Controlling Recombination to Evolve Bacteriophages

Recombination among different phages sometimes facilitates their ability to grow on new hosts. Protocols to direct the evolution of phage host range, as might be used in the application of phage therapy, would then benefit from including steps to enable recombination. Applying mathematical and computational models, in addition to experiments using phages T3 and T7, we consider ways that a protocol may influence recombination levels. We first address coinfection, which is the first step to enabling recombination. The multiplicity of infection (MOI, the ratio of phage to cell concentration) is insufficient for predicting (co)infection levels. The force of infection (the rate at which cells are infected) is also critical but is more challenging to measure. Using both a high force of infection and high MOI (>1) for the different phages ensures high levels of coinfection. We also apply a four-genetic-locus model to study protocol effects on recombinant levels. Recombinants accumulate over multiple generations of phage growth, less so if one phage outgrows the other. Supplementing the phage pool with the low-fitness phage recovers some of this 'lost' recombination. Overall, fine tuning of phage recombination rates will not be practical with wild phages, but qualitative enhancement can be attained with some basic procedures.

Modeling the Therapeutic Potential of Defective Interfering Particles in the Presence of Immunity

Defective interfering particles (DIPs) are naturally occurring viruses that have evolved to parasitize other viruses. They suppress wild-type (WT) virus infections through their role as intracellular parasites. Because most encode few or no viral proteins, they have been entertained as possible safe antiviral therapies—something that might be given to patients infected with the WT virus. Adding to their safety, they cannot reproduce except when co-infecting the same cells as the WT, so they pose no danger of evolving into independent disease agents. But this dependence on the WT also limits their therapeutic utility by restricting the timing at which their administration can be effective. To develop a qualitative sense of these constraints for acute viral infections, we use ordinary differential equation models to study the mass-action dynamics of DIPs and WT virus in the presence of adaptive and innate immunity that will otherwise clear the infection. Our goal is to understand whether the therapeutic administration of DIPs will augment or interfere with the immune response and, in the former case, we seek to provide guidance on how virus suppression is affected by infection and clearance parameters, as well as by the timing of DIP introduction. Consistent with previous theoretical work, we find that DIPs can significantly suppress viral load. When immunity is present, the timing of DIP administration matters, with an intermediate optimum. When successful at viral suppression, DIPs even slow the immune response, but the combined effect of DIPs and immunity is still beneficial. Outcomes depend somewhat on whether immunity is elicited by and clears DIPs, but timing appears to have the greater effect.

The coalescent with replication-independent mutations

We develop the mathematical structure of the neutral coalescent with both replication-dependent and replication-independent mutations. This allows us to explain and quantify empirical results that explore differences in genetic diversity in bacterial cultures with different growth rates. We also derive an unbiased and consistent estimator for the replication-independent mutation rate that is based on a comparison of total single nucleotide polymorphism counts for two independent well-mixed cultures with different growth rates. In addition to explaining differences in genetic diversity between well-mixed cultures with different (but constant) growth rates, our coalescent also quantifies the effects of fluctuating growth rates-a situation that can be common in natural populations.

A white noise approach to evolutionary ecology

Although the evolutionary response to random genetic drift is classically modelled as a sampling process for populations with fixed abundance, the abundances of populations in the wild fluctuate over time. Furthermore, since wild populations exhibit demographic stochasticity and since random genetic drift is in part due to demographic stochasticity, theoretical approaches are needed to understand the role of demographic stochasticity in eco-evolutionary dynamics. Here we close this gap for quantitative characters evolving in continuously reproducing populations by providing a framework to track the stochastic dynamics of abundance density across phenotypic space using stochastic partial differential equations. In the process we develop a set of heuristics to operationalize the powerful, but abstract theory of white noise and diffusion-limits of individual-based models. Applying these heuristics, we obtain stochastic ordinary differential equations that generalize classical expressions of ecological quantitative genetics. In particular, by supplying growth rate and reproductive variance as functions of abundance densities and trait values, these equations track population size, mean trait and additive genetic variance responding to mutation, demographic stochasticity, random genetic drift, deterministic selection and noise-induced selection. We demonstrate the utility of our approach by formulating a model of diffuse coevolution mediated by exploitative competition for a continuum of resources. In addition to trait and abundance distributions, this model predicts interaction networks defined by niche-overlap, competition coefficients, or selection gradients. Using a high-richness approximation, we find linear selection gradients and competition coefficients are uncorrelated, but magnitudes of linear selection gradients and quadratic selection gradients are both positively correlated with competition coefficients. Hence, competing species that strongly affect each other's abundance tend to also impose selection on one another, but the directionality is not predicted. This approach contributes to the development of a synthetic theory of evolutionary ecology by formalizing first principle derivations of stochastic models tracking feedbacks of biological processes and the patterns of diversity they produce.

