Reconstructing a point source from diffusion fluxes to narrow windows in three dimensions

Dobramysl, Ulrich; Holcman, David

doi:10.1098/rspa.2021.0271

Cited by 5 publications

(8 citation statements)

References 24 publications

(34 reference statements)

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“…In this section, we show how to extend this asymptotic analysis to obtain the full time-dependent arrival statistics. After applying the Laplace transform p(x; s) = R 1 t=0 e st p(x; t)dt to (2), we obtain the modified Helmholtz problem (18) We now discuss the inversion from Laplace space to physical time. One quantity of interest is the flux (3) through each of the receptors.…”

Section: Receptor Arrival Statisticsmentioning

confidence: 99%

“…1) and the distribution of arrivals across the set of receptors, known as the splitting probabilities [29], encodes information on the source location. In the scenario of planar di↵usion, or three dimension di↵usion to a spherical cell with surface receptors, Dobramysl and Holcman [17,18] demonstrated that a unique source location can be inferred from the splitting probabilities, provided the number of receptors is at least N = 3. Biological receptor numbers vary considerably between systems with examples including N ⇡ 10 4 in budding yeast [25] and N ⇡ 10 4 10 5 in lymphocytes [44].…”

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confidence: 99%

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Short time diffusive fluxes over membrane receptors yields the direction of a signaling source

Lindsay

Bernoff

Hernández

2023

Preprint

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An essential ability of many cell types is to detect stimuli in the form of shallow chemical gradients. Such cues may indicate the direction that new growth should occur, or the location of a mate. Amplification of these faint signals is due to intra cellular mechanisms while the cue itself is generated by the noisy arrival of signaling molecules to surface bound membrane receptors. We employ a new hybrid numerical-asymptotic technique coupling matched asymptotic analysis and numerical inverse Laplace transform to rapidly and accurately solve the parabolic exterior problem describing the dynamic diffusive fluxes to receptors. We observe that equilibration occurs on long timescales, potentially limiting the usefulness of steady state quantities for localization at practical biological timescales. We demonstrate that directional information is encoded primarily in early arrivals to the receptors, while equilibrium quantities inform on source distance. We develop a new homogenization result showing that complex receptor configurations can be replaced by a uniform effective condition. In the extreme scenario where the cell adopts the angular direction of the first impact, we show this estimate to be surprisingly accurate.

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Section: Receptor Arrival Statisticsmentioning

confidence: 99%

mentioning

confidence: 99%

Short time diffusive fluxes over membrane receptors yields the direction of a signaling source

Lindsay

Bernoff

Hernández

2023

Preprint

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“…These facts render the computational effort quite prohibitive: Naive simulations become inefficient due to the very large excursions of Brownian trajectories before hitting targets of interest. We note however, that although the mean time is infinite, the splitting probability of hitting one of several windows is finite and thus estimating it can provide relevant information for many applications [28,27]. In this context, hybrid stochastic simulations circumnavigates these issues of naive Brownian simulations and avoids to simulate explicitly long trajectories with large excursions and thus it circumvents the need for an arbitrary cutoff distance for our infinite domain.…”

Section: Computing the Fluxes Of Brownian Particles To Small Targets ...mentioning

confidence: 99%

“…In analogy to equation 5.6, the sensitivity function for three windows in three dimensions is expressed as the maximum of the differences between the splitting probabilities computed from the fluxes [28] S 123 (x 0 ;…”

Section: Sensitivity Analysis In 3dmentioning

confidence: 99%

“…These processes involve multiple navigational mechanisms: positive attraction to cues, negative repulsion by cues or a combination thereof. Recently, we focused on the problem of neuronal guidance in the brain by answering the question: Is it possible to triangulate the point source of a molecular gradient using the arrival rate of Brownian particles in a domain containing many absorbing windows on its boundary (both in two and three dimensions) [26,13,27,28]? Here, we review recent computational approaches (modeling, asymptotics and hybrid simulations) designed to estimate the position of a point source that emits stochastic particles from the steady-state fluxes collected at narrow windows, located on a bounded surface.…”

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Computational methods and fast hybrid stochastic simulations for triangulation sensing and identifying principles of cell navigation in the brain

Dobramysl,

Holcman

2020

Preprint

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Brownian simulations can be used to generate statistics relevant for studying molecular interactions or trafficking. However, the concurrent simulation of many Brownian trajectories at can become computationally intractable. Replacing detailed Brownian simulations by a rate model was the basis of Gillespie's algorithm, but requires one to disregard spatial information. However, this information is crucial in molecular and cellular biology. Alternatively one can use a hybrid approach, generating Brownian paths only in a small region where the spatial organization is relevant and avoiding it in the remainder of the domain. Here we review the recent progress of hybrid methods and simulations in the context of cell sensing and guidance via external chemical gradients. Specifically, we highlight the reconstruction of the location of a point source in 2D and 3D from diffusion fluxes arriving at narrow windows located on the cell. We discuss cases in which these windows are located on the boundary of the 2D or 3D half-space, on a disk in free space, inside a 2D corridor, or a 3D ball. The hybrid method in question performs Brownian simulations only inside a region of interest. It uses the Neumann-Green's function for the mentioned geometries to generate exact mappings exit and entry points when the trajectory leaves the region, thus avoiding the explicit computation of Brownian paths in an infinite domain. Matched asymptotics is used to compute the probability fluxes to small windows and we review how such an approach can be used to reconstruct the location of a point source and estimating the uncertainty in the source reconstruction due to an additive perturbation present in the fluxes. We also review the influence of various window configurations on the source position recovery. Finally, we discuss potential applications in developmental cell biology and possible computational principles.

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Single-cell directional sensing from just a few receptor binding events

Bernoff¹,

Jilkine²,

Hernández³

et al. 2023

Biophysical Journal

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Reconstructing a point source from diffusion fluxes to narrow windows in three dimensions

Cited by 5 publications

References 24 publications

Short time diffusive fluxes over membrane receptors yields the direction of a signaling source

Short time diffusive fluxes over membrane receptors yields the direction of a signaling source

Computational methods and fast hybrid stochastic simulations for triangulation sensing and identifying principles of cell navigation in the brain

Single-cell directional sensing from just a few receptor binding events

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