recognized, differences in the number of these palindromic repeats had been used as markers for studies of the genetic (molecular) epidemiology and forensics of pathogenic bacteria [13], [14], [15], [16]. Isolates that would be classified as identical by standard typing procedures, like Serotyping, Multi-Locus Sequence Typing, and Pulse Field Gel Electrophoresis, can differ in their CRISPR regions, particularly in their spacer sequences [17], [18]. Genes, and clusters of genes, borne on plasmids, phage, or chromosomal genes, and acquired by horizontal transfer, code for the virulence and antibiotic resistance of many pathogenic bacteria. And, as one might anticipate, strains of at least some species of pathogenic bacteria that lack functional CRISPR–cas systems are more likely to be virulent and resistant to antibiotics than those that bear these systems [19]. A number of biotechnological and industrial processes rely on the use of bacteria that are subject to infection by phage, and therefore risk disruption of production, low-quality products and economic loss. Can CRISPR–cas systems be harnessed to counter antibiotic resistance and virulence of pathogenic bacteria and/or prevent phage contamination of industrially important bacteria (8)? Central to understanding these practical implications and potential applications of CRISPR–cas systems are the population and evolutionary dynamics of bacteria bearing these adaptive immune systems and those of the phage and plasmids that infect them. Prior to the experimental demonstration that CRISPR is part of an adaptive immune system, and motivating those experiments, were observations that the spacer sequences in the CRISPR regions of bacteria and archaea from natural communities were identical to sequences in the genomes of viruses in their communities [15], [20], [21]. These and more recent studies of bacteria and archaea and their viruses [10], [11], [12], [22], and metagenomic studies of microbiomes [23], provided compelling support for the hypothesis that CRISPR–cas systems play an active role in the population dynamics and co-evolution of these organisms in natural communities. However, the inferences one can draw from these investigations are limited. They provide little of the necessary quantitative information about the nature and contribution of CRISPR–cas to these dynamics and co-evolution and its consequential role in determining the structure of their communities. Studies using mathematical models and computer simulations suggest that bacteria with CRISPR–cas systems can invade communities of phage and bacteria without this immune system and can also prevent phage from invading their communities [24]. Modeling studies also suggest that, in theory, a CRISPR–cas-mediated BIM-CEM co-evolutionary arms race can maintain considerable diversity in the interacting populations of bacteria and phage [25] and account for spacer changes in the CRISPR elements in the metagenomic structure of natural communities of archaea and bacteria and their phage [26]. But, as compelling as these mathematical models and computer simulation studies may be, their limitations are also apparent. The models used are based on simplifying assumptions about the phage infection process and CRISPR–cas adaptive immunity that have not been formally tested in the systems under study. The computer simulations and other numerical analyses of the properties of these models use parameter values for which there are few independent estimates. In this study we use mathematical models, computer simulations, parameter estimation, and population dynamic and other microbiological experiments with the Gram-positive bacteria Streptococcus thermophilus and its virulent phage 2972 (Siphoviridae family) to explore the nature and contribution of CRISPR–cas adaptive immunity to the population and evolutionary dynamics of bacteria and phage. The results of these experiments and the molecular characterization of the bacteria and phage employed demonstrate that the qualitative Lamarckian (iterative acquisition of spacers) [27] and Darwinian (mutation in protospacers) genetic conditions for a CRISPR–mediated arms race are met with this phage-host system. BIMs are readily formed and CEMs capable of growing on these BIMs are generated at high rates. On the other hand, our experimental results and theoretical analyses indicate that even at a qualitative level, models based on existing assumptions about phage infection and CRISPR–mediated resistance cannot account for the population and evolutionary dynamics of the interactions between S. thermophilus and phage 2972. Findings contrary to what is anticipated from the analysis of the properties of these models include: (i) the invasion of phage into populations of BIMs resistant due to the acquisition of one (but not two) spacers; (ii) the survival of substantial numbers of sensitive bacteria despite the presence of high densities of phage; and (iii) the maintenance of phage-limited communities due to the failure of even two-spacer BIMs to become established in populations with wild type bacteria and phage. We postulate that the latter two results can be attributed to the phage infection-associated production of enzymes or other compounds that kill resistant BIMs and generate phenotypically phage-resistant “persister” populations of sensitive bacteria. By modifying the model to account for these chemical processes we can explain the observed dynamics. The predictions of this extended model are supported by longer-term population dynamic experiments.