Vector dynamics influence spatially imperfect genetic interventions against disease

Background and objectives

Genetic engineering and similar technologies offer promising new approaches to controlling human diseases by blocking transmission from vectors. However, in spatially structured populations, imperfect coverage of the vector will leave pockets in which the parasite may persist. Movement by humans may disrupt this local persistence and facilitate eradication when these pockets are small, spreading parasite reproduction outside unprotected areas and into areas that block its reproduction. Here, we consider the sensitivity of this process to biological details: do simple generalities emerge that may facilitate interventions?

Methodology

We develop formal mathematical models of this process similar to standard Ross–Macdonald models, but (i) specifying spatial structure of two patches, with vector transmission blocked in one patch but not in the other, (ii) allowing temporary human movement (travel instead of migration) and (iii) considering two different modes of mosquito biting.

Results

We find that there is no invariant effect of disrupting spatial structure with travel. For both biting models, travel out of the unprotected patch has different consequences than travel by visitors into the patch, but the effects are reversed between the two biting models.

Conclusions and implications

Overall, the effect of human travel on the maintenance of vector-borne diseases in structured habitats must be considered in light of the actual biology of mosquito abundances, biting dynamics and human movement patterns.

Lay summary: Genetic interventions against pathogens transmitted by insect vectors are promising methods of controlling infectious diseases. These interventions may be imperfect, leaving pockets where the parasite persists. How will human movement between protected and unprotected areas affect persistence? Mathematical models developed here show that the answer is ecology-dependent, depending on vector biting behavior.

The Timing and Nature of Behavioural Responses Affect the Course of an Epidemic

During an epidemic, the interplay of disease and opinion dynamics can lead to outcomes that are different from those predicted based on disease dynamics alone. Opinions and the behaviours they elicit are complex, so modelling them requires a measure of abstraction and simplification. Here, we develop a differential equation model that couples SIR-type disease dynamics with opinion dynamics. We assume a spectrum of opinions that change based on current levels of infection as well as interactions that to some extent amplify the opinions of like-minded individuals. Susceptibility to infection is based on the level of prophylaxis (disease avoidance) that an opinion engenders. In this setting, we observe how the severity of an epidemic is influenced by the distribution of opinions at disease introduction, the relative rates of opinion and disease dynamics, and the amount of opinion amplification. Some insight is gained by considering how the effective reproduction number is influenced by the combination of opinion and disease dynamics.

Spatial structure undermines parasite suppression by gene drive cargo

Gene drives may be used in two ways to curtail vectored diseases. Both involve engineering the drive to spread in the vector population. One approach uses the drive to directly depress vector numbers, possibly to extinction. The other approach leaves intact the vector population but suppresses the disease agent during its interaction with the vector. This second application may use a drive engineered to carry a genetic cargo that blocks the disease agent. An advantage of the second application is that it is far less likely to select vector resistance to block the drive, but the disease agent may instead evolve resistance to the inhibitory cargo. However, some gene drives are expected to spread so fast and attain such high coverage in the vector population that, if the disease agent can evolve resistance only gradually, disease eradication may be feasible. Here we use simple models to show that spatial structure in the vector population can greatly facilitate persistence and evolution of resistance by the disease agent. We suggest simple approaches to avoid some types of spatial structure, but others may be intrinsic to the populations being challenged and difficult to overcome.

Gene-drive-mediated extinction is thwarted by population structure and evolution of sib mating

Background and objectives

Genetic engineering combined with CRISPR technology has developed to the point that gene drives can, in theory, be engineered to cause extinction in countless species. Success of extinction programs now rests on the possibility of resistance evolution, which is largely unknown. Depending on the gene-drive technology, resistance may take many forms, from mutations in the nuclease target sequence (e.g. for CRISPR) to specific types of non-random population structures that limit the drive (that may block potentially any gene-drive technology).

Methodology

We develop mathematical models of various deviations from random mating to consider escapes from extinction-causing gene drives. A main emphasis here is sib mating in the face of recessive-lethal and Y-chromosome drives.

Results

Sib mating easily evolves in response to both kinds of gene drives and maintains mean fitness above 0, with equilibrium fitness depending on the level of inbreeding depression. Environmental determination of sib mating (as might stem from population density crashes) can also maintain mean fitness above 0. A version of Maynard Smith’s haystack model shows that pre-existing population structure can enable drive-free subpopulations to be maintained against gene drives.

