Time-delayed spread of viruses in growing plaques

Pattern formation by bacteria-phage interactions

The interactions between bacteria and phages-viruses that infect bacteria-play critical roles in agriculture, ecology, and medicine; however, how these interactions influence the spatial organization of both bacteria and phages remain largely unexplored. Here, we address this gap in knowledge by developing a theoretical model of motile, proliferating bacteria that aggregate via motility-induced phase separation (MIPS) and encounter phage that infect and lyse the cells. We find that the non-reciprocal predator-prey interactions between phage and bacteria strongly alter spatial organization, in some cases giving rise to a rich array of finite-scale stationary and dynamic patterns in which bacteria and phage coexist. We establish principles describing the onset and characteristics of these diverse behaviors, thereby helping to provide a biophysical basis for understanding pattern formation in bacteria-phage systems, as well as in a broader range of active and living systems with similar predator-prey or other non-reciprocal interactions.

Superinfection exclusion: A viral strategy with short-term benefits and long-term drawbacks

Viral superinfection occurs when multiple viral particles subsequently infect the same host. In nature, several viral species are found to have evolved diverse mechanisms to prevent superinfection (superinfection exclusion) but how this strategic choice impacts the fate of mutations in the viral population remains unclear. Using stochastic simulations, we find that genetic drift is suppressed when superinfection occurs, thus facilitating the fixation of beneficial mutations and the removal of deleterious ones. Interestingly, we also find that the competitive (dis)advantage associated with variations in life history parameters is not necessarily captured by the viral growth rate for either infection strategy. Putting these together, we then show that a mutant with superinfection exclusion will easily overtake a superinfecting population even if the latter has a much higher growth rate. Our findings suggest that while superinfection exclusion can negatively impact the long-term adaptation of a viral population, in the short-term it is ultimately a winning strategy.

Phage–Antibiotic Therapy as a Promising Strategy to Combat Multidrug-Resistant Infections and to Enhance Antimicrobial Efficiency

Infections caused by multidrug-resistant (MDR) bacteria have highlighted the importance of the development of new antimicrobial agents. While bacteriophages (phages) are widely studied as alternative agents to antibiotics, combined treatments using phages and antibiotics have exhibited Phage–Antibiotic Synergy (PAS), in which antibiotics promote phage replication and extraordinary antimicrobial efficacy with reduced development of bacterial resistance. This review paper on the current progress of phage–antibiotic therapy includes aspects of the mechanisms of PAS and the therapeutic performance of PAS in combating multidrug-resistant bacterial infections. The choice of phages and antibiotics, the administration time and sequence, and the concentrations of the two agents impact the bacterial inhibitory effects to different extents.

Diffusive instability in hyperbolic reaction-diffusion equation with different inertia

This work considers a two-dimensional hyperbolic reaction-diffusion system with different inertia and explores criteria for various instabilities, like a wave, Turing, and Hopf, both theoretically and numerically. It is proven that wave instability may occur in a two-species hyperbolic reaction-diffusion system with identical inertia if the diffusion coefficients of the species are nonidentical but cannot occur if diffusion coefficients are identical. Wave instability may also arise in a two-dimensional hyperbolic reaction-diffusion system if the diffusivities of the species are equal, which is never possible in a parabolic reaction-diffusion system, provided the inertias are different. Interestingly, Turing instability is independent of inertia, but the stability of the corresponding local system depends on the inertia. Theoretical results are demonstrated with an example where the local interaction is represented by the Schnakenberg system.

Mathematical Model of SARS-Cov-2 Propagation Versus ACE2 Fits COVID-19 Lethality Across Age and Sex and Predicts That of SARS

The fatality rate of Covid-19 escalates with age and is larger in men than women. I show that these variations correlate strongly with the level of the viral receptor protein ACE2 in rat lungs, which is consistent with the still limited data on human ACE2. Surprisingly, lower receptor levels correlate with higher fatality. I propose two possible explanations of this negative correlation: First, a previous mathematical model predicts that the velocity of viral progression in the organism as a function of the receptor level has a maximum and declines for abundant receptor. Secondly, degradation of ACE2 by the virus may cause the runaway inflammatory response that characterizes severe CoViD-19. I present here a mathematical model that predicts the lethality as a function of ACE2 protein level based on the two above hypothesis. The model fits Covid-19 fatality rate across age and sex in three countries with high accuracy (r 2 > 0.9 ) under the hypothesis that the speed of viral progression in the infected organism is a decreasing function of the ACE2 level. Moreover, rescaling the fitted parameters by the ratio of the binding rates of the spike proteins of SARS-CoV and SARS-CoV-2 allows predicting the fatality rate of SARS-CoV across age and sex, thus linking the molecular and epidemiological levels.

A Two-Dimensional Model of Potassium Signaling and Oscillatory Growth in a Biofilm

Biofilms are complex communities of bacteria that exhibit a variety of collective behaviors. These behaviors improve their ability to survive in many different environments. One of these collective behaviors seen in Bacillus subtilis is the ability for starving cells to stop the growth of other cells using potassium signaling and voltage changes. This signaling produces an oscillatory growth pattern so that during periods of low growth the nutrients diffuse deeper into the biofilm and reach the nutrient-starved, interior regions of the biomass. In this paper, we develop a mathematical model to describe this oscillatory behavior, and we use this model to develop a two-dimensional simulation that reproduces many of the important features seen in the experimental data. This simulation allows us to examine the spatial patterning of the oscillatory behavior to better understand the relationships between the various regions of the biofilm. Studying the spatial components of the metabolic and voltage oscillations could allow for the development of new control techniques for biofilms with complex shapes.

