A mathematical model for spreading cortical depression.

A mathematical model is derived from physiological considerations for slow potential waves (called spreading depression) in cortical neuronal structures. The variables taken into account are the intra- and extracellular concentrations of Na+, Cl-, K+, and Ca++, together with excitatory and inhibitor transmitter substances. The general model includes conductance changes for these various ions, which may occur at nonsynaptic and synaptic membrane together with active transport mechanisms (pumps). A detailed consideration of only the conductance changes due to transmitter release leads to a system of nonlinear diffusion equations coupled with a system or ordinary differential equations. We obtain numerical solutions of a set of simplified model equations involving only K+ and Ca++ concentrations. The solutions agree qualitatively with experimentally obtained time-courses of these two ionic concentrations during spreading depression. The numerical solutions exhibit the observed phenomena of solitary waves and annihilation of colliding waves.     

Virulence evolution at the front line of spreading epidemics

Understanding and predicting the spatial spread of emerging pathogens is a major challenge for the public health management of infectious diseases. Theoretical epidemiology shows that the speed of an epidemic is governed by the life‐history characteristics of the pathogen and its ability to disperse. Rapid evolution of these traits during the invasion may thus affect the speed of epidemics. Here we study the influence of virulence evolution on the spatial spread of an epidemic. At the edge of the invasion front, we show that more virulent and transmissible genotypes are expected to win the competition with other pathogens. Behind the front line, however, more prudent exploitation strategies outcompete virulent pathogens. Crucially, even when the presence of the virulent mutant is limited to the edge of the front, the invasion speed can be dramatically altered by pathogen evolution. We support our analysis with individual‐based simulations and we discuss the additional effects of demographic stochasticity taking place at the front line on virulence evolution. We confirm that an increase of virulence can occur at the front, but only if the carrying capacity of the invading pathogen is large enough. These results are discussed in the light of recent empirical studies examining virulence evolution at the edge of spreading epidemics.     

Common source epidemics I: A stochastic model

A discrete time stochastic model is formulated for the spread of a disease which is transmitted to an uninfected but susceptible individual through an environmental source and not through contact (either direct or indirect) with infected individuals. The model incorporates both exposure and infection components. The exposure component includes consideration of the introduction of an infectious agent into the environment and the subsequent diffusion of the agent. It also includes time and location patterns for visits by individuals in the target population to the affected environment. The infection component incorporates physiological responses of exposed individuals to the infectious agent. The goal of the model is to provide a method for developing a predicted epidemic curve. Comments are given on an application of the model to the study of an outbreak of toxoplasmosis in Atlanta, Georgia, in 1977. This work was partially supported by BRSG Grant S07 RR0731 awarded by the Biomedical Research Support Grant Program, Division of Research Resources, National Institutes of Health.     

A minimal biophysical model for an excitable and oscillatory neuron

Presented here is a minimal biophysical cell model, based on work by Hodgkin and Huxley and by Rinzel, that can exhibit both excitable and oscillatory behavior. Two versions of the model are studied, which conform to data for squid and lobster giant axons.     

Cell spreading and motility: A model lamellipod

Cells moving in vitro do so by means of a motile appendage, the lamellipod. This is a broad, flat sheet of cytogel which spreads in front of the cell and pulls the cell forward. We present here a mathematical model for lamellipodial motion based on the physical chemistry of actomyosin gels.     

A minimal model of metabolism-based chemotaxis

Since the pioneering work by Julius Adler in the 1960's, bacterial chemotaxis has been predominantly studied as metabolism-independent. All available simulation models of bacterial chemotaxis endorse this assumption. Recent studies have shown, however, that many metabolism-dependent chemotactic patterns occur in bacteria. We hereby present the simplest artificial protocell model capable of performing metabolism-based chemotaxis. The model serves as a proof of concept to show how even the simplest metabolism can sustain chemotactic patterns of varying sophistication. It also reproduces a set of phenomena that have recently attracted attention on bacterial chemotaxis and provides insights about alternative mechanisms that could instantiate them. We conclude that relaxing the metabolism-independent assumption provides important theoretical advances, forces us to rethink some established pre-conceptions and may help us better understand unexplored and poorly understood aspects of bacterial chemotaxis.     

A general model for stochastic SIR epidemics with two levels of mixing

This paper is concerned with a general stochastic model for susceptible→infective→removed epidemics, among a closed finite population, in which during its infectious period a typical infective makes both local and global contacts. Each local contact of a given infective is with an individual chosen independently according to a contact distribution ‘centred’ on that infective, and each global contact is with an individual chosen independently and uniformly from the whole population. The asymptotic situation in which the local contact distribution remains fixed as the population becomes large is considered. The concepts of local infectious clump and local susceptibility set are used to develop a unified approach to the threshold behaviour of this class of epidemic models. In particular, a threshold parameter R_* governing whether or not global epidemics can occur, the probability that a global epidemic occurs and the mean proportion of initial susceptibles ultimately infected by a global epidemic are all determined. The theory is specialised to (i) the households model, in which the population is partitioned into households and local contacts are chosen uniformly within an infective’s household; (ii) the overlapping groups model, in which the population is partitioned in several ways, with local uniform mixing within the elements of the partitions; and (iii) the great circle model, in which individuals are equally spaced on a circle and local contacts are nearest-neighbour.     

