A stochastic metapopulation model accounting for habitat dynamics

A stochastic metapopulation model accounting for habitat dynamics is presented. This is the stochastic SIS logistic model with the novel aspect that it incorporates varying carrying capacity. We present results of Kurtz and Barbour, that provide deterministic and diffusion approximations for a wide class of stochastic models, in a form that most easily allows their direct application to population models. These results are used to show that a suitably scaled version of the metapopulation model converges, uniformly in probability over finite time intervals, to a deterministic model previously studied in the ecological literature. Additionally, they allow us to establish a bivariate normal approximation to the quasi-stationary distribution of the process. This allows us to consider the effects of habitat dynamics on metapopulation modelling through a comparison with the stochastic SIS logistic model and provides an effective means for modelling metapopulations inhabiting dynamic landscapes.     

Modeling the Dynamics of Disease States in Depression

Major depressive disorder (MDD) is a common and costly disorder associated with considerable morbidity, disability, and risk for suicide. The disorder is clinically and etiologically heterogeneous. Despite intense research efforts, the response rates of antidepressant treatments are relatively low and the etiology and progression of MDD remain poorly understood. Here we use computational modeling to advance our understanding of MDD. First, we propose a systematic and comprehensive definition of disease states, which is based on a type of mathematical model called a finite-state machine. Second, we propose a dynamical systems model for the progression, or dynamics, of MDD. The model is abstract and combines several major factors (mechanisms) that influence the dynamics of MDD. We study under what conditions the model can account for the occurrence and recurrence of depressive episodes and how we can model the effects of antidepressant treatments and cognitive behavioral therapy within the same dynamical systems model through changing a small subset of parameters. Our computational modeling suggests several predictions about MDD. Patients who suffer from depression can be divided into two sub-populations: a high-risk sub-population that has a high risk of developing chronic depression and a low-risk sub-population, in which patients develop depression stochastically with low probability. The success of antidepressant treatment is stochastic, leading to widely different times-to-remission in otherwise identical patients. While the specific details of our model might be subjected to criticism and revisions, our approach shows the potential power of computationally modeling depression and the need for different type of quantitative data for understanding depression.     

The role of cytoplasmic nanospaces in smooth muscle cell Ca2+ signalling

We address the importance of cytoplasmic nanospaces in Ca(2+) transport and signalling in smooth muscle cells and how quantitative modelling can shed significant light on the understanding of signalling mechanisms. Increasingly more convincing evidence supports the view that these nanospaces--nanometre-scale spaces between organellar membranes, hosting cell signalling machinery--are key to Ca(2+) signalling as much as Ca(2+) transporters and Ca(2+) storing organelles. Our research suggests that the origin of certain diseases is to be sought in the disruption of the proper functioning of cytoplasmic nanospaces. We begin with a historical perspective on the study of smooth muscle cell plasma membrane-sarcoplasmic reticulum nanospaces, including experimental evidence of their role in the generation of asynchronous Ca(2+) waves. We then summarize how stochastic modelling approaches have aided and guided our understanding of two basic functional steps leading to healthy smooth muscle cell contraction. We furthermore outline how more sophisticated and realistic quantitative stochastic modelling is now being employed not only to deepen our understanding but also to aid in the hypothesis generation for further experimental investigation.     

Multi-scale motility amplitude associated with suicidal thoughts in major depression

Major depression occurs at high prevalence in the general population, often starts in juvenile years, recurs over a lifetime, and is strongly associated with disability and suicide. Searches for biological markers in depression may have been hindered by assuming that depression is a unitary and relatively homogeneous disorder, mainly of mood, rather than addressing particular, clinically crucial features or diagnostic subtypes. Many studies have implicated quantitative alterations of motility rhythms in depressed human subjects. Since a candidate feature of great public-health significance is the unusually high risk of suicidal behavior in depressive disorders, we studied correlations between a measure (vulnerability index [VI]) derived from multi-scale characteristics of daily-motility rhythms in depressed subjects (n?=?36) monitored with noninvasive, wrist-worn, electronic actigraphs and their self-assessed level of suicidal thinking operationalized as a wish to die. Patient-subjects had a stable clinical diagnosis of bipolar-I, bipolar-II, or unipolar major depression (n?=?12 of each type). VI was associated inversely with suicidal thinking (r?=?-0.61 with all subjects and r?=?-0.73 with bipolar disorder subjects; both p<0.0001) and distinguished patients with bipolar versus unipolar major depression with a sensitivity of 91.7% and a specificity of 79.2%. VI may be a useful biomarker of characteristic features of major depression, contribute to differentiating bipolar and unipolar depression, and help to detect risk of suicide. An objective biomarker of suicide-risk could be advantageous when patients are unwilling or unable to share suicidal thinking with clinicians.     

