Stochastic Models and Numerical Algorithms for a Class of Regulatory Gene Networks

Regulatory gene networks contain generic modules, like those involving feedback loops, which are essential for the regulation of many biological functions (Guido et al. in Nature 439:856–860, 2006). We consider a class of self-regulated genes which are the building blocks of many regulatory gene networks, and study the steady-state distribution of the associated Gillespie algorithm by providing efficient numerical algorithms. We also study a regulatory gene network of interest in gene therapy, using mean-field models with time delays. Convergence of the related time-nonhomogeneous Markov chain is established for a class of linear catalytic networks with feedback loops.     

Global Stability for a Class of Virus Models with Cytotoxic T Lymphocyte Immune Response and Antigenic Variation

We study the global stability of a class of models for in-vivo virus dynamics that take into account the Cytotoxic T Lymphocyte immune response and display antigenic variation. This class includes a number of models that have been extensively used to model HIV dynamics. We show that models in this class are globally asymptotically stable, under mild hypothesis, by using appropriate Lyapunov functions. We also characterise the stable equilibrium points for the entire biologically relevant parameter range. As a by-product, we are able to determine what is the diversity of the persistent strains.     

On the Modelling of Biological Patterns with Mechanochemical Models: Insights from Analysis and Computation

The diversity of biological form is generated by a relatively small number of underlying mechanisms. Consequently, mathematical and computational modelling can, and does, provide insight into how cellular level interactions ultimately give rise to higher level structure. Given cells respond to mechanical stimuli, it is therefore important to consider the effects of these responses within biological self-organisation models. Here, we consider the self-organisation properties of a mechanochemical model previously developed by three of the authors in Acta Biomater. 4, 613–621 (2008), which is capable of reproducing the behaviour of a population of cells cultured on an elastic substrate in response to a variety of stimuli. In particular, we examine the conditions under which stable spatial patterns can emerge with this model, focusing on the influence of mechanical stimuli and the interplay of non-local phenomena. To this end, we have performed a linear stability analysis and numerical simulations based on a mixed finite element formulation, which have allowed us to study the dynamical behaviour of the system in terms of the qualitative shape of the dispersion relation. We show that the consideration of mechanotaxis, namely changes in migration speeds and directions in response to mechanical stimuli alters the conditions for pattern formation in a singular manner. Furthermore without non-local effects, responses to mechanical stimuli are observed to result in dispersion relations with positive growth rates at arbitrarily large wavenumbers, in turn yielding heterogeneity at the cellular level in model predictions. This highlights the sensitivity and necessity of non-local effects in mechanically influenced biological pattern formation models and the ultimate failure of the continuum approximation in their absence.     

Conditions for Global Dynamic Stability of a Class of Resource-Bounded Model Ecosystems

This paper studies a class of dynamical systems that model multi-species ecosystems. These systems are ‘resource bounded’ in the sense that species compete to utilize an underlying limiting resource or substrate. This boundedness means that the relevant state space can be reduced to a simplex, with coordinates representing the proportions of substrate utilized by the various species. If the vector field is inward pointing on the boundary of the simplex, the state space is forward invariant under the system flow, a requirement that can be interpreted as the presence of non-zero exogenous recruitment. We consider conditions under which these model systems have a unique interior equilibrium that is globally asymptotically stable. The systems we consider generalize classical multi-species Lotka–Volterra systems, the behaviour of which is characterized by properties of the community (or interaction) matrix. However, the more general systems considered here are not characterized by a single matrix, but rather a family of matrices. We develop a set of ‘explicit conditions’ on the basis of a notion of ‘uniform diagonal dominance’ for such a family of matrices, that allows us to extract a set of sufficient conditions for global asymptotic stability based on properties of a single, derived matrix. Examples of these explicit conditions are discussed.     

