Hybrid simulations of stochastic reaction-diffusion processes for modeling intracellular signaling pathways

Close agreement between deterministic versus stochastic modeling of first-passage time to vesicle fusion

Ca2+-dependent cell processes, such as neurotransmitter or endocrine vesicle fusion, are inherently stochastic due to large fluctuations in Ca2+ channel gating, Ca2+ diffusion, and Ca2+ binding to buffers and target sensors. However, previous studies revealed closer-than-expected agreement between deterministic and stochastic simulations of Ca2+ diffusion, buffering, and sensing if Ca2+ channel gating is not Ca2+ dependent. To understand this result more fully, we present a comparative study complementing previous work, focusing on Ca2+ dynamics downstream of Ca2+ channel gating. Specifically, we compare deterministic (mean-field/mass-action) and stochastic simulations of vesicle exocytosis latency, quantified by the probability density of the first-passage time (FPT) to the Ca2+-bound state of a vesicle fusion sensor, following a brief Ca2+ current pulse. We show that under physiological constraints, the discrepancy between FPT densities obtained using the two approaches remains small even if as few as ∼50 Ca2+ ions enter per single channel-vesicle release unit. Using a reduced two-compartment model for ease of analysis, we illustrate how this close agreement arises from the smallness of correlations between fluctuations of the reactant molecule numbers, despite the large magnitude of fluctuation amplitudes. This holds if all relevant reactions are heteroreaction between molecules of different species, as is the case for bimolecular Ca2+ binding to buffers and downstream sensor targets. In this case, diffusion and buffering effectively decorrelate the state of the Ca2+ sensor from local Ca2+ fluctuations. Thus, fluctuations in the Ca2+ sensor's state underlying the FPT distribution are only weakly affected by the fluctuations in the local Ca2+ concentration around its average, deterministically computable value.

In silico Prediction on the PI3K/AKT/mTOR Pathway of the Antiproliferative Effect of O. joconostle in Breast Cancer Models

The search for new cancer treatments from traditional medicine involves developing studies to understand at the molecular level different cell signaling pathways involved in cancer development. In this work, we present a model of the PI3K/Akt/mTOR pathway, which plays a key role in cell cycle regulation and is related to cell survival, proliferation, and growth in cancer, as well as resistance to antitumor therapies, so finding drugs that act on this pathway is ideal to propose a new adjuvant treatment. The aim of this work was to model, simulate and predict in silico using the Big Data-Cellulat platform the possible targets in the PI3K/Akt/mTOR pathway on which the Opuntia joconostle extract acts, as well as to indicate the concentration range to be used to find the mean lethal dose in in vitro experiments on breast cancer cells. The in silico results show that, in a cancer cell, the activation of JAK and STAT, as well as PI3K and Akt is related to the effect of cell proliferation, angiogenesis, and inhibition of apoptosis, and that the extract of O. joconostle has an antiproliferative effect on breast cancer cells by inhibiting cell proliferation, regulating the cell cycle and inhibiting apoptosis through this signaling pathway . In vitro it was demonstrated that the extract shows an antiproliferative effect, causing the arrest of cells in the G2/M phase of the cell cycle. Therefore, it is concluded that the use of in silico tools is a valuable method to perform virtual experiments and discover new treatments. The use of this type of model supports in vitro experimentation, reducing the costs and number of experiments in the real laboratory.