Beyond the pair approximation: Modeling colonization population dynamics

The process of range expansion (colonization) is one of the basic types of biological dynamics, whereby a species grows and spreads outwards, occupying new territories. Spatial modeling of this process is naturally implemented as a stochastic cellular automaton, with individuals occupying nodes on a rectangular grid, births and deaths occurring probabilistically, and individuals only reproducing onto unoccupied neighboring spots. In this paper we derive several approximations that allow prediction of the expected range expansion dynamics, based on the reproduction and death rates. We derive several approximations, where the cellular automaton is described by a system of ordinary differential equations that preserves correlations among neighboring spots (up to a distance). This methodology allows us to develop accurate approximations of the population size and the expected spatial shape, at a fraction of the computational time required to simulate the original stochastic system. In addition, we provide simple formulas for the steady-state population densities for von Neumann and Moore neighborhoods. Finally, we derive concise approximations for the speed of range expansion in terms of the reproduction and death rates, for both types of neighborhoods. The methodology is generalizable to more complex scenarios, such as different interaction ranges and multiple-species systems.

Hitchhiking, collapse, and contingency in phage infections of migrating bacterial populations

Natural bacterial populations are subjected to constant predation pressure by bacteriophages. Bacteria use a variety of molecular mechanisms to defend themselves from phage predation. However, since phages are nonmotile, perhaps the simplest defense against phage is for bacteria to move faster than phages. In particular, chemotaxis, the active migration of bacteria up attractant gradients, may help the bacteria escape slowly diffusing phages. Here we study phage infection dynamics in migrating bacterial populations driven by chemotaxis through low viscosity agar plates. We find that expanding phage-bacteria populations supports two moving fronts, an outermost bacterial front driven by nutrient uptake and chemotaxis and an inner phage front at which the bacterial population collapses due to phage predation. We show that with increasing adsorption rate and initial phage population, the speed of the moving phage front increases, eventually overtaking the bacterial front and driving the system across a transition from a regime where bacterial front speed exceeds that of the phage front to one where bacteria must evolve phage resistance to survive. Our data support the claim that this process requires phage to hitchhike with moving bacteria. A deterministic model recapitulates the transition under the assumption that phage virulence declines with host growth rate which we confirm experimentally. Finally, near the transition between regimes we observe macroscopic fluctuations in bacterial densities at the phage front. Our work opens a new, spatio-temporal, line of investigation into the eco-evolutionary struggle between bacteria and phage.

Chaos in a nonautonomous eco-epidemiological model with delay

In this paper, we propose and analyze a nonautonomous predator-prey model with disease in prey, and a discrete time delay for the incubation period in disease transmission. Employing the theory of differential inequalities, we find sufficient conditions for the permanence of the system. Further, we use Lyapunov's functional method to obtain sufficient conditions for global asymptotic stability of the system. We observe that the permanence of the system is unaffected due to presence of incubation delay. However, incubation delay affects the global stability of the positive periodic solution of the system. To reinforce the analytical results and to get more insight into the system's behavior, we perform some numerical simulations of the autonomous and nonautonomous systems with and without time delay. We observe that for the gradual increase in the magnitude of incubation delay, the autonomous system develops limit cycle oscillation through a Hopf-bifurcation while the corresponding nonautonomous system shows chaotic dynamics through quasi-periodic oscillations. We apply basic tools of non-linear dynamics such as Poincaré section and maximum Lyapunov exponent to confirm the chaotic behavior of the system.

Sustainability of spatially distributed bacteria-phage systems

Virulent phages can expose their bacterial hosts to devastating epidemics, in principle leading to complete elimination of their hosts. Although experiments indeed confirm a large reduction of susceptible bacteria, there are no reports of complete extinctions. We here address this phenomenon from the perspective of spatial organization of bacteria and how this can influence the final survival of them. By modelling the transient dynamics of bacteria and phages when they are introduced into an environment with finite resources, we quantify how time delayed lysis, the spatial separation of initial bacterial positions, and the self-protection of bacteria growing in spherical colonies favour bacterial survival. Our results suggest that spatial structures on the millimetre and submillimetre scale play an important role in maintaining microbial diversity.

Modeling Phage-Bacteria Dynamics

Phage-bacteria interaction is a classic example of competitive coevolution in nature. Mathematical modeling of such interactions furnishes new insight into the dynamics of phage and bacteria. Besides its intrinsic value, a somewhat underutilized aspect of such insight is that it can provide beneficial inputs toward better experimental design. In this chapter, we discuss several modeling techniques that can be used to study the dynamics between phages and their host bacteria. Monte Carlo simulations and differential equations (both ordinary and delay differential equations) can be used to successfully model phage-bacteria dynamics in well-mixed populations. The presence of spatial restrictions in the interaction media significantly affects the dynamics of phage-bacteria interactions. For such cases, techniques like cellular automata and reaction-diffusion equations can be used to capture these effects adequately. We discuss details of the modeling techniques with specific examples.

How pirate phage interferes with helper phage: Comparison of the two distinct strategies

Pirate phages use the structural proteins encoded by unrelated helper phages to propagate. The best-studied example is the pirate P4 and helper P2 of coliphages, and it has been known that the Staphylococcus aureus pathogenicity islands (SaPIs) that can encode virulence factors act as pirate phages, too. When alone in the host, the pirate phages act as a prophage, but when the helper phage gene is also in the same host cell, the pirate phage has ability to exploit the helper phages structural proteins to produce pirate phage particles and spread, interfering with the helper phage production. The known helper phages in these systems are temperate phages. Interestingly, the interference of the pirate phage to the helper phage occurs in a different manner between the SaPI-helper system and the P4-P2 system. SaPIs cannot lyse a helper lysogen upon infection, while when a helper phage lyse a SaPI lysogen, most of the phage particles produced are the SaPI particles. On the contrary, in the P4-P2 system, a pirate phage P4 can lyse a helper P2 lysogen to produce mostly the P4 particles, while when P2 phage lyses a P4 lysogen, most of the produced phages are the P2 particles. Here, the consequences of these different strategies in the pirate and helper phage spreading among uninfected host is analyzed by using mathematical models. It is found that SaPI's strategy interferes with the helper phage spreading significantly more than the P4's strategy, because SaPI interferes with the helper phage's main reproduction step, while P4 interferes only by forcing the helper lysogens to lyse. However, the interference is found to be weaker in the spatially structured environment than in the well-mixed environment. This is because, in the spatial setting, the system tends to self-organize so that the helper phages take over the front of propagation due to the need of helper phage for the pirate phage spreading.

