Kinetics of cell-mediated cytotoxicity: Stochastic and deterministic multistage models

One of the body's major defenses against viral diseases and tumors is the killing of abnormal cells by host defense cells, such as T lymphocytes. The mechanism by which killing is accomplished is unknown. Here we develop both stochastic and deterministic models for the kinetics of killing in aggregates which contain a single lymphocyte and multiple target cells (LT_n conjugates), as might be seen early in an immune response, and in aggregates containing multiple lymphocytes and a single target cell (L_nT conjugates), which is characteristic of the late phase of a successful immune response. Comparing our models with data, we rule out the possibility of certain classes of lytic mechanisms and draw attention to the characteristics of likely mechanisms. Our stochastic model can be viewed as a specialized application of queueing theory to cell biology. For certain choices of arrival-time and service-time distributions, we find an exact correspondence between our stochastic and deterministic models.     

Analysis of the activity of single neurons in stochastic settings

This paper presents a new way of modeling the activity of single neurons in stochastic settings. It incorporates in a natural way many physiological mechanisms not usually found in stochastic models, such as spatial integration, non-linear membrane characteristics and non-linear interactions between excitation and inhibition. The model is based on the fact that most of the neuronal inputs have a finite lifetime. Thus, the stochastic input can be modeled as a simple finite markov chain, and the membrane potential becomes a function of the state of this chain. Firing occurs at states whose membrane potential is above threshold. The main mathematical results of the model are: (i) the input-output firing rate curve is convex at low firing rates and is saturated at high firing rates, and (ii) at low firing rates, firing usually occurs when there is synchronous convergence of many excitatory events.     

A population‐based temporal logic gate for timing and recording chemical events

Engineered bacterial sensors have potential applications in human health monitoring, environmental chemical detection, and materials biosynthesis. While such bacterial devices have long been engineered to differentiate between combinations of inputs, their potential to process signal timing and duration has been overlooked. In this work, we present a two‐input temporal logic gate that can sense and record the order of the inputs, the timing between inputs, and the duration of input pulses. Our temporal logic gate design relies on unidirectional DNA recombination mediated by bacteriophage integrases to detect and encode sequences of input events. For an E. coli strain engineered to contain our temporal logic gate, we compare predictions of Markov model simulations with laboratory measurements of final population distributions for both step and pulse inputs. Although single cells were engineered to have digital outputs, stochastic noise created heterogeneous single‐cell responses that translated into analog population responses. Furthermore, when single‐cell genetic states were aggregated into population‐level distributions, these distributions contained unique information not encoded in individual cells. Thus, final differentiated sub‐populations could be used to deduce order, timing, and duration of transient chemical events.     

Population extinction and quasi-stationary behavior in stochastic density-dependent structured models

Density-independent and density-dependent, stochastic and deterministic, discrete-time, structured models are formulated, analysed and numerically simulated. A special case of the deterministic, density-independent, structured model is the well-known Leslie age-structured model. The stochastic, density-independent model is a multitype branching process. A review of linear, density-independent models is given first, then nonlinear, density-dependent models are discussed. In the linear, density-independent structured models, transitions between states are independent of time and state. Population extinction is determined by the dominant eigenvalue λ of the transition matrix. If λ ≤ 1, then extinction occurs with probability one in the stochastic and deterministic models. However, if λ > 1, then the deterministic model has exponential growth, but in the stochastic model there is a positive probability of extinction which depends on the fixed point of the system of probability generating functions. The linear, density-independent, stochastic model is generalized to a nonlinear, density-dependent one. The dependence on state is in terms of a weighted total population size. It is shown for small initial population sizes that the density-dependent, stochastic model can be approximated by the density-independent, stochastic model and thus, the extinction behavior exhibited by the linear model occurs in the nonlinear model. In the deterministic models there is a unique stable equilibrium. Given the population does not go extinct, it is shown that the stochastic model has a quasi-stationary distribution with mean close to the stable equilibrium, provided the population size is sufficiently large. For small values of the population size, complete extinction can be observed in the simulations. However, the persistence time increases rapidly with the population size. This author received partial support by the National Science Foundation grant # DMS-9626417.     

