Mobility and its sensitivity to fitness differences determine consumer-resource distributions.

An animal's movement rate (mobility) and its ability to perceive fitness gradients (fitness sensitivity) determine how well it can exploit resources. Previous models have examined mobility and fitness sensitivity separately and found that mobility, modelled as random movement, prevents animals from staying in high-quality patches, leading to a departure from an ideal free distribution (IFD). However, empirical work shows that animals with higher mobility can more effectively collect environmental information and better sense patch quality, especially when the environment is frequently changed by human activities. Here, we model, for the first time, this positive correlation between mobility and fitness sensitivity and measure its consequences for the populations of a consumer and its resource. In the absence of consumer demography, mobility alone had no effect on system equilibria, but a positive correlation between mobility and fitness sensitivity could produce an IFD. In the presence of consumer demography, lower levels of mobility prevented the system from approaching an IFD due to the mixing of consumers between patches. However, when positively correlated with fitness sensitivity, high mobility led to an IFD. Our study demonstrates that the expected covariation of animal movement attributes can drive broadly theorized consumer-resource patterns across space and time and could underlie the role of consumers in driving spatial heterogeneity in resource abundance.

Habitat-dependent movement rate can determine the efficacy of marine protected areas.

Theoretical studies of marine protected areas (MPAs) suggest that more mobile species should exhibit reduced local effects (defined as the ratio of the density inside vs. outside of the MPA). However, empirical studies have not supported the expected negative relationship between the local effect and mobility. We propose that differential, habitat-dependent movement (i.e., a higher movement rate in the fishing grounds than in the MPA) might explain the disparity between theoretical expectations and empirical results. We evaluate this hypothesis by building two-patch box and stepping-stone models and show that increasing disparity in the habitat-specific movement rates shifts the relationship between the local effect and mobility from negative (the previous theoretical results) to neutral or positive (the empirical pattern). This shift from negative to positive occurs when differential movement offsets recruitment and mortality differences between the two habitats. Thus, local effects of MPAs might be caused by behavioral responses via differential movement rather than by, or in addition to, reductions in mortality. In addition, the benefits of MPAs, in terms of regional abundance and fishing yields, can be altered by the magnitude of differential movement. Thus, our study points to a need for empirical investigations that disentangle the interactions among mobility, differential movement, and protection.

The effectiveness of marine protected areas for predator and prey with varying mobility.

Marine protected areas (MPAs) are regions in the ocean where fishing is restricted or prohibited. Although several measures for MPA performance exist, here we focus on a specific one, namely the ratio of the steady state fish densities inside and outside the MPA. Several 2 patch models are proposed and analyzed mathematically. One patch represents the MPA, whereas the second patch represents the fishing ground. Fish move freely between both regions in a diffusive manner. Our main objective is to understand how fish mobility affects MPA performance. We show that MPA effectiveness decreases with fish mobility for single species models with logistic growth, and that densities inside and outside the MPA tend to equalize. This suggests that MPA performance is highest for the least mobile species. We then consider a 2 patch Lotka-Volterra predator-prey system. When one of the species moves, and the other does not, the ratio of the moving species first remains constant, and ultimately decreases with increased fish mobility, again with a tendency of equalization of the density in both regions. This suggests that MPA performance is not only highest for slow, but also for moderately mobile species. The discrepancy in MPA performance for single species models and for predator-prey models, confirms that MPA design requires an integrated, ecosystem-based approach. The mathematical approaches advocated here complement and enhance the numerical and theoretical approaches that are commonly applied to more complex models in the context of MPA design.

Analysis of a mathematical model for tumor therapy with a fusogenic oncolytic virus.

Oncolytic virotherapy is a tumor treatment which uses viruses to selectively target and destroy cancer cells. Fusogenic viruses, capable of causing cell-to-cell fusion upon infection of a tumor cell, have shown promise in experimental studies. Fusion causes the formation of large, multinucleated syncytia which eventually leads to cell death. We formulate a partial differential equations model with a moving boundary to describe the treatment of a spherical tumor with a fusogenic oncolytic virus. Fusion, lysis, and budding are incorporated as mechanisms of viral spread, resulting in nonlocal integral terms. A proof is presented for existence and uniqueness of global solutions to the nonlinear hyperbolic-parabolic system. Numerical simulations demonstrate convergence to spatially homogeneous solutions and exponential growth or decay of the tumor radius depending on viral burst size and rate of fusion. Long-term tumor radius is shown to decrease with increasing values of viral burst size while the effect of the rate of fusion on tumor growth is demonstrated to be nonmonotonic.