Mathematical Modelling in Biomedicine: A Primer for the Curious and the Skeptic

In most disciplines of natural sciences and engineering, mathematical and computational modelling are mainstay methods which are usefulness beyond doubt. These disciplines would not have reached today's level of sophistication without an intensive use of mathematical and computational models together with quantitative data. This approach has not been followed in much of molecular biology and biomedicine, however, where qualitative descriptions are accepted as a satisfactory replacement for mathematical rigor and the use of computational models is seen by many as a fringe practice rather than as a powerful scientific method. This position disregards mathematical thinking as having contributed key discoveries in biology for more than a century, e.g., in the connection between genes, inheritance, and evolution or in the mechanisms of enzymatic catalysis. Here, we discuss the role of computational modelling in the arsenal of modern scientific methods in biomedicine. We list frequent misconceptions about mathematical modelling found among biomedical experimentalists and suggest some good practices that can help bridge the cognitive gap between modelers and experimental researchers in biomedicine. This manuscript was written with two readers in mind. Firstly, it is intended for mathematical modelers with a background in physics, mathematics, or engineering who want to jump into biomedicine. We provide them with ideas to motivate the use of mathematical modelling when discussing with experimental partners. Secondly, this is a text for biomedical researchers intrigued with utilizing mathematical modelling to investigate the pathophysiology of human diseases to improve their diagnostics and treatment.

The blending region hybrid framework for the simulation of stochastic reaction-diffusion processes

The simulation of stochastic reaction–diffusion systems using fine-grained representations can become computationally prohibitive when particle numbers become large. If particle numbers are sufficiently high then it may be possible to ignore stochastic fluctuations and use a more efficient coarse-grained simulation approach. Nevertheless, for multiscale systems which exhibit significant spatial variation in concentration, a coarse-grained approach may not be appropriate throughout the simulation domain. Such scenarios suggest a hybrid paradigm in which a computationally cheap, coarse-grained model is coupled to a more expensive, but more detailed fine-grained model, enabling the accurate simulation of the fine-scale dynamics at a reasonable computational cost. In this paper, in order to couple two representations of reaction–diffusion at distinct spatial scales, we allow them to overlap in a ‘blending region’. Both modelling paradigms provide a valid representation of the particle density in this region. From one end of the blending region to the other, control of the implementation of diffusion is passed from one modelling paradigm to another through the use of complementary ‘blending functions’ which scale up or down the contribution of each model to the overall diffusion. We establish the reliability of our novel hybrid paradigm by demonstrating its simulation on four exemplar reaction–diffusion scenarios.

Analysis and remedy of negativity problem in hybrid stochastic simulation algorithm and its application

BACKGROUND

The hybrid stochastic simulation algorithm, proposed by Haseltine and Rawlings (HR), is a combination of differential equations for traditional deterministic models and Gillespie's algorithm (SSA) for stochastic models. The HR hybrid method can significantly improve the efficiency of stochastic simulations for multiscale biochemical networks. Previous studies on the accuracy analysis for a linear chain reaction system showed that the HR hybrid method is accurate if the scale difference between fast and slow reactions is above a certain threshold, regardless of population scales. However, the population of some reactant species might be driven negative if they are involved in both deterministic and stochastic systems.

RESULTS

This work investigates the negativity problem of the HR hybrid method, analyzes and tests it with several models including a linear chain system, a nonlinear reaction system, and a realistic biological cell cycle system. As a benchmark, the second slow reaction firing time is used to measure the effect of negative populations on the accuracy of the HR hybrid method. Our analysis demonstrates that usually the error caused by negative populations is negligible compared with approximation errors of the HR hybrid method itself, and sometimes negativity phenomena may even improve the accuracy. But for systems where negative species are involved in nonlinear reactions or some species are highly sensitive to negative species, the system stability will be influenced and may lead to system failure when using the HR hybrid method. In those circumstances, three remedies are studied for the negativity problem.

CONCLUSION

The results of different models and examples suggest that the Zero-Reaction rule is a good remedy for nonlinear and sensitive systems considering its efficiency and simplicity.

