Modelling and parameter inference of predator-prey dynamics in heterogeneous environments using the direct integral approach

The environment to the rescue: can physics help predict predator-prey interactions?

Understanding the factors that determine the occurrence and strength of ecological interactions under specific abiotic and biotic conditions is fundamental since many aspects of ecological community stability and ecosystem functioning depend on patterns of interactions among species. Current approaches to mapping food webs are mostly based on traits, expert knowledge, experiments, and/or statistical inference. However, they do not offer clear mechanisms explaining how trophic interactions are affected by the interplay between organism characteristics and aspects of the physical environment, such as temperature, light intensity or viscosity. Hence, they cannot yet predict accurately how local food webs will respond to anthropogenic pressures, notably to climate change and species invasions. Herein, we propose a framework that synthesises recent developments in food-web theory, integrating body size and metabolism with the physical properties of ecosystems. We advocate for combination of the movement paradigm with a modular definition of the predation sequence, because movement is central to predator-prey interactions, and a generic, modular model is needed to describe all the possible variation in predator-prey interactions. Pending sufficient empirical and theoretical knowledge, our framework will help predict the food-web impacts of well-studied physical factors, such as temperature and oxygen availability, as well as less commonly considered variables such as wind, turbidity or electrical conductivity. An improved predictive capability will facilitate a better understanding of ecosystem responses to a changing world.

Exploiting predatory bacteria as biocontrol agents across ecosystems

Predatory bacteria have been increasingly known for their ubiquity in environments and great functional potentials in controlling unwanted microorganisms. Fundamental understanding of the predation mechanisms, population dynamics, and interaction patterns underlying bacterial predation is required for wise exploitation of predatory bacteria for enhancing ecoenvironmental, animal, and human health. Here, we review the recent achievements on applying predatory bacteria in different systems as biocontrol agents and living antibiotics as well as new findings in their phylogenetic diversity and predation mechanisms. We finally propose critical issues that deserve priority research and highlight the necessity to combine classic culture-based and advanced culture-independent approaches to push research frontiers of bacterial predation across ecosystems for promising biocontrol and therapy strategies towards a sustainable ecoenvironment and health.

Direct Estimation of Parameters in ODE Models Using WENDy: Weak-Form Estimation of Nonlinear Dynamics

We introduce the Weak-form Estimation of Nonlinear Dynamics (WENDy) method for estimating model parameters for non-linear systems of ODEs. Without relying on any numerical differential equation solvers, WENDy computes accurate estimates and is robust to large (biologically relevant) levels of measurement noise. For low dimensional systems with modest amounts of data, WENDy is competitive with conventional forward solver-based nonlinear least squares methods in terms of speed and accuracy. For both higher dimensional systems and stiff systems, WENDy is typically both faster (often by orders of magnitude) and more accurate than forward solver-based approaches. The core mathematical idea involves an efficient conversion of the strong form representation of a model to its weak form, and then solving a regression problem to perform parameter inference. The core statistical idea rests on the Errors-In-Variables framework, which necessitates the use of the iteratively reweighted least squares algorithm. Further improvements are obtained by using orthonormal test functions, created from a set of [Formula: see text] bump functions of varying support sizes.We demonstrate the high robustness and computational efficiency by applying WENDy to estimate parameters in some common models from population biology, neuroscience, and biochemistry, including logistic growth, Lotka-Volterra, FitzHugh-Nagumo, Hindmarsh-Rose, and a Protein Transduction Benchmark model. Software and code for reproducing the examples is available at https://github.com/MathBioCU/WENDy .

From the distributions of times of interactions to preys and predators dynamical systems

We consider a stochastic individual based model where each predator searches and then manipulates its prey or rests during random times. The time distributions may be non-exponential and density dependent. An age structure allows to describe these interactions and get a Markovian setting. The process is characterized by a measure-valued stochastic differential equation. We prove averaging results in this infinite dimensional setting and get the convergence of the slow-fast macroscopic prey predator process to a two dimensional dynamical system. We recover classical functional responses. We also get new forms arising in particular when births and deaths of predators are affected by the lack of food.

