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Nasty Viruses, Plush Plasmids, Population Dynamics, and the Conditions for Establishing and Maintaining CRISPR-Mediated Adaptive Immunity in Bacteria

Nasty Viruses, Costly Plasmids, Population Dynamics, and the Conditions for Establishing and Maintaining CRISPR-Mediated Adaptive Immunity in Bacteria

• Bruce R. Levin

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• Published: Oct 28, 2010
• https://doi.org/10.1371/periodical.pgen.1001171

Figures

Abstract

Amassed, Regularly Interspaced Short Palindromic Repeats (CRISPR) abound in the genomes of almost all archaebacteria and nearly half the eubacteria sequenced. Through a genetic interference mechanism, bacteria with CRISPR regions carrying copies of the Dna of previously encountered phage and plasmids abort the replication of phage and plasmids with these sequences. Thus it would seem that protection against infecting phage and plasmids is the selection pressure responsible for establishing and maintaining CRISPR in bacterial populations. But is it? To address this question and provide a framework and hypotheses for the experimental report of the ecology and evolution of CRISPR, I utilize mathematical models of the population dynamics of CRISPR-encoding bacteria with lytic phage and conjugative plasmids. The results of the numerical (computer simulation) assay of the properties of these models with parameters in the ranges estimated for Escherichia coli and its phage and conjugative plasmids point: (i) In the presence of lytic phage there are broad weather where leaner with CRISPR-mediated amnesty will have an advantage in competition with non-CRISPR bacteria with otherwise college Malthusian fettle. (2) These conditions for the existence of CRISPR are narrower when there is envelope resistance to the phage. (3) While there are situations where CRISPR-mediated immunity can provide bacteria an advantage in competition with higher Malthusian fitness bacteria bearing deleterious conjugative plasmids, the conditions for this to obtain are relatively narrow and the intensity of selection favoring CRISPR weak. The parameters of these models tin be independently estimated, the assumption behind their structure validated, and the hypotheses generated from the analysis of their properties tested in experimental populations of leaner with lytic phage and conjugative plasmids. I suggest protocols for estimating these parameters and outline the blueprint of experiments to evaluate the validity of these models and exam these hypotheses.

Writer Summary

CRISPR is the acronym for the adaptive allowed system that has been found in well-nigh all archaebacteria and nearly half the eubacteria examined. Unlike the other defenses bacteria have for protection from phage and other deleterious DNAs, CRISPR has the virtues of specificity, retentivity, and the capacity to arrest infections with a almost indefinite diverseness of deleterious DNAs. In this report, mathematical models of the population dynamics of bacteria, phage, and plasmids are used to decide the atmospheric condition under which CRISPR can become established and will be maintained in bacterial populations and the contribution of this adaptive immune organization to the ecology and (co)evolution of bacteria and bacteriophage. The models predict realistic and broad conditions under which leaner begetting CRISPR regions tin invade and be maintained in populations of college fitness bacteria confronted with bacteriophage and narrower atmospheric condition when the confrontation is with competitors carrying conjugative plasmids. The models predict that CRISPR can facilitate long-term co-evolutionary arms races between phage and bacteria and betwixt phage- rather than resource-limited bacterial communities. The parameters of these models tin can be independently estimated, the assumptions backside their construction validated, and the hypotheses generated from the analysis of their properties tested with experimental populations of bacteria.

Commendation: Levin BR (2010) Nasty Viruses, Plush Plasmids, Population Dynamics, and the Conditions for Establishing and Maintaining CRISPR-Mediated Adaptive Immunity in Bacteria. PLoS Genet 6(x): e1001171. https://doi.org/10.1371/periodical.pgen.1001171

Editor: David Southward. Guttman, Academy of Toronto, Canada

Received: June 2, 2010; Accepted: September 21, 2010; Published: October 28, 2010

Copyright: © 2010 Bruce R. Levin. This is an open-access article distributed under the terms of the Creative Eatables Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Funding: This work was supported by US National Institutes of Health GM091875 (BRL). The funders had no role in study design, information drove and assay, decision to publish, or preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist.

Introduction

For many species of bacteria, adaptive development is through the expression of chromosomal and extrachromosomal (plasmid- and prophage - borne) genes or clusters of genes (pathogenicity and nicer islands) caused by horizontal gene transfer (HGT) from the same or fifty-fifty quite afar species [1], [2]. Thus, on commencement consideration information technology may seem that leaner and their accessory genetic elements would have mechanism to promote the conquering, incorporation and expression of genes from without. And, indeed there are mechanisms like integrons [3]–[7] that appear to take that role. On the other side, Dna acquired from external sources may be deleterious. This is certainly the case when that DNA is borne on lytic bacteriophage, but also for plasmids that engender fitness costs [viii], [nine] or chromosomal Dna from the wrong source [x], [eleven]. To deal with these contingencies, information technology would seem that bacteria would have mechanisms to protect themselves confronting infection by deleterious foreign DNA [12]. And indeed in that location are systems like brake-modification (restriction endonucleases) which appear to have that role [thirteen], [14].

