Sheet excitability and nonlinear wave propagation

Wave failure at strong coupling in intracellular Ca^{2+} signaling system with clustered channels

As an important intracellular signal, Ca^{2+} ions control diverse cellular functions. In this paper, we discuss the Ca^{2+} signaling with a two-dimensional model in which the inositol 1,4,5-trisphosphate (IP_{3}) receptor channels are distributed in clusters on the endoplasmic reticulum membrane. The wave failure at large Ca^{2+} diffusion coupling is discussed in detail in the model. We show that with varying model parameters the wave failure is a robust behavior with either deterministic or stochastic channel dynamics. We suggest that the wave failure should be a general behavior in inhomogeneous diffusing systems with clustered excitable regions and may occur in biological Ca^{2+} signaling systems.

Nuclear pores enable sustained perinuclear calcium oscillations

Background

Calcium signalling relies on the flux of calcium ions across membranes yet how signals in different compartments are related remains unclear. In particular, similar calcium signals on both sides of the nuclear envelope have been reported and attributed to passive diffusion through nuclear pores. However, observed differing cytosolic and nucleosolic calcium signatures suggest that the signalling machinery in these compartments can act independently.

Results

We adapt the fire-diffuse-fire model to investigate the generation of perinuclear calcium oscillations. We demonstrate that autonomous spatio-temporal calcium patterns are still possible in the presence of nuclear and cytosolic coupling via nuclear pores. The presence or absence of this autonomy is dependent upon the strength of the coupling and the maximum firing rate of an individual calcium channel. In all cases, coupling through the nuclear pores enables robust signalling with respect to changes in the diffusion constant.

Conclusions

We show that contradictory interpretations of experimental data with respect to the autonomy of nuclear calcium oscillations can be reconciled within one model, with different observations being a consequence of varying nuclear pore permeabilities for calcium and refractory conditions of channels. Furthermore, our results provide an explanation for why calcium oscillations on both sides of the nuclear envelope may be beneficial for sustained perinuclear signaling.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-016-0289-9) contains supplementary material, which is available to authorized users.

Law of mass action, detailed balance, and the modeling of calcium puffs

Using deterministic-stochastic simulations we show that for intracellular calcium puffs the mixing assumption for reactants does not hold within clusters of receptor channels. Consequently, the law of mass action does not apply and useful definitions of averaged calcium concentrations in the cluster are not obvious. Effective reaction kinetics can be derived, however, by separating concentrations for self-coupling of channels and coupling to different channels, thus eliminating detailed balance in the reaction scheme. A minimal Markovian model can be inferred, describing well calcium puffs in neuronal cells and allowing insight into the functioning of calcium puffs.

Puff-wave transition in an inhomogeneous model for calcium signals

In many cell types, calcium ion channels on the endoplasmic reticulum membrane occur in a clustered distribution. The channels generate either localized puffs, each comprising channels of only one cluster, or global calcium waves. In this work we model the calcium system as a two-dimensional lattice of active elements distributed regularly in an otherwise passive space. We address an important feature of the puff-wave transition, which is the difference in lifetime of puffs at a few hundred milliseconds and long-lived global waves with periods of several seconds. We show that such a lifetime difference between puffs and waves can be understood with strongly reduced ordinary differential equations modified by a time-scale factor that takes into account the coupling strength of active and passive regions determined by the Ca2+ diffusion coefficient. Furthermore, we show that the point model can also describe very well the dependence of Ca2+ oscillation characteristics on the cluster-cluster distance in the case of large diffusivity.

Global noise and oscillations in clustered excitable media

We study the effects of global noise on waves in heterogeneous, spatially clustered, reaction-diffusion systems with possible applications to calcium signaling. We first discuss how clustering of the excitability determines the dynamics by shifting bifurcation points and creating new oscillatory solutions. We then consider the specific situation, where intrinsic noise, due to the smallness of the excitable patches, destroys the global oscillatory state. We show that additional small global fluctuations, however, can partially restore temporal and spatial coherence of the oscillatory signal.