Conclusions and implications

Translation of mean fitness into population size depends on ecological details, so understanding mean fitness evolution and dynamics is merely the first step in predicting extinction. Nonetheless, these results point to possible escapes from gene-drive-mediated extinctions that lie beyond the control of genome engineering.

Lay summary

Recent gene drive technologies promise to suppress and even eradicate pests and disease vectors. Simple models of gene-drive evolution in structured populations show that extinction-causing gene drives can be thwarted both through the evolution of sib mating as well as from purely demographic processes that cluster drive-free individuals.

Spatial opinion dynamics and the effects of two types of mixing

Spatially situated opinions that can be held with different degrees of conviction lead to spatiotemporal patterns such as clustering (homophily), polarization, and deadlock. Our goal is to understand how sensitive these patterns are to changes in the local nature of interactions. We introduce two different mixing mechanisms, spatial relocation and nonlocal interaction ("telephoning"), to an earlier fully spatial model (no mixing). Interestingly, the mechanisms that create deadlock in the fully spatial model have the opposite effect when there is a sufficient amount of mixing. With telephoning, not only is polarization and deadlock broken up, but consensus is hastened. The effects of mixing by relocation are even more pronounced. Further insight into these dynamics is obtained for selected parameter regimes via comparison to the mean-field differential equations.

Phage-Bacterial Dynamics with Spatial Structure: Self Organization around Phage Sinks Can Promote Increased Cell Densities

Bacteria growing on surfaces appear to be profoundly more resistant to control by lytic bacteriophages than do the same cells grown in liquid. Here, we use simulation models to investigate whether spatial structure per se can account for this increased cell density in the presence of phages. A measure is derived for comparing cell densities between growth in spatially structured environments versus well mixed environments (known as mass action). Maintenance of sensitive cells requires some form of phage death; we invoke death mechanisms that are spatially fixed, as if produced by cells. Spatially structured phage death provides cells with a means of protection that can boost cell densities an order of magnitude above that attained under mass action, although the effect is sometimes in the opposite direction. Phage and bacteria self organize into separate refuges, and spatial structure operates so that the phage progeny from a single burst do not have independent fates (as they do with mass action). Phage incur a high loss when invading protected areas that have high cell densities, resulting in greater protection for the cells. By the same metric, mass action dynamics either show no sustained bacterial elevation or oscillate between states of low and high cell densities and an elevated average. The elevated cell densities observed in models with spatial structure do not approach the empirically observed increased density of cells in structured environments with phages (which can be many orders of magnitude), so the empirical phenomenon likely requires additional mechanisms than those analyzed here.

Evolution and the duration of a doomed population

Many populations are doomed to extinction, but little is known about how evolution contributes to their longevity. We address this by modeling an asexual population consisting of genotypes whose abundances change independently according to a system of continuous branching diffusions. Each genotype is characterized by its initial abundance, growth rate, and reproductive variance. The latter two components determine the genotype's "risk function" which describes its per capita probability of extinction at any time. We derive the probability distribution of extinction times for a polymorphic population, which can be expressed in terms of genotypic risk functions. We use this to explore how spontaneous mutation, abrupt environmental change, or population supplementation and removal affect the time to extinction. Results suggest that evolution based on new mutations does little to alter the time to extinction. Abrupt environmental changes that affect all genotypes can have more substantial impact, but, curiously, a beneficial change does more to extend the lifetime of thriving than threatened populations of the same initial abundance. Our results can be used to design policies that meet specific conservation goals or management strategies that speed the elimination of agricultural pests or human pathogens.

Planning horizon affects prophylactic decision-making and epidemic dynamics

The spread of infectious diseases can be impacted by human behavior, and behavioral decisions often depend implicitly on a planning horizon—the time in the future over which options are weighed. We investigate the effects of planning horizons on epidemic dynamics. We developed an epidemiological agent-based model (along with an ODE analog) to explore the decision-making of self-interested individuals on adopting prophylactic behavior. The decision-making process incorporates prophylaxis efficacy and disease prevalence with the individuals’ payoffs and planning horizon. Our results show that for short and long planning horizons individuals do not consider engaging in prophylactic behavior. In contrast, individuals adopt prophylactic behavior when considering intermediate planning horizons. Such adoption, however, is not always monotonically associated with the prevalence of the disease, depending on the perceived protection efficacy and the disease parameters. Adoption of prophylactic behavior reduces the epidemic peak size while prolonging the epidemic and potentially generates secondary waves of infection. These effects can be made stronger by increasing the behavioral decision frequency or distorting an individual’s perceived risk of infection.

