Local accumulation times for source, diffusion, and degradation models in two and three dimensions

Stationary Ca2+ nanodomains in the presence of buffers with two binding sites

We examine closed-form approximations for the equilibrium Ca²⁺ and buffer concentrations near a point Ca²⁺ source representing a Ca²⁺ channel, in the presence of a mobile buffer with two Ca²⁺ binding sites activated sequentially and possessing distinct binding affinities and kinetics. This allows us to model the impact on Ca²⁺ nanodomains of realistic endogenous Ca²⁺ buffers characterized by cooperative Ca²⁺ binding, such as calretinin. The approximations we present involve a combination or rational and exponential functions, whose parameters are constrained using the series interpolation method that we recently introduced for the case of simpler Ca²⁺ buffers with a single Ca²⁺ binding site. We conduct extensive parameter sensitivity analysis and show that the obtained closed-form approximations achieve reasonable qualitative accuracy for a wide range of buffer's Ca²⁺ binding properties and other relevant model parameters. In particular, the accuracy of the derived approximants exceeds that of the rapid buffering approximation in large portions of the relevant parameter space.

New homogenization approaches for stochastic transport through heterogeneous media

The diffusion of molecules in complex intracellular environments can be strongly influenced by spatial heterogeneity and stochasticity. A key challenge when modelling such processes using stochastic random walk frameworks is that negative jump coefficients can arise when transport operators are discretized on heterogeneous domains. Often this is dealt with through homogenization approximations by replacing the heterogeneous medium with an effective homogeneous medium. In this work, we present a new class of homogenization approximations by considering a stochastic diffusive transport model on a one-dimensional domain containing an arbitrary number of layers with different jump rates. We derive closed form solutions for the kth moment of particle lifetime, carefully explaining how to deal with the internal interfaces between layers. These general tools allow us to derive simple formulae for the effective transport coefficients, leading to significant generalisations of previous homogenization approaches. Here, we find that different jump rates in the layers give rise to a net bias, leading to a non-zero advection, for the entire homogenized system. Example calculations show that our generalized approach can lead to very different outcomes than traditional approaches, thereby having the potential to significantly affect simulation studies that use homogenization approximations.

Theoretical analysis of degradation mechanisms in the formation of morphogen gradients

Fundamental biological processes of development of tissues and organs in multicellular organisms are governed by various signaling molecules, which are called morphogens. It is known that spatial and temporal variations in the concentration profiles of signaling molecules, which are frequently referred as morphogen gradients, lead to a cell differentiation via activating specific genes in a concentration-dependent manner. It is widely accepted that the establishment of the morphogen gradients involves multiple biochemical reactions and diffusion processes. One of the critical elements in the formation of morphogen gradients is a degradation of signaling molecules. We develop a new theoretical approach that provides a comprehensive description of the degradation mechanisms. It is based on the idea that the degradation works as an effective potential that drives the signaling molecules away from the source region. Utilizing the method of first-passage processes, the dynamics of the formation of morphogen gradients for various degradation mechanisms is explicitly evaluated. It is found that linear degradation processes lead to a dynamic behavior specified by times to form the morphogen gradients that depend linearly on the distance from the source. This is because the effective potential due to the degradation is quite strong. At the same time, nonlinear degradation mechanisms yield a quadratic scaling in the morphogen gradients formation times since the effective potentials are much weaker. Physical-chemical explanations of these phenomena are presented.

New Model for Understanding Mechanisms of Biological Signaling: Direct Transport via Cytonemes

Biological signaling is a crucial natural process that governs the formation of all multicellular organisms. It relies on efficient and fast transfer of information between different cells and tissues. It has been presumed for a long time that these long-distance communications in most systems can take place only indirectly via the diffusion of signaling molecules, also known as morphogens, through the extracellular fluid; however, recent experiments indicate that there is also an alternative direct delivery mechanism. It utilizes dynamic tubular cellular extensions, called cytonemes, that directly connect cells, supporting the flux of morphogens to specific locations. We present a first quantitative analysis of the cytoneme-mediated mechanism of biological signaling. Dynamics of the formation of signaling molecule profiles, which are also known as morphogen gradients, is discussed. It is found that the direct-delivery mechanism is more robust with respect to fluctuations in comparison with the passive diffusion mechanism. In addition, we show that the direct transport of morphogens through cytonemes simultaneously delivers the information to all cells, which is also different from the diffusional indirect delivery; however, it requires energy dissipation and it might be less efficient at large distances due to intermolecular interactions of signaling molecules.

Three-Dimensional Gradients of Cytokine Signaling between T Cells

Immune responses are regulated by diffusible mediators, the cytokines, which act at sub-nanomolar concentrations. The spatial range of cytokine communication is a crucial, yet poorly understood, functional property. Both containment of cytokine action in narrow junctions between immune cells (immunological synapses) and global signaling throughout entire lymph nodes have been proposed, but the conditions under which they might occur are not clear. Here we analyze spatially three-dimensional reaction-diffusion models for the dynamics of cytokine signaling at two successive scales: in immunological synapses and in dense multicellular environments. For realistic parameter values, we observe local spatial gradients, with the cytokine concentration around secreting cells decaying sharply across only a few cell diameters. Focusing on the well-characterized T-cell cytokine interleukin-2, we show how cytokine secretion and competitive uptake determine this signaling range. Uptake is shaped locally by the geometry of the immunological synapse. However, even for narrow synapses, which favor intrasynaptic cytokine consumption, escape fluxes into the extrasynaptic space are expected to be substantial (≥20% of secretion). Hence paracrine signaling will generally extend beyond the synapse but can be limited to cellular microenvironments through uptake by target cells or strong competitors, such as regulatory T cells. By contrast, long-range cytokine signaling requires a high density of cytokine producers or weak consumption (e.g., by sparsely distributed target cells). Thus in a physiological setting, cytokine gradients between cells, and not bulk-phase concentrations, are crucial for cell-to-cell communication, emphasizing the need for spatially resolved data on cytokine signaling.