A stochastic model for extinction and recurrence of epidemics: estimation and inference for measles outbreaks

Epidemic dynamics pose a great challenge to stochastic modelling because chance events are major determinants of the size and the timing of the outbreak. Reintroduction of the disease through contact with infected individuals from other areas is an important latent stochastic variable. In this study we model these stochastic processes to explain extinction and recurrence of epidemics observed in measles. We develop estimating functions for such a model and apply the methodology to temporal case counts of measles in 60 cities in England and Wales. In order to estimate the unobserved spatial contact process we suggest a method based on stochastic simulation and marginal densities. The estimation results show that it is possible to consider a unified model for the UK cities where the parameters depend on the city size. Stochastic realizations from the dynamic model realistically capture the transitions from an endemic cyclic pattern in large populations to irregular epidemic outbreaks in small human host populations.     

A minimal kinetic model for a viral DNA packaging machine

Terminase enzymes are common to both eukaryotic and prokaryotic double-stranded DNA viruses. These enzymes possess ATPase and nuclease activities that work in concert to "package" a viral genome into an empty procapsid, and it is likely that terminase enzymes from disparate viruses utilize a common packaging mechanism. Bacteriophage lambda terminase possesses a site-specific nuclease activity, a so-called helicase activity, a DNA translocase activity, and multiple ATPase catalytic sites that function to package viral DNA. Allosteric interactions between the multiple catalytic sites have been reported. This study probes these catalytic interactions using enzyme kinetic, photoaffinity labeling, and vanadate inhibition studies. The ensemble of data forms the basis for a minimal kinetic model for lambda terminase. The model incorporates an ADP-driven conformational reorganization of the terminase subunits assembled on viral DNA, which is central to the activation of a catalytically competent packaging machine. The proposed model provides a unifying mechanism for allosteric interaction between the multiple catalytic sites of the holoenzyme and explains much of the kinetic data in the literature. Given that similar packaging mechanisms have been proposed for viruses as dissimilar as lambda and the herpes viruses, the model may find general utility in our global understanding of the enzymology of virus assembly.     

Physical model for membrane protrusions during spreading

During cell spreading onto a substrate, the kinetics of the contact area is an observable quantity. This paper is concerned with a physical approach to modeling this process in the case of ameboid motility where the membrane detaches itself from the underlying cytoskeleton at the leading edge. The physical model we propose is based on previous reports which highlight that membrane tension regulates cell spreading. Using a phenomenological feedback loop to mimic stress-dependent biochemistry, we show that the actin polymerization rate can be coupled to the stress which builds up at the margin of the contact area between the cell and the substrate. In the limit of small variation of membrane tension, we show that the actin polymerization rate can be written in a closed form. Our analysis defines characteristic lengths which depend on elastic properties of the membrane-cytoskeleton complex, such as the membrane-cytoskeleton interaction, and on molecular parameters, the rate of actin polymerization. We discuss our model in the case of axi-symmetric and non-axi-symmetric spreading and we compute the characteristic time scales as a function of fundamental elastic constants such as the strength of membrane-cytoskeleton adherence.     

Percolation on heterogeneous networks as a model for epidemics

We consider a spatial model related to bond percolation for the spread of a disease that includes variation in the susceptibility to infection. We work on a lattice with random bond strengths and show that with strong heterogeneity, i.e. a wide range of variation of susceptibility, patchiness in the spread of the epidemic is very likely, and the criterion for epidemic outbreak depends strongly on the heterogeneity. These results are qualitatively different from those of standard models in epidemiology, but correspond to real effects. We suggest that heterogeneity in the epidemic will affect the phylogenetic distance distribution of the disease-causing organisms. We also investigate small world lattices, and show that the effects mentioned above are even stronger.     

A minimal model of non-hyperbolic enzyme and receptor kinetics

A simplified version of P.W. Kühl's Recovery Model [Biochem. J. 298 (1994) 171-180] has been developed in which the duration of the recovery phase of receptor or enzyme (macro)molecule was assumed to be a random value distributed exponentially like other model parameters. The model has been shown to retain all the properties of the original Recovery Model except for its ability to yield asymmetric dose-response curves (if plotted in semi-logarithmic scale). Due to its simplicity, the present model is applicable for routine fitting to experimental data. In enzyme kinetics, the model yields a diversity of non-hyperbolic dose-response curves both with higher and lower steepness than that of Henri-type ones. In receptor kinetics, the diversity of dose-response curves is further increased due to virtually no restraints being imposed on the efficacies of any state of the macromolecule.     