Opinion strength influences the spatial dynamics of opinion formation

Opinions are rarely binary; they can be held with different degrees of conviction, and this expanded attitude spectrum can affect the influence one opinion has on others. Our goal is to understand how different aspects of influence lead to recognizable spatio-temporal patterns of opinions and their strengths. To do this, we introduce a stochastic spatial agent-based model of opinion dynamics that includes a spectrum of opinion strengths and various possible rules for how the opinion strength of one individual affects the influence that this individual has on others. Through simulations, we find that even a small amount of amplification of opinion strength through interaction with like-minded neighbors can tip the scales in favor of polarization and deadlock.

The coalescent process in a population with stochastically varying size

We study the genealogical structure of a population with stochastically fluctuating size. If such fluctuations, after suitable rescaling, can be approximated by a nice continuous-time process, we prove weak convergence in the Skorokhod topology of the scaled ancestral process to a stochastic time change of Kingman's coalescent, the time change being given by an additive functional of the limiting backward size process.

Coalescent theory for seed bank models

We study the genealogical structure of samples from a population for which any given generation is made up of direct descendants from several previous generations. These occur in nature when there are seed banks or egg banks allowing an individual to leave offspring several generations in the future. We show how this temporal structure in the reproduction mechanism causes a decrease in the coalescence rate. We also investigate the effects of age-dependent neutral mutations. Our main result gives weak convergence of the scaled ancestral process, with the usual diffusion scaling, to a coalescent process which is equivalent to a time-changed version of Kingman's coalescent.

A spatial model of range-dependent succession

We consider an interacting particle system in which each site of the d-dimensional integer lattice can be in state 0, 1, or 2. Our aim is to model the spread of disease in plant populations, so think of 0 = vacant, 1 = healthy plant, 2 = infected plant. A vacant site becomes occupied by a plant at a rate which increases linearly with the number of plants within range R, up to some saturation level, F1, above which the rate is constant. Similarly, a plant becomes infected at a rate which increases linearly with the number of infected plants within range M, up to some saturation level, F2. An infected plant dies (and the site becomes vacant) at constant rate δ. We discuss coexistence results in one and two dimensions. These results depend on the relative dispersal ranges for plants and disease.

A Computational Model of the Rainbow Trout Hypothalamus-Pituitary-Ovary-Liver Axis

Reproduction in fishes and other vertebrates represents the timely coordination of many endocrine factors that culminate in the production of mature, viable gametes. In recent years there has been rapid growth in understanding fish reproductive biology, which has been motivated in part by recognition of the potential effects that climate change, habitat destruction and contaminant exposure can have on natural and cultured fish populations. New approaches to understanding the impacts of these stressors are being developed that require a systems biology approach with more biologically accurate and detailed mathematical models. We have developed a multi-scale mathematical model of the female rainbow trout hypothalamus-pituitary-ovary-liver axis to use as a tool to help understand the functioning of the system and for extrapolation of laboratory findings of stressor impacts on specific components of the axis. The model describes the essential endocrine components of the female rainbow trout reproductive axis. The model also describes the stage specific growth of maturing oocytes within the ovary and permits the presence of sub-populations of oocytes at different stages of development. Model formulation and parametrization was largely based on previously published in vivo and in vitro data in rainbow trout and new data on the synthesis of gonadotropins in the pituitary. Model predictions were validated against several previously published data sets for annual changes in gonadotropins and estradiol in rainbow trout. Estimates of select model parameters can be obtained from in vitro assays using either quantitative (direct estimation of rate constants) or qualitative (relative change from control values) approaches. This is an important aspect of mathematical models as in vitro, cell-based assays are expected to provide the bulk of experimental data for future risk assessments and will require quantitative physiological models to extrapolate across biological scales.

Reproduction in fishes and other vertebrates represents the timely coordination of many endocrine factors that culminate in the production of mature, viable gametes. Improving the ability to estimate reproductive performance in fish is important, due to the growth of the aquaculture industry and the need to maintain adequate broodstock and concerns over the effects of anthropogenic stressors on feral fish populations. We present here a quantitative, mathematical model of the female rainbow trout reproductive cycle. We show how the model is able to accurately describe experimentally measured data associated with pituitary, ovarian and liver reproductive performance. We also use the model to describe similar data sets collected in rainbow trout by other researchers. An important value of quantitative biological models is the ability to simulate various physiological conditions, real or hypothetical. We demonstrate this by predicting the effects of exposure to an endocrine disruptor on oocyte growth. The need to limit cost and animal usage will encourage future experimental studies to use in vitro methods. The model presented here can assist with the extrapolation of in vitro effects to the whole fish.

