Mutators in Space: The Dynamics of High-Mutability Clones in a Two-Patch Model

Corresponding author: E. R. Travis, IBLS, University of Glasgow, Glasgow G12 8QQ, United Kingdom., emmatravis{at}cantab.net (E-mail)

Communicating editor: Y.-X. FU

	ABSTRACT

TOP ABSTRACT MODEL RESULTS DISCUSSION LITERATURE CITED

Clones of bacteria possessing high-mutability rates (or mutators) are being observed in an increasing number of species. In a constant environment most mutations are deleterious, and hence the spontaneous mutation rate is generally low. However, mutators may play an important role in the adaptation of organisms to changing environments. To date, theoretical work has focused on temporal variability in the environment, implicitly assuming that environmental conditions are constant through space. Here, we develop a two-patch model to investigate how spatiotemporal environmental variability and dispersal might influence mutator dynamics. Environmental conditions in each patch fluctuate between two states; the rate of fluctuation varies in each patch at differing phase angles. We find that at low and intermediate rates of fluctuation, an increase in dispersal results in a decrease in the density of mutators. However, at high rates of environmental change, dispersal causes an increase in mutator density. For all frequencies of environmental fluctuation these trends are enhanced as the phase angle approaches 180°. We argue that future work, both empirical and theoretical, is needed to improve our understanding of how spatiotemporal variability impacts on mutator densities and dynamics.

RECENT years have seen considerable attention paid to the dynamics of mutators—populations with high mutation rates (for a recent review see SNIEGOWSKI et al. 2000 ). This work has been stimulated by the increased frequency with which mutators are being found within different bacterial species (LECLERC et al. 1996 ; MATIC et al. 1997 ; OLIVER et al. 2000 ; GIRAUD et al. 2001 ). Evidence suggesting that mutators may increase the likelihood of the evolution of drug resistance (SNIEGOWSKI et al. 2000 ) seems likely to further accelerate attempts to understand both the causes and the consequences of mutator clones. High frequencies of mutator strains have been observed among pathogenic (disease-causing) bacteria (LECLERC et al. 1996 ; DENAMUR et al. 2002 ; RICHARDSON et al. 2002 ), although it is unclear whether the hypermutability contributes to the pathogenicity of the bacterial strain (MATIC 2000 ; DENAMUR et al. 2002 ).

It has long been recognized that in constant environments most mutations are deleterious, and hence mutation occurs at a low rate that is constrained only by the costs of error avoidance and error repair (KIMURA 1967 ; DRAKE 1991 ). However, several theoretical studies have demonstrated that when environmental conditions are temporally variable then higher mutation rates can be selected for (LEIGH 1970 , LEIGH 1973 ; GILLESPIE 1981 ; ISHII et al. 1989 ). These studies tended to seek the single optimal mutation strategy under a particular set of environmental conditions, rather than considering the temporal dynamics of competing clones with different mutation rates.

More recently, models have been developed to investigate the dynamics of mutator clones that increase the mutation rate by a particular factor (TADDEI et al. 1997 ; TENAILLON et al. 1999 , TENAILLON et al. 2000 ; TRAVIS and TRAVIS 2002 ). In these models a wild-type clone with a low mutation rate competes with a mutator clone that has a higher mutation rate. Results demonstrate that even when a population experiences a single shift in its environmental conditions a mutator clone may dramatically increase its density within the population. The predictions of these models are in agreement with results from laboratory studies. MAO et al. 1997 and SNIEGOWSKI et al. 1997 both demonstrated that mutator density increases in laboratory populations of Escherichia coli subjected to a single shift in the environment. Mutator clones are at a selective advantage following an environmental change as they enable more rapid adaptation to the novel conditions. In a recent article, TANAKA et al. 2003 use both a simulation and a mathematical model to investigate the dynamics of a changing environment where a single adaptive change increases adaptation to a particular environment. They demonstrate that the success of mutator clones is dependent on the initial frequency of mutators in the environment and show that mutators can climb to dominance in stages, even from low initial levels.