Travelling fronts in time-delayed reaction-diffusion systems

We review a series of key travelling front problems in reaction-diffusion systems with a time-delayed feedback, appearing in ecology, nonlinear optics and neurobiology. For each problem, we determine asymptotic approximations for the wave shape and its speed. Particular attention is devoted to their validity and all analytical solutions are compared to solutions obtained numerically. We also extend the work by Erneux et al. (Erneux et al. 2010 Phil. Trans. R. Soc. A 368, 483-493 (doi:10.1098/rsta.2009.0228)) by considering the case of a slowly propagating front subject to a weak delayed feedback. The delay may either speed up the front in the same direction or reverse its direction. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Reaction fronts in persistent random walks with demographic stochasticity

Standard reaction-diffusion systems are characterized by infinite velocities and no persistence in the movement of individuals, two conditions that are violated when considering living organisms. Here we consider a discrete particle model in which individuals move following a persistent random walk with finite speed and grow with logistic dynamics. We show that, when the number of individuals is very large, the individual-based model is well described by the continuous reactive Cattaneo equation (RCE), but for smaller values of the carrying capacity important finite-population effects arise. The effects of fluctuations on the propagation speed are investigated both considering the RCE with a cutoff in the reaction term and by means of numerical simulations of the individual-based model. Finally, a more general Lévy walk process for the transport of individuals is examined and an expression for the front speed of the resulting traveling wave is proposed.

CRISPR-based herd immunity can limit phage epidemics in bacterial populations

Herd immunity, a process in which resistant individuals limit the spread of a pathogen among susceptible hosts has been extensively studied in eukaryotes. Even though bacteria have evolved multiple immune systems against their phage pathogens, herd immunity in bacteria remains unexplored. Here we experimentally demonstrate that herd immunity arises during phage epidemics in structured and unstructured Escherichia coli populations consisting of differing frequencies of susceptible and resistant cells harboring CRISPR immunity. In addition, we develop a mathematical model that quantifies how herd immunity is affected by spatial population structure, bacterial growth rate, and phage replication rate. Using our model we infer a general epidemiological rule describing the relative speed of an epidemic in partially resistant spatially structured populations. Our experimental and theoretical findings indicate that herd immunity may be important in bacterial communities, allowing for stable coexistence of bacteria and their phages and the maintenance of polymorphism in bacterial immunity.

Incorporating inductances in tissue-scale models of cardiac electrophysiology

In standard models of cardiac electrophysiology, including the bidomain and monodomain models, local perturbations can propagate at infinite speed. We address this unrealistic property by developing a hyperbolic bidomain model that is based on a generalization of Ohm's law with a Cattaneo-type model for the fluxes. Further, we obtain a hyperbolic monodomain model in the case that the intracellular and extracellular conductivity tensors have the same anisotropy ratio. In one spatial dimension, the hyperbolic monodomain model is equivalent to a cable model that includes axial inductances, and the relaxation times of the Cattaneo fluxes are strictly related to these inductances. A purely linear analysis shows that the inductances are negligible, but models of cardiac electrophysiology are highly nonlinear, and linear predictions may not capture the fully nonlinear dynamics. In fact, contrary to the linear analysis, we show that for simple nonlinear ionic models, an increase in conduction velocity is obtained for small and moderate values of the relaxation time. A similar behavior is also demonstrated with biophysically detailed ionic models. Using the Fenton-Karma model along with a low-order finite element spatial discretization, we numerically analyze differences between the standard monodomain model and the hyperbolic monodomain model. In a simple benchmark test, we show that the propagation of the action potential is strongly influenced by the alignment of the fibers with respect to the mesh in both the parabolic and hyperbolic models when using relatively coarse spatial discretizations. Accurate predictions of the conduction velocity require computational mesh spacings on the order of a single cardiac cell. We also compare the two formulations in the case of spiral break up and atrial fibrillation in an anatomically detailed model of the left atrium, and we examine the effect of intracellular and extracellular inductances on the virtual electrode phenomenon.

Population Dynamics of Phage and Bacteria in Spatially Structured Habitats Using Phage λ and Escherichia coli

Bacteria living in physically structured habitats are exposed heterogeneously to both resources and different types of phages. While there have been numerous experimental approaches to examine spatially distributed bacteria exposed to phages, there is little theory to guide the design of these experiments, interpret their results, or expand the inferences drawn to a broader ecological and evolutionary context. Plaque formation provides a window into understanding phage-bacterium interactions in physically structured populations, including surfaces, semisolids, and biofilms. We develop models to address the plaque dynamics for a temperate phage and its virulent mutants. The models are compared with phage λ-Escherichia coli system to quantify their applicability. We found that temperate phages gave an increasing number of gradually smaller colonies as the distance increased from the plaque center. For low-lysogen frequency this resulted in plaques with most of the visible colonies at an intermediate distance between the center and periphery. Using spot inoculation, where phages in excess of bacteria were inoculated in a circular area, we measured the frequency and spatial distribution of lysogens. The spot morphology of cII-negative (cII(-)) and cIII(-) mutants of phage λ displays concentric rings of high-density lysogenic colonies. The simplest of these ring morphologies was reproduced by including multiplicity of infection (MOI) sensitivity in lysis-lysogeny decisions, but its failure to explain the occasional observation of multiple rings in cIII(-) mutant phages highlights unknown features of this phage. Our findings demonstrated advantages of temperate phages over virulent phages in exploiting limited resources in spatially distributed microbial populations.