The most recently discovered machinery postulated to provide bacteria immunity to infectious genetic elements are Clustered Regularly Interspaced Short Palindromic Repeats (CRISPR). For recent reviews see [15], [16]. CRISPR is particularly intriguing because of its ubiquity, appearing in ∼90% and ∼forty% of archaeal and eubacterial sequenced genomes, respectively, and because of the adaptive machinery past which it provides immunity to infections by a virtually indefinite diverseness of bacteriophage and plasmids. Dna from infecting phage and plasmids is incorporated into the CRISPR assortment. Through a yet to be fully elucidated machinery, bacteria abort the replication of infecting phage [17] or the establishment of conjugative plasmids [18] bearing copies of the DNA incorporated into their CRISPR arrays, also see [19]. Further support for CRISPR being an adaptive allowed system that is maintained because information technology protects bacteria from infection with phage comes from studies of the community environmental of bacteria and phage; DNA in the CRISPR regions of the bacteria from those communities corresponds to that in the co-existing phage [xx]–[23]. For an intriguing perspective on CRISPR as a witness to the coevolutionary history of bacteria and phage, see [24].

CRISPR-mediated amnesty has been likened to a Lamarckian mechanism [25], because the option pressure, the infecting phage and plasmids, determine the genotype. This analogy yet does non account for the evolution and maintenance of the mechanism responsible for taking upwardly the infecting phage and plasmid Dna and the machinery employed to preclude the replication or establishment of infecting genetic elements with those sequences. Under what weather will adaptive immunity to phage and plasmid infection be the selection pressure level responsible for establishing and maintaining CRISPR-mediated immunity in populations of archeae and bacteria? What about other mechanisms of resistance, similar structural modification blocking phage adsorption (envelope resistance) and restriction-modification? How do these mechanisms interact with CRISPR – acquired immunity and contribute to its establishment and maintenance?

To accost these questions and provide a framework and hypotheses for their study experimentally, I use mathematical models of the population dynamics of bacteria, phage and plasmids to explore the conditions nether which a CRISPR–similar adaptive allowed machinery will provide bacteria a selective advantage in competition with leaner without this immune system. The results of the numerical analysis of the properties of these models suggest that with bacterial replication and phage infection parameters in realistic ranges, there are wide merely non universal conditions where a CRISPR–like adaptive allowed organization tin can be favored and will exist maintained in populations of bacteria confronted with lytic phage. While this model predicts conditions where CRISPR-mediated immunity will be favored when leaner compete with populations bearing conjugative plasmids, these conditions are relatively restrictive. The parameters of these models tin be independently estimated, the validity of the assumptions behind their construction and the hypotheses generated from the analysis of the backdrop can exist tested in experimental populations of bacteria with lytic phage and conjugative plasmids. Procedures for doing these experiments are outlined and their potential outcomes described and/or speculated upon. Also discussed are the broader implications of CRISR-mediated adaptive immunity to the population and evolutionary biology and ecology of bacteria and phage.

Model

Bacterial growth and population maintenance

Both the lytic phage and conjugative plasmid models used here assume a chemostat-like habitat. The bacteria abound at a rate that is a monotonically increasing part of the concentration of a limiting resources, R µg/ml [26]. where V_i hour⁻¹ is the maximum growth rate of the i^thursday strain of leaner and thou the concentration of the resource when the growth rate is half its maximum value (the "Monod abiding"). The populations are maintained in a vessel of unit book, (1ml) into which medium containing the limiting resource from a reservoir where it is maintained at a concentration A µg/ml flows in at a charge per unit due west per hour. Excess resources and wastes are removed from the vessel at the aforementioned rate. As in [27], the charge per unit of uptake of the resource by the leaner is proportional to the density, the resource concentration-dependent growth rates of the different populations of bacteria and a conversion efficiency parameter, e µg/per jail cell.

The phage model

The model developed here is an extension of that in [28]. In that location are four populations of leaner. 2 are sensitive to the phage, N, non–CRISPR and C, CRISPR and 2 that are either fully resistant (envelope resistance), or immune because of CRISPR, N_R and C_R , respectively. The variables N, C, N_R and C_R are the both the densities (bacteria per ml) of these populations and used as their designations. At that place is one population of phage, with density and designation, P particles per ml.

The phage adsorb to the N and C and C_R bacteria with rate constants, δ_{Due north} and δ_C (ml per phage per cell per hour) respectively. Phage exercise not adsorb to bacteria with envelope resistant, i.e. the North_R cells. To account for a possible multiplicity of infection (MOI) effect on survival of phage-infected C_R, the effective killing charge per unit constant for phage adsorption to CRISPR can be an increasing function of the ratio of free phage and C_R cells, M = P/C_R . (one) where δ_MIN and δ_MAX are the minimum and maximum adsorption rates. The parameter x is a coefficient (0≤x≤i) that specifies the magnitude of the MOI effect, q is the MOI where the adsorption rate is one-half its maximum value and n is an exponent which contributes to the shape of the distribution. At low multiplicities, δ_CR (M) the CRISPR cells would be finer allowed (resistant) (Effigy 1). At loftier multiplicities, however, immune CRISPR cells tin can be overburdened by phage, their immunity would exist overridden, and the phage would replicate, killing the cells. On the other side, we assume that the phage are removed from the population by adsorption to allowed CRISPR cells at the maximum adsorption rate, δ_MAX.