Spatially extended hybrid methods: a review

Many biological and physical systems exhibit behaviour at multiple spatial, temporal or population scales. Multiscale processes provide challenges when they are to be simulated using numerical techniques. While coarser methods such as partial differential equations are typically fast to simulate, they lack the individual-level detail that may be required in regions of low concentration or small spatial scale. However, to simulate at such an individual level throughout a domain and in regions where concentrations are high can be computationally expensive. Spatially coupled hybrid methods provide a bridge, allowing for multiple representations of the same species in one spatial domain by partitioning space into distinct modelling subdomains. Over the past 20 years, such hybrid methods have risen to prominence, leading to what is now a very active research area across multiple disciplines including chemistry, physics and mathematics. There are three main motivations for undertaking this review. Firstly, we have collated a large number of spatially extended hybrid methods and presented them in a single coherent document, while comparing and contrasting them, so that anyone who requires a multiscale hybrid method will be able to find the most appropriate one for their need. Secondly, we have provided canonical examples with algorithms and accompanying code, serving to demonstrate how these types of methods work in practice. Finally, we have presented papers that employ these methods on real biological and physical problems, demonstrating their utility. We also consider some open research questions in the area of hybrid method development and the future directions for the field.

Multiplicity of Mathematical Modeling Strategies to Search for Molecular and Cellular Insights into Bacteria Lung Infection

Even today two bacterial lung infections, namely pneumonia and tuberculosis, are among the 10 most frequent causes of death worldwide. These infections still lack effective treatments in many developing countries and in immunocompromised populations like infants, elderly people and transplanted patients. The interaction between bacteria and the host is a complex system of interlinked intercellular and the intracellular processes, enriched in regulatory structures like positive and negative feedback loops. Severe pathological condition can emerge when the immune system of the host fails to neutralize the infection. This failure can result in systemic spreading of pathogens or overwhelming immune response followed by a systemic inflammatory response. Mathematical modeling is a promising tool to dissect the complexity underlying pathogenesis of bacterial lung infection at the molecular, cellular and tissue levels, and also at the interfaces among levels. In this article, we introduce mathematical and computational modeling frameworks that can be used for investigating molecular and cellular mechanisms underlying bacterial lung infection. Then, we compile and discuss published results on the modeling of regulatory pathways and cell populations relevant for lung infection and inflammation. Finally, we discuss how to make use of this multiplicity of modeling approaches to open new avenues in the search of the molecular and cellular mechanisms underlying bacterial infection in the lung.

A hybrid continuous-discrete method for stochastic reaction-diffusion processes

Stochastic fluctuations in reaction-diffusion processes often have substantial effect on spatial and temporal dynamics of signal transductions in complex biological systems. One popular approach for simulating these processes is to divide the system into small spatial compartments assuming that molecules react only within the same compartment and jump between adjacent compartments driven by the diffusion. While the approach is convenient in terms of its implementation, its computational cost may become prohibitive when diffusive jumps occur significantly more frequently than reactions, as in the case of rapid diffusion. Here, we present a hybrid continuous-discrete method in which diffusion is simulated using continuous approximation while reactions are based on the Gillespie algorithm. Specifically, the diffusive jumps are approximated as continuous Gaussian random vectors with time-dependent means and covariances, allowing use of a large time step, even for rapid diffusion. By considering the correlation among diffusive jumps, the approximation is accurate for the second moment of the diffusion process. In addition, a criterion is obtained for identifying the region in which such diffusion approximation is required to enable adaptive calculations for better accuracy. Applications to a linear diffusion system and two nonlinear systems of morphogens demonstrate the effectiveness and benefits of the new hybrid method.

Discrete-continuous reaction-diffusion model with mobile point-like sources and sinks

In many applications in soft and biological physics, there are multiple time and length scales involved but often with a distinct separation between them. For instance, in enzyme kinetics, enzymes are relatively large, move slowly and their copy numbers are typically small, while the metabolites (being transformed by these enzymes) are often present in abundance, are small in size and diffuse fast. It seems thus natural to apply different techniques to different time and length levels and couple them. Here we explore this possibility by constructing a stochastic-deterministic discrete-continuous reaction-diffusion model with mobile sources and sinks. Such an approach allows in particular to separate different sources of stochasticity. We demonstrate its application by modelling enzyme-catalysed reactions with freely diffusing enzymes and a heterogeneous source of metabolites. Our calculations suggest that using a higher amount of less active enzymes, as compared to fewer more active enzymes, reduces the metabolite pool size and correspondingly the lag time, giving rise to a faster response to external stimuli. The methodology presented can be extended to more complex systems and offers exciting possibilities for studying problems where spatial heterogeneities, stochasticity or discreteness play a role.

