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.

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.

Paul Joyce and the infinite alleles model.

Paul Joyce's work touched on a variety of topics in population genetics-from mathematical models of idealized systems to working closely with biologists on experimental evolution and landscape genetics. I will focus on his earlier mathematical/statistical work that centered on the infinite alleles model.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Modeling the Directed Evolution of Broad Host Range Phages.

Background: The host ranges of individual phages tend to be narrow, yet many applications of phages would benefit from expanded host ranges. Empirical methods have been developed to direct the evolution of phages to attack new strains, but the methods have not been evaluated or compared for their consequences. In particular, how do different methods favor generalist (broad host range) phages over specialist phages? All methods involve exposing phages to two or more novel bacterial strains, but the methods differ in the order in which those hosts are presented through time: Parallel presentation, Sequential presentation, and Mixed presentation. Methods: We use a combination of simple analytical methods and numerical analyses to study the effect of these different protocols on the selection of generalist versus specialist phages. Results: The three presentation protocols have profoundly different consequences for the evolution of generalist versus specialist phages. Sequential presentation favors generalists almost to the exclusion of specialists, whereas Parallel presentation does the least so. However, other protocol attributes (the nature of dilution between transfers of phages to new cultures) also have effects on selection and phage maintenance. It is also noted that protocols can be designed to enhance recombination to augment evolution and to reduce stochastic loss of newly arisen mutants.

Increased transfer of a multidrug resistance plasmid in Escherichia coli biofilms at the air-liquid interface.

Although biofilms represent a common bacterial lifestyle in clinically and environmentally important habitats, there is scant information on the extent of gene transfer in these spatially structured populations. The objective of this study was to gain insight into factors that affect transfer of the promiscuous multidrug resistance plasmid pB10 in Escherichia coli biofilms. Biofilms were grown in different experimental settings, and plasmid transfer was monitored using laser scanning confocal microscopy and plate counting. In closed flow cells, plasmid transfer in surface-attached submerged biofilms was negligible. In contrast, a high plasmid transfer efficiency was observed in a biofilm floating at the air-liquid interface in an open flow cell with low flow rates. A vertical flow cell and a batch culture biofilm reactor were then used to detect plasmid transfer at different depths away from the air-liquid interface. Extensive plasmid transfer occurred only in a narrow zone near that interface. The much lower transfer frequencies in the lower zones coincided with rapidly decreasing oxygen concentrations. However, when an E. coli csrA mutant was used as the recipient, a thick biofilm was obtained at all depths, and plasmid transfer occurred at similar frequencies throughout. These results and data from separate aerobic and anaerobic matings suggest that oxygen can affect IncP-1 plasmid transfer efficiency, not only directly but also indirectly, through influencing population densities and therefore colocalization of donors and recipients. In conclusion, the air-liquid interface can be a hot spot for plasmid-mediated gene transfer due to high densities of juxtaposed donor and recipient cells.

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.

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.

Accounting for mating pair formation in plasmid population dynamics.

Plasmids are important vehicles for horizontal gene transfer and rapid adaptation in bacteria, including the spread of antibiotic resistance genes. Conjugative transfer of a plasmid from a plasmid-bearing to a plasmid-free bacterial cell requires contact and attachment of the cells followed by plasmid DNA transfer prior to detachment. We introduce a system of differential equations for plasmid transfer in well-mixed populations that accounts for attachment, DNA transfer, and detachment dynamics. These equations offer advantages over classical mass-action models that combine these three processes into a single "bulk" conjugation rate. By decomposing the process of plasmid transfer into its constituent parts, this new model provides a framework that facilitates meaningful comparisons of plasmid transfer rates in surface and liquid environments. The model also allows one to account for experimental and environmental effects such as mixing intensity. To test the adequacy of the model and further explore the effects of mixing on plasmid transfer, we performed batch culture experiments using three different plasmids and a range of different mixing intensities. The results show that plasmid transfer is optimized at low to moderate shaking speeds and that vigorous shaking negatively affects plasmid transfer. Using reasonable assumptions on attachment and detachment rates, the mathematical model predicts the same behavior.

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.

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.