Autocrine signal transmission with extracellular ligand degradation

Traveling waves of cell signaling in epithelial layers orchestrate a number of important processes in developing and adult tissues. These waves can be mediated by positive feedback autocrine loops, a mode of cell signaling where binding of a diffusible extracellular ligand to a cell surface receptor can lead to further ligand release. We formulate and analyze a biophysical model that accounts for ligand-induced ligand release, extracellular ligand diffusion and ligand-receptor interaction. We focus on the case when the main mode for ligand degradation is extracellular and analyze the problem with the sharp threshold positive feedback nonlinearity. We derive expressions that link the speed of propagation and other characteristics of traveling waves to the parameters of the biophysical processes, such as diffusion rates, receptor expression level, etc. Analyzing the derived expressions we found that traveling waves in such systems can exhibit a number of unusual properties, e.g. non-monotonic dependence of the speed of propagation on ligand diffusivity. Our results for the fully developed traveling fronts can be used to analyze wave initiation from localized perturbations, a scenario that frequently arises in the in vitro models of epithelial wound healing, and guide future modeling studies of cell communication in epithelial layers.

Compensated optimal grids for elliptic boundary-value problems

A method is proposed which allows to efficiently treat elliptic problems on unbounded domains in two and three spatial dimensions in which one is only interested in obtaining accurate solutions at the domain boundary. The method is an extension of the optimal grid approach for elliptic problems, based on optimal rational approximation of the associated Neumann-to-Dirichlet map in Fourier space. It is shown that, using certain types of boundary discretization, one can go from second-order accurate schemes to essentially spectrally accurate schemes in two-dimensional problems, and to fourth-order accurate schemes in three-dimensional problems without any increase in the computational complexity. The main idea of the method is to modify the impedance function being approximated to compensate for the numerical dispersion introduced by a small finite-difference stencil discretizing the differential operator on the boundary. We illustrate how the method can be efficiently applied to nonlinear problems arising in modeling of cell communication.

Self-sustained spatiotemporal oscillations induced by membrane-bulk coupling

We propose a novel mechanism leading to spatiotemporal oscillations in extended systems that does not rely on local bulk instabilities. Instead, oscillations arise from the interaction of two subsystems of different spatial dimensionality. Specifically, we show that coupling a passive diffusive bulk of dimension d with an excitable membrane of dimension d-1 produces a self-sustained oscillatory behavior. An analytical explanation of the phenomenon is provided for d=1. Moreover, in-phase and antiphase synchronization of oscillations are found numerically in one and two dimensions. This novel dynamic instability could be used by biological systems such as cells, where the dynamics on the cellular membrane is necessarily different from that of the cytoplasmic bulk.

Membrane-bound Turing patterns

Motivated by recent observations in biological cells, we study Turing patterns in bounded regions where the nonlinear chemical reactions occur on the boundary and where reagent transport occurs in the bulk. Within a generic model, we formulate the stability problem and discuss the conditions for the occurrence of a Turing instability. By choosing other model parameters to be unequal, we find that Turing patterns exist even in the case of equal diffusion constants. Finally, a recently introduced computation technique is utilized to follow the nascent pattern into the highly nonlinear regime.

Cortically restricted production of IP3 leads to propagation of the fertilization Ca2+ wave along the cell surface in a model of the Xenopus egg

The fertilization Ca2+ wave in Xenopus laevis is a single, large wave of elevated free cytosolic Ca2+ concentration that emanates from the point of sperm-egg fusion and traverses the entire diameter of the egg. This phenomenon appears to involve an increase in inositol-1,4,5-trisphosphate (IP3) resulting from interaction of the sperm and egg, which then results in the activation of the endoplasmic reticulum Ca2+ release machinery. We have proposed models based on a static elevated distribution of IP3, and dynamic [IP3], however, these models have suggested that the fertilization wave passes through the center of the egg. Complementing these earlier models, we propose a more detailed model of the fertilization Ca2+ wave in Xenopus eggs to explore the hypothesis that IP3 is produced only at or near the plasma membrane. In this case, we find that the wave propagates primarily through the cortex of the egg, and that Ca2+ -induced production of IP3 at the plasma membrane allows IP3 to propagate in advance of the wave. Our model includes Ca2+ -dependent production of IP3 at the plasma membrane and IP3 degradation. Simulations in 1 dimension and axi-symmetric 3 dimensions illustrate the basic features of the wave.