A computational kinetic model of diffusion for molecular systems

Regulation of biomolecular transport in cells involves intra-protein steps like gating and passage through channels, but these steps are preceded by extra-protein steps, namely, diffusive approach and admittance of solutes. The extra-protein steps develop over a 10-100 nm length scale typically in a highly particular environment, characterized through the protein's geometry, surrounding electrostatic field, and location. In order to account for solute energetics and mobility of solutes in this environment at a relevant resolution, we propose a particle-based kinetic model of diffusion based on a Markov State Model framework. Prerequisite input data consist of diffusion coefficient and potential of mean force maps generated from extensive molecular dynamics simulations of proteins and their environment that sample multi-nanosecond durations. The suggested diffusion model can describe transport processes beyond microsecond duration, relevant for biological function and beyond the realm of molecular dynamics simulation. For this purpose the systems are represented by a discrete set of states specified by the positions, volumes, and surface elements of Voronoi grid cells distributed according to a density function resolving the often intricate relevant diffusion space. Validation tests carried out for generic diffusion spaces show that the model and the associated Brownian motion algorithm are viable over a large range of parameter values such as time step, diffusion coefficient, and grid density. A concrete application of the method is demonstrated for ion diffusion around and through the Eschericia coli mechanosensitive channel of small conductance ecMscS.

Computational modeling reveals optimal strategy for kinase transport by microtubules to nerve terminals

Intracellular transport of proteins by motors along cytoskeletal filaments is crucial to the proper functioning of many eukaryotic cells. Since most proteins are synthesized at the cell body, mechanisms are required to deliver them to the growing periphery. In this article, we use computational modeling to study the strategies of protein transport in the context of JNK (c-JUN NH2-terminal kinase) transport along microtubules to the terminals of neuronal cells. One such strategy for protein transport is for the proteins of the JNK signaling cascade to bind to scaffolds, and to have the whole protein-scaffold cargo transported by kinesin motors along microtubules. We show how this strategy outperforms protein transport by diffusion alone, using metrics such as signaling rate and signal amplification. We find that there exists a range of scaffold concentrations for which JNK transport is optimal. Increase in scaffold concentration increases signaling rate and signal amplification but an excess of scaffolds results in the dilution of reactants. Similarly, there exists a range of kinesin motor speeds for which JNK transport is optimal. Signaling rate and signal amplification increases with kinesin motor speed until the speed of motor translocation becomes faster than kinase/scaffold-motor binding. Finally, we suggest experiments that can be performed to validate whether, in physiological conditions, neuronal cells do indeed adopt such an optimal strategy. Understanding cytoskeletal-assisted protein transport is crucial since axonal and cell body accumulation of organelles and proteins is a histological feature in many human neurodegenerative diseases. In this paper, we have shown that axonal transport performance changes with altered transport component concentrations and transport speeds wherein these aspects can be modulated to improve axonal efficiency and prevent or slowdown axonal deterioration.

Multi-scale stochastic simulation of diffusion-coupled agents and its application to cell culture simulation

Many biological systems consist of multiple cells that interact by secretion and binding of diffusing molecules, thus coordinating responses across cells. Techniques for simulating systems coupling extracellular and intracellular processes are very limited. Here we present an efficient method to stochastically simulate diffusion processes, which at the same time allows synchronization between internal and external cellular conditions through a modification of Gillespie's chemical reaction algorithm. Individual cells are simulated as independent agents, and each cell accurately reacts to changes in its local environment affected by diffusing molecules. Such a simulation provides time-scale separation between the intra-cellular and extra-cellular processes. We use our methodology to study how human monocyte-derived dendritic cells alert neighboring cells about viral infection using diffusing interferon molecules. A subpopulation of the infected cells reacts early to the infection and secretes interferon into the extra-cellular medium, which helps activate other cells. Findings predicted by our simulation and confirmed by experimental results suggest that the early activation is largely independent of the fraction of infected cells and is thus both sensitive and robust. The concordance with the experimental results supports the value of our method for overcoming the challenges of accurately simulating multiscale biological signaling systems.