Stochastic flowering phenology in Dactylis glomerata populations described by Markov chain modelling

Understanding the relationship between flowering patterns and pollen dispersal is important in climate change modelling, pollen forecasting, forestry and agriculture. Enhanced understanding of this connection can be gained through detailed spatial and temporal flowering observations on a population level, combined with modelling simulating the dynamics. Species with large distribution ranges, long flowering seasons, high pollen production and naturally large populations can be used to illustrate these dynamics. Revealing and simulating species-specific demographic and stochastic elements in the flowering process will likely be important in determining when pollen release is likely to happen in flowering plants. Spatial and temporal dynamics of eight populations of Dactylis glomerata were collected over the course of two years to determine high-resolution demographic elements. Stochastic elements were accounted for using Markov chain approaches in order to evaluate tiller-specific contribution to overall population dynamics. Tiller-specific developmental dynamics were evaluated using three different RV matrix correlation coefficients. We found that the demographic patterns in population development were the same for all populations with key phenological events differing only by a few days over the course of the seasons. Many tillers transitioned very quickly from non-flowering to full flowering, a process that can be replicated with Markov chain modelling. Our novel approach demonstrates the identification and quantification of stochastic elements in the flowering process of D. glomerata, an element likely to be found in many flowering plants. The stochastic modelling approach can be used to develop detailed pollen release models for Dactylis, other grass species and probably other flowering plants.

     

Estimating similarity of communities: a parametric approach to spatio‐temporal analysis of species diversity

Several stochastic models with environmental noise generate spatio‐temporal Gaussian fields of log densities for the species in a community. Combinations of such models for many species often lead to lognormal species abundance distributions. In spatio‐temporal analysis it is often realistic to assume that the same species are expected to occur at different times and/or locations because extinctions are rare events. Spatial and temporal β‐diversity can then be analyzed by studying pairs of communities at different times or locations defined by a bivariate lognormal species abundance model in which a single correlation occurs. This correlation, which is a measure of similarity between two communities, can be estimated from samples even if the sampling intensities vary and are unknown, using the bivariate Poisson lognormal distribution. The estimators are approximately unbiased, although each specific correlation may be rather uncertain when the sampling effort is low with only a small fraction of the species represented in the samples. An important characteristic of this community correlation is that it relates to the classical Jaccard‐ or the Sørensen‐indices of similarity based on the number of species present or absent in two communities. However, these indices calculated from samples of species in a community do not necessarily reflect similarity of the communities because the observed number of species depends strongly on the sampling intensities. Thus, we propose that our community correlation should be considered as an alternative to these indices when comparing similarity of communities. We illustrate the application of the correlation method by computing the similarity between temperate bird communities.     

Weak convergence of discrete time non-markovian processes related to selection models in population genetics

We consider discrete time stochastic processes defined by solutions to some non-linear difference equations whose coefficients are autocorrelated random sequences. It is proved that these processes converge weakly in D[0, T] to diffusion processes, under the assumption that the random sequences satisfy some mixing condition. Diffusion approximation for stochastic selection models in population genetics is discussed, as the application of this limit theorem.     

Spatial data replacing temporal data in population viability analyses: An empirical investigation for plants

In conservation management, there is an urgent need for estimates of population viability and for knowledge of the contributions of different life-history stages to population growth rates. Collection of long-term demographic data from a study population is time-consuming and may considerably delay the start of proper management actions. We examined the possibility of replacing a long-term temporal data set (demographic data from several years within a population) with a short-term spatial data set (demographic data from different populations for the same subset of two continuous years) for stochastic estimates of population viability. Using matrix population models for ten perennial plant species, we found that the matrix elements of spatial data sets often deviated from those of temporal data sets and that matrix elements generally varied more spatially than temporally. The appropriateness of replacing temporal data with spatial data depended on the subset of years and populations used to estimate stochastic population growth rates (log λ_s). Still, the precision of log λ_s estimates measured as variation in the yearly change of logarithmic population size rarely differed significantly between the spatial and temporal data sets. Since a spatiotemporal comparison of matrix elements and their variation cannot be used to assess whether spatial and temporal data sets are interchangeable, we recommend further research on the topic.     