A minimal gating model for the cardiac calcium release channel.

A Markovian model of the cardiac Ca release channel, based on experimental single-channel gating data, was constructed to understand the transient nature of Ca release. The rate constants for a minimal gating scheme with one Ca-free resting state, and with two open and three closed states with one bound Ca2+, were optimized to simulate the following experimental findings. In steady state the channel displays three modes of activity: inactivated 1 mode without openings, low-activity L mode with single openings, and high-activity H mode with bursts of openings. At the onset of a Ca2+ step, the channel first activates in H mode and then slowly relaxes to a mixture of all three modes, the distribution of which depends on the new Ca2+. The corresponding ensemble current shows rapid activation, which is followed by a slow partial inactivation. The transient reactivation of the channel (increment detection) in response to successive additions of Ca2+ is then explained by the model as a gradual recruitment of channels from the extant pool of channels in the resting state. For channels in a living cell, the model predicts a high level of peak activation, a high extent of inactivation, and rapid deactivation, which could underlie the observed characteristics of the elementary release events (calcium sparks).     

A minimal model for decoding of time-limited Ca2+ oscillations

Calcium oscillations regulate several cellular processes by activating particular proteins. Most theoretical studies focused on the idealized situation of infinitely long oscillations. Here we analyze information transfer by time-limited calcium spike trains. We show that proteins can be selectively activated in a resonance-like manner by time-limited spike trains of different frequencies, while infinitely long oscillations do not show this resonance phenomenon. We found that proteins are activated more specifically by shorter oscillatory signals with narrower spikes.     

Monitoring for multiple malformations in the detection of epidemics of birth defects

Although most known human teratogens often produce a combination of birth defects in an affected infant, surveillance programs aimed at detecting epidemics of birth defects usually only monitor rates of individual defects. A drawback to this approach is that an increase in the rate of infants affected with a specific combination of defects may lead to little or no increase in the rates of component defects. Using the Poisson distribution, we show that, compared with monitoring for individual defects, monitoring for combinations of two and three defects may require fewer numbers of births to detect an epidemic. In general, an increase can be detected more rapidly by monitoring the rates of defect combinations than by monitoring the rates of individual defects if most affected infants have combinations of defects rather than isolated defects. For example, in the case of Congenital rubella syndrome (CRS), monitoring for the combination of cataracts with deafness and/or patent ductus arteriosus could have led to earlier detection of an epidemic than could monitoring for cataracts alone. In contrast, in the case of thalidomide embryopathy, monitoring for reduction defects of upper limbs in combination with reduction defects of lower limbs and/or microtia/anotia would not have led to earlier detection of an epidemic than would monitoring for reduction defects of upper limbs alone. This is due mainly to the low frequency of defect combinations among affected cases. When used with regular monitoring for individual defects, surveillance of defect combinations can enhance the ability of monitoring programs to detect epidemics of birth defects.     

A minimal continuous model for simulating growth and development of plant root systems

Aims: This paper proposes a general and minimal continuous model of root growth that aggregates architectural and developmental information and that can be used at different spatial scales. Methods: The model is described by advection, diffusion and reaction operators, which are related to growth processes such as primary growth, branching, mortality and root death. Output variable is the number of root tips per unit volume of soil. Operator splitting techniques are used to fit, solve and analyze the model with regards to ontogeny. The modeling approach is illustrated on a 2D case study concerning a part of Eucalyptus root system. Results: Operator splitting is helpful to fit the model. Basic knowledge on root architecture and development allows decreasing the number of unknown parameters and defining ontogenic phases on which specific calibrations must be carried out. Simulation results reproduce quantitatively the dynamic evolution of root density distribution with a good accuracy. Conclusion: The proposed root growth model is based on a continuous formalism that can be easily coupled with other physical models, e.g. nutrient and water transfer. The equations are generic and allow simulating different root architectures and growth strategies. They can be efficiently solved using adapted numerical methods.     

A minimal mathematical model of human periodic breathing

Numerous mathematical models of periodic breathing (PB) currently exist. These models suggest mechanisms that may underlie many known causes of PB. However, each model that has been shown to simulate PB under reasonable conditions contains greater than 15 physiological parameters. Because some parameters exhibit a wide range of values in a population, such simulations cannot test a model's ability to account for the breathing patterns of individuals. Furthermore it is impractical to perform a direct experimental validation study that would require the estimation of each of 15 or more parameters for each subject. A minimal model of PB is presented that is suitable for direct validation. Analytic expressions are given that define the conditions for PB in terms of the following: 1) CO2 sensitivity, 2) Cardiac output, 3) Mixed venous CO2, 4) Circulation time, and 5) Mean lung volume for CO2. This model is shown to be consistent with previous models and experimental data regarding the degree of hypoxia or congestive heart failure required to produce PB. A quantitative measure of relative stability is defined as a metric of comparison to the human studies described in the accompanying paper (J. Appl. Physiol. 65: 1389-1399, 1988).