Hybrid stochastic and deterministic simulations of calcium blips

Intracellular calcium release is a prime example for the role of stochastic effects in cellular systems. Recent models consist of deterministic reaction-diffusion equations coupled to stochastic transitions of calcium channels. The resulting dynamics is of multiple time and spatial scales, which complicates far-reaching computer simulations. In this article, we introduce a novel hybrid scheme that is especially tailored to accurately trace events with essential stochastic variations, while deterministic concentration variables are efficiently and accurately traced at the same time. We use finite elements to efficiently resolve the extreme spatial gradients of concentration variables close to a channel. We describe the algorithmic approach and we demonstrate its efficiency compared to conventional methods. Our single-channel model matches experimental data and results in intriguing dynamics if calcium is used as charge carrier. Random openings of the channel accumulate in bursts of calcium blips that may be central for the understanding of cellular calcium dynamics.     

Return map approach to description of the deterministic chaos in cytosolic calcium oscillations

A modified method of the return map reconstruction is proposed. The method is applied to the analysis of intracellular calcium oscillations. On the basis of the approach developed, these oscillations are recognized as low-dimensional deterministic chaotic process.     

Establishing the stochastic nature of intracellular calcium oscillations from experimental data

Calcium has been established as a key messenger in both intra- and intercellular signaling. Experimentally observed intracellular calcium responses to different agonists show a variety of behaviors from simple spiking to complex oscillatory regimes. Here we study typical experimental traces of calcium oscillations in hepatocytes obtained in response to phenylephrine and ATP. The traces were analyzed with methods of nonlinear time series analysis in order to determine the stochastic/deterministic nature of the intracellular calcium oscillations. Despite the fact that the oscillations appear, visually, to be deterministic yet perturbed by noise, our analyses provide strong evidence that the measured calcium traces in hepatocytes are prevalently of stochastic nature. In particular, bursting calcium oscillations are temporally correlated Gaussian series distorted by a monotonic, instantaneous, time-independent function, whilst the spiking behavior appears to have a dynamical nonlinear component whereby the overall determinism level is still low. The biological importance of this finding is discussed in relation to the mechanisms incorporated in mathematical models as well as the role of stochasticity and determinism at cellular and tissue levels which resemble typical statistical and thermodynamic effects in physics.     

A mesoscopic stochastic mechanism of cytosolic calcium oscillations

Based on a model of intracellular calcium (Ca(2+)) oscillation with self-modulation of inositol 1,4,5-trisphosphate signal, the mesoscopic stochastic differential equations for the intracellular Ca(2+) oscillations are theoretically derived by using the chemical Langevin equation method. The effects of the finite biochemical reaction molecule number on both simple and complex cytosolic Ca(2+) oscillations are numerically studied. In the case of simple intracellular Ca(2+) oscillation, it is found that, with the increase of molecule number, the coherence resonance or autonomous resonance phenomena can occur for some external stimulation parameter values. In the cases of complex cytosolic Ca(2+) oscillations, each extremum of concentration of cytosolic Ca(2+) oscillations corresponds to a peak in the histogram of Ca(2+) concentration, and the most probability appeared during the bursting plateau level for bursting, but at the largest minimum of Ca(2+) concentration for chaos. For quasi-periodicity, however, there are only two peaks in the histogram of Ca(2+) concentration, and the most probability is located at low concentration state.     

Oligomeric protein associations: transition from stochastic to deterministic equilibrium

Transfer of electronic excitation energy (sensitized fluorescence) between donor and acceptor fluorophores separately attached to dimer or tetramer proteins is used to demonstrate the exchange of subunits among the undissociated particles. In dimers subjected to a pressure that produces half-dissociation, the exchange occurs at a rate that approaches the rate of dissociation. In the tetramers of glyceraldehydephosphate dehydrogenase and lactate dehydrogenase at 0 degrees C, the times for subunit exchange are nearly 2 orders of magnitude, and at room temperature 5-10 times longer than the time required to reach the dissociation equilibrium. By application of a novel method, pressure is shown to preferentially increase the rate of dissociation in dimers and decrease the rate of association in tetramers. From these observations, we conclude that the tetramers constitute a heterogeneous population, the members of which are dissociated by pressure according to individual molecular properties that can be retained over periods of time much longer than the time for equilibration of the dissociation. The dissociation of dimers exhibits the characteristics of the classical stochastic chemical equilibria, while those of the tetramers, like the more complex protein aggregates, must already be considered similar to the deterministic mechanical equilibria of macroscopic bodies.     