IMPORTANCE

Phages are the most abundant organisms on earth, and yet little is known about how phages and bacterial hosts are influencing each other in density and evolution. Phages can be either virulent or temperate, a difference that is highlighted when a spatially structured bacterial population is infected. Phage λ is a temperate phage, with a capacity for dormancy that can be modified by single gene knockouts. The stochastic bias in the lysis-lysogeny decision's probability is reflected in plaque morphologies on bacterial lawns. We present a model for plaque morphology of both virulent and temperate phages, taking into account the underlying survival of bacterial microcolonies. It reproduces known plaque morphologies and speaks to advantages of temperate phages in a spatially structured environment.

A mathematical approach to virus therapy of glioblastomas

BACKGROUND

It is widely believed that the treatment of glioblastomas (GBM) could benefit from oncolytic virus therapy. Clinical research has shown that Vesicular Stomatitis Virus (VSV) has strong oncolytic properties. In addition, mathematical models of virus treatment of tumors have been developed in recent years. Some experiments in vitro and in vivo have been done and shown promising results, but have been never compared quantitatively with mathematical models. We use in vitro data of this virus applied to glioblastoma.

RESULTS

We describe three increasingly realistic mathematical models for the VSV-GBM in vitro experiment with progressive incorporation of time-delay effects. For the virus dynamics, we obtain results consistent with the in vitro experimental speed data only when applying the more complex and comprehensive model, with time-delay effects both in the reactive and diffusive terms. The tumor speed is given by the minimum of a very simple function that nonetheless yields results within the experimental measured range.

CONCLUSIONS

We have improved a previous model with new ideas and carefully incorporated concepts from experimental results. We have shown that the delay time τ is the crucial parameter in this kind of models. We have demonstrated that our new model can satisfactorily predict the front speed for the lytic action of oncolytic VSV on glioblastoma observed in vitro. We provide a basis that can be applied in the near future to realistically simulate in vivo virus treatments of several cancers.

Front propagation speeds of T7 virus mutants

We propose a new reaction-diffusion model with an eclipse time to study the spread of viruses on bacterial populations. This new model is both biologically and physically sound, unlike previous ones. We determine important parameter values from experimental data, such as the one-step growth. We verify the proposed model by comparing theoretical and experimental data of the front propagation speed for several T7 virus strains.

Growth and dispersal with inertia: hyperbolic reaction-transport systems

We investigate the behavior of five hyperbolic reaction-diffusion equations most commonly employed to describe systems of interacting organisms or reacting particles where dispersal displays inertia. We first discuss the macroscopic or mesoscopic foundation, or lack thereof, of these reaction-transport equations. This is followed by an analysis of the temporal evolution of spatially uniform states. In particular, we determine the uniform steady states of the reaction-transport systems and their stability properties. We then address the spatiotemporal behavior of pure death processes. We end with a unified treatment of the front speed for hyperbolic reaction-diffusion equations with Kolmogorov-Petrosvskii-Piskunov kinetics. In particular, we obtain an exact expression for the front speed of a general class of reaction correlated random walk systems. Our results establish that three out of the five hyperbolic reaction-transport equations provide physically acceptable models of biological and chemical systems.

Closing the loop: lamellipodia dynamics from the perspective of front propagation

We develop a simple physical model that captures the large-scale lamellipodia dynamics in crawling cells and explains the observed spectrum of fish keratocytes behavior. The main ingredients in this description are the geometrical evolution of the lamellipodium leading edge, the dynamic remodeling of the actin network, and the interconnection between them. We deviate from existing theoretical works and consider the lamellipodium leading edge as a propagating front. The agreement of our model with experimental works suggests that the large-scale morphological and migration features exhibited by keratocyte cells are a direct consequence of the closed feedback loop between the shape of the leading edge and the density of the actin network.

Dynamics of localized structures in reaction-diffusion systems induced by delayed feedback

We are interested in stability properties of a single localized structure in a three-component reaction-diffusion system subjected to the time-delayed feedback. We shall show that variation in the product of the delay time and the feedback strength leads to complex dynamical behavior of the system, including formation of target patterns, spontaneous motion, and spontaneous breathing as well as various complex structures, arising from combination of different oscillatory instabilities. In the case of spontaneous motion, we provide a bifurcation analysis of the delayed system and derive an order parameter equation for the position of the localized structure, explicitly describing its temporal evolution in the vicinity of the bifurcation point. This equation is a subject to a nonlinear delay differential equation, which can be transformed to the normal form of the pitchfork drift bifurcation.

Instabilities of localized structures in dissipative systems with delayed feedback

We report on a novel behavior of solitary localized structures in a real Swift-Hohenberg equation subjected to a delayed feedback. We shall show that variation in the product of the delay time and the feedback strength leads to nontrivial instabilities resulting in the formation of oscillons, soliton rings, labyrinth patterns, or moving structures. We provide a bifurcation analysis of the delayed system and derive a system of order parameter equations explicitly describing the temporal behavior of the localized structure in the vicinity of the bifurcation point. We demonstrate that a normal form of the bifurcation, responsible for the emergence of moving solitary structures, can be obtained and show that spontaneous motion to the lowest order occurs without change of the shape.