The natural environment is often complex, characterized by many different patterns of spatial and temporal variability (HERRERASILVEIRA 1994 ; MILLER et al. 1995 ; PICKETT and CADENASSO 1995 ; CHAZDON 1996 ), and this may have important implications for mutators. As de Visser points out in his recent review (DE VISSER 2002 ), the indirect selective benefit of mutator clones is dependent on opportunities for adaptation. Novel environmental conditions that can provide such opportunities can arise either by the environment rapidly changing in time or through a spatially heterogeneous environment. In a recent article we began to address the question of temporal complexity (TRAVIS and TRAVIS 2002 ) by investigating the dynamics of mutators within environments where conditions fluctuated with different frequencies. A key result to emerge is that mutators are most prevalent at an intermediate frequency of fluctuation. When the environment oscillates more rapidly, mutators are found to be unable to provide sufficient adaptability to keep pace with the frequent changes in selection pressure, and instead a population of "environmental generalists" dominates. This work, along with most other theoretical work concerned with the evolution of mutation rate or mutator dynamics, implicitly assumes that the environment is spatially homogeneous, an assumption unlikely to be true for the environment encountered by many species.

The role that spatial variation plays in the evolution of hypermutability is, as yet, a relatively unexplored area, with no models developed. An important recent article showing the colonization of the lungs of cystic fibrosis patients with a hypermutable strain of Pseudomonas aeruginosa (OLIVER et al. 2000 ) emphasizes the need for clearer understanding of the role of spatial heterogeneity in the development of mutator clones. In this article it is argued that the highly compartmentalized, anatomically deteriorating nature of the lungs, as well as the challenges presented by the immune system and antibiotic drug therapy, provides conditions suitable for the selection of the mutator clones. Thus, there is a need to consider mutator dynamics in a spatially variable environment. In recent years, a considerable body of work has been devoted to improving our understanding of the consequences of spatial heterogeneity for both ecological and genetical processes (PICKETT and CADENASSO 1995 ; LLOYD and MAY 1996 ; BARTON and WHITLOCK 1997 ; BARTON 2001 ; WHITLOCK 2002 ). Perhaps the simplest theoretical spatial extension that allows for the incorporation of spatial heterogeneity is a two-patch model (MCPEEK and HOLT 1992 ; HOLT and HASSELL 1993 ). Here we develop such a model to investigate how mutator dynamics are influenced by both spatial and temporal environmental variability.

	MODEL

TOP ABSTRACT MODEL RESULTS DISCUSSION LITERATURE CITED

The model developed here aims to explore the interactions between dispersal and mutators during adaptation to constantly fluctuating environments in a two-patch system. This study is parameterized using mutation rates, selective advantages, and costs of deleterious mutations that are consistent with those observed in long-term experiments with E. coli (NINIO 1991 ; KIBOTA and LYNCH 1996 ). In each patch, the environment can be in one of two states (0 and 1) and it regularly fluctuates between these states. Here, it is assumed that there are 11 possible genotypes (0–10) and that type 0 is perfectly adapted to environment state 0 while type 10 is perfectly adapted to environment state 1. Conversely, type 10 is poorly adapted to environment 0, and type 0 has low fitness in environment state 1. Genotype 5 has equal fitness in both environments. Each genotype reproduces according to its fitness relative to the mean fitness of the total population in its patch. This competition between genotypes is simulated according to the equation

where g^a,d,m_t refers to the densities of the different genotypes and f^a,d,m_t to their fitness. Indices a and d indicate, respectively, the number of well-adapted and the number of deleterious alleles possessed by a genotype, and m indicates whether the genotype is wild type or mutator. D is the total population density.

Mutations occur at a rate determined by the state of the mutator allele. The initial density of the mutator allele is set to 0. Mutations can lead to genotypes that are better (or more poorly) adapted to the current environmental condition. A mutation results in a shift of ±1 to the genotype: thus mutations occurring to genotype 3 increase the density of genotypes 2 and 4. Ten mutations are required for a shift from a genotype that is perfectly adapted to one environment to a genotype that is ideally suited to the other environment. The rates of mutation for wild-type clones are shown in Table 1. These rates are multiplied by the mutator strength (m) in the mutator clones; the mutation rates for the baseline mutator strength (m = 100) are also shown in Table 1.