Trans-membrane area asymmetry controls the shape of cellular organelles.

Membrane organelles often have complicated shapes and differ in their volume, surface area and membrane curvature. The ratio between the surface area of the cytosolic and luminal leaflets (trans-membrane area asymmetry (TAA)) determines the membrane curvature within different sites of the organelle. Thus, the shape of the organelle could be critically dependent on TAA. Here, using mathematical modeling and stereological measurements of TAA during fast transformation of organelle shapes, we present evidence that suggests that when organelle volume and surface area are constant, TAA can regulate transformation of the shape of the Golgi apparatus, endosomal multivesicular bodies, and microvilli of brush borders of kidney epithelial cells. Extraction of membrane curvature by small spheres, such as COPI-dependent vesicles within the Golgi (extraction of positive curvature), or by intraluminal vesicles within endosomes (extraction of negative curvature) controls the shape of these organelles. For instance, Golgi tubulation is critically dependent on the fusion of COPI vesicles with Golgi cisternae, and vice versa, for the extraction of membrane curvature into 50-60 nm vesicles, to induce transformation of Golgi tubules into cisternae. Also, formation of intraluminal ultra-small vesicles after fusion of endosomes allows equilibration of their TAA, volume and surface area. Finally, when microvilli of the brush border are broken into vesicles and microvilli fragments, TAA of these membranes remains the same as TAA of the microvilli. Thus, TAA has a significant role in transformation of organelle shape when other factors remain constant.

How sticky should a virus be? The impact of virus binding and release on transmission fitness using influenza as an example.

Budding viruses face a trade-off: virions need to efficiently attach to and enter uninfected cells while newly generated virions need to efficiently detach from infected cells. The right balance between attachment and detachment-the right amount of stickiness-is needed for maximum fitness. Here, we design and analyse a mathematical model to study in detail the impact of attachment and detachment rates on virus fitness. We apply our model to influenza, where stickiness is determined by a balance of the haemagglutinin (HA) and neuraminidase (NA) proteins. We investigate how drugs, the adaptive immune response and vaccines impact influenza stickiness and fitness. Our model suggests that the location in the 'stickiness landscape' of the virus determines how well interventions such as drugs or vaccines are expected to work. We discuss why hypothetical NA enhancer drugs might occasionally perform better than the currently available NA inhibitors in reducing virus fitness. We show that an increased antibody or T-cell-mediated immune response leads to maximum fitness at higher stickiness. We further show that antibody-based vaccines targeting mainly HA or NA, which leads to a shift in stickiness, might reduce virus fitness above what can be achieved by the direct immunological action of the vaccine. Overall, our findings provide potentially useful conceptual insights for future vaccine and drug development and can be applied to other budding viruses beyond influenza.

Mathematical modeling of citrus groves infected by huanglongbing.

Huanglongbing (citrus greening) is a bacterial disease that is significantly impacting the citrus industry in Florida and poses a risk to the remaining citrus-producing regions of the United States. A mathematical model of a grove infected by citrus greening is developed. An equilibrium stability analysis is presented. The basic reproductive number and its relation to the persistence of the disease is discussed. A numerical study is performed to illustrate the theoretical findings.

Periodic multidrug therapy in a within-host virus model.

Floquet theory and perturbation techniques are used to analyze a classical within-host virus model with periodic drug treatment. Both single and multidrug treatment strategies are investigated. Specifically, the effects of both RT-inhibitors and P-inhibitors on the stability of the infection-free steady state are studied. It is found that when both classes of drugs have periodic drug efficacy functions, then shifting the phase of these functions can critically affect the stability of the infection-free steady state. A numerical study is conducted to illustrate the theoretical results and provide additional insights.

Senescence and antibiotic resistance in an age-structured population model.