Small-scale spatial and temporal variation in the demographic processes underlying the large-scale decline of eiders in the Baltic Sea

The application of uniform conservation schemes often fails to account for small-scale spatial variation in the drivers of population decline. Demographic comparisons of imperilled populations across locations are therefore crucial for successful conservation, but progress is hampered by lack of long-term data from more than a single population. The recent large-scale decline of eider ducks (Somateria mollissima) in the Baltic Sea is ideal for determining to what extent mechanisms underlying population decline can be extrapolated over larger areas. We utilized stochastic demographic methods incorporating both environmental and sampling variation to assess small-scale spatial and temporal variation in the population dynamics of eiders at Söderskär (eastern range-margin) and Tvärminne (core breeding area), situated 130 km apart. The stochastic growth rate models accurately predicted the observed differences in the rate of decline between sites and time periods. At Söderskär, established breeder survival had by far the greatest elasticity, whereas elasticity was more evenly distributed among vital rates at Tvärminne. Although the study sites showed the single largest difference in fecundity, stochastic life table response experiment analyses revealed that reduced adult female survival at Tvärminne mainly determined the observed difference in growth rates between sites. In contrast, reduced fecundity primarily differentiated the past population increase from the present population decline at Söderskär. Our results demonstrate that different mechanisms may be associated with population decline across adjacent geographic locations, and indicate that dispersal of first-time breeders may be important for population dynamics. Safeguarding adult female survival and/or fecundity should be prioritized in management efforts.     

Structured population dynamics: continuous size and discontinuous stage structures

A nonlinear stochastic model for the dynamics of a population with either a continuous size structure or a discontinuous stage structure is formulated in the Eulerian formalism. It takes into account dispersion effects due to stochastic variability of the development process of the individuals. The discrete equations of the numerical approximation are derived, and an analysis of the existence and stability of the equilibrium states is performed. An application to a copepod population is illustrated; numerical results of Eulerian and Lagrangian models are compared.      

Integrated single‐cell analysis shows Pichia pastoris secretes protein stochastically

The production of heterologous proteins by secretion from cellular hosts is an important determinant for the cost of biotherapeutics. A single‐cell analytical method called microengraving was used to examine the heterogeneity in secretion by the methylotrophic yeast Pichia pastoris. We show that constitutive secretion of a human Fc fragment by P. pastoris is not cell‐cycle dependent, but rather fluctuates between states of high and low productivity in a stochastic manner. Biotechnol. Bioeng. 2010;106: 319–325. © 2010 Wiley Periodicals, Inc.     

On the survival of populations in a heterogeneous and variable environment

Summary The survival time of small and isolated populations will often be relatively low, by which the survival of species living in such a way will depend on powers of dispersal sufficiently high to result in a rate of population foundings that about compensates the rate of population extinctions. The survival time of composite populations uninterruptedly inhabiting large and heterogeneous areas, highly depends on the extent to which the numbers fluctuate unequally in the different subpopulations. The importance of this spreading of the risk of extinction over differently fluctuating subpopulations is demonstrated by comparing over 19 years the fluctuation patterns of the composite populations of two carabid species, Pterostichus versicolor with unequally fluctuating subpopulations, and Calathus melanocephalus with subpopulations fluctuating in parallel, both uninterruptedly occupying the same large heath area. The conclusions from the field data are checked by simulating the fluctuation patterns of these populations, and thus directly estimating survival times. It thus appeared that the former species can be expected to survive more than ten times better than the latter (other things staying the same). These simulations could also be used to study the possible influence of various density restricting processes in populations already fluctuating according to some pattern. As could be expected, the survival time of a population, which shows a tendency towards an upward trend in numbers, will be favoured by some kind of density restriction, but the degree to which these restrictions are density-dependent appeared to be immaterial. Density reductions that are about adequate on the average need even not occur at high densities only, if only the chance of occurrence at very low densities is low. The density-level at which a population is generally fluctuating appeared to be less important for survival than the fluctuation pattern itself, except for very low density levels, of course. The different ways in which deterministic and stochastic processes may interact and thus determine the fluctuations of population numbers are discussed. It is concluded that some stochastic processes will operate everywhere and will thus necessarily result in density fluctuations; such an omnipresence is much less imperative, however, for density-dependent processes, by which population models should primarily be stochastic models. However, if density-dependent processes are added to model populations, that are already fluctuating stochastically the effects are taken up into the general, stochastic fluctuation pattern, without altering it fundamentally.Communication No. 228 of the Biological Station WijsterDedicated to Professor Michael Evenari     