For convenience I neglect the latent periods of the phage infection merely presume that the phage accept potentially different outburst sizes, β_Northward , β _C, and β_CR particles per cell, for North, C and C_R cells, respectively.

Phage-immune CRISPR cells, C_R are produced from C at a rate proportional to the charge per unit at which the phage adsorb to them and a constant grand (0≤m≤i) which is the probability that a phage infection volition be aborted and a CRISPR strain volition be produced. At a rate five per prison cell per hour, CRISPR lose their immunity, C_R→C. For the North and C populations the loss of the adsorbed phage is subsumed in the value of the flare-up size (which is one less than the number of phage produced). For the C_R population, the loss of the phage due to adsorption is specifically considered because merely a small-scale fraction of the adsorbed phage replicate when the MOI is low.

In Table i, I separately define these parameters and in Figure 2, illustrate the interactions between the different populations of bacteria and the phage. The equations for this model follow.

Effigy 2. Model of the population dynamics of lytic phage with CRISPR-mediated adaptive immunity and envelope resistance in continuous civilization: P – phage, N – phage sensitive non–CRISPR bacteria, N_R – envelope resistant, non–CRISPR bacteria C - phage sensitive CRISPR bacteria, C_R - phage immune CRISPR bacteria.

The δs are the adsorption rate constants, m is the fraction of C to which phage are adsorbed that enter the immune state, ν is the charge per unit at which allowed CRISPR cells lose their immunity, and μ is the rate of mutation to envelope resistance. While the phage adsorb to immune CRISPR cells at the maximum rate and are removed from the phage population, their replication on CRISPR cells and the rate of mortality of immune CRISPR is either 0 or a monotonically increasing function of the multiplicity of infection (equation (i)). The leaner reproduce at a rate proportional to the concentration of a limiting resources and their maximum rates of replication. Phage replication is through the killing of adsorbed leaner and their burst size, β, on that cell line. The limiting resource in the reservoir is at concentration A µg/ml and enters the vessel at a rate, w, which is the same charge per unit at which the phage and bacterial populations and backlog resource, R, are removed from the vessel. For more details see the text.

https://doi.org/ten.1371/journal.pgen.1001171.g002

The conjugative plasmid model

The model developed here is an extension of that in [29]. In that location are five bacterial populations. 2 populations exercise not code for CRISPR, N and N_P , and 3 populations code for CRISPR, C and C_P and C_X . The N_P and C_P populations acquit the conjugative plasmid and C_X , carries CRISPR and plasmid sequences that brand it completely allowed to the receipt of these plasmids. Plasmids are transferred by conjugation at rates proportional to the product of the densities of the plasmid-begetting and plasmid-free populations and rate constants, γ_NN , γ_NC , γ_CN and γ_CC (ml per jail cell per 60 minutes) respectively for the transfer of the plasmid from N_P to N, N_P to C, C_P to Due north and C_P to C., respectively. Plasmids are lost by vegetative segregation at rates τ_North and τ_C per cell per hour, with Northward_P→Due north and C_p→C. C are converted to C_X at a rate proportional to the rate at which C acquires the plasmid and a probability m (0≤thousand≤1). C_x lose the CRISPR plasmid immunity region and become C at rate ν per cell per hour. Each of the jail cell lines, have a maximum growth rate, 5_N , V_NP , 5_C , and 5_CP , and Five_X per hr. In Figure three, I illustrate the interactions between the different cell lines in this model, and, in Table 2, I separately ascertain the parameters and variables. The equations for this model are:

Figure 3. Model of the population dynamics of a conjugative plasmid with CRISPR-mediated adaptive amnesty in continuous culture.

N - plasmid-free not–CRISPR, Northward_P - plasmid-bearing not–CRISPR, C - plasmid-complimentary CRISPR, C _P - plasmid-bearing CRISPR, C ₁₀ - allowed CRISPR. The γs are the rate constants of plasmid transfer, g is the fraction of C_P that enter the immune land C_X upon receiving the plasmid from an N_P or C_P, ν is the charge per unit at which allowed CRISPR cells lose their immunity and z the rate at which the CRISPR cells lose the CRISPR element and get N or N_P. The bacteria reproduce at a rate proportional to the concentration of a limiting resources and their maximum rates of replication. The limiting resources in the reservoir is at concentration A µg/ml and enters the vessel at the rate, w, which is the same as the rate at which the phage and bacterial populations and excess resource, R, are removed from the vessel. For more details run into the text.

https://doi.org/ten.1371/journal.pgen.1001171.g003

Numerical solutions

For the numerical solutions to these equations (computer simulations) I use a differential equation-solving software package, Berkeley Madonna. For the phage simulations in that location is a refuge density, beneath which the phage are unable to adsorb to the bacteria. The purpose of this is to control the system from aquiver without limits, see [thirty]. In these simulations, if the phage density falls below 10⁻ⁱ particles per ml, the phage are considered to be lost. Copies of these simulations are available online, www.eclf.net/programs.

Results

The population dynamics and development of CRISPR bacteria with phage

The bacterial growth, resource-uptake, phage adsorption parameters and burst sizes used in these simulations (Table 1) are in a range similar to that which we observed for E. coli and the phages T2 and T7 [28], [31].

Invasion and maintenance of CRISPR in the absence of envelope resistance.