Different theories have been proposed to understand the growing problem of antibiotic resistance of microbial populations. Here we investigate a model that is based on the hypothesis that senescence is a possible explanation for the existence of so-called persister cells which are resistant to antibiotic treatment. We study a chemostat model with a microbial population which is age-structured and show that if the growth rates of cells in different age classes are sufficiently close to a scalar multiple of a common growth rate, then the population will globally stabilize at a coexistence steady state. This steady state persists under an antibiotic treatment if the level of antibiotics is below a certain threshold; if the level exceeds this threshold, the washout state becomes a globally attracting equilibrium.

The chemostat with lateral gene transfer.

We investigate the standard chemostat model when lateral gene transfer is taken into account. We will show that when the different genotypes have growth rate functions that are sufficiently close to a common growth rate function, and when the yields of the genotypes are sufficiently close to a common value, then the population evolves to a globally stable steady state, at which all genotypes coexist. These results can explain why the antibiotic-resistant strains persist in the pathogen population.

The diffusive influx and carrier efflux have a strong effect on the bistability of the lac operon in Escherichia coli.

In the presence of gratuitous inducers, the lac operon of Escherichia coli exhibits bistability. Most models in the literature assume that the inducer enters the cell via the carrier (permease), and exits by a diffusion-like process. The diffusive influx and carrier efflux are neglected. However, analysis of the data shows that in non-induced cells, the diffusive influx is comparable to the carrier influx, and in induced cells, the carrier efflux is comparable to the diffusive efflux. Since bistability entails the coexistence of steady states corresponding to both non-induced and induced cells, neither one of these fluxes can be ignored. We present a model accounting for both fluxes, and show that: (1) The thresholds (i.e., the extracellular inducer levels at which transcription turns on or off) are profoundly affected by both fluxes. The diffusive influx reduces the on threshold, and eliminates irreversible bistability, a phenomenon that is inconsistent with data. The carrier efflux increases the off threshold, and abolishes bistability at large permease activities, a conclusion that can be tested experimentally. (2) The thresholds are well approximated by simple analytical expressions obtained by considering two limiting cases (no carrier efflux and no diffusive influx). (3) The simulations are in good agreement with the data for isopropyl thiogalactoside (IPTG), but somewhat discrepant with respect to the data for thiomethyl galactoside (TMG). We discuss the potential sources of the discrepancy.

Sharing the burden: antigen transport and firebreaks in immune responses.

Communication between cells is crucial for immune responses. An important means of communication during viral infections is the presentation of viral antigen on the surface of an infected cell. Recently, it has been shown that antigen can be shared between infected and uninfected cells through gap junctions, connexin-based channels, that allow the transport of small molecules. The uninfected cell receiving antigen can present it on its surface. Cells presenting viral antigen are detected and killed by cytotoxic T lymphocytes. The killing of uninfected cells can lead to increased immunopathology. However, the immune response might also profit from killing those uninfected bystander cells. One benefit might be the removal of future 'virus factories'. Another benefit might be through the creation of 'firebreaks', areas void of target cells, which increase the diffusion time of free virions, making their clearance more likely. Here, we use theoretical models and simulations to explore how the mechanism of gap junction-mediated antigen transport (GMAT) affects the dynamics of the virus and immune response. We show that under the assumption of a well-mixed system, GMAT leads to increased immunopathology, which always outweighs the benefit of reduced virus production due to the removal of future virus factories. By contrast, a spatially explicit model leads to quite different results. Here we find that the firebreak mechanism reduces both viral load and immunopathology. Our study thus shows the potential benefits of GMAT and illustrates how spatial effects may be crucial for the quantitative understanding of infection dynamics and immune responses.

Multistrain virus dynamics with mutations: a global analysis.

We consider within-host virus models with n >or= 2 strains and allow mutation between the strains. If there is no mutation, a Lyapunov function establishes global stability of the steady state corresponding to the fittest strain. For small perturbations, this steady state persists, perhaps with small concentrations of some or all other strains, depending on the connectivity of the graph describing all possible mutations. Moreover, using a perturbation result due to Smith & Waltman (1999), we show that this steady state also preserves global stability.

Bistability of the lac operon during growth of Escherichia coli on lactose and lactose+glucose.