Genetic analysis of the two active CRISPR loci of ten first-order BIMs revealed that they each had acquired a novel spacer in either the CRISPR1 or CRISPR3 locus. Five BIMs acquired one unique novel spacer in the CRISPR1 locus and 5 BIMs acquired one unique novel spacer in the CRISPR3 locus (Figure 3). Similar analysis of the five second-order BIMs revealed that they each had also acquired an additional novel spacer in the CRISPR1 or CRISPR3 locus. Two of these second-order BIMs acquired one additional unique novel spacer in the CRISPR1 locus, indicating iterative addition into the CRISPR1 locus (BIM32 and BIM52 in Figure 3). Likewise, three second-order BIMs acquired novel spacers in the CRISPR3 locus, including two variants with iterative build-up of the CRISPR3 locus (BIM22 and BIM82 in Figure 3). Interestingly, BIM72 acquired a CRISPR1 spacer to become a first-order CEM and a CRISPR3 spacer to become a second-order CEM, illustrating the duality of active CRISPR loci in Streptococcus thermophilus. Sequence analysis of all the CRISPR1 and CRISPR3 spacers acquired in the first and second rounds of phage challenges indicated perfect matches to the WT genome of phage 2972 used in the challenge (Table S1). Further, all new CRISPR1 and CRISPR3 spacers were associated with protospacer-adjacent motifs (PAMs): NNAGAAW and NGGNG, respectively (Table S1). These findings are consistent with previous results indicating that spacer sequences are directly derived from phage DNA and are associated with PAMs [7], [8], [35], [36]. The results of our sequence analysis of the CEM protospacer and PAM regions are fully consistent with previous findings that single mutations in the matching protospacer or PAM are sufficient to circumvent CRISPR–mediated immunity (Figure 4). Of the 10 CEMs that evaded the immunity of the first-order BIMs, one had a mutation in the protospacer region and the remaining nine CEMs had mutations in PAM sequences (four in the CRISPR1 PAM and five in the CRISPR3 PAM). Each of the five second-order CEMs that evaded the immunity of the second-order BIMs had acquired a second mutation, as expected. One had acquired a second mutation in a protospacer, two had mutations in the PAM, one contained one mutation in the protospacer and another in the PAM, and the last one had an insertion in the PAM. Taken together, these sequencing data are consistent with previous results indicating that CEMs are the product of mutations in the protospacers and/or PAMs which enable these viruses to circumvent CRISPR–cas acquired immunity [7], [8]. At best, the mathematical model and computer simulations employed here are simplistic caricatures of the population dynamic and other processes occurring in these experiments. They are not intended or anticipated to provide quantitatively precise predictions of the changes in density of bacteria and phage observed in our experiments. Some quantitative deviation between experiments and theory is expected. Not expected, however, are major qualitative deviations. There were two of these. One is the failure of the phage to eliminate the sensitive bacteria. The other is the failure of BIMs to ascend and dominate cultures with an abundance of resources. We postulate these failures can be attributed to either the phage or the products of phage replication generating phenotypic immunity among sensitive cells (persistence) [37] and killing emerging BIMs. One line of evidence in support of the hypothesis that something (s) in the phage lysate is preventing BIMs from ascending and also generating persisters comes from results obtained with 24-hour cell-free (filtered, rather than chloroformed) lysates of WT phage. S. thermophilus cells resistant to the phage (BIM22 and SMQ-301) were added to these lysates and the optical and viable cell densities were followed with frequent sampling. In this experiment, 40 µl of BIM22 or SMQ-301 from overnight cultures were added to 4 ml of the phage lysate. As phage-free controls, these bacteria were added to 4 ml LM17Ca (Figure 9). Cells of SMQ-301 were able to grow in these WT phage lysates, thereby indicating that there were plenty of resources. However, the viable cell density of the BIM22 cells declined. At 24 and 31 hours, the optical density of the BIM22 culture in the phage lysate was less than 0. 02 and the viable cell densities were, respectively, 4. 7×105 cfu/ml and 1. 2×106 cfu/ml. At 48 and 54 hours, these viable cell densities were, respectively, 5. 6×105 cfu/ml and 2. 