Stochastic hybrid modeling of intracellular calcium dynamics

Hybrid models of chemotaxis with application to leukocyte migration

Many chemical and biological systems involve reacting species with vastly different numbers of molecules/agents. Hybrid simulations model such phenomena by combining discrete (e.g., agent-based) and continuous (e.g., partial differential equation- or PDE-based) descriptors of the dynamics of reactants with small and large numbers of molecules/agents, respectively. We present a stochastic hybrid algorithm to model a stage of the immune response to inflammation, during which leukocytes reach a pathogen via chemotaxis. While large numbers of chemoattractant molecules justify the use of a PDE-based model to describe the spatiotemporal evolution of its concentration, relatively small numbers of leukocytes and bacteria involved in the process undermine the veracity of their continuum treatment by masking the effects of stochasticity and have to be treated discretely. Motility and interactions between leukocytes and bacteria are modeled via random walk and a stochastic simulation algorithm, respectively. Since the latter assumes the reacting species to be well mixed, the discrete component of our hybrid algorithm deploys stochastic operator splitting, in which the sequence of the diffusion and reaction operations is determined autonomously during each simulation step. We conduct a series of numerical experiments to ascertain the accuracy and computational efficiency of our hybrid simulations and, then, to demonstrate the importance of randomness for predicting leukocyte migration and fate during the immune response to inflammation.

Chemical Langevin equation: A path-integral view of Gillespie's derivation

In 2000, Gillespie rehabilitated the chemical Langevin equation (CLE) by describing two conditions that must be satisfied for it to yield a valid approximation of the chemical master equation (CME). In this work, we construct an original path-integral description of the CME and show how applying Gillespie's two conditions to it directly leads to a path-integral equivalent to the CLE. We compare this approach to the path-integral equivalent of a large system size derivation and show that they are qualitatively different. In particular, both approaches involve converting many sums into many integrals, and the difference between the two methods is essentially the difference between using the Euler-Maclaurin formula and using Riemann sums. Our results shed light on how path integrals can be used to conceptualize coarse-graining biochemical systems and are readily generalizable.

Order Through Disorder: The Characteristic Variability of Systems

Randomness characterizes many processes in nature, and therefore its importance cannot be overstated. In the present study, we investigate examples of randomness found in various fields, to underlie its fundamental processes. The fields we address include physics, chemistry, biology (biological systems from genes to whole organs), medicine, and environmental science. Through the chosen examples, we explore the seemingly paradoxical nature of life and demonstrate that randomness is preferred under specific conditions. Furthermore, under certain conditions, promoting or making use of variability-associated parameters may be necessary for improving the function of processes and systems.

Efficient Multiscale Models of Polymer Assembly

Protein polymerization and bundling play a central role in cell physiology. Predictive modeling of these processes remains an open challenge, especially when the proteins involved become large and their concentrations high. We present an effective kinetics model of filament formation, bundling, and depolymerization after GTP hydrolysis, which involves a relatively small number of species and reactions, and remains robust over a wide range of concentrations and timescales. We apply this general model to study assembly of FtsZ protein, a basic element in the division process of prokaryotic cells such as Escherichia coli, Bacillus subtilis, or Caulobacter crescentus. This analysis demonstrates that our model outperforms its counterparts in terms of both accuracy and computational efficiency. Because our model comprises only 17 ordinary differential equations, its computational cost is orders-of-magnitude smaller than the current alternatives consisting of up to 1000 ordinary differential equations. It also provides, to our knowledge, a new insight into the characteristics and functioning of FtsZ proteins at high concentrations. The simplicity and versatility of our model render it a powerful computational tool, which can be used either as a standalone descriptor of other biopolymers' assembly or as a component in more complete kinetic models.

Numerical Approach to Spatial Deterministic-Stochastic Models Arising in Cell Biology

Hybrid deterministic-stochastic methods provide an efficient alternative to a fully stochastic treatment of models which include components with disparate levels of stochasticity. However, general-purpose hybrid solvers for spatially resolved simulations of reaction-diffusion systems are not widely available. Here we describe fundamentals of a general-purpose spatial hybrid method. The method generates realizations of a spatially inhomogeneous hybrid system by appropriately integrating capabilities of a deterministic partial differential equation solver with a popular particle-based stochastic simulator, Smoldyn. Rigorous validation of the algorithm is detailed, using a simple model of calcium 'sparks' as a testbed. The solver is then applied to a deterministic-stochastic model of spontaneous emergence of cell polarity. The approach is general enough to be implemented within biologist-friendly software frameworks such as Virtual Cell.