The lac operon of Escherichia coli can exhibit bistability. Early studies showed that bistability occurs during growth on TMG/succinate and lactose+glucose, but not during growth on lactose. More recently, studies with lacGFP-transfected cells show bistability during growth on TMG/succinate, but not during growth on lactose and lactose+glucose. In the literature, these results are invariably attributed to variations in the destabilizing effect of the positive feedback generated by induction. Specifically, during growth on TMG/succinate, lac induction generates strong positive feedback because the permease stimulates the accumulation of intracellular TMG, which in turn, promotes the synthesis of even more permease. This positive feedback is attenuated during growth on lactose because hydrolysis of intracellular lactose by beta-galactosidase suppresses the stimulatory effect of the permease. It is attenuated even more during growth on lactose + glucose because glucose inhibits the uptake of lactose. But it is clear that the stabilizing effect of dilution also changes dramatically as a function of the medium composition. For instance, during growth on TMG/succinate, the dilution rate of lac permease is proportional to its activity, e, because the specific growth rate is independent of e (it is completely determined by the concentration of succinate). However, during growth on lactose, the dilution rate of the permease is proportional to e2 because the specific growth rate is proportional to the specific lactose uptake rate, which in turn, proportional to e. We show that: (a) This dependence on e2 creates such a strong stabilizing effect that bistability is virtually impossible during growth on lactose, even in the face of the intense positive feedback generated by induction. (b) This stabilizing effect is weakened during growth on lactose+glucose because the specific growth rate on glucose is independent of e, so that the dilution rate once again contains a term that is proportional to e. These results imply that the lac operon is much more prone to bistability if the medium contains carbon sources that cannot be metabolized by the lac enzymes, e.g., succinate during growth on TMG/succinate and glucose during growth on lactose+glucose. We discuss the experimental data in the light of these results.

Immune response to a malaria infection: properties of a mathematical model.

We establish some properties of a within host mathematical model of malaria proposed by Recker et al. [M. Recker et al., Transient cross-reactive immune responses can orchestrate antigenic variation in malaria, Lett. Nature 429 (2004), pp. 555-558; M. Recker and S. Gupta, Conflicting immune responses can prolong the length of infection in Plasmodium falciparum malaria, Bull. Math. Biol. 68 (2006), pp. 821-835.], which includes the role of the immune system during the infection. The model accounts for the antigenic variation exhibited by the malaria parasite (Plasmodium falciparum). We show that the model can exhibit a wide variety of dynamical behaviours. We provide criteria for global stability, competitive exclusion and persistence. We also demonstrate that the disease equilibrium can be destabilized by non-symmetric cross-reactive responses.

Gap junction-mediated antigen transport in immune responses.

Communication between cells is a crucial part of the immune response. The importance of cytokines and immunological synapses for this purpose has long been recognized. Connexin-based gap junctions that allow exchange of molecules between adjacent cells also seem to have an important role. Recent work suggests that gap junction-mediated antigen transport might be a mechanism of immune-response regulation. Here, we discuss this idea in more detail and suggest possible ways in which this mechanism might have both positive and negative impacts during an immune response.

Bacterial gene regulation in diauxic and non-diauxic growth.

When bacteria are grown in a batch culture containing a mixture of two growth-limiting substrates, they exhibit a rich spectrum of substrate consumption patterns including diauxic growth, simultaneous consumption, and bistable growth. In previous work, we showed that a minimal model accounting only for enzyme induction and dilution captures all the substrate consumption patterns [Narang, A., 1998a. The dynamical analogy between microbial growth on mixtures of substrates and population growth of competing species. Biotechnol. Bioeng. 59, 116-121, Narang, A., 2006. Comparitive analysis of some models of gene regulation in mixed-substrate microbial growth, J. Theor. Biol. 242, 489-501]. In this work, we construct the bifurcation diagram of the minimal model, which shows the substrate consumption pattern at any given set of parameter values. The bifurcation diagram explains several general properties of mixed-substrate growth. (1) In almost all the cases of diauxic growth, the "preferred" substrate is the one that, by itself, supports a higher specific growth rate. In the literature, this property is often attributed to the optimality of regulatory mechanisms. Here, we show that the minimal model, which accounts for induction and growth only, displays the property under fairly general conditions. This suggests that the higher growth rate of the preferred substrate is an intrinsic property of the induction and dilution kinetics. It can be explained mechanistically without appealing to optimality principles. (2) The model explains the phenotypes of various mutants containing lesions in the regions encoding for the operator, repressor, and peripheral enzymes. A particularly striking phenotype is the "reversal of the diauxie" in which the wild-type and mutant strains consume the very same two substrates in opposite order. This phenotype is difficult to explain in terms of molecular mechanisms, such as inducer exclusion or CAP activation, but it turns out to be a natural consequence of the model. We show furthermore that the model is robust. The key property of the model, namely, the competitive dynamics of the enzymes, is preserved even if the model is modified to account for various regulatory mechanisms. Finally, the model has important implications for the problem of size regulation in development. It suggests that protein dilution may be the mechanism coupling patterning and growth.