The adaptive immune system fights pathogens through the activation of immune cell clones that specifically recognize a particular pathogen. Tight contacts, so-called immunological synapses, of immune cells with cells that present ‘digested’ pathogen molecules are pivotal for ensuring specificity. The discovery that immune responses are regulated by small diffusible proteins – the cytokines – has been surprising because cytokine diffusion to ‘bystander’ cells might compromise specificity. It has therefore been argued that cytokines are trapped in immunological synapses, whereas other authors have found that cytokines act on a larger scale through entire lymph nodes. Measurements of cytokine concentrations with fine spatial resolution have not been achieved. Here, we study the spatio-temporal dynamics of cytokines through mathematical analysis and three-dimensional numerical simulation and identify key parameters that control signaling range. We predict that even tight immunological synapses leak a substantial portion of the secreted cytokines. Nevertheless, rapid cellular uptake will render cytokine signals short-range and thus incidental activation of bystander cells can be limited. Long-range signals will only occur with multiple secreting cells or/and slow consumption by sparse target cells. Thus our study identifies key determinants of the spatial range of cytokine communication in realistic multicellular geometries.

The role of source delocalization in the development of morphogen gradients

Successful biological development via spatial regulation of cell differentiation relies on the action of multiple signaling molecules that are known as morphogens. It is now well-established that signaling molecules create non-uniform concentration profiles, morphogen gradients, that activate different genes, leading to patterning in the developing embryos. The current view of the formation of morphogen gradients is that it is a result of complex reaction-diffusion processes that include the strongly localized production, diffusion and uniform degradation of signaling molecules. However, multiple experimental studies also suggest that the production of morphogen in many cases is delocalized. We develop a theoretical method that allows us to investigate the role of the delocalization in the formation of morphogen gradients. The approach is based on discrete-state stochastic models that can be solved exactly for arbitrary production lengths and production rates of morphogen molecules. Our analysis shows that the delocalization might have a strong effect on mechanisms of the morphogen gradient formation. The physical origin of this effect is discussed.

Development of morphogen gradient: the role of dimension and discreteness

The fundamental processes of biological development are governed by multiple signaling molecules that create non-uniform concentration profiles known as morphogen gradients. It is widely believed that the establishment of morphogen gradients is a result of complex processes that involve diffusion and degradation of locally produced signaling molecules. We developed a multi-dimensional discrete-state stochastic approach for investigating the corresponding reaction-diffusion models. It provided a full analytical description for stationary profiles and for important dynamic properties such as local accumulation times, variances, and mean first-passage times. The role of discreteness in developing of morphogen gradients is analyzed by comparing with available continuum descriptions. It is found that the continuum models prediction about multiple time scales near the source region in two-dimensional and three-dimensional systems is not supported in our analysis. Using ideas that view the degradation process as an effective potential, the effect of dimensionality on establishment of morphogen gradients is also discussed. In addition, we investigated how these reaction-diffusion processes are modified with changing the size of the source region.

Computational and mathematical methods for morphogenetic gradient analysis, boundary formation and axonal targeting

Morphogenesis and axonal targeting are key processes during development that depend on complex interactions at molecular, cellular and tissue level. Mathematical modeling is essential to bridge this multi-scale gap in order to understand how the emergence of large structures is controlled at molecular level by interactions between various signaling pathways. We summarize mathematical modeling and computational methods for time evolution and precision of morphogenetic gradient formation. We discuss tissue patterning and the formation of borders between regions labeled by different morphogens. Finally, we review models and algorithms that reveal the interplay between morphogenetic gradients and patterned activity for axonal pathfinding and the generation of the retinotopic map in the visual system.

Simplified approach for calculating moments of action for linear reaction-diffusion equations

The mean action time is the mean of a probability density function that can be interpreted as a critical time, which is a finite estimate of the time taken for the transient solution of a reaction-diffusion equation to effectively reach steady state. For high-variance distributions, the mean action time underapproximates the critical time since it neglects to account for the spread about the mean. We can improve our estimate of the critical time by calculating the higher moments of the probability density function, called the moments of action, which provide additional information regarding the spread about the mean. Existing methods for calculating the nth moment of action require the solution of n nonhomogeneous boundary value problems which can be difficult and tedious to solve exactly. Here we present a simplified approach using Laplace transforms which allows us to calculate the nth moment of action without solving this family of boundary value problems and also without solving for the transient solution of the underlying reaction-diffusion problem. We demonstrate the generality of our method by calculating exact expressions for the moments of action for three problems from the biophysics literature. While the first problem we consider can be solved using existing methods, the second problem, which is readily solved using our approach, is intractable using previous techniques. The third problem illustrates how the Laplace transform approach can be used to study coupled linear reaction-diffusion equations.