The role of cytoplasmic nanospaces in smooth muscle cell Ca<Superscript>2+</Superscript> signalling

We address the importance of cytoplasmic nanospaces in Ca^2?+? transport and signalling in smooth muscle cells and how quantitative modelling can shed significant light on the understanding of signalling mechanisms. Increasingly more convincing evidence supports the view that these nanospaces—nanometre-scale spaces between organellar membranes, hosting cell signalling machinery—are key to Ca^2?+? signalling as much as Ca^2?+? transporters and Ca^2?+? storing organelles. Our research suggests that the origin of certain diseases is to be sought in the disruption of the proper functioning of cytoplasmic nanospaces. We begin with a historical perspective on the study of smooth muscle cell plasma membrane–sarcoplasmic reticulum nanospaces, including experimental evidence of their role in the generation of asynchronous Ca^2?+? waves. We then summarize how stochastic modelling approaches have aided and guided our understanding of two basic functional steps leading to healthy smooth muscle cell contraction. We furthermore outline how more sophisticated and realistic quantitative stochastic modelling is now being employed not only to deepen our understanding but also to aid in the hypothesis generation for further experimental investigation.     

Neuroblastoma signalling models unveil combination therapies targeting feedback-mediated resistance

Very high risk neuroblastoma is characterised by increased MAPK signalling, and targeting MAPK signalling is a promising therapeutic strategy. We used a deeply characterised panel of neuroblastoma cell lines and found that the sensitivity to MEK inhibitors varied drastically between these cell lines. By generating quantitative perturbation data and mathematical modelling, we determined potential resistance mechanisms. We found that negative feedbacks within MAPK signalling and via the IGF receptor mediate re-activation of MAPK signalling upon treatment in resistant cell lines. By using cell-line specific models, we predict that combinations of MEK inhibitors with RAF or IGFR inhibitors can overcome resistance, and tested these predictions experimentally. In addition, phospho-proteomic profiling confirmed the cell-specific feedback effects and synergy of MEK and IGFR targeted treatment. Our study shows that a quantitative understanding of signalling and feedback mechanisms facilitated by models can help to develop and optimise therapeutic strategies. Our findings should be considered for the planning of future clinical trials introducing MEKi in the treatment of neuroblastoma.     

Towards a quantitative model of the post-synaptic proteome

The postsynaptic compartment of the excitatory glutamatergic synapse contains hundreds of distinct polypeptides with a wide range of functions (signalling, trafficking, cell-adhesion, etc.). Structural dynamics in the post-synaptic density (PSD) are believed to underpin cognitive processes. Although functionally and morphologically diverse, PSD proteins are generally enriched with specific domains, which precisely define the mode of clustering essential for signal processing. We applied a stochastic calculus of domain binding provided by a rule-based modelling approach to formalise the highly combinatorial signalling pathway in the PSD and perform the numerical analysis of the relative distribution of protein complexes and their sizes. We specified the combinatorics of protein interactions in the PSD by rules, taking into account protein domain structure, specific domain affinity and relative protein availability. With this model we interrogated the critical conditions for the protein aggregation into large complexes and distribution of both size and composition. The presented approach extends existing qualitative protein-protein interaction maps by considering the quantitative information for stoichiometry and binding properties for the elements of the network. This results in a more realistic view of the postsynaptic proteome at the molecular level.     