Direct Estimation of Parameters in ODE Models Using WENDy: Weak-form Estimation of Nonlinear Dynamics

We introduce the Weak-form Estimation of Nonlinear Dynamics (WENDy) method for estimating model parameters for non-linear systems of ODEs. Without relying on any numerical differential equation solvers, WENDy computes accurate estimates and is robust to large (biologically relevant) levels of measurement noise. For low dimensional systems with modest amounts of data, WENDy is competitive with conventional forward solver-based nonlinear least squares methods in terms of speed and accuracy. For both higher dimensional systems and stiff systems, WENDy is typically both faster (often by orders of magnitude) and more accurate than forward solver-based approaches. The core mathematical idea involves an efficient conversion of the strong form representation of a model to its weak form, and then solving a regression problem to perform parameter inference. The core statistical idea rests on the Errors-In-Variables framework, which necessitates the use of the iteratively reweighted least squares algorithm. Further improvements are obtained by using orthonormal test functions, created from a set of C-infinity bump functions of varying support sizes. We demonstrate the high robustness and computational efficiency by applying WENDy to estimate parameters in some common models from population biology, neuroscience, and biochemistry, including logistic growth, Lotka-Volterra, FitzHugh-Nagumo, Hindmarsh-Rose, and a Protein Transduction Benchmark model. Software and code for reproducing the examples is available at (https://github.com/MathBioCU/WENDy).

Digital twins and the future of precision mental health

Science faces challenges in developing much-needed precision mental health treatments to accurately identify and diagnose mental health problems and the optimal treatment for each individual. Digital twins (DTs) promise to revolutionize the field of mental health, as they are doing in other fields of science, including oncology and cardiology, where they have been successfully deployed. The use of DTs in mental health is yet to be explored. In this Perspective, we lay the conceptual foundations for mental health DTs (MHDT). An MHDT is a virtual representation of an individual’s mental states and processes. It is continually updated from data collected over the lifespan of the individual, and guides mental health professionals in diagnosing and treating patients based on mechanistic models and statistical and machine learning tools. The merits of MHDT are demonstrated through the example of the working alliance between the therapist and the patient, which is one of the most consistent mechanisms predicting treatment outcome.

The role of mathematical modelling in understanding prokaryotic predation

With increasing levels of antimicrobial resistance impacting both human and animal health, novel means of treating resistant infections are urgently needed. Bacteriophages and predatory bacteria such as Bdellovibrio bacteriovorus have been proposed as suitable candidates for this role. Microbes also play a key environmental role as producers or recyclers of nutrients such as carbon and nitrogen, and predators have the capacity to be keystone species within microbial communities. To date, many studies have looked at the mechanisms of action of prokaryotic predators, their safety in in vivo models and their role and effectiveness under specific conditions. Mathematical models however allow researchers to investigate a wider range of scenarios, including aspects of predation that would be difficult, expensive, or time-consuming to investigate experimentally. We review here a history of modelling in prokaryote predation, from simple Lotka-Volterra models, through increasing levels of complexity, including multiple prey and predator species, and environmental and spatial factors. We consider how models have helped address questions around the mechanisms of action of predators and have allowed researchers to make predictions of the dynamics of predator-prey systems. We examine what models can tell us about qualitative and quantitative commonalities or differences between bacterial predators and bacteriophage or protists. We also highlight how models can address real-world situations such as the likely effectiveness of predators in removing prey species and their potential effects in shaping ecosystems. Finally, we look at research questions that are still to be addressed where models could be of benefit.