Models for the directed evolution of bacterial allelopathy: bacteriophage lysins

Microbes produce a variety of compounds that are used to kill or suppress other species. Traditional antibiotics have their origins in these natural products, as do many types of compounds being pursued today in the quest for new antibacterial drugs. When a potential toxin can be encoded by and exported from a species that is not harmed, the opportunity exists to use directed evolution to improve the toxin's ability to kill other species-allelopathy. In contrast to the typical application of directed evolution, this case requires the co-culture of at least two species or strains, a host that is unharmed by the toxin plus the intended target of the toxin. We develop mathematical and computational models of this directed evolution process. Two contexts are considered, one with the toxin encoded on a plasmid and the other with the toxin encoded in a phage. The plasmid system appears to be more promising than the phage system. Crucial to both designs is the ability to co-culture two species/strains (host and target) such that the host is greatly outgrown by the target species except when the target species is killed. The results suggest that, if these initial conditions can be satisfied, directed evolution is feasible for the plasmid-based system. Screening with a plasmid-based system may also enable rapid improvement of a toxin.

The impact of spatial structure on viral genomic diversity generated during adaptation to thermal stress

BACKGROUND

Most clinical and natural microbial communities live and evolve in spatially structured environments. When changes in environmental conditions trigger evolutionary responses, spatial structure can impact the types of adaptive response and the extent to which they spread. In particular, localized competition in a spatial landscape can lead to the emergence of a larger number of different adaptive trajectories than would be found in well-mixed populations. Our goal was to determine how two levels of spatial structure affect genomic diversity in a population and how this diversity is manifested spatially.

METHODOLOGY/PRINCIPAL FINDINGS

We serially transferred bacteriophage populations growing at high temperatures (40°C) on agar plates for 550 generations at two levels of spatial structure. The level of spatial structure was determined by whether the physical locations of the phage subsamples were preserved or disrupted at each passage to fresh bacterial host populations. When spatial structure of the phage populations was preserved, there was significantly greater diversity on a global scale with restricted and patchy distribution. When spatial structure was disrupted with passaging to fresh hosts, beneficial mutants were spread across the entire plate. This resulted in reduced diversity, possibly due to clonal interference as the most fit mutants entered into competition on a global scale. Almost all substitutions present at the end of the adaptation in the populations with disrupted spatial structure were also present in the populations with structure preserved.

CONCLUSIONS/SIGNIFICANCE

Our results are consistent with the patchy nature of the spread of adaptive mutants in a spatial landscape. Spatial structure enhances diversity and slows fixation of beneficial mutants. This added diversity could be beneficial in fluctuating environments. We also connect observed substitutions and their effects on fitness to aspects of phage biology, and we provide evidence that some substitutions exclude each other.

Influence of humans on evolution and mobilization of environmental antibiotic resistome

The clinical failure of antimicrobial drugs that were previously effective in controlling infectious disease is a tragedy of increasing magnitude that gravely affects human health. This resistance by pathogens is often the endpoint of an evolutionary process that began billions of years ago in non–disease-causing microorganisms. This environmental resistome, its mobilization, and the conditions that facilitate its entry into human pathogens are at the heart of the current public health crisis in antibiotic resistance. Understanding the origins, evolution, and mechanisms of transfer of resistance elements is vital to our ability to adequately address this public health issue.

Fitness benefits of low infectivity in a spatially structured population of bacteriophages

For a parasite evolving in a spatially structured environment, an evolutionarily advantageous strategy may be to reduce its transmission rate or infectivity. We demonstrate this empirically using bacteriophage (phage) from an evolution experiment where spatial structure was maintained over 550 phage generations on agar plates. We found that a single substitution in the major capsid protein led to slower adsorption of phage to host cells with no change in lysis time or burst size. Plaques formed by phage isolates containing this mutation were not only larger but also contained more phage per unit area. Using a spatially explicit, individual-based model, we showed that when there is a trade-off between adsorption and diffusion (i.e. less 'sticky' phage diffuse further), slow adsorption can maximize plaque size, plaque density and overall productivity. These findings suggest that less infective pathogens may have an advantage in spatially structured populations, even when well-mixed models predict that they will not.

Invasion of E. coli biofilms by antibiotic resistance plasmids

In spite of the contribution of plasmids to the spread of antibiotic resistance in human pathogens, little is known about the transferability of various drug resistance plasmids in bacterial biofilms. The goal of this study was to compare the efficiency of transfer of 19 multidrug resistance plasmids into Escherichia coli recipient biofilms and determine the effects of biofilm age, biofilm-donor exposure time, and donor-to-biofilm attachment on this process. An E. coli recipient biofilm was exposed separately to 19 E. coli donors, each with a different plasmid, and transconjugants were determined by plate counting. With few exceptions, plasmids that transferred well in a liquid environment also showed the highest transferability in biofilms. The difference in transfer frequency between the most and least transferable plasmid was almost a million-fold. The 'invasibility' of the biofilm by plasmids, or the proportion of biofilm cells that acquired plasmids within a few hours, depended not only on the type of plasmid, but also on the time of biofilm exposure to the donor and on the ability of the plasmid donor to attach to the biofilm, yet not on biofilm age. The efficiency of donor strain attachment to the biofilm was not affected by the presence of plasmids. The most invasive plasmid was pHH2-227, which based on genome sequence analysis is a hybrid between IncU-like and IncW plasmids. The wide range in transferability in an E. coli biofilm among plasmids needs to be taken into account in our fight against the spread of drug resistance.