Space competition and time delays in human range expansions. Application to the neolithic transition

Space competition effects are well-known in many microbiological and ecological systems. Here we analyze such an effect in human populations. The Neolithic transition (change from foraging to farming) was mainly the outcome of a demographic process that spread gradually throughout Europe from the Near East. In Northern Europe, archaeological data show a slowdown on the Neolithic rate of spread that can be related to a high indigenous (Mesolithic) population density hindering the advance as a result of the space competition between the two populations. We measure this slowdown from a database of 902 Early Neolithic sites and develop a time-delayed reaction-diffusion model with space competition between Neolithic and Mesolithic populations, to predict the observed speeds. The comparison of the predicted speed with the observations and with a previous non-delayed model show that both effects, the time delay effect due to the generation lag and the space competition between populations, are crucial in order to understand the observations.

Early infection and spread of a conditionally replicating adenovirus under conditions of plaque formation

Conditionally-replicating adenoviruses (CRAds) and other oncolytic viruses replicate selectively in tumor cells, presenting a potential cancer treatment approach. To optimize application of these viruses, understanding of early spread of these viruses in target cells is important. Here we used a recombinant adenovirus expressing enhanced jellyfish green fluorescent protein (EGFP) in place of the EIA and EIB genes (AdEGFPuci). Infection of susceptible cells (AD-293) under plaque formation conditions (MOI<<1) on gridded culture dishes and daily monitoring allowed visualization of initially infected cells, as well as spread to neighboring cells. We determined key parameters of early infection, including the rate and efficiency of spread from the initially infected cell to other cells. It was noteworthy that a minority of initially infected cells ultimately resulted in plaques. The approaches elucidated here will be useful for determining early infection parameters for CRAds of therapeutic interest.

Vertical cultural transmission effects on demic front propagation: theory and application to the Neolithic transition in Europe

It is shown that Lotka-Volterra interaction terms are not appropriate to describe vertical cultural transmission. Appropriate interaction terms are derived and used to compute the effect of vertical cultural transmission on demic front propagation. They are also applied to a specific example, the Neolithic transition in Europe. In this example, it is found that the effect of vertical cultural transmission can be important (about 30%). On the other hand, simple models based on differential equations can lead to large errors (above 50%). Further physical, biophysical, and cross-disciplinary applications are outlined.

Assessing the in vitro fitness of an oseltamivir-resistant seasonal A/H1N1 influenza strain using a mathematical model

In 2007, the A/Brisbane/59/2007 (H1N1) seasonal influenza virus strain acquired the oseltamivir-resistance mutation H275Y in its neuraminidase (NA) gene. Although previous studies had demonstrated that this mutation impaired the replication capacity of the influenza virus in vitro and in vivo, the A/Brisbane/59/2007 H275Y oseltamivir-resistant mutant completely out-competed the wild-type (WT) strain and was, in the 2008–2009 influenza season, the primary A/H1N1 circulating strain. Using a combination of plaque and viral yield assays, and a simple mathematical model, approximate values were extracted for two basic viral kinetics parameters of the in vitro infection. In the ST6GalI-MDCK cell line, the latent infection period (i.e., the time for a newly infected cell to start releasing virions) was found to be 1–3 h for the WT strain and more than 7 h for the H275Y mutant. The infecting time (i.e., the time for a single infectious cell to cause the infection of another one) was between 30 and 80 min for the WT, and less than 5 min for the H275Y mutant. Single-cycle viral yield experiments have provided qualitative confirmation of these findings. These results, though preliminary, suggest that the increased fitness success of the A/Brisbane/59/2007 H275Y mutant may be due to increased infectivity compensating for an impaired or delayed viral release, and are consistent with recent evidence for the mechanistic origins of fitness reduction and recovery in NA expression. The method applied here can reconcile seemingly contradictory results from the plaque and yield assays as two complementary views of replication kinetics, with both required to fully capture a strain's fitness.

Virus infection speeds: theory versus experiment

In order to explain the speed of Vesicular Stomatitis Virus (VSV) infections, we develop a simple model that improves previous approaches to the propagation of virus infections. For VSV infections, we find that the delay time elapsed between the adsorption of a viral particle into a cell and the release of its progeny has a very important effect. Moreover, this delay time makes the adsorption rate essentially irrelevant in order to predict VSV infection speeds. Numerical simulations are in agreement with the analytical results. Our model satisfactorily explains the experimentally measured speeds of VSV infections.

Anomalous transport and nonlinear reactions in spiny dendrites

We present a mesoscopic description of the anomalous transport and reactions of particles in spiny dendrites. As a starting point we use two-state Markovian model with the transition probabilities depending on residence time variable. The main assumption is that the longer a particle survives inside spine, the smaller becomes the transition probability from spine to dendrite. We extend a linear model presented in Fedotov [Phys. Rev. Lett. 101, 218102 (2008)] and derive the nonlinear Master equations for the average densities of particles inside spines and parent dendrite by eliminating residence time variable. We show that the flux of particles between spines and parent dendrite is not local in time and space. In particular the average flux of particles from a population of spines through spines necks into parent dendrite depends on chemical reactions in spines. This memory effect means that one cannot separate the exchange flux of particles and the chemical reactions inside spines. This phenomenon does not exist in the Markovian case. The flux of particles from dendrite to spines is found to depend on the transport process inside dendrite. We show that if the particles inside a dendrite have constant velocity, the mean particle's position <x(t)> increases as t(μ) with μ<1 (anomalous advection). We derive a fractional advection-diffusion equation for the total density of particles.