Stochastic models of neuronal dynamics

Cortical activity is the product of interactions among neuronal populations. Macroscopic electrophysiological phenomena are generated by these interactions. In principle, the mechanisms of these interactions afford constraints on biologically plausible models of electrophysiological responses. In other words, the macroscopic features of cortical activity can be modelled in terms of the microscopic behaviour of neurons. An evoked response potential (ERP) is the mean electrical potential measured from an electrode on the scalp, in response to some event. The purpose of this paper is to outline a population density approach to modelling ERPs.We propose a biologically plausible model of neuronal activity that enables the estimation of physiologically meaningful parameters from electrophysiological data. The model encompasses four basic characteristics of neuronal activity and organization: (i) neurons are dynamic units, (ii) driven by stochastic forces, (iii) organized into populations with similar biophysical properties and response characteristics and (iv) multiple populations interact to form functional networks. This leads to a formulation of population dynamics in terms of the Fokker-Planck equation. The solution of this equation is the temporal evolution of a probability density over state-space, representing the distribution of an ensemble of trajectories. Each trajectory corresponds to the changing state of a neuron. Measurements can be modelled by taking expectations over this density, e.g. mean membrane potential, firing rate or energy consumption per neuron. The key motivation behind our approach is that ERPs represent an average response over many neurons. This means it is sufficient to model the probability density over neurons, because this implicitly models their average state. Although the dynamics of each neuron can be highly stochastic, the dynamics of the density is not. This means we can use Bayesian inference and estimation tools that have already been established for deterministic systems. The potential importance of modelling density dynamics (as opposed to more conventional neural mass models) is that they include interactions among the moments of neuronal states (e.g. the mean depolarization may depend on the variance of synaptic currents through nonlinear mechanisms).Here, we formulate a population model, based on biologically informed model-neurons with spike-rate adaptation and synaptic dynamics. Neuronal sub-populations are coupled to form an observation model, with the aim of estimating and making inferences about coupling among sub-populations using real data. We approximate the time-dependent solution of the system using a bi-orthogonal set and first-order perturbation expansion. For didactic purposes, the model is developed first in the context of deterministic input, and then extended to include stochastic effects. The approach is demonstrated using synthetic data, where model parameters are identified using a Bayesian estimation scheme we have described previously.     

Modelling spatial dynamics of fish

Our ability to model spatial distributions of fish populations is reviewed by describing the available modelling tools. Ultimate models of the individual's motivation for behavioural decisions are derived from evolutionary ecology. Mechanistic models for how fish sense and may respond to their surroundings are presented for vision, olfaction, hearing, the lateral line and other sensory organs. Models for learning and memory are presented, based both upon evolutionary optimization premises and upon neurological information processing and decision making. Functional tools for modelling behaviour and life histories can be categorized as belonging to an optimization or an adaptation approach. Among optimization tools, optimal foraging theory, life history theory, ideal free distribution, game theory and stochastic dynamic programming are presented. Among adaptation tools, genetic algorithms and the combination with artificial neural networks are described. The review advocates the combination of evolutionary and neurological approaches to modelling spatial dynamics of fish.     

Trait Substitution Sequence process and Canonical Equation for age-structured populations

We are interested in a stochastic model of trait and age-structured population undergoing mutation and selection. We start with a continuous time, discrete individual-centered population process. Taking the large population and rare mutations limits under a well-chosen time-scale separation condition, we obtain a jump process that generalizes the Trait Substitution Sequence process describing Adaptive Dynamics for populations without age structure. Under the additional assumption of small mutations, we derive an age-dependent ordinary differential equation that extends the Canonical Equation. These evolutionary approximations have never been introduced to our knowledge. They are based on ecological phenomena represented by PDEs that generalize the Gurtin–McCamy equation in Demography. Another particularity is that they involve an establishment probability, describing the probability of invasion of the resident population by the mutant one, that cannot always be computed explicitly. Examples illustrate how adding an age-structure enrich the modelling of structured population by including life history features such as senescence. In the cases considered, we establish the evolutionary approximations and study their long time behavior and the nature of their evolutionary singularities when computation is tractable. Numerical procedures and simulations are carried.      

Integration of Boolean models exemplified on hepatocyte signal transduction

The number of mathematical models for biological pathways is rapidly growing. In particular, Boolean modelling proved to be suited to describe large cellular signalling networks. Systems biology is at the threshold to holistic understanding of comprehensive networks. In order to reach this goal, connection and integration of existing models of parts of cellular networks into more comprehensive network models is necessary. We discuss model combination approaches for Boolean models. Boolean modelling is qualitative rather than quantitative and does not require detailed kinetic information. We show that these models are useful precursors for large-scale quantitative models and that they are comparatively easy to combine. We propose modelling standards for Boolean models as a prerequisite for smooth model integration. Using these standards, we demonstrate the coupling of two logical models on two different examples concerning cellular interactions in the liver. In the first example, we show the integration of two Boolean models of two cell types in order to describe their interaction. In the second example, we demonstrate the combination of two models describing different parts of the network of a single cell type. Combination of partial models into comprehensive network models will take systems biology to the next level of understanding. The combination of logical models facilitated by modelling standards is a valuable example for the next step towards this goal.     