The Series Solution for a Class of Nonstationary Biological Population Systems and Their Stability

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Continuum Gating Current Models Computed with Consistent Interactions

The action potential of nerve and muscle is produced by voltage-sensitive channels that include a specialized device to sense voltage. The voltage sensor depends on the movement of charges in the changing electric field as suggested by Hodgkin and Huxley. Gating currents of the voltage sensor are now known to depend on the movements of positively charged arginines through the hydrophobic plug of a voltage sensor domain. Transient movements of these permanently charged arginines, caused by the change of transmembrane potential V, further drag the S4 segment and induce opening/closing of the ion conduction pore by moving the S4-S5 linker. This moving permanent charge induces capacitive current flow everywhere. Everything interacts with everything else in the voltage sensor and protein, and so it must also happen in its mathematical model. A Poisson-Nernst-Planck (PNP)-steric model of arginines and a mechanical model for the S4 segment are combined using energy variational methods in which all densities and movements of charge satisfy conservation laws, which are expressed as partial differential equations in space and time. The model computes gating current flowing in the baths produced by arginines moving in the voltage sensor. The model also captures the capacitive pile up of ions in the vestibules that link the bulk solution to the hydrophobic plug. Our model reproduces the signature properties of gating current: 1) equality of ON and OFF charge Q in integrals of gating current, 2) saturating voltage dependence in the Q(charge)-voltage curve, and 3) many (but not all) details of the shape of gating current as a function of voltage. Our results agree qualitatively with experiments and can be improved by adding more details of the structure and its correlated movements. The proposed continuum model is a promising tool to explore the dynamics and mechanism of the voltage sensor.     

On the Evolution of the Timing of Reproduction with Non-equilibrium Resident Dynamics

We study the evolution of an individual’s reproductive strategy in a mechanistic modeling framework. We assume that the total number of juveniles one adult individual can produce is a finite constant, and we study how this number should be distributed during the season, given the types of inter-individual interactions and mortality processes included in the model. The evolution of the timing of reproduction in this modeling framework has already been studied earlier in the case of equilibrium resident dynamics, but we generalize the situation to also fluctuating population dynamics. We find that, as in the equilibrium case, the presence or absence of inter-juvenile aggression affects the functional form of the evolutionarily stable reproductive strategy. If an ESS exists, it can have an absolutely continuous part only if inter-juvenile aggression is included in the model. If inter-juvenile aggression is not included in the model, an ESS can have no continuous parts, and only Dirac measures are possible.     

On 20 years of Lophotrochozoa

Lophotrochozoa is a protostome clade that includes disparate animals such as molluscs, annelids, bryozoans, and flatworms, giving it the distinction of including the most body plans of any of the three major clades of Bilateria. This extreme morphological disparity has prompted numerous conflicting phylogenetic hypotheses about relationships among lophotrochozoan phyla. Here, I review the current understanding of lophotrochozoan phylogeny with emphasis on recent insights gained through approaches taking advantage of high-throughput DNA sequencing (phylogenomics). Of significance, Platyzoa, a hypothesized clade of mostly small-bodied animals, appears to be an artifact of long-branch attraction. Recent studies recovered Gnathifera (Syndermata, Gnathostomulida, and Micrognathozoa) sister to all other lophotrochozoans and a clade called Rouphozoa (Platyhelminthes and Gastrotricha) sister to the remaining non-gnathiferan lophotrochozoans. Although Bryozoa was traditionally grouped with Brachiopoda and Phoronida (Lophophorata), most molecular studies have supported a clade including Entoprocta, Cycliophora, and Bryozoa (Polyzoa). However, recent phylogenomic work has shown that entoprocts and bryozoans have compositionally heterogeneous genomes that may cause systematic artifacts affecting their phylogenetic placement. Lastly, relationships within Trochozoa (Mollusca, Annelida, and relatives) largely remain ambiguous. Recent work has shown that phylogenomic studies must identify and reduce sources of systematic error, such as amino acid compositional heterogeneity and long-branch attraction. Still, other approaches such as the analysis of rare genomic changes may be needed to overcome challenges to standard phylogenomic approaches. Resolving lophotrochozoan phylogeny will provide important insight into how these complex and diverse body plans evolved and provide a much-needed framework for comparative studies.     

Optimal Timing of Disease Transmission in an Age-Structured Population

It is a common medical folk-practice for parents to encourage their children to contract certain infectious diseases while they are young. This folk-practice is controversial, in part, because it contradicts the long-term public health goal of minimizing disease incidence. We study an epidemiological model of infectious disease in an age-structured population where virulence is age-dependent and show that, in some cases, the optimal behavior will increase disease transmission. This provides a rigorous justification of the concept of “endemic stability,” and demonstrates that folk-practices may have been historically justified.     