Predation strategies of the bacterium Bdellovibrio bacteriovorus result in overexploitation and bottlenecks

With increasing antimicrobial resistance, alternatives for treating infections or removing resistant bacteria are urgently needed, such as the bacterial predator Bdellovibrio bacteriovorus or bacteriophage. Therefore, we need to better understand microbial predator-prey dynamics. We developed mass-action mathematical models of predation for chemostats, which capture the low substrate concentration and slow growth typical for intended application areas of the predators such as wastewater treatment, aquaculture or the gut. Our model predicted that predator survival required a minimal prey cell size, explaining why Bdellovibrio is much smaller than its prey. A too good predator (attack rate too high, mortality too low) overexploited its prey leading to extinction (tragedy of the commons). Surprisingly, a predator taking longer to produce more offspring outcompeted a predator producing fewer offspring more rapidly (rate versus yield trade-off). Predation was only efficient in a narrow region around optimal parameters. Moreover, extreme oscillations under a wide range of conditions led to severe bottlenecks. These could be avoided when two prey species became available in alternating seasons. A bacteriophage outcompeted Bdellovibrio due to its higher burst size and faster life cycle. Together, results suggest that Bdellovibrio would struggle to survive on a single prey, explaining why it must be a generalist predator and suggesting it is better suited than phage to environments with multiple prey. Importance The discovery of antibiotics led to a dramatic drop in deaths due to infectious disease. Increasing levels of antimicrobial resistance, however, threaten to reverse this progress. There is thus a need for alternatives, such as therapies based on phage and predatory bacteria that kill bacteria regardless of whether they are pathogens or resistant to antibiotics. To best exploit them, we need to better understand what determines their effectiveness. By using a mathematical model to study bacterial predation in realistic slow growth conditions, we found that the generalist predator Bdellovibrio is most effective within a narrow range of conditions for each prey. For example, a minimum prey cell size is required, and the predator should not be too good as this would result in over-exploitation risking extinction. Together these findings give insights into the ecology of microbial predation and help explain why Bdellovibrio needs to be a generalist predator.

Effects of the prey landscape on the fitness of the bacterial predators Bdellovibrio and like organisms

Bdellovibrio and like organisms (BALOs) are obligate predatory bacteria commonly encountered in the environment. In dual predator-prey cultures, prey accessibility ensures optimal feeding and replication and rapid BALO population growth. However, the environmental prey landscape is complex, as it also incorporates non-prey cells and other particles. These may act as decoys, generating unproductive encounters which in turn may affect both predator and prey population dynamics. In this study, we hypothesized that increasing decoy:prey ratios would bring about increasing costs on the predator's reproductive fitness. We also tested the hypothesis that different BALOs and decoys would have different effects. To this end, we constructed prey landscapes including periplasmic or epibiotic predators including two types of decoy under a large range of initial decoy:prey ratio, and mixed cultures containing multiple predators and prey. We show that as decoy:prey ratios increase, the maximal predator population sizes is reduced and the time to reach it significantly increases. We found that BALOs spent less time handling non-prey (including superinfection-immune invaded prey) than prey cells, and did not differentiate between efficient and less efficient prey. This may explain why in multiple predator and prey cultures, less preferred prey appear to act as decoy.

To hunt or to rest: prey depletion induces a novel starvation survival strategy in bacterial predators

The small size of bacterial cells necessitates rapid adaption to sudden environmental changes. In Bdellovibrio bacteriovorus, an obligate predator of bacteria common in oligotrophic environments, the non-replicative, highly motile attack phase (AP) cell must invade a prey to ensure replication. AP cells swim fast and respire at high rates, rapidly consuming their own contents. How the predator survives in the absence of prey is unknown. We show that starvation for prey significantly alters swimming patterns and causes exponential decay in prey-searching cells over hours, until population-wide swim-arrest. Swim-arrest is accompanied by changes in energy metabolism, enabling rapid swim-reactivation upon introduction of prey or nutrients, and a sweeping change in gene expression and gene regulation that largely differs from those of the paradigmatic stationary phase. Swim-arrest is costly as it imposes a fitness penalty in the form of delayed growth. We track the control of the swim arrest-reactivation process to cyclic-di-GMP (CdG) effectors, including two motility brakes. CRISPRi transcriptional inactivation, and in situ localization of the brakes to the cell pole, demonstrated their essential role for effective survival under prey-induced starvation. Thus, obligate predators evolved a unique CdG-controlled survival strategy, enabling them to sustain their uncommon lifestyle under fluctuating prey supply.