Fixed points and limit cycles in the population dynamics of lysogenic viruses and their hosts

Starting with stochastic rate equations for the fundamental interactions between microbes and their viruses, we derive a mean-field theory for the population dynamics of microbe-virus systems, including the effects of lysogeny. In the absence of lysogeny, our model is a generalization of that proposed phenomenologically by Weitz and Dushoff. In the presence of lysogeny, we analyze the possible states of the system, identifying a limit cycle, which we interpret physically. To test the robustness of our mean-field calculations to demographic fluctuations, we have compared our results with stochastic simulations using the Gillespie algorithm. Finally, we estimate the range of parameters that delineate the various steady states of our model.

Low-temperature T4-like coliphages vB_EcoM-VR5, vB_EcoM-VR7 and vB_EcoM-VR20

Bacteriophages vB_EcoM-VR5, vB_EcoM-VR7 and vB_EcoM-VR20, showing an unusual low-temperature plating profile and producing constantly growing plaques, were isolated from aquatic environments of Lithuania. Although vB_EcoM-VR5, vB_EcoM-VR7 and vB_EcoM-VR20 resembled phage T4 both in their genome size and in their major structural protein (gp23) pattern, physiological properties of all three phages tested differed significantly from those of T4. With an optimum temperature for plating around 24 degrees C and a high efficiency of plating in the range 7-30 degrees C, bacteriophages vB_EcoM-VR7 and vB_EcoM-VR20 failed to plate at 37 degrees C, whereas phage vB_EcoM-VR5 could not be plated at 40 degrees C. Sequence analysis of diagnostic g23 PCR products revealed that g23 of vB_EcoM-VR5, vB_EcoM-VR7 and vB_EcoM-VR20 differed from the corresponding T4 g23 DNA sequence by 21, 21 and 20%, respectively.

Bifurcation to fronts due to delay

The stability of a steady-state front (kink) subject to a time-delayed feedback control (TDFC) is examined in detail. TDFC is based on the use of the difference between system variables at the current moment of time and their values at some time in the past. We first show that there exists a bifurcation to a moving front. We then investigate the limit of large delays but weak feedback and obtain a global bifurcation diagram for the propagation speed. Finally, we examine the case of a two-dimensional front with radial symmetry and determine the critical radius above which propagation is possible.

Non-Markovian random walks and nonlinear reactions: subdiffusion and propagating fronts

The main aim of the paper is to incorporate the nonlinear kinetic term into non-Markovian transport equations described by a continuous time random walk (CTRW) with nonexponential waiting time distributions. We consider three different CTRW models with reactions. We derive nonlinear Master equations for the mesoscopic density of reacting particles corresponding to CTRW with arbitrary jump and waiting time distributions. We apply these equations to the problem of front propagation in the reaction-transport systems with Kolmogorov-Petrovskii-Piskunov kinetics and anomalous diffusion. We have found an explicit expression for the speed of a propagating front in the case of subdiffusive transport.

Fronts from two-dimensional dispersal kernels: Beyond the nonoverlapping-generations model

Most integrodifference models of biological invasions are based on the nonoverlapping-generations approximation. However, the effect of multiple reproduction events (overlapping generations) on the front speed can be very important (especially for species with a long life spam). Only in one-dimensional space has this approximation been relaxed previously, although almost all biological invasions take place in two dimensions. Here we present a model that takes into account the overlapping generations effect (or, more generally, the stage structure of the population), and we analyze the main differences with the corresponding nonoverlapping-generations results.

Time-delayed reaction-diffusion fronts

A time-delayed second-order approximation for the front speed in reaction-dispersion systems was obtained by Fort and Méndez [Phys. Rev. Lett. 82, 867 (1999)]. Here we show that taking proper care of the effect of the time delay on the reactive process yields a different evolution equation and, therefore, an alternate equation for the front speed. We apply the new equation to the Neolithic transition. For this application the new equation yields speeds about 10% slower than the previous one.

Bacteriophage plaques: theory and analysis

Laboratory characterization of bacteriophage growth traditionally is done either in broth cultures or in semisolid agar media. These two environments may be distinguished in terms of their spatial structure, i.e., the degree to which they limit diffusion, motility, and environmental mixing. Well-mixed broth, for example, represents the microbiological ideal of a non-spatially structured environment. Agar, by contrast, imposes significant limitations on phage and bacterial movement and therefore gives rise to spatial structure. The study of phage growth within spatially structured environments, such as that seen during phage plaque formation, is important for three reasons. First, a large fraction of environmental bacteria live within spatially structured environments such as within biofilms, within soil, or when growing in or on the tissues of plants and animals. Second, phage growth as plaques is a central technique to phage studies, yet appears to be under appreciated by phage workers in terms of its complexity. Third, selective pressures acting on phage during plaque growth differ from those seen during broth growth. In this chapter we will discuss just what a plaque is, how one forms, and what can affect plaque size. We will describe methods, both experimental and theoretical, that have been employed to study plaque growth. As caveats we will discuss why plaque formation failure is not necessarily equivalent to virion inviability (Note 1). We also will consider problems with inferring phage broth growth fitness as a function of plaque characteristics (Note 2).

Simple method for plating Escherichia coli bacteriophages forming very small plaques or no plaques under standard conditions

The use of low concentrations (optimally 2.5 to 3.5 microg/ml, depending on top agar thickness) of ampicillin in the bottom agar of the plate allows for formation of highly visible plaques of bacteriophages which otherwise form extremely small plaques or no plaques on Escherichia coli lawns. Using this method, we were able to obtain plaques of newly isolated bacteriophages, propagated after induction of prophages present in six E. coli O157:H(-) strains which did not form plaques when standard plating procedures were employed.

