A spatially discrete model for aggregating populations

 In this paper we introduce a spatially discrete model for aggregating populations described by a system of ODEs. We study the long time behavior of the solutions and we show that the model contains mechanisms by which individuals in the population aggregate at particular points in space. Received: 29 June 1996 / Revised version: 5 August 1997     

A simple stochastic model of spatially complex neurons

A method for studying the coding properties of a multicompartmental integrate-and-fire neuron of arbitrary geometry is presented. Depolarization at each compartment evolves like a leaky integrator with an after-firing reset imposed only at the trigger zone. The frequency of firing at the steady-state regime is related to the properties of the multidimensional input. The decreasing variability of subthreshold depolarization from the dendritic tree to the trigger zone is shown for an input that is corrupted by a white noise. The role of a Poissonian noise is also investigated. The proposed method gives an estimate of the mean interspike interval that can be used to study the input output transfer function of the system. Both types of the stochastic inputs result in broadening the transfer function with respect to the deterministic case.     

Persistence of Tb in possums: Insights from a spatially explicit, stochastic model

The majority of models for predicting the dynamics of bovine Tb in brushtail possums in New Zealand owe much to the pioneering work of the late Nigel Barlow. These non-spatial, deterministic models subsumed local disease dynamics by using a heterogeneous mixing term that assumed Tb was confined to a fraction of the population (in patches). However, the underlying mechanism(s) that could result in this heterogeneity of infection risk were obscure. We present a new individual-based, spatial, stochastic simulation model of Tb in possums that provides an explicit mechanism for simulating heterogeneous risk of infection based on a model of individual home range utilisation and disease susceptibility. The manipulation of parameters governing individual utilisation of space also means that processes such as non-linear contact structure can be handled naturally. We use the model to predict the persistence of Tb in possums under scenarios currently implemented for the management of bovine Tb in wildlife and determine conditions under which Tb might be predicted to persist despite control efforts.     

A discrete time stochastic model of a two prey, one predator species interaction

A stochastic discrete time model of a two prey, one predator interaction, an extension of one and two species models proposed by Leslie (1958) and Leslie and Gower, 1958, Leslie and Gower, 1960, is studied. Monte Carlo simulations and the stability properties of the analogous continuous time deterministic model suggest the following hypotheses. (1) The two prey, one predator interaction is in general unstable. The range of parameters allowing coexistence of all three species is small. (2) Deterministically the predator always survives. (3) If the parameters defining the effects of density on the rates of population growth are large, the simulations lead to the rapid extinction of all three species or all but one of the prey species even if the interaction is deterministically stable. (4) The outcome of this three species interaction is largely probabilistic over a wide range of parameters. (5) A prey species with a competitive advantage over a second prey species may still find it difficult to invade and displace the second prey species if the density of the second prey species is high. Increasing the density of the predator offsets this numerical advantage somewhat. (6) The introduction of a predator common to two noncompeting species of prey usually leads to the extinction of one of the prey species. (7) In a stable two prey, one predator interaction the fluctuations of the two prey species are nonperiodic and erratic. The fluctuations of the rarer prey species are damped relative to the commoner species and the fluctuations of the rarer prey species behave as if the series has no fixed mean abundance. The predator population fluctuates with a remarkably constant period. The relevance of these hypotheses to the problem of relating population stability and persistence with the number of species in a community is discussed.     

Patterns of mesenchymal condensation in a multiscale, discrete stochastic model

Cells of the embryonic vertebrate limb in high-density culture undergo chondrogenic pattern formation, which results in the production of regularly spaced “islands” of cartilage similar to the cartilage primordia of the developing limb skeleton. The first step in this process, in vitro and in vivo, is the generation of “cell condensations,” in which the precartilage cells become more tightly packed at the sites at which cartilage will form. In this paper we describe a discrete, stochastic model for the behavior of limb bud precartilage mesenchymal cells in vitro. The model uses a biologically motivated reaction–diffusion process and cell-matrix adhesion (haptotaxis) as the bases of chondrogenic pattern formation, whereby the biochemically distinct condensing cells, as well as the size, number, and arrangement of the multicellular condensations, are generated in a self-organizing fashion. Improving on an earlier lattice-gas representation of the same process, it is multiscale (i.e., cell and molecular dynamics occur on distinct scales), and the cells are represented as spatially extended objects that can change their shape. The authors calibrate the model using experimental data and study sensitivity to changes in key parameters. The simulations have disclosed two distinct dynamic regimes for pattern self-organization involving transient or stationary inductive patterns of morphogens. The authors discuss these modes of pattern formation in relation to available experimental evidence for the in vitro system, as well as their implications for understanding limb skeletal patterning during embryonic development.     