1×105 cfu/ml. The sensitivity of BIM22 to the phage and/or molecules in the WT lysates was not shown by SMQ-301. A second line of evidence in support of the hypothesis that phage or the products of their replication or lysis of infected cells prevent the ascent of BIMs is the results of our experiments with WT phage in cultures with first and second-order BIMs. Batch cultures inoculated with low densities of WT phage and bacteria that are resistant to those phages by one or two spacers become resource-limited. However, if we allow for considerable phage replication by adding WT cells, cultures with first- and second-order BIMs become phage-limited (Figure S4). We have yet to characterize the agent (s) responsible for killing and/or inhibiting the growth of infected BIMs. However, preliminary results from experiments with lysates prepared from French Press extracts suggest that this agent is likely to be present within the cells and need not be a product of phage infection. The growth rates of resistant BIMs are markedly reduced when cultured in mixtures of medium and French Press extracts of phage-free cultures of wild type cells (Text S1). In this interpretation, this agent is released as a product of lysis of the cells by the phage. To explore, in a more quantitative way, whether the mechanisms postulated above, where products of phage replication generate phenotypically resistant persisters and kill BIMs, can account for the observed dynamics, we use an extension of the basic model described in the Theoretical Framework. In this extension, we consider three bacterial populations with densities B0, B1, and B2, and two populations of phage, P0 and P1, respectively corresponding to WT, first- and second- order BIM bacteria, and WT and first-order CEM phage. For convenience, we assume there are agents of collective concentration LY µg/ml that kill resistant cells and provide transient immunity to sensitive bacteria (persistence effect [38], [39]). Since the effects of LY in killing resistant cells and generating persisters are governed by separate parameters, they can be at least two different compounds. These agents are produced at a rate proportional to the product of the densities of replicating phage and the corresponding sensitive bacteria: where vL is a rate parameter for the production of LY. We assume that the rates of growth of the bacteria now depend on the concentration of LY, as well as the resources, R. For example, the per capita growth of B0 is now where ν is the maximum growth rate of this strain, k the Monod constant, η the maximum rate of killing by LY, and KL the concentration of LY where killing is at its maximum rate. In the simulation LY/KL is set equal to 1 when LY exceeds KL. To account for phenotypically resistant B0 and B1 bacteria, we assume there are populations of non-dividing, persistent cells of densities BP0 and BP1 bacteria per ml that are produced from their respective dividing populations at a rate proportional to LY/KL and a constant (g) and return to the dividing state at a constant rate (h). For example, the rate of change in the density of the wild type (B0) population is nowThe Berkeley Madonna simulation used for this extended model can be obtained from www. eclf. net/programs. In Figure 10, we present the results of simulation experiments initiated with wild type phage (P0), wild type bacteria (B0), and a second-order BIM (B2). The population growth, initial resource concentration and phage infection parameters used in this simulation are same as those used in earlier simulations and thus in the range estimated in these experiments. The parameters for the production of LY and its action are arbitrary and chosen to illustrate the principle rather than fit specific data in any but a qualitative way. In the absence of the killing effect, Figure 10A, the B2 population ascends and the phage are eliminated. The B0 population continues to be maintained, B1 bacteria and P1 phage are produced, and within short order the population becomes resource-limited. During the first transfer of the simulations with the killing effect (Figure 10B), the P0 phage ascend and the densities of the B0 and B2 populations decline and the community becomes phage limited. While the second transfer (Figure 10C) also ends in a phage - limited state, the relative density of B2 is now greater. In subsequent transfers (Figure 10D), B2 ascends to dominance, phage are eliminated, and the community becomes resource-limited. As a consequence of the serial passages, the molecule killing BIMs is diluted out and the BIMs ascend. For the arms race to continue, the phage would have to generate CEMs capable of replicating on the dominant population of BIMs. If the above postulated effect of dilution in serial passage is valid, in successive transfers of experiments initiated with WT cells and WT phage, BIMs should emerge. And, unless they are countered by CEM phage, they will ascend to dominance and the phage will be lost. To test this hypothesis, we mixed WT cells with phage and each day transferred the culture to fresh medium (40 µl into 4 ml). We initiated six separate cultures from the same mixture of approximately 2×106 cells and phage. As a control we serially transferred a phage-free WT culture. At the end of each transfer, we estimated optical densities of these cultures and, by serial dilution and plating, the viable densities of bacteria and free phage. The results of this experiment are presented in Figure 11. We interpret the results of this experiment to be consistent with the hypothesis that as a consequence of successive dilutions of the compound killing BIMs, the resistant mutants ascend. Moreover, in 5 of the 6 cultures in which BIMs emerged, phage were lost and could not be recovered even with enrichment with WT cells (i. e. the arms race was terminated). Additional serial transfer experiments should be performed with different starting conditions and possible different media (perhaps milk) to explore the generality of this observation about the limit to a BIM-CEM arms race.