Adaptive regulatory substitutions affect multiple stages in the life cycle of the bacteriophage φX174

BACKGROUND

Previously, we showed that adaptive substitutions in one of the three promoters of the bacteriophage φX174 improved fitness at high-temperature by decreasing transcript levels three- to four-fold. To understand how such an extreme change in gene expression might lead to an almost two-fold increase in fitness at the adaptive temperature, we focused on stages in the life cycle of the phage that occur before and after the initiation of transcription. For both the ancestral strain and two single-substitution strains with down-regulated transcription, we measured seven phenotypic components of fitness (attachment, ejection, eclipse, virion assembly, latent period, lysis rate and burst size) during a single cycle of infection at each of two temperatures. The lower temperature, 37°C, is the optimal temperature at which phages are cultivated in the lab; the higher temperature, 42°C, exerts strong selection and is the condition under which these substitutions arose in evolution experiments. We augmented this study by developing an individual-based stochastic model of this same life cycle to explore potential explanations for our empirical results.

RESULTS

Of the seven fitness parameters, three showed significant differences between strains that carried an adaptive substitution and the ancestor, indicating the presence of pleiotropy in regulatory evolution. 1) Eclipse was longer in the adaptive strains at both the optimal and high-temperature environments. 2) Lysis rate was greater in the adaptive strains at the high temperature. 3) Burst size for the mutants was double that of the ancestor at the high temperature, but half that at the lower temperature. Simulation results suggest that eclipse length and latent period variance can explain differences in burst sizes and fitness between the mutant and ancestral strains.

CONCLUSIONS

Down-regulating transcription affects several steps in the phage life cycle, and all of these occur after the initiation of transcription. We attribute the apparent tradeoff between delayed progeny production and faster progeny release to improved host resource utilization at high temperature.

On the meaning and estimation of plasmid transfer rates for surface-associated and well-mixed bacterial populations

Conjugative plasmid transfer is key to the ability of bacteria to rapidly adapt to new environments, but there is no agreement on a single quantitative measure of the rate of plasmid transfer. Some studies derive estimates of transfer rates from mass-action differential equation models of plasmid population biology. The often-used 'endpoint method' is such an example. Others report measures of plasmid transfer efficiency that simply represent ratios of plasmid-bearing and plasmid-free cell densities and do not correspond to parameters in any mathematical model. Unfortunately, these quantities do not measure the same thing - sometimes differing by orders of magnitude - and their use is often clouded by a lack of specificity. Moreover, they do not distinguish between bulk transfer rates that are only relevant in well-mixed populations and the 'intrinsic' rates between individual cells. This leads to problems for surface-associated populations, which are not well-mixed but spatially structured. We used simulations of a spatially explicit mathematical model to evaluate the effectiveness of these various plasmid transfer efficiency measures when they are applied to surface-associated populations. The simulation results, supported by some experimental findings, showed that these measures can be affected by initial cell densities, donor-to-recipient ratios and initial cell cluster size, and are therefore flawed as universal measures of plasmid transfer efficiency. The simulations also allowed us to formulate some guiding principles on when these estimates are appropriate for spatially structured populations and how to interpret the results. While we focus on plasmid transfer, the general lessons of this study should apply to any measures of horizontal spread (e.g., infection rates in epidemiology) that are based on simple mass-action models (e.g., SIR models in epidemiology) but applied to spatial settings.

Dual reporter system for in situ detection of plasmid transfer under aerobic and anaerobic conditions

We designed a new genetic tool to detect plasmid transfer under anaerobic and aerobic conditions. The system is based on the T7 RNA polymerase gene and a T7 promoter-driven oxygen-independent green fluorescent protein, evoglow, alone or in combination with red fluorescent protein DsRed. Constructs are available as plasmids and mini-mariner transposons.