A test of a model for planktivorous filter feeding by bream Abramis brama

Synopsis The planktivorous feeding of bream, Abramis brama on Daphnia hyalina and Bosmina coregoni was analyzed in a stepwise regression analysis with the average size (and standard deviation) of consumed organisms as dependent variable and the size of the fish, the average size (and standard deviation) of the organisms and their density in the environment as independent variables. Three basic predictions on filter feeding were formulated and tested. It was predicted that the (average) prey size should increase with fish size, but that the standard deviation should decline. Secondly the prey size should be strongly correlated with the prey size available and thirdly the prey density should have little effect on the size selection. These hypotheses could not be rejected for bream>10 cm feeding on B. coregoni and for bream>20 cm feeding on D. hyalina. The hypotheses were not valid for smaller bream as these acted as particulate or combined filter- and particulate feeders.     

Determination of cellular rate distributions in microbial cell populations: feeding rates of ciliated protozoa

A novel procedure is proposed for determining distributions of rate properties and correlations of rate with state properties of microbial cell populations. The procedure is novel in that it uses transient data, and thus, it does not require that the population be in balanced growth, although it requires that the population structure does not change during the short transient experiment. The procedure is applied to populations of the ciliated protozoan Tetrahymena to determine ingestion rate variability. The number of ingested microspheres per cell and the single-cell protein content-an indicator of cell size-were directly determined with dual-color flow cytometry. The proposed technique revealed the correlation pattern of the particle ingestion rate with cell size. In particular, ingestion rate was found to be positively correlated with cell size for the smaller feeding cells and to be uncorrelated with size for the larger cells. Using the fact that particle uptake from dilute particle suspensions is a Poisson random process, we determined that the coefficient of variation of the distribution of ingestion rates within the feeding population is about 50%. It was concluded that the dynamics of particle ingestion can be accurately described only if it is realized that particle ingestion rates are distributed. (c) 1993 John Wiley & Sons, Inc.     

A spatially extended stochastic model of the bacterial chemotaxis signalling pathway

We have combined two distinct but related stochastic approaches to model the Escherichia coli chemotaxis pathway. Reactions involving cytosolic components of the pathway were assumed to obey the laws of conventional stochastic chemical kinetics, while the clustered membrane receptors were represented in two-dimensional arrays similar to the Ising model. Receptors were assumed to flip between an active and an inactive state with probabilities dependent upon three energy inputs: ligand binding, methylation level due to adaptation, and the activity of neighbouring receptors. Examination of models with different lattice size and geometry showed that the sensitivity to stimuli increases with lattice size and the nearest-neighbour coupling strength up to a critical point, but this amplification was also accompanied by a proportional increase in steady-state noise. Multiple methylation of receptors resulted in diminished signal-to-noise ratio, but showed improved stability to variation in the coupling strength and increased gain. Under the best conditions the simulated output of a coupled lattice of receptors closely matched the time-course and amplitude found experimentally in living bacteria. The model also has some of the properties of a cellular automaton and shows an unexpected emergence of spatial patterns of methylation within the receptor lattice.     

A data-driven model of the generation of human EEG based on a spatially distributed stochastic wave equation

We discuss a model for the dynamics of the primary current density vector field within the grey matter of human brain. The model is based on a linear damped wave equation, driven by a stochastic term. By employing a realistically shaped average brain model and an estimate of the matrix which maps the primary currents distributed over grey matter to the electric potentials at the surface of the head, the model can be put into relation with recordings of the electroencephalogram (EEG). Through this step it becomes possible to employ EEG recordings for the purpose of estimating the primary current density vector field, i.e. finding a solution of the inverse problem of EEG generation. As a technique for inferring the unobserved high-dimensional primary current density field from EEG data of much lower dimension, a linear state space modelling approach is suggested, based on a generalisation of Kalman filtering, in combination with maximum-likelihood parameter estimation. The resulting algorithm for estimating dynamical solutions of the EEG inverse problem is applied to the task of localising the source of an epileptic spike from a clinical EEG data set; for comparison, we apply to the same task also a non-dynamical standard algorithm.     