Resetting biological oscillators—A stochastic approach

We present a stochastic approach to phase-resetting of an ensemble of oscillators. In order to describe stimulation-induced dynamical phenomena we develop a stochastic model which consists of an ensemble of phase oscillators interacting via random forces. Every single oscillator is submitted to a phase stimulus. The ensemble's dynamics is determined by a Fokker-Planck equation. The stationary states are calculated explicitly, whereas the transients are analysed numerically. If the stimulus of a given (non-vanishing) intensity is administered at a critical initial cluster phase for a critical duration T _crit the ensemble's synchronized oscillation is annihilated. A transition from type 1 resetting to type 0 resetting occurs when the stimulation duration exceeds T _crit. Stimulation causes a shift of the mean frequency of every single oscillator. This frequency shift is explicitly calculated by deriving the mean first passage time. The model shows that there is a subcritical intensity which is connected with an enhanced vulnerability to stimulation. The desynchronized states, the so-called black holes, are typically associated with a double peak in the ensemble's phase distribution. This is important for analysing experimental data because simple peak-detection algorithms are not able to extract the underlying dynamics.Our results are discussed from the experimentator's point of view so that the insights derived from our model can improve data analysis and design of stimulation experiments.     

A dynamic theory of coordination of discrete movement

The concepts of pattern dynamics and their adaptation through behavioral information, developed in the context of rhythmic movement coordination, are generalized to describe discrete movements of single components and the coordination of multiple components in discrete movement. In a first step we consider only one spatial component and study the temporal order inherent in discrete movement in terms of stable, reproducible space-time relationships. The coordination of discrete movement is captured in terms of relative timing. Using an exactly solvable nonlinear oscillator as a mathematical model, we show how the timing properties of discrete movement can be described by these pattern dynamics and discuss the relation of the pattern variables to observable end-effector movement. By coupling several such component dynamics in a fashion analogous to models of rhythmic movement coordination we capture the coordination of discrete movements of two components. We find the tendency to synchronize the component movements as the discrete analogon of in-phase locking and study its breakdown when the components become too different in their dynamic properties. The concept of temporal stability leads to the prediction that remote compensatory responses occur such as the restore synchronization when one component is perturbed. This prediction can be used to test the theory. We find that the discrete analogon to antiphase locking in rhythmic movement is a tendency to move sequentially, a finding that can also be subjected to empirical test.     

CRANIAL SIZE VARIATION AND LINEAGE DIVERSITY IN EARLY PLEISTOCENE HOMO

A recent article in this journal concluded that a sample of early Pleistocene hominin crania assigned to genus Homo exhibits a pattern of size variation that is time dependent, with specimens from different time periods being more different from each other, on average, than are specimens from the same time period. The authors of this study argued that such a pattern is not consistent with the presence of multiple lineages within the sample, but rather supports the hypothesis that the fossils represent an anagenetically evolving lineage (i.e., an evolutionary species). However, the multiple‐lineage models considered in that study do not reflect the multiple‐species alternatives that have been proposed for early Pleistocene Homo. Using simulated data sets, I show that fossil assemblages that contain multiple lineages can exhibit the time‐dependent pattern of variation specified for the single‐lineage model under certain conditions, particularly when temporal overlap among fossil specimens attributed to the lineages is limited. These results do not reject the single‐lineage hypothesis, but they do indicate that rejection of multiple lineages in the early Pleistocene Homo fossil record is premature, and that other sources of variation, such as differences in cranial shape, should be considered.     

A Stochastic Metapopulation Model with Variability in Patch Size and Position

Analytically tractable metapopulation models usually assume that every patch is identical, which limits their application to real metapopulations. We describe a new single species model of metapopulation dynamics that allows variation in patch size and position. The state of the metapopulation is defined by the presence or absence of the species in each patch. For a system of n patches, this gives 2ⁿ possible states. We show how to construct and analyse a matrix describing transitions between all possible states by first constructing separate extinction and colonisation matrices. We illustrate the model′s application to metapopulations by considering an example of malleefowl, Leipoa ocellata, in southern Australia, and calculate extinction probabilities and quasi-stationary distributions. We investigate the relative importance of modelling the particular arrangement of patches and the variation in patch sizes for this metapopulation and we use the model to examine the effects of further habitat loss on extinction probabilities.     