An accelerated algorithm for discrete stochastic simulation of reaction-diffusion systems using gradient-based diffusion and tau-leaping

Stochastic simulation of reaction-diffusion systems enables the investigation of stochastic events arising from the small numbers and heterogeneous distribution of molecular species in biological cells. Stochastic variations in intracellular microdomains and in diffusional gradients play a significant part in the spatiotemporal activity and behavior of cells. Although an exact stochastic simulation that simulates every individual reaction and diffusion event gives a most accurate trajectory of the system's state over time, it can be too slow for many practical applications. We present an accelerated algorithm for discrete stochastic simulation of reaction-diffusion systems designed to improve the speed of simulation by reducing the number of time-steps required to complete a simulation run. This method is unique in that it employs two strategies that have not been incorporated in existing spatial stochastic simulation algorithms. First, diffusive transfers between neighboring subvolumes are based on concentration gradients. This treatment necessitates sampling of only the net or observed diffusion events from higher to lower concentration gradients rather than sampling all diffusion events regardless of local concentration gradients. Second, we extend the non-negative Poisson tau-leaping method that was originally developed for speeding up nonspatial or homogeneous stochastic simulation algorithms. This method calculates each leap time in a unified step for both reaction and diffusion processes while satisfying the leap condition that the propensities do not change appreciably during the leap and ensuring that leaping does not cause molecular populations to become negative. Numerical results are presented that illustrate the improvement in simulation speed achieved by incorporating these two new strategies.

Models at the single cell level

Many cellular behaviors cannot be completely captured or appropriately described at the cell population level. Noise induced by stochastic chemical reactions, spatially polarized signaling networks, and heterogeneous cell-cell communication are among the many phenomena that require fine-grained analysis. Accordingly, the mathematical models used to describe such systems must be capable of single cell or subcellular resolution. Here, we review techniques for modeling single cells, including models of stochastic chemical kinetics, spatially heterogeneous intracellular signaling, and spatial stochastic systems. We also briefly discuss applications of each type of model.

Towards a matrix mechanics framework for dynamic protein network

Protein–protein interaction networks are currently visualized by software generated interaction webs based upon static experimental data. Current state is limited to static, mostly non-compartmental network and non time resolved protein interactions. A satisfactory mathematical foundation for particle interactions within a viscous liquid state (situation within the cytoplasm) does not exist nor do current computer programs enable building dynamic interaction networks for time resolved interactions. Building mathematical foundation for intracellular protein interactions can be achieved in two increments (a) trigger and capture the dynamic molecular changes for a select subset of proteins using several model systems and high throughput time resolved proteomics and, (b) use this information to build the mathematical foundation and computational algorithm for a compartmentalized and dynamic protein interaction network. Such a foundation is expected to provide benefit in at least two spheres: (a) understanding physiology enabling explanation of phenomenon such as incomplete penetrance in genetic disorders and (b) enabling several fold increase in biopharmaceutical production using impure starting materials.

Stochastic simulation of signal transduction: impact of the cellular architecture on diffusion

The transduction of signals depends on the translocation of signaling molecules to specific targets. Undirected diffusion processes play a key role in the bridging of spaces between different cellular compartments. The diffusion of the molecules is, in turn, governed by the intracellular architecture. Molecular crowding and the cytoskeleton decrease macroscopic diffusion. This article shows the use of a stochastic simulation method to study the effects of the cytoskeleton structure on the mobility of macromolecules. Brownian dynamics and single particle tracking were used to simulate the diffusion process of individual molecules through a model cytoskeleton. The resulting average effective diffusion is in line with data obtained in the in vitro and in vivo experiments. It shows that the cytoskeleton structure strongly influences the diffusion of macromolecules. The simulation method used also allows the inclusion of reactions in order to model complete signaling pathways in their spatio-temporal dynamics, taking into account the effects of the cellular architecture.