A discrete, size-structured model of microbial growth and competition in the chemostat

A nonlinear matrix model of a size-structured microbial population growing on a scarce nutrient in a chemostat, derived by Gage et al. [6], is modified to include two competing populations. It is shown that competitive exclusion results. The winner is the population able to grow at the lower nutrient concentration.Research partially supported by NSF Grant DMS 9300974     

A geometric model of mortality and crop protection for insects feeding on discrete toxicant deposits

Current theory governing the biological effectiveness of toxicants stresses the dose-response relationship and focuses on uniform toxicant distributions in the insect's environment. However, toxicants are seldom uniformly dispersed under field conditions. Toxicant distribution affects bioavailability, but the mechanics of such interactions is not well documented. We present a geometric model of the interactions between insects and heterogeneously distributed toxicants. From the model, we conclude the following: 1) There is an optimal droplet size, and droplets both smaller and larger than this optimum will decrease efficacy. 2) There is an ideal droplet distribution. Droplets should be spaced based on two criteria: calculate the allowable damage, double this quantity, and one lethal deposit should be placed in this area; and define the quantity of leaf the larva could eat before the toxicant decays below the lethal level and place one lethal deposit within this area. 3) Distributions of toxicant where deposits are sublethal will often be ineffective, but the application is wasteful if deposits contain more than a lethal dose. 4) Insect behavior both as individuals and collectively influences the level of crop production provided by an application. This conclusion has implications for both crop protection and natural plant-insect interactions. The effective utilization of new more environmentally sensitive toxicants may depend on how well we understand how heterogeneous toxicant distributions interact with insect behavior to determine the biological outcome.     

A stochastic discrete generation birth,continuous death population growth model and its approximate solution

This paper describes a single species growth model with a stochastic population size dependent number of births occurring at discrete generation times and a continuous population size dependent death rate. An integral equation for a suitable transformation of the limiting population size density function is not in general soluble, but a Gram-Charlier representation procedure, previously used in storage theory, is successfully extended to cover this problem. Examples of logistic and Gompertz type growth are used to illustrate the procedure, and to compare with growth models in random environments. Comments on the biological consequences of these models are also given.Currently at Department of Mathematics, University of MarylandWork partially supported by the Danish Natural Science Research Council and Monash University     

A correct formulation for a spatially implicit predator-prey metacommunity model

In order to mitigate the problem of increasing model complexity with increasing number of occupation states in spatially implicit metacommunity models, the assumption of independency among species distributions is often required. In the present paper, we show that this approach only works correctly if set relations among patch occupancy states are considered adequately. This is illustrated by means of a well-known, although incorrectly formulated, predator-prey metacommunity model devised by Bascompte and Solé [1]. We demonstrate that this model shows anomalous dynamical behavior caused by inconsistence between the model formulation and its assumptions. In order to formalize our finding we develop a corrected model formulation that accounts for the principles of set theory so that the sum of the system compartments change rate is nulled. Applying this method successfully rules out the occurrence of anomalous dynamical behavior found in the original model. Finally we discuss the implications of our findings for the accuracy of model predictions.     

A stochastic model for a progressive chronic disease

For many progressive chronic diseases, there exist useful prognostic indicators for the course of the disease and the survival of the patient. The evolution of such an indicator is modelled as a monotone transformation of a pure birth process with killing. Explicit formulas are derived for the probability distribution of this process at an arbitrary time, the distribution of the first-passage times, the joint distribution of the survival time and the maximum of the process, and the marginals of this joint distribution. In two examples, the general formulas are evaluated in closed form.     

A model of TLR4 signaling and tolerance using a qualitative, particle-event-based method: introduction of spatially configured stochastic reaction chambers (SCSRC)

Introduction

There have been great advances in the examination and characterization of intracellular signaling and synthetic pathways. However, these pathways are generally represented using static diagrams when in reality they exist with considerable dynamic complexity. In addition to the expansion of existing mathematical pathway representation tools (many utilizing systems biology markup language format), there is a growing recognition that spatially explicit modeling methods may be necessary to capture essential aspects of intracellular dynamics. This paper introduces spatially configured stochastic reaction chambers (SCSRC), an agent-based modeling (ABM) framework that incorporates an abstracted molecular ‘event’ rule system with a spatially explicit representation of the relationship between signaling and synthetic compounds. Presented herein is an example of the SCSRC as applied to Toll-like receptor (TLR) 4 signaling and the inflammatory response.