Temporal clustering of tropical cyclones on the Great Barrier Reef and its ecological importance

Tropical cyclones have been a major cause of reef coral decline during recent decades, including on the Great Barrier Reef (GBR). While cyclones are a natural element of the disturbance regime of coral reefs, the role of temporal clustering has previously been overlooked. Here, we examine the consequences of different types of cyclone temporal distributions (clustered, stochastic or regular) on reef ecosystems. We subdivided the GBR into 14 adjoining regions, each spanning roughly 300 km, and quantified both the rate and clustering of cyclones using dispersion statistics. To interpret the consequences of such cyclone variability for coral reef health, we used a model of observed coral population dynamics. Results showed that clustering occurs on the margins of the cyclone belt, being strongest in the southern reefs and the far northern GBR, which also has the lowest cyclone rate. In the central GBR, where rates were greatest, cyclones had a relatively regular temporal pattern. Modelled dynamics of the dominant coral genus, Acropora, suggest that the long-term average cover might be more than 13 % greater (in absolute cover units) under a clustered cyclone regime compared to stochastic or regular regimes. Thus, not only does cyclone clustering vary significantly along the GBR but such clustering is predicted to have a marked, and management-relevant, impact on the status of coral populations. Additionally, we use our regional clustering and rate results to sample from a library of over 7000 synthetic cyclone tracks for the GBR. This allowed us to provide robust reef-scale maps of annual cyclone frequency and cyclone impacts on Acropora. We conclude that assessments of coral reef vulnerability need to account for both spatial and temporal cyclone distributions.

     

A Stochastic Compartmental Model with Long Lasting Infusion

Most of the compartmental models in current use to model pharmacokinetic systems are deterministic. Stochastic formulations of pharmacokinetic compartmental models introduce stochasticity through either a probabilistic transfer mechanism or the randomization of the transfer rate constants. In this paper we consider a linear stochastic differential equation (LSDE) which represents a stochastic version of a one‐compartment linear model when input function undergoes random fluctuations. The solution of the LSDE, its mean value and covariance structure are derived. An explicit likelihood function is obtained either when the process is observed continuously over a period of time or when sampled data are available, as it is generally feasible. We discuss some asymptotic properties of the maximum likelihood estimators for the model parameters. Furthermore we develop expressions for two random variables of interest in pharmacokinetics: the area under the time‐concentration curve, M₀(T), and the plateau concentration, x_ss. Finally the estimation procedure is illustrated by an application to real data.     

Integrating species distribution models and interacting particle systems to predict the spread of an invasive alien plant

Aim We demonstrate how to integrate two widely used tools for modelling the spread of invasive plants, and compare the performance of the combined model with that of its individual components using the recent range dynamics of the invasive annual weed Ambrosia artemisiifolia L. Location Austria. Methods Species distribution models, which deliver habitat‐based information on potential distributions, and interacting particle systems, which simulate spatio‐temporal range dynamics as dependent on neighbourhood configurations, were combined into a common framework. We then used the combined model to simulate the invasion of A. artemisiifolia in Austria between 1990 and 2005. For comparison, simulations were also performed with models that accounted only for habitat suitability or neighbourhood configurations. The fit of the three models to the data was assessed by likelihood ratio tests, and simulated invasion patterns were evaluated against observed ones in terms of predictive discrimination ability (area under the receiver operating characteristic curve, AUC) and spatial autocorrelation (Moran’s I). Results The combined model fitted the data significantly better than the single‐component alternatives. Simulations relying solely on parameterized spread kernels performed worst in terms of both AUC and spatial pattern formation. Simulations based only on habitat information correctly predicted infestation of susceptible areas but reproduced the autocorrelated patterns of A. artemisiifolia expansion less adequately than did the integrated model. Main conclusions Our integrated modelling approach offers a flexible tool for forecasts of spatio‐temporal invasion patterns from landscape to regional scales. As a further advantage, scenarios of environmental change can be incorporated consistently by appropriately updating habitat suitability layers. Given the susceptibility of many alien plants, including A. artemisiifolia, to both land use and climate changes, taking such scenarios into account will increasingly become relevant for the design of proactive management strategies.     

Microstructure of common reef-building coral genera Acropora, Pocillopora, Goniastrea and Porites: constraints on spatial resolution in geochemical sampling

Scleractinian corals are increasingly used as recorders of modern and paleoclimates. The microstructure of four common reef-building coral genera is documented here: Acropora, Pocillopora, Goniastrea, and Porites. This study highlights the complexity and spatial variability of skeletal growth in different coral genera and suggests that a single growth model is too generalized to allow the accurate depiction of the variability observed in the four genera studied. New models must be introduced in order for coral skeletogenesis to be understood adequately to allow coral skeletons to serve as repositories of temporally constrained geochemical data. Owing to differences in microstructural patterns in different genera, direct observation of microstructural elements and growth lines may be necessary to allow microsamples to be placed into series that represent temporal sequences with known degrees of time averaging. Such data are critical for constraining microsampling strategies aimed at developing true time series geochemical data at very fine spatial and temporal scales.