The two patches fluctuate at the same frequencies but out of phase with each other. The phase shift (degrees by which patch 2 is behind patch 1) is set to between 0° and 360°. Dispersal occurs between the two patches at a fixed rate, set between 0 and 0.2. Fitness of the population is calculated as the proportion of the possible advantageous alleles that are present. Baseline parameters are fixed: each advantageous allele confers a 0.05 benefit, each deleterious allele costs 0.05, and mutator strength (m) is set to 100.

	RESULTS

TOP ABSTRACT MODEL RESULTS DISCUSSION LITERATURE CITED

The effect of mutator clones on a temporally variable environment has been investigated in TRAVIS and TRAVIS 2002 . Mutator density was shown to be dependent on the frequency of fluctuation of the environment, as illustrated in Fig 1A. At rapid frequencies of fluctuation only a small proportion of the population are mutator clones. At intermediate frequencies of fluctuation the highest percentage of mutators is observed. Interesting dynamics, such as chaos and limit cycles, are observed within this parameter space (see TRAVIS and TRAVIS 2002 for details). At lower frequencies of fluctuation the mutator density is again lower.

View larger version (27K): In this window In a new window Download PPT slide   Figure 1. Effect of varying the frequency of environmental fluctuation on mutator density. (A) Fluctuation frequency vs. mutator density. B, C, and D show time courses with the environment type (top), percentage of mutator density (middle), and percentage of fitness (bottom) against time (gen) represented. (B) A time course at 3000 gen–1. (C) A time course at 1000 gen–1. (D) A time course at 500 gen–1. (E) A time course showing environment type (top) and percentage of mutator density against time for a fluctuation frequency of 500 gen–1. Note that the scale of mutator density is greatly enlarged compared with B, C, and D to illustrate that there is a very slight fluctuation of mutator density at high frequencies of environmental change.

Fig 1B reveals that at low frequencies of fluctuation (e.g., at 3000 gen^–1) during each cycle of fluctuation the mutator density initially rises until all of the population is adapted to the current environment, at which point mutator density drops again, resulting in a relatively low mutator density. Fig 1C illustrates how during each cycle of fluctuation at an intermediate frequency of fluctuation (e.g., 1000 gen^–1) the mutator density rises, with the environment switching before the mutator density has fallen far, resulting in a high average mutator density. Fig 1D shows that at rapid frequencies of fluctuation (e.g., at 500 gen^–1) the environment switches before mutator clones have had time to evolve, resulting in low average mutator density. Fig 1E shows the slight increase in mutator density that occurs each cycle, with a maximal mutator density of <0.15% reached.

This study shows that the role of mutators in evolution can dramatically change when spatial variation of environments is also considered. A two-patch spatial model has been chosen, with each patch experiencing the same rate of environmental fluctuation, although the patches vary in phase. Dispersal occurs between the patches.

Fig 2 shows the effect of varying the rate of dispersal when the two patches are fluctuating out of phase with each other. Frequencies of fluctuation of 3000, 1000, and 500 gen^–1 have been chosen to represent low, intermediate, and rapid rates of change, respectively. The average mutator density in patch 1 is illustrated. In all cases the mutator density observed at zero rate of dispersal represents that achieved with simply temporal rather than spatiotemporal variation. The Roman numerals I–VI indicate the parameter space where dynamics are explored further in the RESULTS section.

View larger version (15K): In this window In a new window Download PPT slide   Figure 2. Effect of varying dispersal on the mutator density of patch 1, when patch 2 is out of phase. Shaded line, 3000 gen–1; solid line, 1000 gen–1; thick dashed line, 500 gen–1. I–VI, the positions at which the time courses are performed. (A) Patch 2 is 72° behind the phase of patch 1. (B) Patch 2 is 216° behind the phase of patch 1.

Fig 2A shows the effect of dispersal on the mutator density of patch 1, when patch 2 lags 72° behind patch 1. The effect of dispersal on mutator density is dependent on the frequency at which the environment is fluctuating. At low levels of fluctuation (e.g., 3000 gen^–1) dispersal does not result in a dramatic change in mutator density, with densities of between 15 and 25% observed. At intermediate rates of fluctuation (e.g., 1000 gen^–1) as dispersal rises to 0.065 the mutator density remains high. As dispersal increases further the mutator density drops to a near-zero value. At high levels of fluctuation (e.g., 500 gen^–1) the mutator density is at a very low level at all dispersal rates.