The results of this jointly theoretical and experimental study make at least five predictions about the contribution of CRISPR–cas immunity to the population and community ecology of bacteria and phage and the (co) evolution of these organisms. Taken at large, this study predicts that the role of CRISPR–cas systems in the day-to-day ecology and evolution of natural communities of prokaryotes and their viruses is at best modest and more restrictive than anticipated from existing theory. As compelling as we may consider these predictions to be, at this juncture we only see them as hypotheses that have not been fully tested. We have restricted this study to the dynamics of the early phase of the interactions between single populations of phage and bacteria with very active CRISPR–cas immune systems. While our models and experiments are appropriate for this phase of the association between these organisms, they are not sufficient for making predictions about the longer-term population and (co) evolutionary dynamics of these interactions. Most importantly, our models and experiments do not formally consider the potential diversity of BIMs and CEMs that can be generated at each level and the interactions between them (see, for example, [25]). How this diversity will play into the longer-term population and (co) evolutionary dynamics of these interactions and whether the bacterial populations will be limited by phage rather than resources remain to be seen. Particularly important in this regard is whether bacteria with other mechanisms of resistance to the phage will emerge and whether the phage will be able to generate mutants capable of replicating on these resistant cells (see [42] but also [43]). It may well be that the failure to observe bacteria with other resistance mechanisms in these experiments is due to the relatively high rate at which CRISPR–mediated immunity is generated in S. thermophilus. Finally, in support of the prediction that CRISPR–mediated immunity plays a limited role in the day-to-day (rather than historical) ecology and evolution of prokaryotes and their viruses, is our choice of experimental system. S. thermophilus and the phage 2972 may well be an exceptional system for in vitro study of the population dynamics and evolution of phage and bacteria with CRISPR–mediated immunity. Not only is it arguably (surely, to those who work with it) the best-characterized naturally occurring CRISPR–phage system; with respect to the rate of acquisition of spacers and possibly the rate of generation of CEMs, it is the most active. It is an ideal system in which to observe CRISPR playing a prominent role in the ecology and evolution of prokaryotes and their viruses. To our knowledge, there is no evidence for naturally occurring CRISPR–cas systems of other organisms being as active in generating BIMs and CEMs as those of S. thermophilus and its phages. For these reasons, we conjecture that the CRISPR–cas systems of other bacteria, archaea and their phages would play an even less active role in the day-to-day ecology and evolution than anticipated from this study.
For our experimental model, we selected the strain S. thermophilus DGCC7710 because it contains arguably the best-characterized CRISPR–cas functional system. S. thermophilus DGCC7710 [1], which bears four different CRISPR–cas systems, two of which are active (CRISPR1 and CRISPR3), was used as a host for the virulent phage 2972 [34]. We also used S. thermophilus SMQ-301 [44], which is resistant to 2972 as a non-host strain. A single medium designated LM17Ca was used for all of the experiments. M17 (Oxoid) was supplemented with 0. 5% lactose (L) and 0. 05 M Calcium Borogluconate (Ca). For the soft/top agar and hard/bottom agar, LM17Ca was supplemented with 0. 6% and 1. 5% agar, respectively. Cell densities were estimated by dilution in 0. 85% saline and plating on LM17Ca. To estimate phage titers, serially diluted samples were added to 1 ml of a late-log culture of S. thermophilus, put into 3 ml soft agar and poured onto LM17Ca plates. About 5×108 S. thermophilus DGCC7710 cells were mixed with about 2. 5×107 of wild type (WT) phage 2972 and held for about 10 minutes. These mixtures were then added to 3 ml of LM17Ca soft agar, which was then poured over LM17Ca plates. Ten experiments identical to the above were run in parallel. All the plates were incubated at 42°C for 30 hours. Single colonies were removed from each plate and streaked twice to purify the BIM clones and remove contaminating phages. Resistance to WT phage 2972 was confirmed by spotting 10 µl (at least 5×105 pfu) of the diluted lysate on soft agar lawns of the BIMs. Clear zones were scored as sensitive while turbid zones or few separate plaques were scored as resistant. From these experiments, we selected 10 BIMs (one per experiment), designated BIM1, BIM2, … BIM9, and BIM10. A similar procedure was used to randomly isolate five second-order BIMs from five of these first-order BIMs by spotting CEMs capable of replicating on these first-order bacteria. These were designated BIM22, BIM32, BIM52, BIM72, and BIM82. CEM phages were isolated by mixing 100 µl of phage 2972 lysate with 1 ml of overnight cultures of first- or second-order BIMs. After 10 minutes these mixtures were added into 3 ml soft agar, poured onto LM17Ca plates, and incubated overnight at 42°C. Single phage plaques were picked from the plates the following day and added to 3 or 10 ml of LM17Ca. A few drops of chloroform were added to each tube to kill the remaining bacteria. These lysates were then vortexed and centrifuged. These single plaque lysates were then filtered (0. 