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Martin Meier-Schellersheim

Publications and source records attributed to Martin Meier-Schellersheim.

16 recordsLinked to original sources
arXiv2024

Timing consistency of T cell receptor activation in a stochastic model combining kinetic segregation and proofreading

T cell receptor signaling must operate reliably under tight time constraints. While assuming quite different mechanisms, two prominent models of T cell receptor activation, kinetic segregation and kinetic proofreading, both introduce a distinct time scale. However, a clear understanding of whether and how those characteristic times give rise to a consistent timing of T cell receptor activation in the presence of stochastic fluctuations has been lacking so far. Here, using a simulation approach capable of modeling molecular interactions between adjacent cell membranes, we explore a stochastic model that combines elements of kinetic segregation and proofreading. Our simulations suggest that the two mechanisms interoperate, thereby rendering the corresponding stochastic times biologically functional. Receptor activation relies on rare molecular events that are not well characterized by the mean of the underlying probability density function. Yet, a consistent timing of receptor activation can be ensured by a modest number of proofreading steps.

q-bio.QM↗
arXiv2023

Grid-free stochastic simulations of reaction-diffusion processes at cell-cell contacts

Biological cells can exchange messages through soluble molecules or membrane-bound receptors. In particular in the latter case, the interaction is usually located in specific regions of the interacting cells and may depend on or induce local morphological features or reorganizations of the membrane-associated or membrane-proximal biochemistry. Examples are interactions among cells that probe their environment and the surfaces of neighboring cells with dendrite-like protrusions. Previously, we introduced an algorithm capable of creating spatially-resolved, stochastic particle-based computational models of cellular biochemistry. However, its applicability was limited to the case of soluble ligands. Here, building on this earlier approach, we introduce a simulation method that accounts for the unique features that distinguish the sites of cell-cell contact from other regions of the cell membrane. Employing a smooth, grid-free computational representation of the contact sites, we extend the applicability of Green's function-based stochastic simulation algorithms to study intermembrane bimolecular interactions between particles on adjacent curved surfaces. We demonstrate the utility of this computational approach by exploring the effects of contact geometry and local biochemistry for the early phase of T-cell receptor activation.

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arXiv2020

Stochastic single-particle based simulations of cellular signaling embedded into computational models of cellular morphology

Cells exhibit a wide variety of different shapes. This diversity poses a challenge for computational approaches that attempt to shed light on the role cell geometry plays in regulating cell physiology and behavior. The simulation platform Simmune is capable of embedding the computational representation of signaling pathways into realistic models of cellular morphology. However, Simmune's current approach to account for the cell geometry is limited to deterministic models of reaction-diffusion processes, thus providing a coarse-grained description that ignores stochastic local fluctuations. Here we present an extension of Simmune that removes these limitations by employing an alternative computational representation of cellular geometry that is smooth and grid-free. These features make it possible to incorporate a fully stochastic, spatially resolved description of the cellular biochemistry. The alternative computational representation is compatible with Simmune's current approach for specifying molecular interactions. This means that a modeler using the approach needs to create a model of cellular biochemistry and morphology only once to be able to use it for both, deterministic and stochastic simulations.

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arXiv2016

Path integral approach to theories of diffusion-influenced reactions

The path decomposition expansion represents the propagator of the irreversible reaction as a convolution of the first-passage, last-passage and rebinding time probability densities. Using path integral technique, we give an elementary, yet rigorous, proof of the path decomposition expansion of the Green's functions describing the non-reactive case and the irreversible reaction of an isolated pair of molecules. To this end, we exploit the connection between boundary value problems and interaction potential problems with $δ$- and $δ'$-function perturbation. In particular, we employ a known exact summation of a perturbation series to derive exact relations between the Green's functions of the perturbed and unperturbed problem. Along the way, we are able to derive a number of additional exact identities that relate the propagators describing the free-space, the non-reactive as well as the completely and partially reactive case.