Insight into the Possible Use of the Predator Bdellovibrio bacteriovorus as a Probiotic

The gut microbiota is a complex microbial ecosystem that coexists with the human organism in the intestinal tract. The members of this ecosystem live together in a balance between them and the host, contributing to its healthy state. Stress, aging, and antibiotic therapies are the principal factors affecting the gut microbiota composition, breaking the mutualistic relationship among microbes and resulting in the overgrowth of potential pathogens. This condition, called dysbiosis, has been linked to several chronic pathologies. In this review, we propose the use of the predator Bdellovibrio bacteriovorus as a possible probiotic to prevent or counteract dysbiotic outcomes and look at the findings of previous research.

Separable Nonlinear Least-Squares Parameter Estimation for Complex Dynamic Systems

Nonlinear dynamic models are widely used for characterizing processes that govern complex biological pathway systems. Over the past decade, validation and further development of these models became possible due to data collected via high-throughput experiments using methods from molecular biology. While these data are very beneficial, they are typically incomplete and noisy, which renders the inference of parameter values for complex dynamic models challenging. Fortunately, many biological systems have embedded linear mathematical features, which may be exploited, thereby improving fits and leading to better convergence of optimization algorithms. In this paper, we explore options of inference for dynamic models using a novel method of separable nonlinear least-squares optimization and compare its performance to the traditional nonlinear least-squares method. The numerical results from extensive simulations suggest that the proposed approach is at least as accurate as the traditional nonlinear least-squares, but usually superior, while also enjoying a substantial reduction in computational time.

Dual Predation by Bacteriophage and Bdellovibrio bacteriovorus Can Eradicate Escherichia coli Prey in Situations where Single Predation Cannot

Bacteria are preyed upon by diverse microbial predators, including bacteriophage and predatory bacteria, such as Bdellovibrio bacteriovorus While bacteriophage are used as antimicrobial therapies in Eastern Europe and are being applied for compassionate use in the United States, predatory bacteria are only just beginning to reveal their potential therapeutic uses. However, predation by either predator type can falter due to different adaptations arising in the prey bacteria. When testing poultry farm wastewater for novel Bdellovibrio isolates on Escherichia coli prey lawns, individual composite plaques were isolated containing both an RTP (rosette-tailed-phage)-like-phage and a B. bacteriovorus strain and showing central prey lysis and halos of extra lysis. Combining the purified phage with a lab strain of B. bacteriovorus HD100 recapitulated haloed plaques and increased killing of the E. coli prey in liquid culture, showing an effective side-by-side action of these predators compared to their actions alone. Using approximate Bayesian computation to select the best fitting from a variety of different mathematical models demonstrated that the experimental data could be explained only by assuming the existence of three prey phenotypes: (i) sensitive to both predators, (ii) genetically resistant to phage only, and (iii) plastic resistant to B. bacteriovorus only. Although each predator reduces prey availability for the other, high phage numbers did not abolish B. bacteriovorus predation, so both predators are competent to coexist and are causing different selective pressures on the bacterial surface while, in tandem, controlling prey bacterial numbers efficiently. This suggests that combinatorial predator therapy could overcome problems of phage resistance.IMPORTANCE With increasing levels of antibiotic resistance, the development of alternative antibacterial therapies is urgently needed. Two potential alternatives are bacteriophage and predatory bacteria. Bacteriophage therapy has been used, but prey/host specificity and the rapid acquisition of bacterial resistance to bacteriophage are practical considerations. Predatory bacteria are of interest due to their broad Gram-negative bacterial prey range and the lack of simple resistance mechanisms. Here, a bacteriophage and a strain of Bdellovibrio bacteriovorus, preyed side by side on a population of E. coli, causing a significantly greater decrease in prey numbers than either alone. Such combinatorial predator therapy may have greater potential than individual predators since prey surface changes selected for by each predator do not protect prey against the other predator.