Homogenization of Large-Scale Movement Models in Ecology

A difficulty in using diffusion models to predict large scale animal population dispersal is that individuals move differently based on local information (as opposed to gradients) in differing habitat types. This can be accommodated by using ecological diffusion. However, real environments are often spatially complex, limiting application of a direct approach. Homogenization for partial differential equations has long been applied to Fickian diffusion (in which average individual movement is organized along gradients of habitat and population density). We derive a homogenization procedure for ecological diffusion and apply it to a simple model for chronic wasting disease in mule deer. Homogenization allows us to determine the impact of small scale (10–100 m) habitat variability on large scale (10–100 km) movement. The procedure generates asymptotic equations for solutions on the large scale with parameters defined by small-scale variation. The simplicity of this homogenization procedure is striking when compared to the multi-dimensional homogenization procedure for Fickian diffusion,and the method will be equally straightforward for more complex models.     

Prey Selection,Vertical Migrations and the Impacts of Harvesting Upon the Population Dynamics of a Predator-Prey System

A model is developed to describe the interaction between a predator and two prey types located in different regions. Conditions for stability and persistence are analysed. The effects of harvesting the predators are investigated by making the predator mortality rate habitat dependent. Results demonstrate that for any given set of parameter values there is a value of the intrinsic preference of the predator for each prey type at which the system undergoes a Hopf bifurcation. Above this critical value the system evolves towards a stable equilibrium, whereas below it, stable limit cycles arise by Hopf bifurcations. Simulations demonstrate that the presence of demographic stochasticity may destabilise oscillatory populations, thereby causing population extinctions. An application of the model to the foraging behaviour of North Sea cod is described. It is shown that if the preferred prey is more productive, it is likely that the equilibrium will be stable, whereas if the less preferred prey is more productive, populations are likely to display cycles and in the stochastic case become extinct. As cod fishing mortality is increased, the point of bifurcation and region of parameter space for which the system is unstable decreases. An increased understanding of how cod behave may enable fish stocks to be managed more successfully, for example by indicating where marine reserves should be placed.     

A Comparison of Bimolecular Reaction Models for Stochastic Reaction–Diffusion Systems

Stochastic reaction–diffusion models have become an important tool in studying how both noise in the chemical reaction process and the spatial movement of molecules influences the behavior of biological systems. There are two primary spatially-continuous models that have been used in recent studies: the diffusion limited reaction model of Smoluchowski, and a second approach popularized by Doi. Both models treat molecules as points undergoing Brownian motion. The former represents chemical reactions between two reactants through the use of reactive boundary conditions, with two molecules reacting instantly upon reaching a fixed separation (called the reaction-radius). The Doi model uses reaction potentials, whereby two molecules react with a fixed probability per unit time, λ, when separated by less than the reaction radius. In this work, we study the rigorous relationship between the two models. For the special case of a protein diffusing to a fixed DNA binding site, we prove that the solution to the Doi model converges to the solution of the Smoluchowski model as λ→∞, with a rigorous $O(\lambda^{-\frac{1}{2} + \epsilon})$ error bound (for any fixed ?>0). We investigate by numerical simulation, for biologically relevant parameter values, the difference between the solutions and associated reaction time statistics of the two models. As the reaction-radius is decreased, for sufficiently large but fixed values of λ, these differences are found to increase like the inverse of the binding radius.     

Effects of Demographic Stochasticity on Population Persistence in Advective Media

Many populations live and disperse in advective media. A fundamental question, known as the “drift paradox” in stream ecology, is how a closed population can survive when it is constantly being transported downstream by the flow. Recent population-level models have focused on the role of diffusive movement in balancing the effects of advection, predicting critical conditions for persistence. Here, we formulate an individual-based stochastic analog of the model described in (Lutscher et al., SIAM Rev. 47(4):749–772, 2005) to quantify the effects of demographic stochasticity on persistence. Population dynamics are modeled as a logistic growth process and dispersal as a position-jump process on a finite domain divided into patches. When there is no correlation in the interpatch movement of residents, stochasticity simply smooths the persistence-extinction boundary. However, when individuals disperse in “packets” from one patch to another and the flow field is memoryless on the timescale of packet transport, the probability of persistence is greatly enhanced. The latter transport mechanism may be characteristic of larval dispersal in the coastal ocean or wind-dispersed seed pods.     