Can complex cellular processes be governed by simple linear rules?

Complex living systems have shown remarkably well-orchestrated, self-organized, robust, and stable behavior under a wide range of perturbations. However, despite the recent generation of high-throughput experimental datasets, basic cellular processes such as division, differentiation, and apoptosis still remain elusive. One of the key reasons is the lack of understanding of the governing principles of complex living systems. Here, we have reviewed the success of perturbation-response approaches, where without the requirement of detailed in vivo physiological parameters, the analysis of temporal concentration or activation response unravels biological network features such as causal relationships of reactant species, regulatory motifs, etc. Our review shows that simple linear rules govern the response behavior of biological networks in an ensemble of cells. It is daunting to know why such simplicity could hold in a complex heterogeneous environment. Provided physical reasons can be explained for these phenomena, major advancement in the understanding of basic cellular processes could be achieved.

Hybrid stochastic simulations of intracellular reaction-diffusion systems

With the observation that stochasticity is important in biological systems, chemical kinetics have begun to receive wider interest. While the use of Monte Carlo discrete event simulations most accurately capture the variability of molecular species, they become computationally costly for complex reaction-diffusion systems with large populations of molecules. On the other hand, continuous time models are computationally efficient but they fail to capture any variability in the molecular species. In this study a hybrid stochastic approach is introduced for simulating reaction-diffusion systems. We developed an adaptive partitioning strategy in which processes with high frequency are simulated with deterministic rate-based equations, and those with low frequency using the exact stochastic algorithm of Gillespie. Therefore the stochastic behavior of cellular pathways is preserved while being able to apply it to large populations of molecules. We describe our method and demonstrate its accuracy and efficiency compared with the Gillespie algorithm for two different systems. First, a model of intracellular viral kinetics with two steady states and second, a compartmental model of the postsynaptic spine head for studying the dynamics of Ca+2 and NMDA receptors.

Monte Carlo simulation and linear stability analysis of Turing pattern formation in reaction-subdiffusion systems

Subdiffusion is an important physical phenomenon observed in many systems. However, numerical techniques to study it, especially when coupled to reactions, are lacking. In this paper, we develop an efficient Monte Carlo algorithm based on the Gillespie algorithm and the continuous-time random walk to simulate reaction-subdiffusion systems. Using this algorithm, we investigate Turing pattern formation in the Schnakenberg model with subdiffusion. First, we show that, as the system becomes more subdiffusive, the homogeneous state becomes more difficult to destablize and Turing patterns form less easily. Second, we show that, as the number of particles in the system decreases, the magnitude of fluctuations increases and again the Turing patterns form less easily. Third, we show that, as the system becomes more subdiffusive, the ratio between the two diffusive constants must be higher in order to observe Turing patterns. Finally, we also carry out linear stability analysis to validate the results obtained from our algorithm.

FERN - a Java framework for stochastic simulation and evaluation of reaction networks

Background

Stochastic simulation can be used to illustrate the development of biological systems over time and the stochastic nature of these processes. Currently available programs for stochastic simulation, however, are limited in that they either a) do not provide the most efficient simulation algorithms and are difficult to extend, b) cannot be easily integrated into other applications or c) do not allow to monitor and intervene during the simulation process in an easy and intuitive way. Thus, in order to use stochastic simulation in innovative high-level modeling and analysis approaches more flexible tools are necessary.