Methods

The underlying rationale for the architecture of the SCSRC is described. A SCSRC model of TLR-4 signaling was developed after a review of the literature regarding TLR-4 signaling and downstream synthetic events. The TLR-4 SCSRC was implemented in the free-ware software platform, Netlogo. A series of in silico experiments were performed to evaluate the response of the TLR-4 SCSRC with respect to response to simulated administration of lipopolysaccharide (LPS). The pro-inflammatory response was represented by simulated secretion of tumor necrosis factor (TNF). Subsequent in silico experiments examined the response to of the TLR-4 SCSRC in terms of a simulated preconditioning effect represented as tolerance of pro-inflammatory signaling to a second dose of LPS.

Results

The SCSRC produces simulated dynamics of TLR-4 signaling in response to LPS stimulation that are qualitatively similar to that reported in the literature. The expression of various components of the signaling cascade demonstrated stochastic noise, consistent with molecular expression data reported in the literature. There is a dose dependent pro-inflammatory response effect seen with increasing initial doses of LPS, and there was also a dose dependent response with respect to preconditioning effect and the establishment of tolerance. Both of these dynamics are consistent with published responses to LPS.

Conclusions

The particle-based, spatially oriented SCSRC model of TLR-4 signaling captures the essential dynamics of the TLR-4 signal transduction cascade, including stochastic signal behavior, dose dependent response, negative feedback control, and preconditioning effect. This is accomplished even given a high degree of molecular event abstraction. The component detail of the SCSRC may allow for sequential parsing of various preconditioning effects, something not possible without computational modeling and simulation, and may give insight into the expected consequences and responses resulting from manipulation of one or many of these modulating factors. The SCSRC is admittedly a work in evolution, and future work will sequentially incorporate additional regulatory mechanisms, both intracellular and paracrine/autocrine, and improved mapping between the spatial chamber configuration and molecular event rules, and experimentally define biochemical reaction rate constants. However, the SCSRC has promise as a highly modular and flexible modeling method that is suited to the dynamic knowledge representation of intracellular processes.     

A general population-genetic model for the production by population structure of spurious genotype-phenotype associations in discrete, admixed or spatially distributed populations

In linkage disequilibrium mapping of genetic variants causally associated with phenotypes, spurious associations can potentially be generated by any of a variety of types of population structure. However, mathematical theory of the production of spurious associations has largely been restricted to population structure models that involve the sampling of individuals from a collection of discrete subpopulations. Here, we introduce a general model of spurious association in structured populations, appropriate whether the population structure involves discrete groups, admixture among such groups, or continuous variation across space. Under the assumptions of the model, we find that a single common principle--applicable to both the discrete and admixed settings as well as to spatial populations--gives a necessary and sufficient condition for the occurrence of spurious associations. Using a mathematical connection between the discrete and admixed cases, we show that in admixed populations, spurious associations are less severe than in corresponding mixtures of discrete subpopulations, especially when the variance of admixture across individuals is small. This observation, together with the results of simulations that examine the relative influences of various model parameters, has important implications for the design and analysis of genetic association studies in structured populations.     

Calcium waves in a model with a random spatially discrete distribution of Ca2+ release sites

We study the propagation of intracellular calcium waves in a model that features Ca2+ release from discrete sites in the endoplasmic reticulum membrane and random spatial distribution of these sites. The results of our simulations qualitatively reproduce the experimentally observed behavior of the waves. When the level of the channel activator inositol trisphosphate is low, the wave undergoes fragmentation and eventually vanishes at a finite distance from the region of initiation, a phenomenon we refer to as an abortive wave. With increasing activator concentration, the mean distance of propagation increases. Above a critical level of activator, the wave becomes stable. We show that the heterogeneous distribution of Ca2+ channels is the cause of this phenomenon.     

A stochastic model of a temperature-dependent population

The theoretical basis is developed for a population model which allows the use of constant temperature experimental data in predicting the size of an insect population for any variable temperature environment. The model is based on a stochastic analysis of an insect's mortality, development, and reproduction response to temperature. The key concept in the model is the utilization of a physiological time scale. Different temperatures affect the population by increasing an individual's physiological age by differing rates. Conditions for the temperature response properties are given which establish the validity of the model for variable temperature regimes. These conditions refer to the relationship between chronological and physiological age. Reasonable agreement between the model and field populations demonstrates the practicality of this approach.