In a chemostat with susceptible bacteria at an equilibrium density N*, a lytic phage can become established and will maintain a population with sensitive bacteria every bit long as the rate of phage production exceeds the rate of washout, δ_Nβ_NNorth*>westward [28]. With the parameters used in these simulations, N^* ∼x^viii (see [32]). As long as δ_Nβ_NN^*>2×10⁻⁹, the phage will go established and can maintain a population past replicating on sensitive bacteria (Figure 4A). The oscillations in the densities of bacteria and phage in these and the post-obit simulations are those anticipated for the predator-casualty nature of these dynamics.

Figure 4. Population dynamics of lytic phage, P, with sensitive non–CRISPR bacteria, North, non-immune and allowed CRISPR-encoding cells, C and C_R, respectively.

Changes in the densities of the bacterial and phage populations and the concentration of the limiting resource, R. In this and the other simulations, A = fifty µg/m, westward = 0.2 per hr, east = five×ten⁻⁷µg, thou = 0.25 µg. In these phage simulations, β_N = β_C =β_CP . (a) The dynamics of sensitive bacteria and phage in the absence of CRISPR, V_N = 1.0 hr⁻ⁱ, δ_Northward = v×10⁻⁹. (b) Invasion of CRISPR in the presence of phage, no MOI upshot (x = 0), V_N = 1.0. Five_C = 0.95, V_CR = 0.90, δ_N  =δ_C  = 5×10^−ix, δ_CP  =δ_MIN  =ten^−xiv (δ_MAX = 5×10^−nine ) (c) Invasion of CRISPR with presence of phage V_Northward = 1.0. V_C = 0.95, V_CR = 0.90, δ_N  =δ_C  = 5×10⁻⁹, Strong MOI issue (10 = 0.5, north = ii.0, q = 10², δ_MIN  = x⁻¹⁴, δ_MAX  = 5×10⁻⁹). (d) Invasion of CRISPR with presence of phage, Five_{Due north} = 1.0. V_C = 0.95, V_CR = 0.ninety, δ_N  =δ_C  = 5×10^−nine, Pocket-size MOI effect (x = 0.2, north = 2.0, q = 10^two, δ_MIN  = 10^−xiv, δ_MAX  = v×10^−ix).

https://doi.org/x.1371/journal.pgen.1001171.g004

To explore the conditions nether which a CRISPR population will become established and be maintained in the presence of phage, I consider situations where the C and C_R populations accept an intrinsic selective disadvantage relative to N (Five_N>V_C , V_CR ) and therefore cannot invade an established N population in the absence of these bacterial viruses. Because of the immunity of C_R , with phage present and in the absence of a multiplicity effect, an initially rare CRISPR population will invade and ascend to dominance despite its lower intrinsic fitness (Figure 4B). With these parameters, the phage are maintained along with N and C, the latter beingness continually generated by the loss of amnesty by the dominant C_R population. The Due north population is maintained because of its higher intrinsic fitness (growth rate) relative to C_R , and resources, rather than phage predation, limit the bacteria at large. The phage continue to exist maintained past replicating on the North and C cells. Although the oscillations are damped and in fourth dimension would no longer be noticed, that fourth dimension would be considerably greater than would be viable to study experimentally with chemostats. If we let for a strong multiplicity issue (10 = 0.5), the CRISPR population becomes established, and both immune and not-immune CRISPR cells maintain their populations with sensitive non–CRISPR in a phage- rather than resource- express community (Figure 4C). When the magnitude of the multiplicity result is reduced (10 = 0.2), the phage continue to be maintained but immune CRISPR cells arise to dominance and the community with three populations of bacteria, N, C and C_R are maintained in a resource- rather than a phage-express state (Figure 4D).

The invasion and maintenance of CRISPR in the presence of envelope resistance.

In addition to CRISPR immunity, when confronted with phage, leaner may generate mutants to which phage are unable to adsorb or are resistant by other mechanisms [33]. To explore how this envelope resistance will bear upon the conditions for the establishment and maintenance of CRISPR, we consider the invasion of an envelope resistant strain of N, Due north_R , into a population of N and phage. In these simulations, the C and C_R are less fit than Due north (V_N>V_C , V_CR ) and N_R are less fit than C and C_R , (V_NR<V_C , 5_CR ). Were the N_R cells more than fit than C and C_R , they would dominate and the CRISPR population would not invade an would not exist established. Whether this fettle human relationship will exist seen with existent bacteria and what those fitness will be is an empirical question.

As can be seen in Figure 5A, although the resistant, N_R strain is the least intrinsically fit leaner in the community (everyman maximum growth rate), in the presence of phage information technology ascends chop-chop and achieves dominance. During this initial phase, as a upshot of the product of immune C_R cells, the CRISPR population also increases in density, only remains a minority population relative to the resistant non–CRISPR North_R. With these parameters, the phage density declines afterward the ascent of resistance and the densities of both the N and C populations increase. Before long after the phage are eliminated the highest fitness N population ascends and lower fitness C, C_R and N_R decline. If the phage resistant population is substantially less fit than the other bacterial populations, the C_R population ascends to say-so and continues to co-exist with the phage, Northward, and C populations (Figure 5B).

Figure v. Population dynamics of lytic phage, P, with sensitive and resistant non–CRISPR leaner, N and N_R, non-immune and immune CRISPR-encoding cells, C and C_R, respectively.