The transition between stochastic and deterministic behavior in an excitable gene circuit

We explore the connection between a stochastic simulation model and an ordinary differential equations (ODEs) model of the dynamics of an excitable gene circuit that exhibits noise-induced oscillations. Near a bifurcation point in the ODE model, the stochastic simulation model yields behavior dramatically different from that predicted by the ODE model. We analyze how that behavior depends on the gene copy number and find very slow convergence to the large number limit near the bifurcation point. The implications for understanding the dynamics of gene circuits and other birth-death dynamical systems with small numbers of constituents are discussed.     

Switching from simple to complex oscillations in calcium signaling

We present a new model for calcium oscillations based on experiments in hepatocytes. The model considers feedback inhibition on the initial agonist receptor complex by calcium and activated phospholipase C, as well as receptor type-dependent self-enhanced behavior of the activated G(alpha) subunit. It is able to show simple periodic oscillations and periodic bursting, and it is the first model to display chaotic bursting in response to agonist stimulations. Moreover, our model offers a possible explanation for the differences in dynamic behavior observed in response to different agonists in hepatocytes.     

Transition between stochastic evolution and deterministic evolution in the presence of selection: general theory and application to virology.

We present here a self-contained analytic review of the role of stochastic factors acting on a virus population. We develop a simple one-locus, two-allele model of a haploid population of constant size including the factors of random drift, purifying selection, and random mutation. We consider different virological experiments: accumulation and reversion of deleterious mutations, competition between mutant and wild-type viruses, gene fixation, mutation frequencies at the steady state, divergence of two populations split from one population, and genetic turnover within a single population. In the first part of the review, we present all principal results in qualitative terms and illustrate them with examples obtained by computer simulation. In the second part, we derive the results formally from a diffusion equation of the Wright-Fisher type and boundary conditions, all derived from the first principles for the virus population model. We show that the leading factors and observable behavior of evolution differ significantly in three broad intervals of population size, N. The "neutral limit" is reached when N is smaller than the inverse selection coefficient. When N is larger than the inverse mutation rate per base, selection dominates and evolution is "almost" deterministic. If the selection coefficient is much larger than the mutation rate, there exists a broad interval of population sizes, in which weakly diverse populations are almost neutral while highly diverse populations are controlled by selection pressure. We discuss in detail the application of our results to human immunodeficiency virus population in vivo, sampling effects, and limitations of the model.     

Sustained oscillations in stochastic systems

Many non-linear deterministic models for interacting populations present damped oscillations towards the corresponding equilibrium values. However, simulations produced with related stochastic models usually present sustained oscillations which preserve the natural frequency of the damped oscillations of the deterministic model but showing non-vanishing amplitudes. The relation between the amplitude of the stochastic oscillations and the values of the equilibrium populations is not intuitive in general but scales with the square root of the populations when the ratio between different populations is kept fixed. In this work, we explain such phenomena for the case of a general epidemic model. We estimate the stochastic fluctuations of the populations around the equilibrium point in the epidemiological model showing their (approximated) relation with the mean values.     

Constructing stochastic models from deterministic process equations by propensity adjustment

Background

Models of biochemical systems are typically complex, which may complicate the discovery of cardinal biochemical principles. It is therefore important to single out the parts of a model that are essential for the function of the system, so that the remaining non-essential parts can be eliminated. However, each component of a mechanistic model has a clear biochemical interpretation, and it is desirable to conserve as much of this interpretability as possible in the reduction process. Furthermore, it is of great advantage if we can translate predictions from the reduced model to the original model.

Results

In this paper we present a novel method for model reduction that generates reduced models with a clear biochemical interpretation. Unlike conventional methods for model reduction our method enables the mapping of predictions by the reduced model to the corresponding detailed predictions by the original model. The method is based on proper lumping of state variables interacting on short time scales and on the computation of fraction parameters, which serve as the link between the reduced model and the original model. We illustrate the advantages of the proposed method by applying it to two biochemical models. The first model is of modest size and is commonly occurring as a part of larger models. The second model describes glucose transport across the cell membrane in baker's yeast. Both models can be significantly reduced with the proposed method, at the same time as the interpretability is conserved.

Conclusions

We introduce a novel method for reduction of biochemical models that is compatible with the concept of zooming. Zooming allows the modeler to work on different levels of model granularity, and enables a direct interpretation of how modifications to the model on one level affect the model on other levels in the hierarchy. The method extends the applicability of the method that was previously developed for zooming of linear biochemical models to nonlinear models.     