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arXiv2015

Exact propagation without analytical solutions

We present a simulation algorithm that accurately propagates a molecule pair using large time steps without the need to invoke the full exact analytical solutions of the Smoluchowski diffusion equation. Because the proposed method only uses uniform and Gaussian random numbers, it allows for position updates that are two to three orders of magnitude faster than those of a corresponding scheme based on full solutions, while mantaining the same degree of accuracy. Neither simplifying nor ad hoc assumptions that are foreign to the underlying Smoluchowski theory are employed, instead, the algorithm faithfully incorporates the individual elements of the theoretical model. The method is flexible and applicable in 1, 2 and 3 dimensions, suggesting that it may find broad usage in various stochastic simulation algorithms. We demonstrate the algorithm for the case of a non-reactive, irreversible and reversible reacting molecule pair.

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arXiv2015

Quantum mechanical inspired factorization of the molecule pair propagator in theories of diffusion-influenced reactions

Building on mathematical similarities between quantum mechanics and theories of diffusion-influenced reactions, we discuss how the propagator of a reacting molecule pair can be represented as a product of three factors in the Laplace domain. This representation offers several advantages. First, the full propagator can be calculated without ever having to solve the corresponding partial differential equation or path integral. Second, the representation is quite general and capable of capturing not only the classical Smoluchowski-Collins-Kimball model, but also alternative theories, as is here exemplified by the case of a delta- and step-function potential in one and two dimensions, respectively. Third, the three factors correspond to physical quantities that feature prominently in stochastic spatially-resolved simulation algorithms and hence the interpretation of current and the design of future algorithms may benefit. Finally, the representation may serve as a suitable starting point for numerical approximations that could be employed to enhance the efficiency of stochastic simulations.

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arXiv2015

Interplay of receptor memory and ligand rebinding

Rapid rebinding of molecular interaction partners that are in close proximity after dissociation leads to a dissociation and association kinetics that can profoundly differ from predictions based on bulk reaction models. The cause of this effect can be traced back to the non-Markovian character of the ligand's rebinding time probability density function, reflecting the fact that, for a certain time span, the ligand still 'remembers' the receptor it was bound to previously. In this manuscript, we explore the consequences of the hypothesis that initial binding and consecutive rebinding give rise to a bond lifetime density that is non-Markovian as well. We study the combined effect of the two non-Markovian waiting time probability densities and show that even for very short times the decay of the fraction of occupied receptors deviates from an exponential. For long times, dissociation is slower than an exponential and the fate of the the steady-state bound receptor fraction critically depends on the extent of the deviation, relative to the rebinding time density, from the Markovian limit: The population of occupied receptors may either decay completely or assume a non-vanishing value for small and strong deviations, respectively. Furthermore, we point out the important role played by fractional calculus and demonstrate that the short- and long-time dynamics of the occupied receptors can be naturally expressed as well as easily obtained in terms of fractional differential equations involving the Riemann-Liouville derivative. Our analysis shows that cells may exploit receptor memory as mechanism to dynamically widen the range of potential response-patterns to a given signal.

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arXiv2014

General theory of area reactivity models: rate coefficients, binding probabilities and all that

We further develop the general theory of the area reactivity model that provides an alternative description of the diffusion-influenced reaction of an isolated receptor-ligand pair in terms of a generalized Feynman-Kac equation. We analyze both the irreversible and reversible reaction and derive the equation of motion for the survival and separation probability. Furthermore, we discuss the notion of a time-dependent rate coefficient within the alternative model and obtain a number of relations between the rate coefficient, the survival and separation probabilities and the reaction rate. Finally, we calculate asymptotic and approximate expressions for the (irreversible) rate coefficient, the binding probability, the average lifetime of the bound state and discuss on- and off-rates in this context. Throughout our treatment, we will point out similarities and differences between the area and the classical contact reactivity model. The presented analysis and obtained results provide a theoretical framework that will facilitate the comparison of experiment and model predictions.

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arXiv2013

Exact solution of the area reactivity model of an isolated pair

We investigate the reversible diffusion-influenced reaction of an isolated pair in two space dimensions in the context of the area reactivity model. We compute the exact Green's function in the Laplace domain for the initially unbound molecule. Furthermore, we calculate the exact expression for the Green's function in the time domain by inverting the Laplace transform via the Bromwich contour integral. The obtained results should be useful for comparing the behavior of the area reactivity model with more conventional models based on contact reactivity.

q-bio.QM↗
arXiv2013

Exact Green's function of the reversible ABCD reaction in two space dimensions

We derive an exact expression for the Green's functions in the time domain of the reversible diffusion-influenced ABCD reaction $A+B\leftrightarrow C+D$ in two space dimensions. Furthermore, we calculate the corresponding survival and reaction probabilities. The obtained expressions should prove useful for the study of reversible membrane-bound reactions in cell biology and can serve as a useful ingredient of enhanced stochastic particle-based simulation algorithms.