Changes in the densities of the bacterial and phage populations and the concentration of the limiting resource, R. Unless otherwise noted, the parameter values used are those in Figure 4B. (a) Invasion of C and N_R into a population with phage, pocket-size price of resistance, V_NR = 0.85. (b) Invasion of C and N_R into a population with phage, with a greater cost of resistance, V_NR = 0.70.

https://doi.org/10.1371/periodical.pgen.1001171.g005

The population dynamics of CRISPR with conjugative plasmids

In accord with [34], conjugative plasmids volition be maintained every bit long as the rate of infectious transfer exceeds the rates of loss of the plasmid due to pick against the cells carrying it, vegetative segregation, and the rate of flow through the chemostat. In terms of the above parameters, the plasmid will be maintained in an N-NP population as long every bit (ii) where Due north^* is the density of plasmid-free cells at the chemostat equilibrium. For case, if V_N = 1.0, Five_NP = 0.95, w = 0.2, τ_North = x⁻³, the plasmid will exist maintained in a population of density Due north^* = ten⁸ as long as γ_NN >1.ane×10^−ten. If the plasmid augments the growth rate (which in this model is the sole parameter of cell fettle) of the bacteria that deport it, Five_NP >V_{Due north} , as we would anticipate for antibiotic resistance encoding plasmids in the presence of the selecting antibody, bacteria begetting the plasmid will be able to invade even without transfer, every bit long every bit the segregation rate, τ_North, is sufficiently small.

Invasion and maintenance of CRISPR in the presence of a competing population bearing a conjugative plasmid.

The population dynamics of selection and plasmid transfer in an equilibrium chemostat in the absence of CRISPR are presented in Figure 6A. If the conditions specified in equation (2) are met, the plasmid- bearing cells get established and ascend to dominate the N-NP community, whether cells bearing the plasmid are favored or not. If the plasmid is maintained by transfer or selection for the genes it carries and τ_North>0, there will be a stable population of plasmid-gratuitous cells. When the rate constant of plasmid transfer is too low, the deleterious plasmid will be lost.

Effigy 6. Population dynamics of a conjugative plasmid with not–CRISPR, North and N_P and CRISPR, C, C_P and C_Ten populations; changes in the densities of the bacterial populations.

Unless otherwise noted all of the charge per unit constants of plasmid transfer, the γ_ijs = 10⁻⁹ [38], the segregation rates, τ_N and τ_C = 10⁻³, the rate of loss of immunity ν = 10⁻³, upon receiving the plasmid the rate of conversion of C_P to C_X = 0.two, and the rate of conversion of CRISPR cells to N or N_P, z = 10⁻⁸. (a) No CRISPR – Just N and N_P 1 - Deleterious plasmid Five_{Due north} = 1, V_NP = 0.95; 2 - a beneficial plasmid V_{Due north} = one, V_NP = 1.2 and 3- deleterious plasmid V_North = 1, V_NP = 0.95, γ_NN = 10⁻¹¹. (b) Invasion of bacteria carrying a deleterious plasmid into a lower fitness CRISPR, C, population, V_North = ane, Five_NP = 0.95, V_C = 0.97, Five_CP = 0.88, Five₁₀ = 0.96, (c) Invasion of CRISPR 10 into a equilibrium population of plasmid-bearing and plasmid free cells, Northward-NP with a deleterious plasmid (parameters the same as b). (d) Invasion of cells carrying a higher fitness plasmid, NP, into a C population, 5_N = ane, V_NP = i.ii 5_C = 0.97, Five_CP = one.1, 5_x = 0.96.

https://doi.org/x.1371/journal.pgen.1001171.g006

To consider the effects of CRISPR on the population dynamics of bacteria with conjugative plasmids and the conditions under which CRISPR immunity will provide an advantage to bacteria, I let the maximum growth rates of the CRISPR strains (the sole measure of intrinsic, phage-independent fettle) be somewhat lower than the corresponding non–CRISPR cells. In Effigy 6B, the population is initially at equilibrium with a plasmid-free, not-immune CRISPR population and a low density of plasmid-bearing not–CRISPR leaner are introduced. The plasmid spreads rapidly from Northward_P to C producing a C_P population which in turn generates immune CRISPR, C₁₀ . While the C and C _P populations dice out, C_X ascends to dominance and minority populations of North and North_P are maintained. Although the C_X population has a lower growth rate than N, in the presence of a deleterious conjugative plasmid they take an reward because they cannot be infected by that element. They practise not eliminate the North and Due north_P populations due to the loss of the CRISPR region and the conversion into Northward. As can exist seen in Figure 6C, with these parameters and a lower growth rate, C_X can invade an equilibrium Due north-N_P population, but the rate of increase in the density of C₁₀ is low. The invasion rate for C_X would even be farther reduced if, instead of C_X, a plasmid-free C invaded an NP population, because information technology would exist some time before the C_X is produced and, in a finite population, may non be produced at all ("data" not shown). A very different situation obtains when the plasmid confers a growth rate reward to the infected host (Figure 6D). Under these conditions, the C populations and its derivatives, C_P and C_X, are eliminated.

