Asymptotic Analysis of Buffered Calcium Diffusion near a Point Source

The domain calcium (Ca2+) concentration near an open Ca2+ channel can be mod- eled as buffered diffusion from a point source. The concentration profiles can be well approximated by hemispherically symmetric steady-state solutions to a system of reaction-diffusion equations. After nondimensionalizing these equations and scaling space so that both reaction terms and the source amplitude are 0(1), we identify two dimensionless parameters, Cc and Eb, that correspond to the diffusion coefficients of dimensionless Ca2+ and buffer, respectively. Using perturbation methods, we derive approximations for the Ca2+ and buffer profiles in three asymptotic limits: (1) an excess buffer approximation (EBA), where the mobility of buffer exceeds that of Ca2+ (Eb \u3e Ec) and the fast diffusion of buffer toward the Ca2+ channel prevents buffer saturation (cf. Neher [Calcium Electrogenesis and Neuronal Functioning, Exp. Brain Res. 14, Springer-Verlag, Berlin, 1986, pp. 80-96]); (2) a rapid buffer approximation (RBA), where the diffusive time-scale for Ca2+ and buffer are comparable, but slow compared to reaction (ec \u3c\u3c 1, Eb reaction (ec \u3c\u3c 1, Eb reaction (ec \u3c\u3c 1, E

Sensitisation waves in a bidomain fire-diffuse-fire model of intracellular Ca²⁺ dynamics

We present a bidomain threshold model of intracellular calcium (Ca²⁺) dynamics in which, as suggested by recent experiments, the cytosolic threshold for Ca²⁺ liberation is modulated by the Ca²⁺ concentration in the releasing compartment. We explicitly construct stationary fronts and determine their stability using an Evans function approach. Our results show that a biologically motivated choice of a dynamic threshold, as opposed to a constant threshold, can pin stationary fronts that would otherwise be unstable. This illustrates a novel mechanism to stabilise pinned interfaces in continuous excitable systems. Our framework also allows us to compute travelling pulse solutions in closed form and systematically probe the wave speed as a function of physiologically important parameters. We find that the existence of travelling wave solutions depends on the time scale of the threshold dynamics, and that facilitating release by lowering the cytosolic threshold increases the wave speed. The construction of the Evans function for a travelling pulse shows that of the co-existing fast and slow solutions the slow one is always unstable

Efficient approximations for stationary single-channel calcium nanodomains

Mathematical and computational modeling plays an important role in the study of local Ca2+ signals underlying many fundamental physiological processes such as synaptic neurotransmitter release and myocyte contraction. Closed-form approximations describing steady-state distribution of Ca2+ in the vicinity of an open Ca2+ channel have proved particularly useful for the qualitative modeling of local Ca2+ signals. This dissertation presents several simple and efficient approximants for the equilibrium Ca2+ concentration near a point source in the presence of a mobile Ca2+ buffer, which achieve great accuracy over a wide range of model parameters. Such approximations provide an efficient method for estimating Ca2+ and buffer concentrations without resorting to numerical simulations and allow to study the qualitative dependence of nanodomain Ca2+ distribution on the buffer’s Ca2+ binding properties and its diffusivity. The new approximants presented here for the case of a simple, one-to-one Ca2+ buffer have a functional form that combines rational and exponential functions, which is similar to that of the well-known Excess Buffer Approximation and the linear approximation, but with parameters estimated using two novel methods. One of the methods involves interpolation between the short-range Taylor series of the buffer concentration and its long-range asymptotic series in inverse powers of distance from the channel. A second method is based on the variational approach and involves a global minimization of an appropriate functional with respect to parameters of the chosen approximations. Extensive parameter sensitivity analysis is presented, comparing approximants found using these two methods with the previously developed approximants. Apart from increased accuracy, the strength of the new approximants is that they can be extended to more realistic buffers with multiple Ca2+ binding sites, such as calmodulin and calretinin. In the second part of the dissertation, the series interpolation method is extended to buffers with two Ca2+ binding sites, yielding closed-form interpolants combining exponential and rational functions that achieve reasonable accuracy even in the case of buffers characterized by significant Ca2+ binding cooperativity. Finally, open challenges and potential future extensions of this work are discussed in detail

The Influence of Ca2+ Buffers on Free [Ca2+] Fluctuations and the Effective Volume of Ca2+ Microdomains