The effects of distributed life cycles on the dynamics of viral infections

We explore the role of cellular life cycles for viruses and host cells in an infection process. For this purpose, we derive a generalized version of the basic model of virus dynamics (Nowak, M.A., Bangham, C.R.M., 1996. Population dynamics of immune responses to persistent viruses. Science 272, 74-79) from a mesoscopic description. In its final form the model can be written as a set of Volterra integrodifferential equations. We consider the role of distributed lifespans and a intracellular (eclipse) phase. These processes are implemented by means of probability distribution functions. The basic reproductive ratio R(0) of the infection is properly defined in terms of such distributions by using an analysis of the equilibrium states and their stability. It is concluded that the introduction of distributed delays can strongly modify both the value of R(0) and the predictions for the virus loads, so the effects on the infection dynamics are of major importance. We also show how the model presented here can be applied to some simple situations where direct comparison with experiments is possible. Specifically, phage-bacteria interactions are analyzed. The dynamics of the eclipse phase for phages is characterized analytically, which allows us to compare the performance of three different fittings proposed before for the one-step growth curve.

Image-guided modeling of virus growth and spread

Although many tools of cellular and molecular biology have been used to characterize single intracellular cycles of virus growth, few culture methods exist to study the dynamics of spatially spreading viruses over multiple generations. We have previously developed a method that addresses this need by tracking the spread of focal infections using immunocytochemical labeling and digital imaging. Here, we build reaction-diffusion models to account for spatio-temporal patterns formed by the spreading viral infection front as well as data from a single cycle of virus growth (one-step growth). Systems with and without the interferon-mediated antiviral response of the host cells are considered. Dynamic images of the spreading infections guide iterative model refinement steps that lead to reproduction of all of the salient features contained in the images, not just the velocity of the infection front. The optimal fits provide estimates for key parameters such as virus-host binding and the production rate of interferon. For the examined data, highly-lumped infection models that ignore the one-step growth dynamics provide a comparable fit to models that more accurately account for these dynamics, highlighting the fact that increased model complexity does not necessarily translate to improved fit. This work demonstrates how model building can facilitate the interpretation of experiments by highlighting contributions from both biological and methodological factors.

Fronts from integrodifference equations and persistence effects on the Neolithic transition

We extend a previous model of the Neolithic transition in Europe [J. Fort and V. Méndez, Phys. Rev. Lett. 82, 867 (1999)] by taking two effects into account: (i) we do not use the diffusion approximation (which corresponds to second-order Taylor expansions), and (ii) we take proper care of the fact that parents do not migrate away from their children (we refer to this as a time-order effect, in the sense that it implies that children grow up with their parents, before they become adults and can survive and migrate). We also derive a time-ordered, second-order equation, which we call the sequential reaction-diffusion equation, and use it to show that effect (ii) is the most important one, and that both of them should in general be taken into account to derive accurate results. As an example, we consider the Neolithic transition: the model predictions agree with the observed front speed, and the corrections relative to previous models are important (up to 70%).

Bacteriophage evolution given spatial constraint

Spatial structure can impede mixing, diffusion, and motility. In microbiology laboratories, spatial structure is commonly achieved via formation of agar gels, within which bacteriophage (phage) replication results in localized clearings called plaques. Developing a better understanding of phage plaque formation is relevant because of the ubiquity of phage plaquing in the laboratory; because plaque size has been employed as a measure of phage fitness; because many bacteria exist within environments that display significant spatial structure (e.g., biofilms, soils, sediments, and in or on plant or animal tissues); and because spatial structure could impede phage exploitation of bacterial communities. There is, however, a relative dearth of experimentation and analysis considering phage plaque formation from the perspective of selection acting on individual phage growth parameters-latent period, burst size, and adsorption rate. Here we consider the impact of these parameters on rates of plaque wavefront velocity (rates of radial plaque enlargement), especially as functions of existing phage and environmental properties. We do so based on analyses of published equations which predict plaque enlargement rates. These indicate that greater wavefront velocities should be associated with (i) latent period reductions, (ii) larger burst sizes, or (iii) faster virion binding to bacteria. We suggest, however, that deviations could occur, respectively, (i) if virion adsorption is "slow" or if burst sizes are large, (ii) if burst sizes are already large, or (iii) if virion binding rates are already fast, bacterial densities are especially high, or burst sizes are large. Higher initial lawn bacterial densities could also contribute to faster plaque expansion, but only if adsorption is otherwise slow or burst sizes are large. By contrast, faster virion diffusion is always expected to result in greater plaque wavefront velocities. Overall, we provide a snapshot of how phage populations may respond evolutionarily to selection for more-rapid propagation during spatially constrained growth.

Propagating waves of self-assembly in organosilane monolayers

Wavefronts associated with reaction-diffusion and self-assembly processes are ubiquitous in the natural world. For example, propagating fronts arise in crystallization and diverse other thermodynamic ordering processes, in polymerization fronts involved in cell movement and division, as well as in the competitive social interactions and population dynamics of animals at much larger scales. Although it is often claimed that self-sustaining or autocatalytic front propagation is well described by mean-field "reaction-diffusion" or "phase field" ordering models, it has recently become appreciated from simulations and theoretical arguments that fluctuation effects in lower spatial dimensions can lead to appreciable deviations from the classical mean-field theory (MFT) of this type of front propagation. The present work explores these fluctuation effects in a real physical system. In particular, we consider a high-resolution near-edge x-ray absorption fine structure spectroscopy (NEXAFS) study of the spontaneous frontal self-assembly of organosilane (OS) molecules into self-assembled monolayer (SAM) surface-energy gradients on oxidized silicon wafers. We find that these layers organize from the wafer edge as propagating wavefronts having well defined velocities. In accordance with two-dimensional simulations of this type of front propagation that take fluctuation effects into account, we find that the interfacial widths w(t) of these SAM self-assembly fronts exhibit a power-law broadening in time, w(t) approximately t(beta), rather than the constant width predicted by MFT. Moreover, the observed exponent values accord rather well with previous simulation and theoretical estimates. These observations have significant implications for diverse types of ordering fronts that occur under confinement conditions in biological or materials-processing contexts.