Contact tracing in stochastic and deterministic epidemic models

We consider a simple unstructured individual based stochastic epidemic model with contact tracing. Even in the onset of the epidemic, contact tracing implies that infected individuals do not act independent of each other. Nevertheless, it is possible to analyze the embedded non-stationary Galton-Watson process. Based upon this analysis, threshold theorems and also the probability for major outbreaks can be derived. Furthermore, it is possible to obtain a deterministic model that approximates the stochastic process, and in this way, to determine the prevalence of disease in the quasi-stationary state and to investigate the dynamics of the epidemic.     

Extinction thresholds in deterministic and stochastic epidemic models

The basic reproduction number, ?(0), one of the most well-known thresholds in deterministic epidemic theory, predicts a disease outbreak if ?(0)>1. In stochastic epidemic theory, there are also thresholds that predict a major outbreak. In the case of a single infectious group, if ?(0)>1 and i infectious individuals are introduced into a susceptible population, then the probability of a major outbreak is approximately 1-(1/?(0))( i ). With multiple infectious groups from which the disease could emerge, this result no longer holds. Stochastic thresholds for multiple groups depend on the number of individuals within each group, i ( j ), j=1, …, n, and on the probability of disease extinction for each group, q ( j ). It follows from multitype branching processes that the probability of a major outbreak is approximately [Formula: see text]. In this investigation, we summarize some of the deterministic and stochastic threshold theory, illustrate how to calculate the stochastic thresholds, and derive some new relationships between the deterministic and stochastic thresholds.     

Evolution of cardiac calcium waves from stochastic calcium sparks

We present a model that provides a unified framework for studying Ca2+ sparks and Ca2+ waves in cardiac cells. The model is novel in combining 1) use of large currents (approximately 20 pA) through the Ca2+ release units (CRUs) of the sarcoplasmic reticulum (SR); 2) stochastic Ca2+ release (or firing) of CRUs; 3) discrete, asymmetric distribution of CRUs along the longitudinal (separation distance of 2 microm) and transverse (separated by 0.4-0.8 microm) directions of the cell; and 4) anisotropic diffusion of Ca2+ and fluorescent indicator to study the evolution of Ca2+ waves from Ca2+ sparks. The model mimics the important features of Ca2+ sparks and Ca2+ waves in terms of the spontaneous spark rate, the Ca2+ wave velocity, and the pattern of wave propagation. Importantly, these features are reproduced when using experimentally measured values for the CRU Ca2+ sensitivity (approximately 15 microM). Stochastic control of CRU firing is important because it imposes constraints on the Ca2+ sensitivity of the CRU. Even with moderate (approximately 5 microM) Ca2+ sensitivity the very high spontaneous spark rate triggers numerous Ca2+ waves. In contrast, a single Ca2+ wave with arbitrarily large velocity can exist in a deterministic model when the CRU Ca2+ sensitivity is sufficiently high. The combination of low CRU Ca2+ sensitivity (approximately 15 microM), high cytosolic Ca2+ buffering capacity, and the spatial separation of CRUs help control the inherent instability of SR Ca2+ release. This allows Ca2+ waves to form and propagate given a sufficiently large initiation region, but prevents a single spark or a small group of sparks from triggering a wave.     

Pharmacologically induced calcium oscillations protect neurons from increases in cytosolic calcium after trauma

Increases in cytosolic calcium ([Ca(2+)](i)) following mechanical injury are often considered a major contributing factor to the cellular sequelae in traumatic brain injury (TBI). However, very little is known on how developmental changes may affect the calcium signaling in mechanically injured neurons. One key feature in the developing brain that may directly impact its sensitivity to stretch is the reduced inhibition which results in spontaneous [Ca(2+)](i) oscillations. In this study, we examined the mechanism of stretch-induced [Ca(2+)](i) transients in 18-days in vitro (DIV) neurons exhibiting bicuculline-induced [Ca(2+)](i) oscillations. We used an in vitro model of mechanical trauma to apply a defined uniaxial strain to cultured cortical neurons and used increases in [Ca(2+)](i) as a measure of the neuronal response to the stretch insult. We found that stretch-induced increases in [Ca(2+)](i) in 18-DIV neurons were inhibited by pretreatment with either the NMDA receptor antagonist, APV [D(-)-2-Amino-5-phosphonopentanoic acid], or by depolymerizing the actin cytoskeleton prior to stretch. Blocking synaptic NMDA receptors prior to stretch significantly attenuated most of the [Ca(2+)](i) transient. In comparison, cultures with pharmacologically induced [Ca(2+)](i) oscillations showed a substantially reduced [Ca(2+)](i) peak after stretch. We provide evidence showing that a contributing factor to this mechanical desensitization from induced [Ca(2+)](i) oscillations is the PKC-mediated uncoupling of NMDA receptors (NMDARs) from spectrin, an actin-associated protein, thereby rendering neurons insensitive to stretch. These results provide novel insights into how the [Ca(2+)](i) response to stretch is initiated, and how reduced inhibition - a feature of the developing brain - may affect the sensitivity of the immature brain to trauma.     