Space, time, and host evolution facilitate coexistence of competing bacteriophages: theory and experiment

We present a joint experimental/theoretical investigation into the roles of spatial structure and time in the competition between two pathogens for a single host. We suggest a natural mechanism by which competing pathogens can coexist when host evolution and competitive dynamics occur on similar timescales. Our experimental system consisted of a single bacterial host species and two competing bacteriophage strains grown on agar plates, with a serial transfer of samples of the bacteriophage population to fresh host populations after each incubation cycle. The experiments included two incubation times and two transfer protocols that either maintained or disrupted the spatial structure of the viruses at each transfer. The same bacteriophage acted as the dominant competitor under both transfer protocols. A striking difference between the treatments is that the weak competitor was able to persist in the long-incubation experiments but not in the short-incubation experiments. Mathematical and experimental evidence suggest that coexistence is due to the appearance of resistant mutant host cells that provide a transient "spatiotemporal refuge" for the weaker competitor. Our mathematical model is individual based, captures the stochastic spatial dynamics down to the level of individual cells, and helps to explain the differences in behavior under the various experimental conditions.

Spatial structure and nutrients promote invasion of IncP-1 plasmids in bacterial populations

In spite of the importance of plasmids in bacterial adaptation, we have a poor understanding of their dynamics. It is not known if or how plasmids persist in and spread through (invade) a bacterial population when there is no selection for plasmid-encoded traits. Moreover, the differences in dynamics between spatially structured and mixed populations are poorly understood. Through a joint experimental/theoretical approach, we tested the hypothesis that self-transmissible IncP-1 plasmids can invade a bacterial population in the absence of selection when initially very rare, but only in spatially structured habitats and when nutrients are regularly replenished. Using protocols that differed in the degree of spatial structure and nutrient levels, the invasiveness of plasmid pB10 in Escherichia coli was monitored during at least 15 days, with an initial fraction of plasmid-bearing (p(+)) cells as low as 10(-7). To further explore the mechanisms underlying plasmid dynamics, we developed a spatially explicit mathematical model. When cells were grown on filters and transferred to fresh medium daily, the p(+) fraction increased to 13%, whereas almost complete invasion occurred when the population structure was disturbed daily. The plasmid was unable to invade in liquid. When carbon source levels were lower or not replenished, plasmid invasion was hampered. Simulations of the mathematical model closely matched the experimental results and produced estimates of the effects of alternative experimental parameters. This allowed us to isolate the likely mechanisms most responsible for the observations. In conclusion, spatial structure and nutrient availability can be key determinants in the invasiveness of plasmids.

Modelling the spatial dynamics of plasmid transfer and persistence

Bacterial plasmids are extra-chromosomal genetic elements that code for a wide variety of phenotypes in their bacterial hosts and are maintained in bacterial communities through both vertical and horizontal transfer. Current mathematical models of plasmid-bacteria dynamics, based almost exclusively on mass-action differential equations that describe these interactions in completely mixed environments, fail to adequately explain phenomena such as the long-term persistence of plasmids in natural and clinical bacterial communities. This failure is, at least in part, due to the absence of any spatial structure in these models, whereas most bacterial populations are spatially structured in microcolonies and biofilms. To help bridge the gap between theoretical predictions and observed patterns of plasmid spread and persistence, an individual-based lattice model (interacting particle system) that provides a predictive framework for understanding the dynamics of plasmid-bacteria interactions in spatially structured populations is presented here. To assess the accuracy and flexibility of the model, a series of experiments that monitored plasmid loss and horizontal transfer of the IncP-1beta plasmid pB10 : : rfp in Escherichia coli K12 and other bacterial populations grown on agar surfaces were performed. The model-based visual patterns of plasmid loss and spread, as well as quantitative predictions of the effects of different initial parental strain densities and incubation time on densities of transconjugants formed on a 2D grid, were in agreement with this and previously published empirical data. These results include features of spatially structured populations that are not predicted by mass-action differential equation models.

Analyzing animal movements using Brownian bridges

By studying animal movements, researchers can gain insight into many of the ecological characteristics and processes important for understanding population-level dynamics. We developed a Brownian bridge movement model (BBMM) for estimating the expected movement path of an animal, using discrete location data obtained at relatively short time intervals. The BBMM is based on the properties of a conditional random walk between successive pairs of locations, dependent on the time between locations, the distance between locations, and the Brownian motion variance that is related to the animal's mobility. We describe two critical developments that enable widespread use of the BBMM, including a derivation of the model when location data are measured with error and a maximum likelihood approach for estimating the Brownian motion variance. After the BBMM is fitted to location data, an estimate of the animal's probability of occurrence can be generated for an area during the time of observation. To illustrate potential applications, we provide three examples: estimating animal home ranges, estimating animal migration routes, and evaluating the influence of fine-scale resource selection on animal movement patterns.