Optimal conservation effort for a population in a stochastic environment

We study the optimal conservation effort for a population in a fluctuating environment. The survivorship of a population is affected by unpredictable environmental fluctuation (noise) and can be improved by conservation effort accompanied by a cost. The optimal effort level is the one that minimizes the total cost, defined as the weighted sum of the population extinction risk and the economic cost of conservation effort. The optimal effort depends on the variance and the probability distribution of the noise, the relative importance of the population's survival vs. the economic cost, the effectiveness of conservation effort, and the time scope over which we optimize. The analysis of dynamic programming illustrates that the choice of extinction risk function greatly affects the optimal effort level. The conservation effort level that is the best solution of a multiple-year optimization may be higher than that for the corresponding single-year optimization, if the population is relatively safe. However, the conservation level for the multiple-year optimization becomes lower than for the single-year optimization if the population is endangered. In a similar manner, the optimal conservation effort level for the problem with a short time scope is either higher or lower than that for the problem with a long time scope, depending on the extinction risk of the population. Next, for each parameter of the model, we define five different sensitivities of extinction probability or of the total cost. We then study the mean increase in the total cost caused by the uncertainty of parameters. To achieve the best conservation result, we need to invest the limited research effort to the parameter with the largest effect to the optimal effort level, rather than to those with large impacts on the extinction probability or on the total cost. The recommended policy should depend critically on the choice of the criterion to optimize, which shows the importance of theoretical study of the relationship in performing proper decision making in conservation practice.     

Epidemics with general generation interval distributions

We study the spread of susceptible-infected-recovered (SIR) infectious diseases where an individual's infectiousness and probability of recovery depend on his/her “age” of infection. We focus first on early outbreak stages when stochastic effects dominate and show that epidemics tend to happen faster than deterministic calculations predict. If an outbreak is sufficiently large, stochastic effects are negligible and we modify the standard ordinary differential equation (ODE) model to accommodate age-of-infection effects. We avoid the use of partial differential equations which typically appear in related models. We introduce a “memoryless” ODE system which approximates the true solutions. Finally, we analyze the transition from the stochastic to the deterministic phase.     

Stochastic branching model for hemopoietic progenitor cell differentiation

We present algebraic expressions describing the predictions of a stochastic branching model for differentiation of hemopoietic progenitor cells. The model assumes that there is a fixed probability, p (0 less than or equal to p less than or equal to 1), that commitment to a differentiative event occurs per progenitor cell division for each daughter cell. The model describes properties of in vitro hemopoietic cell differentiation including the population structure at the time the first progenitor cell becomes committed, the number of committed progenitor cells engendered by a single progenitor cell, and the probability of eventual commitment of all daughter cells derived from a single progenitor or stem cell. Application of the model to experimental data obtained from erythroid cultures suggests that the observed data can be explained by the stochastic branching model alone without making the deterministic assumption that there is a differentiative hierarchy in the lineage of the progenitors of erythropoiesis (BFU-E). The qualitative and quantitative aspects of the proposed stochastic model are discussed in conjunction with other analogous stochastic branching models.     

Simplification of Stochastic Chemical Reaction Models with Fast and Slow Dynamics

Biological systems often involve chemical reactions occurring in low-molecule-number regimes, where fluctuations are not negligible and thus stochastic models are required to capture the system behaviour. The resulting models are generally quite large and complex, involving many reactions and species. For clarity and computational tractability, it is important to be able to simplify these systems to equivalent ones involving fewer elements. While many model simplification approaches have been developed for deterministic systems, there has been limited work on applying these approaches to stochastic modelling. Here, we describe a method that reduces the complexity of stochastic biochemical network models, and apply this method to the reduction of a mammalian signalling cascade and a detailed model of the process of bacterial gene expression. Our results indicate that the simplified model gives an accurate representation for not only the average numbers of all species, but also for the associated fluctuations and statistical parameters.