Propagation of fluorescent viruses in growing plaques

To study virus propagation, we have developed a method by which the propagation of the Lambda bacteriophage can be observed and quantified. This is done by creating a fusion protein of the capsid protein gpD and the enhanced yellow fluorescent protein (EYFP). We show that this fusion allows capsid formation and that the modified viruses propagate on a surface covered with host bacteria thus forming fluorescent plaques. The intensity of fluorescence in a growing plaque determines the distribution of phages. This provides a new tool to study the propagation of infection at the microscopic level.

Kinetic equations for reaction-subdiffusion systems: derivation and stability analysis

We derive general kinetic equations for reacting and subdiffusing entities based on a nonlinear continuous time random walk formalism proposed by Vlad and Ross [Phys. Rev. E 66, 061908 (2002)]. Reaction and diffusion processes are separable in a typical reaction-diffusion system, and their combined influence on the evolution of the density of a species is a simple sum. Our derivation shows that this is no longer true for subdiffusive entities undergoing reactions. The strong memory effects in the transport process, i.e., the non-Markovian nature of subdiffusion, results in a nontrivial combination of reactions and spatial dispersal, which we discuss in detail. We carry out a linear stability analysis of the derived reaction-subdiffusion system to understand the effects of memory on pattern formation. We find that the Turing instability persists in the subdiffusive system. However, the memory modifies the Turing threshold and the characteristics of the band of unstable modes close to this threshold.

Stimulus-locked responses of two phase oscillators coupled with delayed feedback

For a system of two phase oscillators coupled with delayed self-feedback we study the impact of pulsatile stimulation administered to both oscillators. This system models the dynamics of two coupled phase-locked loops (PLLs) with a finite internal delay within each loop. The delayed self-feedback leads to a rich variety of dynamical regimes, ranging from phase-locked and periodically modulated synchronized states to chaotic phase synchronization and desynchronization. Remarkably, for large coupling strength the two PLLs are completely desynchronized. We study stimulus-locked responses emerging in the different dynamical regimes. Simple phase resets may be followed by a response clustering, which is intimately connected with long poststimulus resynchronization. Intriguingly, a maximal perturbation (i.e., maximal response clustering and maximal resynchronization time) occurs, if the system gets trapped at a stable manifold of an unstable saddle fixed point due to appropriately calibrated stimulus. Also, single stimuli with suitable parameters can shift the system from a stable synchronized state to a stable desynchronized state or vice versa. Our result show that appropriately calibrated single pulse stimuli may cause pronounced transient and/or long-lasting changes of the oscillators' dynamics. Pulse stimulation may, hence, constitute an effective approach for the control of coupled oscillators, which might be relevant to both physical and medical applications.

Fronts with continuous waiting-time distributions: theory and application to virus infections

We generalize to arbitrary waiting-time distributions some results which were previously derived for discrete distributions. We show that for any two waiting-time distributions with the same mean delay time, that with higher dispersion will lead to a faster front. Experimental data on the speed of virus infections in a plaque are correctly explained by the theoretical predictions using a Gaussian delay-time distribution, which is more realistic for this system than the Dirac delta distribution considered previously [J. Fort and V. Méndez, Phys. Rev. Lett. 89, 178101 (2002)].

Spread dynamics of invasive species

Species invasions are a principal component of global change, causing large losses in biodiversity as well as economic damage. Invasion theory attempts to understand and predict invasion success and patterns of spread. However, there is no consensus regarding which species or community attributes enhance invader success or explain spread dynamics. Experimental and theoretical studies suggest that regulation of spread dynamics is possible; however, the conditions for its existence have not yet been empirically demonstrated. If invasion spread is a regulated process, the structure that accounts for this regulation will be a main determinant of invasion dynamics. Here we explore the existence of regulation underlying changes in the rate of new site colonization. We employ concepts and analytical tools from the study of abundance dynamics and show that spread dynamics are, in fact, regulated processes and that the regulation structure is notably consistent among invasions occurring in widely different contexts. We base our conclusions on the analysis of the spread dynamics of 30 species invasions, including birds, amphibians, fish, invertebrates, plants, and a virus, all of which exhibited similar regulation structures. In contrast to current beliefs that species invasions are idiosyncratic phenomena, here we provide evidence that general patterns do indeed exist.

Tracing the origin and spread of agriculture in Europe

The origins of early farming and its spread to Europe have been the subject of major interest for some time. The main controversy today is over the nature of the Neolithic transition in Europe: the extent to which the spread was, for the most part, indigenous and animated by imitation (cultural diffusion) or else was driven by an influx of dispersing populations (demic diffusion). We analyze the spatiotemporal dynamics of the transition using radiocarbon dates from 735 early Neolithic sites in Europe, the Near East, and Anatolia. We compute great-circle and shortest-path distances from each site to 35 possible agricultural centers of origin--ten are based on early sites in the Middle East and 25 are hypothetical locations set at 5 degrees latitude/longitude intervals. We perform a linear fit of distance versus age (and vice versa) for each center. For certain centers, high correlation coefficients (R > 0.8) are obtained. This implies that a steady rate or speed is a good overall approximation for this historical development. The average rate of the Neolithic spread over Europe is 0.6-1.3 km/y (95% confidence interval). This is consistent with the prediction of demic diffusion (0.6-1.1 km/y). An interpolative map of correlation coefficients, obtained by using shortest-path distances, shows that the origins of agriculture were most likely to have occurred in the northern Levantine/Mesopotamian area.