Subthreshold and superthreshold coexistence of pathogen variants: the impact of host age-structure.

It is well known that in the most general epidemic models with multiple pathogen variants a competitive exclusion principle is valid, such that the variant with the highest reproduction number eliminates the rest. Mechanisms such as super-infection, coinfection, and cross-immunity can lead to pathogen polymorphism where multiple strains coexist. It is also known that variability of infectivity with host age can destabilize the endemic equilibrium and cause oscillations. In this article we show that the hosts' chronological age can itself lead to coexistence of microparasites in the most basic model where competitive exclusion will occur without the age structure. Moreover, the host age-structure leads to multiple subthreshold dominance equilibria, and both weakly and strongly subthreshold coexistence. We find that the two pathogens cannot cooperate to persist subthreshold if neither one of them can persist subthreshold by itself. If, however, one of them can persist subthreshold by itself, it can cause the two pathogens to coexist in a strongly subthreshold equilibrium. The second strain that persists subthreshold through the mediation of the first always has a lower virulence. Our results show that age structure in infectivity can permit the coexistence of competing pathogens when the incidence is of proportionate mixing type (frequency-dependent transmission) and at least one of the strains is virulent.

How does cross-reactive stimulation affect the longevity of CD8+ T cell memory?

Immunological memory--the ability to "remember" previously encountered pathogens and respond faster upon re-exposure is a central feature of the immune response in vertebrates. The cross-reactive stimulation hypothesis for the maintenance of memory proposes that memory cells specific for a given pathogen are maintained by cross-reactive stimulation following infections with other (unrelated) pathogens. We use mathematical models to examine the cross-reactive stimulation hypothesis. We find that: (i) the direct boosting of cross-reactive lineages only provides a very small increase in the average longevity of immunological memory; (ii) the expansion of cross-reactive lineages can indirectly increase the longevity of memory by reducing the magnitude of expansion of new naive lineages which occupy space in the memory compartment and are responsible for the decline in memory; (iii) cross-reactive stimulation results in variation in the rates of decline of different lineages of memory cells and enrichment of memory cell population for cells that are cross-reactive for the pathogens to which the individual has been exposed.

Quantifying cell turnover using CFSE data.

The CFSE dye dilution assay is widely used to determine the number of divisions a given CFSE labelled cell has undergone in vitro and in vivo. In this paper, we consider how the data obtained with the use of CFSE (CFSE data) can be used to estimate the parameters determining cell division and death. For a homogeneous cell population (i.e., a population with the parameters for cell division and death being independent of time and the number of divisions cells have undergone), we consider a specific biologically based "Smith-Martin" model of cell turnover and analyze three different techniques for estimation of its parameters: direct fitting, indirect fitting and rescaling method. We find that using only CFSE data, the duration of the division phase (i.e., approximately the S+G2+M phase of the cell cycle) can be estimated with the use of either technique. In some cases, the average division or cell cycle time can be estimated using the direct fitting of the model solution to the data or by using the Gett-Hodgkin method [Gett A. and Hodgkin, P. 2000. A cellular calculus for signal integration by T cells. Nat. Immunol. 1:239-244]. Estimation of the death rates during commitment to division (i.e., approximately the G1 phase of the cell cycle) and during the division phase may not be feasible with the use of only CFSE data. We propose that measuring an additional parameter, the fraction of cells in division, may allow estimation of all model parameters including the death rates during different stages of the cell cycle.