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arXiv2013

Solution to the one-dimensional telegrapher's equation subject to a backreaction boundary condition

We discuss solutions of the one-dimensional telegrapher's equation in the presence of boundary conditions. We revisit the case of a radiation boundary condition and obtain an alternative expression for the already known Green's function. Furthermore, we formulate a backreaction boundary condition, which has been widely used in the context of diffusion-controlled reversible reactions, for a one-dimensional telegrapher's equation and derive the corresponding Green's function.

math-ph↗
arXiv2012

Non-Markovian reversible diffusion-influenced reactions in two dimensions

We investigate the reversible diffusion-influenced reaction of an isolated pair in the presence of a non-Markovian generalization of the backreaction boundary condition in two space dimensions. Following earlier work by Agmon and Weiss, we consider residence time probability densities that decay slower than an exponential and that are characterized by a parameter $0<σ\leq 1$. We calculate an exact expression for the probability $S(t|\ast)$ that the initially bound particle is unbound, which is valid for arbitrary $σ$ and for all times. Furthermore, we derive an approximate solution for long times. We show that the ultimate fate of the bound state is complete dissociation, as in the 2D Markovian case. However, the limiting value is approached quite differently: Instead of a $\sim t^{-1}$ decay, we obtain $1-S(t|\ast)\sim t^{-σ}\ln t$.

math-ph↗
arXiv2012

Note on an integral expression for the average lifetime of the bound state in 2D

Recently, an exact Green's function of the diffusion equation for a pair of spherical interacting particles in two dimensions subject to a backreaction boundary condition was used to derive an exact expression for the average lifetime of the bound state. Here, we show that the corresponding divergent integral may be considered as the formal limit of a Stieltjes transform. Upon analytically calculating the Stieltjes transform one can obtain an exact expression for the finite part of the divergent integral and hence for the average lifetime.

math-ph↗
arXiv2011

Survival probabilities and rates derived from an exact Green's function of the reversible diffusion-influenced reaction for an isolated pair in 2D

Recently, an exact Green's function of the diffusion equation for a pair of spherical interacting particles in two dimensions subject to a backreaction boundary condition was derived. Here, we use the obtained Green's function to calculate exact expressions for the survival probability, the time-dependent reaction rate coefficient for the initially unbound pair and the survival probability of the bound state in the time domain. Moreover, we derive an exact expression for the off-rate.

math-ph↗
arXiv2011

Coarse-Grained Stochastic Particle-based Reaction-Diffusion Simulation Algorithm

In recent years, several particle-based stochastic simulation algorithms (PSSA) have been developed to study the spatially resolved dynamics of biochemical networks at a molecular scale. A challenge all these approaches have to address is to allow for simulations at cell-biologically relevant timescales without neither neglecting important spatial and biochemical properties of the simulated system nor introducing ad-hoc assumptions not based on physical principles. Here we describe a PSSA that permits large time steps while still retaining a high degree of accuracy. The approach addresses the typical disadvantage of Brownian dynamics, namely the need to use small time steps to resolve bimolecular encounters accurately, by estimating the number of otherwise unnoticed encounters with the help of the Green's functions of the diffusion equation incorporating molecular interactions. This method has previously been proposed for purely absorbing boundary conditions and irreversible bimolecular reactions. Building on those ideas, we developed a general-purpose PSSA that is applicable to a broad class of reaction-diffusion problems by incorporating reflective and radiation boundary conditions and reversible reactions. We furthermore discuss how reaction-diffusion systems on 2D membranes can be described and derive small time expansions of the Green's functions that substantially speed up key calculations, particularly in the problematic case of molecules in close proximity. Finally, we point out the formal relationship between our and exact algorithms. The proposed algorithm may serve as an easily implementable and flexible, computationally efficient, coarse-grained description of reaction-diffusion systems in 2D and 3D that nevertheless provides a stochastic, detailed representation at the level of individual particle trajectories in space and time.

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