Spatial self-organization in a cyclic resource–species model

Biological communities are remarkable in their ability to form cooperative ensembles that lead to coexistence through various types of niche partitioning, usually intimately tied to spatial structure. This is especially true in microbial settings where differential expression and regulation of genes allows members of a given species to alter their lifestyle so as to fill a functional role within the community. The resulting species interactions can involve feedback, as in the case of some bacterial consortia that participate in the cooperative degradation of a given resource in a succession of steps and in such a way that certain "later" species provide catalytic support for the primary degrader. We seek to capture the essential features of such spatially extended biological systems by introducing a lattice-based stochastic spatial model (interacting particle system) with cyclic local dynamics. Here, a given site progresses through a sequence of resource and species states in a prescribed order. Furthermore, this succession of states (at a site) is assumed to form a cyclic pattern due to a natural feedback mechanism. We explore conditions under which all the species are able to coexist and consider the extent to which this coexistence requires the development of spatio-temporal patterns, including spiral waves. This self-organization, if it occurs, results when synchronization of the dynamics at the microscopic level leads to macroscopic patterns. These patterns result in consumer-driven resource fluctuations that generate a form of spatio-temporal niche partitioning. As with most models of this complexity, we employ a mixture of mathematical analysis and simulations to develop an understanding of the resulting dynamics.

Spatial invasion by a mutant pathogen

Imagine a pathogen that is spreading radially as a circular wave through a population of susceptible hosts. In the interior of this circular region, the infection dies out due to a subcritical density of susceptibles. If a mutant pathogen, having some advantage over wild-type pathogens, arises in this region it is likely to die out without leaving a noticeable trace. Mutants that arise closer to the infection wavefront have access to more susceptible hosts and thus are more likely to become established and perhaps (locally) out-compete the original pathogen. Among the factors (position, transmission rate, pathogen-induced death rate) that influence the fate of a mutant, which are most important? What does this tell us about the types of mutants that are likely to invade and become established? How do such tendencies serve to steer the evolution of pathogens in a spatial setting? Do different types of models of the same phenomena lead to similar conclusions? We address these issues from the point of view of an individual-based stochastic spatial model of host-pathogen interactions. We consider the probability of a successful invasion by a single mutant as a function of the transmissibility and virulence strengths and the mutant position in the wavefront. Next, for a version of the model in which mutations arise spontaneously, we obtain analytical and simulation results on the mean time to a successful invasion. We also use our model predictions to gain insight into experimental data on bacteriophage plaques. Finally, we compare our results to those based on ordinary and partial differential equations to better understand how different models might influence our predictions on the fate of a mutant pathogen.

Ancestral Processes with Selection

In this paper, we show how to construct the genealogy of a sample of genes for a large class of models with selection and mutation. Each gene corresponds to a single locus at which there is no recombination. The genealogy of the sample is embedded in a graph which we call the ancestral selection graph. This graph contains all the information about the ancestry; it is the analogue of Kingman's coalescent process which arises in the case with no selection. The ancestral selection graph can be easily simulated and we outline an algorithm for simulating samples. The main goal is to analyze the ancestral selection graph and to compare it to Kingman's coalescent process. In the case of no mutation, we find that the distribution of the time to the most recent common ancestor does not depend on the selection coefficient and hence is the same as in the neutral case. When the mutation rate is positive, we give a procedure for computing the probability that two individuals in a sample are identical by descent and the Laplace transform of the time to the most recent common ancestor of a sample of two individuals; we evaluate the first two terms of their respective power series in terms of the selection coefficient. The probability of identity by descent depends on both the selection coefficient and the mutation rate and is different from the analogous expression in the neutral case. The Laplace transform does not have a linear correction term in the selection coefficient. We also provide a recursion formula that can be used to approximate the probability of a given sample by simulating backwards along the sample paths of the ancestral selection graph, a technique developed by Griffiths and Tavare (1994).

The genealogy of samples in models with selection

We introduce the genealogy of a random sample of genes taken from a large haploid population that evolves according to random reproduction with selection and mutation. Without selection, the genealogy is described by Kingman's well-known coalescent process. In the selective case, the genealogy of the sample is embedded in a graph with a coalescing and branching structure. We describe this graph, called the ancestral selection graph, and point out differences and similarities with Kingman's coalescent. We present simulations for a two-allele model with symmetric mutation in which one of the alleles has a selective advantage over the other. We find that when the allele frequencies in the population are already in equilibrium, then the genealogy does not differ much from the neutral case. This is supported by rigorous results. Furthermore, we describe the ancestral selection graph for other selective models with finitely many selection classes, such as the K-allele models, infinitely-many-alleles models. DNA sequence models, and infinitely-many-sites models, and briefly discuss the diploid case.