Spatial heterogeneity stabilizes predator-prey interactions at the microscale while patch connectivity controls their outcome

Natural landscapes are both fragmented and heterogeneous, affecting the distribution of organisms, and their interactions. While predation in homogeneous environments increases the probability of population extinction, fragmentation/heterogeneity promotes coexistence and enhances community stability as shown by experimentation with animals and microorganisms, and supported by theory. Patch connectivity can modulate such effects but how microbial predatory interactions are affected by water-driven connectivity is unknown. In soil, patch habitability by microorganisms, and their connectivity depend upon the water saturation degree (SD). Here, using the obligate bacterial predator Bdellovibrio bacteriovorus, and a Burkholderia prey, we show that soil spatial heterogeneity profoundly affects predatory dynamics, enhancing long-term co-existence of predator and prey in a SD-threshold dependent-manner. However, as patches and connectors cannot be distinguished in these soil matrices, metapopulations cannot be invoked to explain the dynamics of increased persistence. Using a set of experiments combined with statistical and physical models we demonstrate and quantify how under full connectivity, predation is independent of water content but depends on soil microstructure characteristics. In contrast, the SD below which predation is largely impaired corresponds to a threshold below which the water network collapses and water connectivity breaks down, preventing the bacteria to move within the soil matrix.

Probabilistic analytical modelling of predator-prey interactions in fishes

Predation is a fundamental interaction between species, yet it is largely unclear what tactics are successful for the survival or capture of prey. One challenge in this area comes with how to test theoretical ideas about strategy with experimental measurements of features such as speed, flush distance and escape angles. Tactics may be articulated with an analytical model that predicts the motion of predator or prey as they interact. However, it may be difficult to recognize how the predictions of such models relate to behavioural measurements that are inherently variable. Here, we present an alternative approach for modelling predator-prey interactions that uses deterministic dynamics, yet incorporates experimental kinematic measurements of natural variation to predict the outcome of biological events. This technique, called probabilistic analytical modelling (PAM), is illustrated by the interactions between predator and prey fish in two case studies that draw on recent experiments. In the first case, we use PAM to model the tactics of predatory bluefish ( Pomatomus saltatrix) as they prey upon smaller fish ( Fundulus heteroclitus). We find that bluefish perform deviated pure pursuit with a variable pursuit angle that is suboptimal for the time to capture. In the second case, we model the escape tactics of zebrafish larvae ( Danio rerio) when approached by adult predators of the same species. Our model successfully predicts the measured patterns of survivorship using measured probability density functions as parameters. As these results demonstrate, PAM is a data-driven modelling approach that can be predictive, offers analytical transparency, and does not require numerical simulations of system dynamics. Though predator-prey interactions demonstrate the use of this technique, PAM is not limited to studying biological systems and has broad utility that may be applied towards understanding a wide variety of natural and engineered dynamical systems where data-driven modelling is beneficial.

Serralysin family metalloproteases protects Serratia marcescens from predation by the predatory bacteria Micavibrio aeruginosavorus

Micavibrio aeruginosavorus is an obligate Gram-negative predatory bacterial species that feeds on other Gram-negative bacteria by attaching to the surface of its prey and feeding on the prey's cellular contents. In this study, Serratia marcescens with defined mutations in genes for extracellular cell structural components and secreted factors were used in predation experiments to identify structures that influence predation. No change was measured in the ability of the predator to prey on S. marcescens flagella, fimbria, surface layer, prodigiosin and phospholipase-A mutants. However, higher predation was measured on S. marcescens metalloprotease mutants. Complementation of the metalloprotease gene, prtS, into the protease mutant, as well as exogenous addition of purified serralysin metalloprotease, restored predation to wild type levels. Addition of purified serralysin also reduced the ability of M. aeruginosavorus to prey on Escherichia coli. Incubating M. aeruginosavorus with purified metalloprotease was found to not impact predator viability; however, pre-incubating prey, but not the predator, with purified metalloprotease was able to block predation. Finally, using flow cytometry and fluorescent microscopy, we were able to confirm that the ability of the predator to bind to the metalloprotease mutant was higher than that of the metalloprotease producing wild-type. The work presented in this study shows that metalloproteases from S. marcescens could offer elevated protection from predation.