Word

"All models are wrong, some are useful." (George Box)

It has been less than eight years since the ubiquitous clusters of palindromic repeats now known as CRISPR outset caused this moniker [35]. Although there had been compelling circumstantial prove that CRISPR was part of an adaptive immune system that provides protection against infecting phage and plasmids, it has been less than 4 and three years respectively since the publication of the first direct (read experimental) testify that CRISPR tin provide immunity to infection past lytic phage [17] and conjugative plasmids [18].

In the course of this fourth dimension a great deal has been learned about the molecular biology of CRISPR and the mechanisms by which information technology provides adaptive immunity to plasmid and phage infection. Just there remain many unanswered questions about these processes. Almost important for this consideration is a dearth of the quantitative information needed to empathize the population dynamics of CRISPR-mediated adaptive immunity and thereby the conditions for the establishment and maintenance of CRISPR in bacterial populations. To my knowledge, this study is the first formal consideration of these dynamics.

The models

The models developed in this study incorporate what has been learned near CRISPR-mediated adaptive amnesty to phage and conjugative plasmids, primarily from the studies of Barrangou and colleagues [17] and Marraffini and Sontheimer [18], into models of the population dynamics of lytic phage [28] and conjugative plasmids [29]. Although they may announced circuitous, at best they are simplistic caricatures of interactions between these infectious genetic elements and bacteria with CRISPR-mediated adaptive immunity. These models are not intended or anticipated to exist numerically precise analogs of these processes and dynamics.

The role of these mathematical models is similar to that of the diagrammatic models (cartoons) used to illustrate the molecular basis and mode of action of CRISPR, i.e., to provide a framework for agreement these processes, designing experiments, and interpreting their results. In this example, these experiments are on population and evolutionary dynamics of bacteria with CRISPR-mediated immunity confronted with lytic phage and competing bacteria bearing conjugative plasmids. The purpose of these models for this experimental enterprise is: (i) to identify and, in a quantitative way, evaluate the role of the different factors (parameters) contributing to these dynamics and the conditions for the establishment and maintenance of CRISPR in bacterial populations, and (ii) to generate hypotheses about these dynamics and existence conditions that tin be tested (and rejected) in experimental populations.

Predictions and some interpretations/speculations

The results of the analysis of the properties of the phage - CRISPR model are consistent with the proposition that in the presence of lytic bateriophage there are broad weather condition under which a CRISPR–like adaptive immune organisation can become established and volition exist maintained in bacterial populations. With population densities, growth rates, and phage infection parameters in realistic ranges, these models predict that despite a growth rate disadvantage, bacteria with CRISPR–like acquired immunity to infecting phage will increase in frequency when initially rare and will be maintained. The necessary status for this is that the phage population continues to persist at a sufficiently high density for CRISPR-mediated adaptive amnesty to overcome an intrinsic disadvantage associated with the costs of carrying and expressing these genes.

When will the phage maintain their populations at sufficient levels for this outcome? With the parameters used to address this question, the phage will be maintained under wide weather, but may eventually be lost if a population with envelope or other resistance ascends to dominance. I emphasized the discussion may for two reasons. The commencement is theoretical, if the relative growth rate of the resistant population is adequately low, the phage and thereby CRISPR will be maintained. The second is empirical, even when resistant leaner boss experimental populations of bacteria and phage, in general the phage go along to be maintained [30], [31], [36].

The CRISPR plasmid model predicts that considering of the immunity to infection with conjugative plasmids, a lower growth charge per unit (Malthusian fettle) CRISPR population can become established and will be maintained when competing with bacteria with a greater Malthusian fitness simply bearing deleterious (fitness-reducing) conjugative plasmids. Although these conditions are met with plasmid fitness costs in the range estimated for "laboratory" plasmids [9], [37], it is not clear that naturally occurring plasmids would be as burdensome as those maintained in the Lab. The greater the Malthusian fitness burden attributed to the plasmid, the greater the advantage of CRISPR-mediated amnesty.

The rate constants of plasmid transfer used in these simulations are those for plasmids with permanently derepressed conjugative pili synthesis. Wild blazon conjugative plasmids are more than likely to be repressed for the product of these transfer organelles and would take substantially lower rates of transmission than plasmids that are permanently derepressed for plasmid transfer [38], [39]. Indeed, it is not clear whether in natural populations conjugative plasmids that engender fitness cost tin exist maintained by transfer alone. Their persistence may require periodic episodes where bacteria carrying them take an reward [34], [40], but too see [41]. If the rate of infectious transfer is non sufficient to maintain deleterious plasmid in a population and they persist by continually or periodically enhancing the cells Malthusian fitness, immunity to these plasmids would not be sufficient to maintain CRISPR-encoding cells that accept an intrinsic fitness disadvantage.

Evaluating the models: estimating their parameters and testing the validity of their assumptions and predictions

It would be well-nigh impossible to make up one's mind whether the quantitative conditions predicted by these models for the establishment and maintenance of CRISPR-mediated amnesty are met in natural populations. On the other mitt, the values of the parameters of these models can be estimated and the validity of the assumptions behind their structure and hypotheses generated from the analysis of their properties can be tested in laboratory culture using CRISPR–positive and CRISPR–negative bacterial constructs, phage and plasmids of the types used respectively by Barrangou and colleagues [17] and Marraffini and Sontheimer, [xviii] in chemostat culture.