Microdomain calcium fluctuations as a colored noise process

Calcium ions play a key role in subcellular signaling as localized transients of the intracellular calcium concentration modify the activity of ion channels, enzymes and transcription factors, among others. The intracellular calcium concentration is inherently noisy, as diffusion, the transient binding to and dissociation from buffer molecules and stochastically gating calcium channels contribute to the fluctuations of the local copy number of Ca²⁺ ions. We study the properties of the fluctuating calcium concentration in sub-femtoliter volumes using an exact stochastic simulation algorithm and approximations to the exact stochastic solution. It is shown that the time course of the local calcium concentration represents a colored noise process whose autocorrelation time is a function of buffer kinetics and diffusion constants. Using the chemical Langevin description and the excess buffer approximation of the process, fast approximative algorithms and theoretical connections to the Ornstein-Uhlenbeck process are obtained. In a generic example, we show how calcium noise can couple to the dynamics of a single variable moving in a double-well potential, leading to a colored noise induced transition. Our work shows how a multitude of intracellular signaling pathways may be influenced by the inherent stochasticity of calcium signals, a key messenger in virtually any cell type, and how the calcium signal can be implemented efficiently in cellular signaling models.

Stochastic contribution for the coding of agonist induced calcium oscillation in hepatocytes

The influence of stochastic inositol-1,4,5-trisphosphate receptor (IP₃R) dynamics and their clustering have been extensively investigated to explore the mechanism through which the stochastic molecular event finally shape the intracellular calcium signaling. Most of the previous works employed simplified models which take the concentration of IP₃ instead of that of the agonist as the stimulation intensity. However, the IP₃ level is not linearly dependent on the agonist concentration in stimulus induced signaling systems because there are feedback links in the transduction network. In this work, we include both the IP₃R dynamics and the typical agonist induced signaling transduction cascade in the model to investigate the essential influence of stochastic IP₃R dynamics on the coding of the stimulus induced calcium signal. Simulation results reveal two distinct oscillation areas under different stimulation levels. The signal is optimally modulate by the IP₃R cluster number in the weak stimulated area while affected by the stimulus intensity in the strong stimulated area. Different dependences of coefficient of variance (CV) on the number of clusters are obtained in these two areas, which explains the disagreement in the previous reported results. Besides, the transition between these areas explains the significant CV reduction observed in experiments.

Stochastic operator-splitting method for reaction-diffusion systems

Many biochemical processes at the sub-cellular level involve a small number of molecules. The local numbers of these molecules vary in space and time, and exhibit random fluctuations that can only be captured with stochastic simulations. We present a novel stochastic operator-splitting algorithm to model such reaction-diffusion phenomena. The reaction and diffusion steps employ stochastic simulation algorithms and Brownian dynamics, respectively. Through theoretical analysis, we have developed an algorithm to identify if the system is reaction-controlled, diffusion-controlled or is in an intermediate regime. The time-step size is chosen accordingly at each step of the simulation. We have used three examples to demonstrate the accuracy and robustness of the proposed algorithm. The first example deals with diffusion of two chemical species undergoing an irreversible bimolecular reaction. It is used to validate our algorithm by comparing its results with the solution obtained from a corresponding deterministic partial differential equation at low and high number of molecules. In this example, we also compare the results from our method to those obtained using a Gillespie multi-particle (GMP) method. The second example, which models simplified RNA synthesis, is used to study the performance of our algorithm in reaction- and diffusion-controlled regimes and to investigate the effects of local inhomogeneity. The third example models reaction-diffusion of CheY molecules through the cytoplasm of Escherichia coli during chemotaxis. It is used to compare the algorithm's performance against the GMP method. Our analysis demonstrates that the proposed algorithm enables accurate simulation of the kinetics of complex and spatially heterogeneous systems. It is also computationally more efficient than commonly used alternatives, such as the GMP method.