Results

In this article, we present FERN (Framework for Evaluation of Reaction Networks), a Java framework for the efficient simulation of chemical reaction networks. FERN is subdivided into three layers for network representation, simulation and visualization of the simulation results each of which can be easily extended. It provides efficient and accurate state-of-the-art stochastic simulation algorithms for well-mixed chemical systems and a powerful observer system, which makes it possible to track and control the simulation progress on every level. To illustrate how FERN can be easily integrated into other systems biology applications, plugins to Cytoscape and CellDesigner are included. These plugins make it possible to run simulations and to observe the simulation progress in a reaction network in real-time from within the Cytoscape or CellDesigner environment.

Conclusion

FERN addresses shortcomings of currently available stochastic simulation programs in several ways. First, it provides a broad range of efficient and accurate algorithms both for exact and approximate stochastic simulation and a simple interface for extending to new algorithms. FERN's implementations are considerably faster than the C implementations of gillespie2 or the Java implementations of ISBJava. Second, it can be used in a straightforward way both as a stand-alone program and within new systems biology applications. Finally, complex scenarios requiring intervention during the simulation progress can be modelled easily with FERN.

Mechanistic simulations of inflammation: current state and future prospects

Inflammation is a normal, robust physiological process. It can also be viewed as a complex system that senses and attempts to resolve homeostatic perturbations initiated from within the body (for example, in autoimmune disease) or from the outside (for example, in infections). Virtually all acute and chronic diseases are either driven or modulated by inflammation. The complex interplay between beneficial and harmful arms of the inflammatory response may underlie the lack of fully effective therapies for many diseases. Mathematical modeling is emerging as a frontline tool for understanding the complexity of the inflammatory response. A series of articles in this issue highlights various modeling approaches to inflammation in the larger context of health and disease, from intracellular signaling to whole-animal physiology. Here we discuss the state of this emerging field. We note several common features of inflammation models, as well as challenges and prospects for future studies.

Three-dimensional Monte Carlo simulations of intracellular diffusion and reaction of signaling proteins

We show that the Monte Carlo technique makes it possible to perform three-dimensional simulations of intracellular protein-mediated signal transduction with realistic ratio of the rates of protein diffusion and association with genes. Specifically, we illustrate that in the simplest case when the protein degradation and phosphorylation/dephosphorylation are negligible the distribution of the first passage time for this process is close to exponential provided that the number of target genes is between 1 and 100.

Oscillations in intracellular signaling cascades

In this paper, we study the oscillatory dynamics of intracellular signaling cascades. We derive a reaction-diffusion model of the mitogen-activated protein kinase cascade, and use it to show how oscillations of the protein kinase concentrations can occur as a function of the depth of the cascade. We find that only cascades with depths of three or more layers undergo oscillatory instabilities. In addition, the oscillatory instability is spatially uniform. Thus, the oscillations synchronize the protein kinase concentrations and result in them being uniformly distributed in the cytosol, despite the presence of protein kinase diffusion. Finally, we show how the oscillations are perturbed when parallel cascades "crosstalk." We find that the protein kinases in the downstream layers of the cascade are less perturbed than those in the upstream layers. In particular, cascades of three layers are able to maintain the total power of the protein kinase activities at approximately the unperturbed level. Taken together, our results suggest that only cascades of at least three layers can synchronize the oscillations of protein kinases in the cytosol and operate in parallel in the presence of crosstalk without loss of signaling fidelity.

Fluctuation-induced phase transition in a spatially extended model for catalytic CO oxidation

A reaction-diffusion master equation has been introduced in order to model the bistable CO oxidation on single crystal metal surfaces at high pressure where the diffusion length becomes small and local fluctuations are important. Analytical solutions can be found in a reduced one-component nonlinear master equation after applying the Weiss mean-field approximation together with the adiabatic elimination of oxygen. It is shown that the Weiss mean-field approximation predicts a symmetry-breaking bifurcation associated with a phase transition. The corresponding stationary solutions of the nonlinear master equation are supported by Gillespie-type Monte Carlo simulations.