Fig 2B shows the effect of dispersal on the mutator density of patch 1, when patch 2 lags 216° behind patch 1. Again the effect of dispersal on mutator density is observed to be dependent on the frequency at which the environment is fluctuating. At low levels of fluctuation (e.g., 3000 gen^–1) at zero dispersal the mutator density is at 23%. However, at nonzero dispersal the mutator density decreases to <5% at all rates of dispersal. At intermediate rates of fluctuation (e.g., 1000 gen^–1) the mutator density drops by over half as dispersal increases from zero to 0.005. As dispersal rises to 0.035 the mutator density rises slightly, after which it drops erratically to a near-zero value at dispersal of 0.105. Time courses run in this parameter space again reveal limit cycles and chaos (data not shown). At high levels of fluctuation (e.g., 500 gen^–1) the mutator density increases considerably from a near-zero density at zero dispersal to a moderate mutator density of 40% at 0.02 dispersal. As dispersal rates further increase, the mutator density drops gradually to a near-zero value at dispersal rate of 0.08.

Effect of phase shift:
The relationship between phase shift and mutator density was explored at a range of dispersal rates. Fig 3 graphs the results for rates of 0.02 (Fig 3A), 0.1 (Fig 3B), and 0.2 (Fig 3C). The Roman numerals I–VI correspond to the same parameters as in Fig 2 and are explored in more detail later. For both low and intermediate frequencies of fluctuation, at all dispersal rates, the mutator density was highest at phase shifts approaching 0° and 360° and lowest at those phase shifts approaching 180°. As dispersal rates increased, the width of the trough increased further. At high rates of fluctuation (e.g., 500 gen^–1), however, the pattern observed is different. Two peaks of maximal mutator density appear, centered around 180°. The size of these peaks is maximal at the lower dispersal rates and at the highest dispersal rate shown of 0.2 it is not apparent.

View larger version (21K): In this window In a new window Download PPT slide   Figure 3. Effect of varying phase shifts on the mutator density of patch 1, when patch 2 is out of phase. Shaded line, 3000 gen–1; solid line, 1000 gen–1; thick dashed line, 500 gen–1. I–VI, the positions at which the time courses are performed. (A) Dispersal, 0.02. (B) Dispersal, 0.1. (C) Dispersal, 0.2.

Time courses:
To gain a better understanding of the effect of dispersal and phase shift on mutator density, it is necessary to examine the variation of mutator density and fitness during each cycle of environmental fluctuation. We have inspected a wide range of these time courses, and six are shown here to illustrate the findings (Fig 4). These time courses (Fig 4, I–VI) correspond to parameter values illustrated by the Roman numerals I–VI, respectively, in Fig 2 and Fig 3.

View larger version (42K): In this window In a new window Download PPT slide   Figure 4. Time courses of environment type (top), percentage of mutator density (middle), and percentage of fitness (bottom) against time (gen) are shown. Patch 1 is illustrated with a dashed line and patch 2 with a solid line. I. Frequency of fluctuation, 3000 gen–1; dispersal, 0.2; phase shift, 72°. II. Frequency of fluctuation, 3000 gen–1; dispersal, 0.2; phase shift, 216°. III. Frequency of fluctuation, 1000 gen–1; dispersal, 0.02; phase shift, 72°. IV. Frequency of fluctuation, 1000 gen–1; dispersal, 0.1; phase shift, 72°. V. Frequency of fluctuation, 500 gen–1; dispersal, 0.02; phase shift, 216°. VI. Frequency of fluctuation, 500 gen–1; dispersal, 0.1; phase shift, 216°.

Fig 4, I and II, compares the time courses at low frequency of fluctuation at phase shift 72° (Fig 4I; where both patches are of the same environment type for the majority of the time) and phase shift 216° (Fig 4II; where both patches are of differing environment type for the majority of the time). For both time courses the dispersal is high at 0.2.