Intracellular calcium (Ca2+) plays a significant role in many cell signaling pathways, some of which are localized to spatially restricted microdomains. Ca2+ binding proteins (Ca2+ buffers) play an important role in regulating Ca2+ concentration ([Ca2+]). Buffers typically slow [Ca2+] temporal dynamics and increase the effective volume of Ca2+ domains. Because fluctuations in [Oa(2+)] decrease in proportion to the square-root of a domain\u27s physical volume, one might conjecture that buffers decrease [Ca2+] fluctuations and, consequently, mitigate the significance of small domain volume concerning Ca2+ signaling. We test this hypothesis through mathematical and computational analysis of idealized buffer-containing domains and their stochastic dynamics during free Ca2+ influx with passive exchange of both Ca2+ and buffer with bulk concentrations. We derive Langevin equations for the fluctuating dynamics of Ca2+ and buffer and use these stochastic differential equations to determine the magnitude of [Ca2+] fluctuations for different buffer parameters (e.g., dissociation constant and concentration). In marked contrast to expectations based on a naive application of the principle of effective volume as employed in deterministic models of Ca2+ signaling, we find that mobile and rapid buffers typically increase the magnitude of domain [Ca2+] fluctuations during periods of Ca2+ influx, whereas stationary (immobile) Ca2+ buffers do not. Also contrary to expectations, we find that in the absence of Ca2+ influx, buffers influence the temporal characteristics, but not the magnitude, of [Ca2+] fluctuations. We derive an analytical formula describing the influence of rapid Ca2+ buffers on [Ca2+] fluctuations and, importantly, identify the stochastic analog of (deterministic) effective domain volume. Our results demonstrate that Ca2+ buffers alter the dynamics of [Ca2+] fluctuations in a nonintuitive manner. The finding that Ca2+ buffers do not suppress intrinsic domain [Ca2+] fluctuations raises the intriguing question of whether or not [Ca2+] fluctuations are a physiologically significant aspect of local Ca2+ signaling

A multiscale hybrid model for pro-angiogenic calcium signals in a vascular endothelial cell

Cytosolic calcium machinery is one of the principal signaling mechanisms by which endothelial cells (ECs) respond to external stimuli during several biological processes, including vascular progression in both physiological and pathological conditions. Low concentrations of angiogenic factors (such as VEGF) activate in fact complex pathways involving, among others, second messengers arachidonic acid (AA) and nitric oxide (NO), which in turn control the activity of plasma membrane calcium channels. The subsequent increase in the intracellular level of the ion regulates fundamental biophysical properties of ECs (such as elasticity, intrinsic motility, and chemical strength), enhancing their migratory capacity. Previously, a number of continuous models have represented cytosolic calcium dynamics, while EC migration in angiogenesis has been separately approached with discrete, lattice-based techniques. These two components are here integrated and interfaced to provide a multiscale and hybrid Cellular Potts Model (CPM), where the phenomenology of a motile EC is realistically mediated by its calcium-dependent subcellular events. The model, based on a realistic 3-D cell morphology with a nuclear and a cytosolic region, is set with known biochemical and electrophysiological data. In particular, the resulting simulations are able to reproduce and describe the polarization process, typical of stimulated vascular cells, in various experimental conditions.Moreover, by analyzing the mutual interactions between multilevel biochemical and biomechanical aspects, our study investigates ways to inhibit cell migration: such strategies have in fact the potential to result in pharmacological interventions useful to disrupt malignant vascular progressio

Spatial Distribution of Calcium-Gated Chloride Channels in Olfactory Cilia

Background: In vertebrate olfactory receptor neurons, sensory cilia transduce odor stimuli into changes in neuronal membrane potential. The voltage changes are primarily caused by the sequential openings of two types of channel: a cyclic-nucleotide-gated (CNG) cationic channel and a calcium-gated chloride channel. In frog, the cilia are 25 to 200 mm in length, so the spatial distributions of the channels may be an important determinant of odor sensitivity. Principal Findings: To determine the spatial distribution of the chloride channels, we recorded from single cilia as calcium was allowed to diffuse down the length of the cilium and activate the channels. A computational model of this experiment allowed an estimate of the spatial distribution of the chloride channels. On average, the channels were concentrated in a narrow band centered at a distance of 29 % of the ciliary length, measured from the base of the cilium. This matches the location of the CNG channels determined previously. This non-uniform distribution of transduction proteins is consistent with similar findings in other cilia. Conclusions: On average, the two types of olfactory transduction channel are concentrated in the same region of the cilium

Barriers to Diffusion in Dendrites and Estimation of Calcium Spread Following Synaptic Inputs

The motion of ions, molecules or proteins in dendrites is restricted by cytoplasmic obstacles such as organelles, microtubules and actin network. To account for molecular crowding, we study the effect of diffusion barriers on local calcium spread in a dendrite. We first present a model based on a dimension reduction approach to approximate a three dimensional diffusion in a cylindrical dendrite by a one-dimensional effective diffusion process. By comparing uncaging experiments of an inert dye in a spiny dendrite and in a thin glass tube, we quantify the change in diffusion constants due to molecular crowding as Dcyto/Dwater = 1/20. We validate our approach by reconstructing the uncaging experiments using Brownian simulations in a realistic 3D model dendrite. Finally, we construct a reduced reaction-diffusion equation to model calcium spread in a dendrite under the presence of additional buffers, pumps and synaptic input. We find that for moderate crowding, calcium dynamics is mainly regulated by the buffer concentration, but not by the cytoplasmic crowding, dendritic spines or synaptic inputs. Following high frequency stimulations, we predict that calcium spread in dendrites is limited to small microdomains of the order of a few microns (<5 μm)