Predator Versus Pathogen: How Does Predatory Bdellovibrio bacteriovorus Interface with the Challenges of Killing Gram-Negative Pathogens in a Host Setting?

Bdellovibrio bacteriovorus is a small deltaproteobacterial predator that has evolved to invade, reseal, kill, and digest other gram-negative bacteria in soils and water environments. It has a broad host range and kills many antibiotic-resistant, clinical pathogens in vitro, a potentially useful capability if it could be translated to a clinical setting. We review relevant mechanisms of B. bacteriovorus predation and the physiological properties that would influence its survival in a mammalian host. Bacterial pathogens increasingly display conventional antibiotic resistance by expressing and varying surface and soluble biomolecules. Predators coevolved alongside prey bacteria and so encode diverse predatory enzymes that are hard for pathogens to resist by simple mutation. Predators do not replicate outside pathogens and thus express few transport proteins and thus few surface epitopes for host immune recognition. We explain these features, relating them to the potential of predatory bacteria as cellular medicines.

A two-stage approach for estimating the parameters of an age-group epidemic model from incidence data

Age-dependent dynamics is an important characteristic of many infectious diseases. Age-group epidemic models describe the infection dynamics in different age-groups by allowing to set distinct parameter values for each. However, such models are highly nonlinear and may have a large number of unknown parameters. Thus, parameter estimation of age-group models, while becoming a fundamental issue for both the scientific study and policy making in infectious diseases, is not a trivial task in practice. In this paper, we examine the estimation of the so-called next-generation matrix using incidence data of a single entire outbreak, and extend the approach to deal with recurring outbreaks. Unlike previous studies, we do not assume any constraints regarding the structure of the matrix. A novel two-stage approach is developed, which allows for efficient parameter estimation from both statistical and computational perspectives. Simulation studies corroborate the ability to estimate accurately the parameters of the model for several realistic scenarios. The model and estimation method are applied to real data of influenza-like-illness in Israel. The parameter estimates of the key relevant epidemiological parameters and the recovered structure of the estimated next-generation matrix are in line with results obtained in previous studies.

Measuring and modelling the response of Klebsiella pneumoniae KPC prey to Bdellovibrio bacteriovorus predation, in human serum and defined buffer

In worldwide conditions of increasingly antibiotic-resistant hospital infections, it is important to research alternative therapies. Bdellovibrio bacteriovorus bacteria naturally prey on Gram-negative pathogens, including antibiotic-resistant strains and so B. bacteriovorus have been proposed as "living antibiotics" to combat antimicrobially-resistant pathogens. Predator-prey interactions are complex and can be altered by environmental components. To be effective B. bacteriovorus predation needs to work in human body fluids such as serum where predation dynamics may differ to that studied in laboratory media. Here we combine mathematical modelling and lab experimentation to investigate the predation of an important carbapenem-resistant human pathogen, Klebsiella pneumoniae, by B. bacteriovorus in human serum versus buffer. We show experimentally that B. bacteriovorus is able to reduce prey numbers in each environment, on different timescales. Our mathematical model captures the underlying dynamics of the experimentation, including an initial predation-delay at the predator-prey-serum interface. Our research shows differences between predation in buffer and serum and highlights both the potential and limitations of B. bacteriovorus acting therapeutically against K. pneumoniae in serum, informing future research into the medicinal behaviours and dosing of this living antibacterial.