In Fig 4I the mutator density rises when both patches are in the same environment type and results in an associated rise in the rate of increase of fitness. However, in Fig 4II the period when both patches are of the same environment type is shorter and the mutator density remains low. Note that the mutator densities for both patches 1 and 2 are identical. This is consistent with the symmetrical nature of Fig 3C around 180°.

Fig 4, III and IV, compares the time courses at intermediate frequency of fluctuation at both a low rate of dispersal of 0.02 (Fig 4III) and a moderate rate of dispersal of 0.1 (Fig 4IV). For each time course the phase shift is 72°.

In Fig 4III at the lower rate of dispersal the mutator density rises when the two patches are of the same environmental type. This results in an associated increase in the fitness of the patches to nearly 100% fitness. In Fig 4IV, however, where there is a moderate rate of dispersal the mutator density increases only fractionally when both patches are of the same environmental type. The fitness levels fluctuate between 26 and 74%, never reaching maximal fitness.

Fig 4, V and VI, compares the time courses at high frequency of fluctuation at both a low rate of dispersal of 0.02 (Fig 4V) and a moderate rate of dispersal of 0.1 (Fig 4VI). For each time course the phase shift is 216°.

In Fig 4, V and VI, the patterns of both mutator density and fitness appear complex, due in part to the two-part limit cycle occurring in both time courses; however, trends are apparent. In Fig 4V, with a low rate of dispersal, it is again when the two patches are of the same environmental type, in this case environment type 1, that the rise in mutator density occurs and produces an acceleration in the increase in fitness. Patch 1 then switches environment type to type 0, and the mutator density drops until a switch is made back to environment type 1, when a sudden peak of mutator density occurs. Due to the almost perpetual presence of mutator clones in patch 1 the fitness increases rapidly. Patch 2 remains in environment type 1 after patch 1 has switched and experiences a continued high mutator density and associated increase in fitness, until it too switches to environment type 0. In Fig 4VI at moderate rates of dispersal the mutator density remains at negligible levels throughout.

	DISCUSSION

TOP ABSTRACT MODEL RESULTS DISCUSSION LITERATURE CITED

This study examined the role of spatial environmental variability and dispersal on mutator dynamics. The model demonstrated that the density of mutator clones in a two-patch spatiotemporally variable environment is dependent on three factors: frequency of fluctuation of the environment, rate of dispersal, and the period of time that both patches are in phase (i.e., the phase angle). These results emphasize the importance of considering the spatial population structure of mutators before making judgments as to their likely role in, for example, the evolution of drug resistance.

In a nonspatial model, the highest mutator densities are observed at intermediate frequencies, with lower densities of mutators seen at low rates of fluctuation and near-zero densities of mutator clones present at rapid fluctuation rates (see also TRAVIS and TRAVIS 2002 ). For low frequencies of environmental change, the mutators spread through the population following each environmental fluctuation (hitchhiking on alleles that are well adapted to the current environment), allowing the population to adapt more rapidly to the new environment. Once the population is well adapted, the mutator clones are selected against, due to the higher rate at which they suffer from deleterious mutations. With intermediate rates of fluctuation, the environment changes before the mutator density returns to a mutation: selection equilibrium, resulting in the mutator density remaining high throughout time. Interesting mutator dynamics occur, with stable limit cycles and chaos observed at intermediate rates of fluctuation. With even more rapid environmental oscillation, mutator genotypes are unable to provide enough genetic variability to keep track of the changing selective pressures; consequently the mutators are present only at low density within the population. A nonmutator genotype that is reasonably adapted to both environments, an environmental generalist, prevails at high rates of fluctuation.

How does dispersal between two patches alter the density of mutators? If the patches are always in the same environmental state as one another then unsurprisingly the results are identical to the nonspatial model regardless of the degree of dispersal between them. When the patches are not always in the same environmental state, one might intuitively expect dispersal between them to lessen the role of mutators. When one patch changes state, it may receive immigrants from the other patch that are already well adapted to the new conditions, and hence mutators do not have a chance to establish. This is exactly what we find when the environment fluctuates slowly between the two states. Dispersal always reduces mutator density in a two-patch system if the patches fluctuate slowly between two states and are out of phase with one another. As the rate of dispersal increases then in general we find a reduction in mutator density.