Turing Pattern Formation with Two Kinds of Cells and a Diffusive Chemical

We study a three-variable Turing system with two kinds of cells and a diffusive chemical, considering the formation and maintenance of fish skin patterns with multiple pigment cells. The two types of cells are produced from undifferentiated cells. They inhibit the supply rate of the other cell type, forming local clusters of the same cell type. In addition, the cells of one type can be maintained only in the presence of a diffusive chemical produced by the other cell type, resulting in the coexistence of two cell types in heterogeneous spatial patterns. We assume linear interaction among cells and the chemical, and cell supply rates constrained to be positive or zero. We derive the condition for diffusion-driven instability. In one-dimensional model, we examine how the wavelength of the periodic pattern depends on parameters. In the two-dimensional model, we study the condition for spot, stripe or reversed spot pattern to emerge (pattern selection). We discuss heuristic criteria for the pattern selection.     

Parameter Sensitivity Analysis of Stochastic Models Provides Insights into?Cardiac Calcium Sparks

We present a parameter sensitivity analysis method that is appropriate for stochastic models, and we demonstrate how this analysis generates experimentally testable predictions about the factors that influence local Ca²⁺ release in heart cells. The method involves randomly varying all parameters, running a single simulation with each set of parameters, running simulations with hundreds of model variants, then statistically relating the parameters to the simulation results using regression methods. We tested this method on a stochastic model, containing 18 parameters, of the cardiac Ca²⁺ spark. Results show that multivariable linear regression can successfully relate parameters to continuous model outputs such as Ca²⁺ spark amplitude and duration, and multivariable logistic regression can provide insight into how parameters affect Ca²⁺ spark triggering (a probabilistic process that is all-or-none in a single simulation). Benchmark studies demonstrate that this method is less computationally intensive than standard methods by a factor of 16. Importantly, predictions were tested experimentally by measuring Ca²⁺ sparks in mice with knockout of the sarcoplasmic reticulum protein triadin. These mice exhibit multiple changes in Ca²⁺ release unit structures, and the regression model both accurately predicts changes in Ca²⁺ spark amplitude (30% decrease in model, 29% decrease in experiments) and provides an intuitive and quantitative understanding of how much each alteration contributes to the result. This approach is therefore an effective, efficient, and predictive method for analyzing stochastic mathematical models to gain biological insight.     

Management of elderly patients with a non-ST-segment-elevation acute coronary syndrome

Elderly patients with an acute coronary syndrome are underrepresented in randomised controlled trials. Neither the European Society of Cardiology nor the American Heart Association/American College of Cardiology acute coronary syndrome guidelines provide specific recommendations for elderly patients. However, elderly patients are at higher thrombotic and bleeding risk compared with younger patients leading to difficulties in choosing the optimal treatment. In this review, we discuss the uncertainties we encounter in treating elderly patients with non-ST-elevation acute coronary syndrome and suggest treatment options based on the existing literature.     

HLH-2/E2A Expression Links Stochastic and Deterministic Elements of a Cell Fate Decision during C. elegans Gonadogenesis

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Medical Disease or Moral Defect? Stigma Attribution and Cultural Models of Addiction Causality in a University Population

This study examines the knowledge individuals use to make judgments about persons with substance use disorder. First, we show that there is a cultural model of addiction causality that is both shared and contested. Second, we examine how individuals’ understanding of that model is associated with stigma attribution. Research was conducted among undergraduate students at the University of Alabama. College students in the 18–25 age range are especially at risk for developing substance use disorder, and they are, perhaps more than any other population group, intensely targeted by drug education. The elicited cultural model includes different types of causes distributed across five distinct themes: Biological, Self-Medication, Familial, Social, and Hedonistic. Though there was cultural consensus among respondents overall, residual agreement analysis showed that the cultural model of addiction causality is a multicentric domain. Two centers of the model, the moral and the medical, were discovered. Differing adherence to these centers is associated with the level of stigma attributed towards individuals with substance use disorder. The results suggest that current approaches to substance use education could contribute to stigma attribution, which may or may not be inadvertent. The significance of these results for both theory and the treatment of addiction are discussed.