Blowing-up of deterministic fixed points in stochastic population dynamics

We discuss the stochastic dynamics of biological (and other) populations presenting a limit behaviour for large environments (called deterministic limit) and its relation with the dynamics in the limit. The discussion is circumscribed to linearly stable fixed points of the deterministic dynamics, and it is shown that the cases of extinction and non-extinction equilibriums present different features. Mainly, non-extinction equilibria have associated a region of stochastic instability surrounded by a region of stochastic stability. The instability region does not exist in the case of extinction fixed points, and a linear Lyapunov function can be associated with them. Stochastically sustained oscillations of two subpopulations are also discussed in the case of complex eigenvalues of the stability matrix of the deterministic system.     

Superiority of single covalent modification in specificity: From deterministic to stochastic viewpoint

Covalent modification(s) are required in many signaling pathways. It has been discussed from a deterministic viewpoint that dual covalent modification is more favorable than single covalent modification for signaling specificity. However, whether this conclusion is feasible in stochastic situation has not yet been studied. To study the role of covalent modification in the specificity of a stochastic signaling pathway, we here simulate the dynamics of a transiently stimulated signaling pathway, considering the influence of the stochasticity arising from the low molecule number of reactants. It turns out that the specificity of dual covalent modification would be worse than that of single covalent modification when the number of molecules is in some biologically plausible range. We further discuss some factors that have potential influence on specificity, such as the rates of the upstream reaction cycle of the covalent modification(s), the duration and the magnitude of the transient stimulus. Our numerical results indicate that whether dual or single covalent modification(s) is better in specificity also depends on these factors. Superiority of single covalent modification in specificity would arise if the stimulus is weak and transient, or if it is embedded downstream of a reaction whose activation rate is slow while deactivation rate is fast. The relevance of these conclusions to signal transduction is briefly discussed.     

Multi-patch deterministic and stochastic models for wildlife diseases

Spatial heterogeneity and host demography have a direct impact on the persistence or extinction of a disease. Natural or human-made landscape features such as forests, rivers, roads, and crops are important to the persistence of wildlife diseases. Rabies, hantaviruses, and plague are just a few examples of wildlife diseases where spatial patterns of infection have been observed. We formulate multi-patch deterministic and stochastic epidemic models and use these models to investigate problems related to disease persistence and extinction. We show in some special cases that a unique disease-free equilibrium exists. In these cases, a basic reproduction number ?₀ can be computed and shown to be bounded below and above by the minimum and maximum patch reproduction numbers ? _j , j=1, …, n. The basic reproduction number has a simple form when there is no movement or when all patches are identical or when the movement rate approaches infinity. Numerical examples of the deterministic and stochastic models illustrate the disease dynamics for different movement rates between three patches.     

Multi-patch deterministic and stochastic models for wildlife diseases

Spatial heterogeneity and host demography have a direct impact on the persistence or extinction of a disease. Natural or human-made landscape features such as forests, rivers, roads, and crops are important to the persistence of wildlife diseases. Rabies, hantaviruses, and plague are just a few examples of wildlife diseases where spatial patterns of infection have been observed. We formulate multi-patch deterministic and stochastic epidemic models and use these models to investigate problems related to disease persistence and extinction. We show in some special cases that a unique disease-free equilibrium exists. In these cases, a basic reproduction number ?(0) can be computed and shown to be bounded below and above by the minimum and maximum patch reproduction numbers ?(j), j=1, …, n. The basic reproduction number has a simple form when there is no movement or when all patches are identical or when the movement rate approaches infinity. Numerical examples of the deterministic and stochastic models illustrate the disease dynamics for different movement rates between three patches.