However, quite contrary to our initial expectations, across a wide range of parameter space some dispersal actually increases mutator density. This occurs when the environment fluctuates rapidly between the two states (i.e., more than once every 600 generations) and when the patches are partially out of phase. Mutators occur only at very low density in a model without spatial structure at a high frequency of environmental variability. Under these conditions mutators do not have long enough to provide well-adapted alleles before the environment switches state again. However, when space is incorporated then dispersal between the patches may have the effect of extending the period of time that mutators are able to hitchhike on the backs of alleles that are well adapted to one of the two states. Mutators start to increase in one of the patches hitchhiking on alleles that are adapted to the prevailing state. Every generation some of these mutators are dispersing to the other patch—let us assume it is in the other environmental state. When the other patch switches states, then the dispersing mutators are able to continue increasing in density there, even when conditions change in their original patch. Of course mutators originating in the second patch can benefit in the same way through dispersal to the first patch. In general mutators are found at lower densities when the two patches fluctuate well out of phase with one another (Fig 3). When the patches are exactly in phase the mutator dynamics are identical to those observed in a single isolated patch. When one patch lags just a little behind the other mutator density remains relatively high, but a critical phase shift exists at which mutator density declines. The position of this critical phase shift depends upon the degree of dispersal between the two patches. Mutator density declines when the patches are further out of phase, as when a patch changes state the immigrants arriving from the other patch are already better adapted to the new environmental conditions, and there is less opportunity for mutators to hitchhike to high densities.

The complexity of the results, in particular at the high and intermediate frequencies of fluctuation where chaos and limit-cycles exist, justifies the choice of the two-patch system to examine the spatiotemporal dynamics. In a multipatch model the dynamics are liable to be even more complex and the interpretation of the results more challenging. Now that the two-patch model has provided an insight into the dynamics of mutator clones in a spatiotemporal variable environment, this could be expanded to investigate a multipatch environment. The temporal variance of the patches could also be altered so that different patches are varying at different rates.

Our model is deterministic, and it deals in rates of mutation and dispersal. While this formulation can provide some useful predictions, a note of caution should be added. For many microbes the number of individuals in a population is often very large, and a deterministic population can be expected to produce reasonable predictions of a population's trajectory: demographic stochasticity is unlikely to be significant. However, although the population size may be very large, specific mutations can still be rare events, and this implies that genetic stochasticity might have an important role in determining the dynamics of mutators within a population. Recasting the model as an individual-based formulation dealing in probabilities of dispersal and mutation would be the preferred alternative. However, a fully individual-based model of several million individuals living in different patches would be stretching computer power to the limit, although an approach that is a hybrid of individual-based and frequency-based methods could present a useful way forward (TENAILLON et al. 1999 ). Here, we have presented the results of a simple strategic model. This type of exercise is useful for providing some general insights into a biological problem. There is a very real need for this approach to be taken forward in a more applied way through the development of tactical models designed to investigate the dynamics of hypermutable strains in spatiotemporally complex environments. Mutator clones have been implicated in the pathogenicity of bacteria and also in the development of cancer. They may also accelerate the adaptation of bacteria to novel drug regimes, resulting in resistance. Extending the spatial model of mutator dynamics presented in this article has considerable potential. It may be possible to coordinate drug delivery to infected individuals in a manner that reduces the risk of increased pathogenesis or drug resistance. Similarly, the spatiotemporal pattern of pesticide regimes may be managed in such a way as to minimize the risk of mutators accelerating the evolution of pesticide resistance.

The work presented here has clearly demonstrated the need to consider the role played by spatial (as well as temporal) environmental variability in the establishment and persistence of strains with high mutator density. This work has considered only regularly fluctuating environments and has been limited to a simple two-patch model. There remains much to learn about mutators, and we hope that this article will encourage future work—both empirical and theoretical—examining the role of spatial environmental variability as a potentially important determinant of both the prevalence and dynamics of high-mutability clones.

	FOOTNOTES

² Present address: Centre for Ecology & Hydrology Banchory, Hill of Brathens, Banchory, Kincardineshire AB31 4BW, Scotland.

	ACKNOWLEDGMENTS

The authors thank the reviewers for their supportive and constructive comments.

Manuscript received July 11, 2003; Accepted for publication January 16, 2004.

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