Comparison of models for IP3 receptor kinetics using stochastic simulations

Inositol 1,4,5-trisphosphate receptor (IP₃R) is a ubiquitous intracellular calcium (Ca²⁺) channel which has a major role in controlling Ca²⁺ levels in neurons. A variety of computational models have been developed to describe the kinetic function of IP₃R under different conditions. In the field of computational neuroscience, it is of great interest to apply the existing models of IP₃R when modeling local Ca²⁺ transients in dendrites or overall Ca²⁺ dynamics in large neuronal models. The goal of this study was to evaluate existing IP₃R models, based on electrophysiological data. This was done in order to be able to suggest suitable models for neuronal modeling. Altogether four models (Othmer and Tang, 1993; Dawson etal., 2003; Fraiman and Dawson, 2004; Doi etal., 2005) were selected for a more detailed comparison. The selection was based on the computational efficiency of the models and the type of experimental data that was used in developing the model. The kinetics of all four models were simulated by stochastic means, using the simulation software STEPS, which implements the Gillespie stochastic simulation algorithm. The results show major differences in the statistical properties of model functionality. Of the four compared models, the one by Fraiman and Dawson (2004) proved most satisfactory in producing the specific features of experimental findings reported in literature. To our knowledge, the present study is the first detailed evaluation of IP₃R models using stochastic simulation methods, thus providing an important setting for constructing a new, realistic model of IP₃R channel kinetics for compartmental modeling of neuronal functions. We conclude that the kinetics of IP₃R with different concentrations of Ca²⁺ and IP₃ should be more carefully addressed when new models for IP₃R are developed.

Simulation strategies for calcium microdomains and calcium-regulated calcium channels

In this article, we present an overview of simulation strategies in the context of subcellular domains where calcium-dependent signaling plays an important role. The presentation follows the spatial and temporal scales involved and represented by each algorithm. As an exemplary cell type, we will mainly cite work done on striated muscle cells, i.e. skeletal and cardiac muscle. For these cells, a wealth of ultrastructural, biophysical and electrophysiological data is at hand. Moreover, these cells also express ubiquitous signaling pathways as they are found in many other cell types and thus, the generalization of the methods and results presented here is straightforward.The models considered comprise the basic calcium signaling machinery as found in most excitable cell types including Ca(2+) ions, diffusible and stationary buffer systems, and calcium regulated calcium release channels. Simulation strategies can be differentiated in stochastic and deterministic algorithms. Historically, deterministic approaches based on the macroscopic reaction rate equations were the first models considered. As experimental methods elucidated highly localized Ca(2+) signaling events occurring in femtoliter volumes, stochastic methods were increasingly considered. However, detailed simulations of single molecule trajectories are rarely performed as the computational cost implied is too large. On the mesoscopic level, Gillespie's algorithm is extensively used in the systems biology community and with increasing frequency also in models of microdomain calcium signaling. To increase computational speed, fast approximations were derived from Gillespie's exact algorithm, most notably the chemical Langevin equation and the τ-leap algorithm. Finally, in order to integrate deterministic and stochastic effects in multiscale simulations, hybrid algorithms are increasingly used. These include stochastic models of ion channels combined with deterministic descriptions of the calcium buffering and diffusion system on the one hand, and algorithms that switch between deterministic and stochastic simulation steps in a context-dependent manner on the other. The basic assumptions of the listed methods as well as implementation schemes are given in the text. We conclude with a perspective on possible future developments of the field.

Exact and approximate stochastic simulation of intracellular calcium dynamics

In simulations of chemical systems, the main task is to find an exact or approximate solution of the chemical master equation (CME) that satisfies certain constraints with respect to computation time and accuracy. While Brownian motion simulations of single molecules are often too time consuming to represent the mesoscopic level, the classical Gillespie algorithm is a stochastically exact algorithm that provides satisfying results in the representation of calcium microdomains. Gillespie's algorithm can be approximated via the tau-leap method and the chemical Langevin equation (CLE). Both methods lead to a substantial acceleration in computation time and a relatively small decrease in accuracy. Elimination of the noise terms leads to the classical, deterministic reaction rate equations (RRE). For complex multiscale systems, hybrid simulations are increasingly proposed to combine the advantages of stochastic and deterministic algorithms. An often used exemplary cell type in this context are striated muscle cells (e.g., cardiac and skeletal muscle cells). The properties of these cells are well described and they express many common calcium-dependent signaling pathways. The purpose of the present paper is to provide an overview of the aforementioned simulation approaches and their mutual relationships in the spectrum ranging from stochastic to deterministic algorithms.