Parameters.

All of the parameters of these models (Table 1 and Tabular array 2) can be independently estimated and procedures for doing so have been published for the majority of them: (one) for the bacterial growth and resource utilization parameters, the 5 _S , k, and e, run into [26], [28]; (two) for the phage latent periods, adsorption rates δs, and flare-up sizes, the βs, see [28], (three) for the charge per unit constants of plasmid transfer, the γs, see [42], [43], and (4) for the mutation charge per unit to envelope resistance, see [44], [45]. Estimates of the plasmid segregation rate, τ, tin be obtained by plating low-density cultures of plasmid-bearing cells, and testing colonies for the plasmid marking. Still, unless τ is very high (τ>0.005 per jail cell per division), this procedure would be excessively labor intensive. Still, if depression, this parameter would have a negligible contribution to the dynamics of the plasmid and estimating its value would non be worthwhile.

Protocols for isolating leaner with CRISPR-mediated resistance to phage and plasmid infection, can be found in [17] and [18], respectively. I am, however, unaware of published studies providing estimates of the fractions of phage and plasmid infected cells that become immune, the parameter m, or the rates of loss of these immunities, ν, in the models (Figure 2 and Figure 3). In Text S1, I outline potential means to gauge these parameters. I emphasize the give-and-take potential considering without actually doing these experiments, information technology is difficult to anticipate pitfalls and bug with the proposed procedures.

Assumptions and tests of their validity.

In developing the model, I made a serial of assumptions about CRISPR – mediated immunity and the population dynamics of bacteria with lytic phage and conjugative plasmids. In the post-obit, I list these assumptions and briefly describe what would be anticipated experimentally if these assumptions are right.

1. CRISPR immunity to phage infection will have no consequence on the rate at which phage adsorb to immune cells. If this is right, the estimated adsorption rate parameter δ of a lytic phage should be the aforementioned for CRISPR cells of any immune country likewise as cells of that strain for which CRISPR is not-functional.
2. Phage infecting allowed CRISPR cells will be lost. If this is correct, when depression densities of phage are introduced into relatively high densities of exponentially growing populations of allowed CRISPR cells, at that place should exist a decline rather than an increment in the density of phage. In the model, the rate of reject in the density of phage, P, adsorbing to a population of bacteria with CRISPR amnesty to that phage tin be calculated from the estimated adsorption maximum rate parameter δ_MAX and the density and maximum growth rate of bacteria, C_R and V_R , respectively. If as suggested in [17], the level of CRISPR – mediated immunity to the phage varies with the extent and nature of the phage DNA incorporated into the CRISPR region, this should be reflected as variation in the rate of loss of the phage.
3. At that place is a multiplicity of infection (MOI) issue. When CRISPR-encoding cells are confronted with high multiplicities of phage to which they are immune, the phage volition replicate and kill the immune cells. If positive results are obtained in these MOI experiments, by varying the multiplicity, the functional relationship between the MOI and the level of immunity can be determined. In doing these experiments, however, it will exist necessary to rule out the possibility that those that phage that replicate on immune cells are not host range mutants [24].
4. CRISPR amnesty to conjugative plasmid transfer is absolute. If this is right, the estimated charge per unit constant of plasmid transfer γ for mixtures of donor CRISPR cells allowed to that plasmid would be nil independent of the density of the culture and ratio of donors and potential CRISPR recipients. Based on the results reported in [18] as well as [17], information technology may well be that the level of CRISPR – mediated immunity to plasmid infection as measured past the charge per unit abiding of plasmid transfer, δx, would vary with the extent and nature of plasmid Dna incorporated into the CRISPR region.
5. CRISPR immunity to plasmid infection is generated during the transfer process, when the recipient first receives the plasmid, rather than during the grade of plasmid wagon. If this the case, bacteria immune to plasmid transfer, C₁₀, would be rare in cultures of plasmid-bearing CRISPR, the C_P population. That is, they would only be generated, when C_P transfer the plasmid to segregants, C.

Population dynamics and existence conditions predictions.

Ane way to evaluate how well these models serve as analogs of the population dynamics of bacteria with CRISPR adaptive immunity to bacteria and phage is to compare the results of simulations with independently estimated parameters to that observed in chemostat populations. Although it would be gratifying to run into quantitative agreement betwixt the anticipated dynamics and those observed in experimental populations, populations with CRISPR constructs of bacteria, conjugative plasmids and phage, it would also be surprising. These models are far also uncomplicated to expect the predicted and observed dynamics to be numerically ancillary. A more than pocket-size, realistic, and, I believe, more than useful goal is test predictions fabricated from the assay of the properties of these models in a qualitative – semi-quantitative way and place those elements of the model that have to be modified to make the models more realistic and accurate. In the following, I list these predictions.

The phage model.

(i) When mixtures of otherwise isogenic CRISPR positive and negative phage –sensitive constructs are introduced into chemostats in approximately equal frequencies:

1. CRISPR cells with immunity to the phage will emerge and arise to dominance.
2. If the phage are maintained, the CRISPR population will continue to persist.
3. If non–CRISPR mutants with envelope or other resistance to the phage evolve, or are introduced, unless they accept a considerable cost in Malthusian fitness, these resistant leaner will increment in frequency and may replace the CRISPR population.
4. Although not considered in the model, there is the possibility that CRISPR cells C or C_R volition acquire envelope resistance. If so, a CRISPR population with envelope resistance may boss.