45 µm) and stored at 4°C. This single plaque isolation process was repeated two times to obtained purified phages. The CEM phages for the first-order BIMs (BIM1 through BIM10) were designated CEM1 through CEM10. The CEM phages for the second-order BIMs were designated CEM22, CEM32, CEM52, CEM72, and CEM82. Two procedures were used to assay liquid cultures for the presence of WT phage and specific CEMs. One was a spot test: 10 µl of filtered or chloroform-treated cultures were either spotted directly or after serial dilution onto soft agar lawns of WT cells or first- or second-order BIMs. Either clear zones or turbid zones with specific plaques were considered to indicate the presence of phage capable of replicating on that bacterial cell line. In cases where negative results were obtained with this spot test, we used an enrichment procedure where 100 µl of the lysate was added to 4 ml of medium with 100 µl of overnight culture bacteria. The optical densities of these enrichment cultures were followed at 42°C and those that cleared were considered positive for phage capable of replicating on those host bacteria. Two procedures were also used to test for the presence of bacteria resistant to wild type or CEM phages. One was a spot test: single colonies were streaked onto fresh medium to purify them from the phage in the cultures from whence they came. Single colonies were grown overnight at 42°C in liquid medium (1 ml) and then put into soft agar to be used as a lawn for spot testing with phage. For this we used the procedure and criteria for sensitivity/resistance described in the preceding paragraph. In some cases, the presence of resistant cells in samples was tested directly by adding 100 µl of phage and 100 µl of the cell culture to soft agar and looking for the presence of colonies. The active S. thermophilus CRISPR loci, namely CRISPR1 and CRISPR3, were subjected to PCR, in order to test whether novel spacers had been acquired in BIMs following exposure to phage. Primers targeting the leader sequence and a unique WT spacer at the trailer end of CRISPR1, as well as a primer pair targeting the leader and the trailer end of CRISPR3, were used for PCR amplification. The names and sequences of these primers are, respectively: yc70 5′-TGCTGAGACAACCTAGTCTCTC-3′ and 89R6 5′-TCAGCAGATTGTCAAATCGG-3′ for CRISPR1 and F1 5′CTGAGATTAATAGTGCGATTACG-3′ and R2 5′-GCTGGATATTCGTATAACATGTC-3′ for CRISPR3. PCR amplifications were carried out as previously described [36], with Ambion Mastermix and the following PCR cycling: denaturation for 7 minutes at 95 C; followed by 35 cycles of 30 sec denaturation at 95C; 30 sec hybridization at 53 C and 1 min extension at 72 C; followed by a final round of 5 min extension at 72 C. PCR amplicons were subjected to agarose gel electrophoresis for size analysis, purified using a Qiagen Kit, and subsequently subjected to Sanger sequencing using the same primer sets used for amplification. The sequencing results were visualized in EXCEL as previously described [36] in order to assess CRISPR repeat occurrence and visualize spacer content. In order to assess the protospacer sequences in the CEMs, targeted PCR amplifications followed by Sanger sequencing of the PCR amplicons were carried out for first and second generation CEMs as described previously [7], [8].
The evidence that the CRISPR regions of the genomes of archaea and bacteria play a role in the ecology and (co) evolution of these microbes and their viruses is overwhelming: (i) the spacers (variable sequences of 26–72 bp of DNA between the repeats of this region) of these prokaryotes are homologous to the DNA of viruses in their communities; (ii) experimentally, the acquisition and incorporation of spacers of viral DNA can protect these organisms from subsequent infection by these viruses; (iii) experimentally, viruses evade this immunity by mutation in homologous protospacers or protospacer-adjacent motifs (PAMs). Not so clear are the nature and magnitude of the role CRISPR plays in this ecology and evolution. Here, we use mathematical models, experiments with Streptococcus thermophilus and the phage 2972, and DNA sequence analyses to explore the contribution of CRISPR–cas immunity to the ecology and (co) evolution of bacteria and their viruses. The results of this study suggest that the contribution of CRISPR to the ecology of bacteria and phage is more modest and limited, and the conditions for a CRISPR–mediated coevolutionary arms race between these organisms more restrictive, than anticipated from models based on the canonical view of phage infection and CRISPR–cas immunity.
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The Population and Evolutionary Dynamics of Phage and Bacteria with CRISPR–Mediated Immunity