(two) When introduced at low frequencies into chemostats with sensitive non–CRISPR cells in the presence of phage, as long as immune CRISPR cells are produced, the CRISPR population volition increase in frequency. This will not be the case in the absenteeism of phage.

The plasmid model.

When mixtures of non–CRISPR cells bearing fitness reducing conjugative plasmids and plasmid-free CRISPR cells are introduced into chemostats:

1. CRISPR cells with immunity to the plasmid will emerge.
2. the allowed CRISPR population will increase in frequency, fifty-fifty if the CRISPR cells accept lower growth rates than plasmid-costless non–CRISPR.
3. the CRISPR population will pass up in frequency if the environmental conditions changed so that selection favors cells bearing the plasmid. (One way to do this experiment is to apply antibiotic resistance, R- plasmids and periodically add antibiotics to which the plasmid confers resistance).

Caveats, excuses, recognized limitations, extensions, and speculations

In this study, I elected to restrict the model and its analysis to the simplest cases with lowest realistic number of states of bacteria, phage and plasmids. I have washed and then because at this fourth dimension these minimum number of states models and the predictions generated from their analysis are more than acquiescent to evaluating and testing experimentally than models with more than states of leaner, phage and plasmids. Moreover, these tests, and especially the population dynamic experiments, should point the importance of the generation of boosted population states by mutation, like host range phage and host range plasmids, are to these dynamics. Exist that as information technology may, I also realize that this minimum number of states model will not account for what may plow out to be the well-nigh important contributions of CRISPR-mediated immunity to the ecology as well equally the population and evolutionary biology of leaner and phage.

Generalized resistance.

Luciano Marraffini (personal communication) suggested one potentially important contribution of CRISPR to the population and evolutionary dynamics of bacteria and phage. Unlike envelope resistance, which is almost always restricted to phage that utilise single adsorption organelles, [33], CRISPR–immunity tin can be effective against multiple phages with different adsorption organelles (independent resistance). Moreover, envelope resistance is likely to engender a toll in Malthusian fitness, e.k. see [31], [36], [46] and that cost will nigh certainly exist greater if this resistance is for multiple phages that utilize different receptors for infection.

If these interpretations are correct, information technology would seem experimental populations with CRISPR-encoding bacteria with envelope resistance to all the phage will not evolve and CRISPR will prevail in competition with sensitive non–CRISPR cells. If, however, the results of a test of this multi-phage hypothesis Ryzard Koroana and did in a report of the atmospheric condition for the maintenance of restriction endonuclease (restriction-modification, R-M) immunity are general [47], this hypothesis may exist rejected. E. coli begetting an R-M organization conferring immunity to three phage with different organelles were challenged with a mixture of all iii of these phages. As a consequence of a hierarchy of phage replication [48], there was sequential selection for the different resistant states and within a day of exposure, bacteria with envelope resistance to all three phages dominated the community [47].

A CRISPR-mediated arms race and phage-limited communities.

A number of years agone, Richard Lenski and I postulated that the arms race between bacterial resistance and host range phage would be express to few cycles and is likely to end with resistant leaner to which phage would not exist able to generate host range mutations [46]. The empirical basis of our hypothesis was the results of experiments with E. coli and its phage and envelope resistance, [31], [46], [49], [50]. While this estimation was also supported by experiments with V. cholerae and its phage JSF4 [36], experiments with Pseudomonas fluorescens and its phage SBW25 [51] advise extended arms races are possible. Although, to my knowledge, the mechanisms responsible for the continuous changes in resistance and host-range reported in this study with this strain of Pseudomonas and phage accept even so to be elucidated, CRISPR does provide a machinery for long-term arms races betwixt bacteria and phage [21], [22], [24]. By unmarried base changes in sequences of DNA into the spacer regions of CRISPR, a phage can infect and replicate on previously immune CRISPR cells. By incorporating the mutated or other region of that phage into another spacer, CRISPR cells can generate resistance to these host range phages. At this time, it is not at all clear how long or through how many cycles a CRISPR-mediated arms race can proceed. I would it certainly exist interesting, tenable experimentally and fun to observe out. Exist it by CRISPR or by sequential resistance and host-range mutation [52], [53] an extended arms race could provide a fashion for phage, rather than resources, to limit the densities of bacterial populations (come across Text S2), which is an ecological outcome with applied likewise as theoretical implications, e.g. see [54]–[59].

Supporting Information

Acknowledgments

I would like to thank Luciano Marraffini and Joshua Weitz for stimulating discussions and correspondence and useful comments and suggestions on an before draft of this report. I am grateful to Amy Kirby, Amoolya Singh, Klas Udekwu, and Jim Balderdash for helpful suggestions, comments, and corrections. I apologize to the members of the EcLF and others sipping java and eating dejeuner in the "Peoples room" and to the nice Grandmother sitting next to me on an airplane for my ranting on virtually the wonders of CRISPR.

Writer Contributions

Conceived and designed the experiments: BRL. Performed the experiments: BRL. Analyzed the data: BRL. Wrote the newspaper: BRL. Did the programming: BRL.

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