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Bernoff AJ, Jilkine A, Navarro Hernández A, Lindsay AE. Single-cell directional sensing from just a few receptor binding events. Biophys J 2023; 122:3108-3116. [PMID: 37355773 PMCID: PMC10432224 DOI: 10.1016/j.bpj.2023.06.015] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/11/2023] [Revised: 06/08/2023] [Accepted: 06/20/2023] [Indexed: 06/26/2023] Open
Abstract
Identifying the directionality of signaling sources from noisy input to membrane receptors is an essential task performed by many cell types. A variety of models have been proposed to explain directional sensing in cells. However, many of these require significant computational and memory capacities for the cell. We propose and analyze a simple mechanism in which a cell adopts the direction associated with the first few membrane binding events. This model yields an accurate angular estimate to the source long before steady state is reached in biologically relevant scenarios. Our proposed mechanism allows for reliable estimates of the directionality of external signals using temporal information and assumes minimal computational capacities of the cell.
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Affiliation(s)
- Andrew J Bernoff
- Department of Mathematics, Harvey Mudd College, Claremont, California
| | - Alexandra Jilkine
- Department of Applied & Computational Mathematics & Statistics, University of Notre Dame, South Bend, Indiana
| | - Adrián Navarro Hernández
- Department of Applied & Computational Mathematics & Statistics, University of Notre Dame, South Bend, Indiana
| | - Alan E Lindsay
- Department of Applied & Computational Mathematics & Statistics, University of Notre Dame, South Bend, Indiana.
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2
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Lawley SD, Johnson J. Slowest first passage times, redundancy, and menopause timing. J Math Biol 2023; 86:90. [PMID: 37148411 DOI: 10.1007/s00285-023-01921-9] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/20/2022] [Revised: 04/01/2023] [Accepted: 04/17/2023] [Indexed: 05/08/2023]
Abstract
Biological events are often initiated when a random "searcher" finds a "target," which is called a first passage time (FPT). In some biological systems involving multiple searchers, an important timescale is the time it takes the slowest searcher(s) to find a target. For example, of the hundreds of thousands of primordial follicles in a woman's ovarian reserve, it is the slowest to leave that trigger the onset of menopause. Such slowest FPTs may also contribute to the reliability of cell signaling pathways and influence the ability of a cell to locate an external stimulus. In this paper, we use extreme value theory and asymptotic analysis to obtain rigorous approximations to the full probability distribution and moments of slowest FPTs. Though the results are proven in the limit of many searchers, numerical simulations reveal that the approximations are accurate for any number of searchers in typical scenarios of interest. We apply these general mathematical results to models of ovarian aging and menopause timing, which reveals the role of slowest FPTs for understanding redundancy in biological systems. We also apply the theory to several popular models of stochastic search, including search by diffusive, subdiffusive, and mortal searchers.
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Affiliation(s)
- Sean D Lawley
- Department of Mathematics, University of Utah, Salt Lake City, UT, 84112, USA.
| | - Joshua Johnson
- Division of Reproductive Sciences, Reproductive Endocrinology and Infertility, Department of Obstetrics and Gynecology, University of Colorado-Anschutz Medical Campus, Aurora, CO, USA
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Lindsay AE, Bernoff AJ, Navarro Hernández A. Short-time diffusive fluxes over membrane receptors yields the direction of a signalling source. ROYAL SOCIETY OPEN SCIENCE 2023; 10:221619. [PMID: 37122946 PMCID: PMC10130716 DOI: 10.1098/rsos.221619] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/19/2022] [Accepted: 03/31/2023] [Indexed: 05/03/2023]
Abstract
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 signalling 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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Affiliation(s)
- Alan E. Lindsay
- Department of Applied and Computational Math and Statistics, University of Notre Dame, Notre Dame, IN 46617, USA
| | - Andrew J. Bernoff
- Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, USA
| | - Adrián Navarro Hernández
- Department of Applied and Computational Math and Statistics, University of Notre Dame, Notre Dame, IN 46617, USA
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Kim H, Mori Y, Plotkin JB. Optimality of intercellular signaling: Direct transport versus diffusion. Phys Rev E 2022; 106:054411. [PMID: 36559352 DOI: 10.1103/physreve.106.054411] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/18/2022] [Accepted: 10/28/2022] [Indexed: 06/17/2023]
Abstract
Intercellular signaling has an important role in organism development, but not all communication occurs using the same mechanism. Here, we analyze the energy efficiency of intercellular signaling by two canonical mechanisms: Diffusion of signaling molecules and direct transport mediated by signaling cellular protrusions. We show that efficient contact formation for direct transport can be established by an optimal rate of projecting protrusions, which depends on the availability of information about the location of the target cell. The optimal projection rate also depends on how signaling molecules are transported along the protrusion, in particular the ratio of the energy cost for contact formation and molecule synthesis. Also, we compare the efficiency of the two signaling mechanisms, under various model parameters. We find that direct transport is favored over diffusion when transporting a large amount of signaling molecules. There is a critical number of signaling molecules at which the efficiencies of the two mechanisms are the same. The critical number is small when the distance between cells is far, which helps explain why protrusion-based mechanisms are observed in long-range cellular communications.
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Affiliation(s)
- Hyunjoong Kim
- Center for Mathematical Biology and Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Yoichiro Mori
- Center for Mathematical Biology and Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
- Department of Biology, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Joshua B Plotkin
- Center for Mathematical Biology and Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
- Department of Biology, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
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Park S, Kim H, Wang Y, Eom DS, Allard J. Zebrafish airinemes optimize their shape between ballistic and diffusive search. eLife 2022; 11:75690. [PMID: 35467525 PMCID: PMC9098217 DOI: 10.7554/elife.75690] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/19/2021] [Accepted: 04/25/2022] [Indexed: 11/13/2022] Open
Abstract
In addition to diffusive signals, cells in tissue also communicate via long, thin cellular protrusions, such as airinemes in zebrafish. Before establishing communication, cellular protrusions must find their target cell. Here we demonstrate that the shapes of airinemes in zebrafish are consistent with a finite persistent random walk model. The probability of contacting the target cell is maximized for a balance between ballistic search (straight) and diffusive search (highly curved, random). We find that the curvature of airinemes in zebrafish, extracted from live cell microscopy, is approximately the same value as the optimum in the simple persistent random walk model. We also explore the ability of the target cell to infer direction of the airineme's source, finding that there is a theoretical trade-off between search optimality and directional information. This provides a framework to characterize the shape, and performance objectives, of non-canonical cellular protrusions in general.
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Affiliation(s)
- Sohyeon Park
- Center for Complex Biological Systems, University of California, Irvine, Irvine, United States
| | - Hyunjoong Kim
- Department of Mathematics, University of Pennsylvania, Pennsylvania, United States
| | - Yi Wang
- Center for Complex Biological Systems, University of California, Irvine, Irvine, United States
| | - Dae Seok Eom
- Center for Complex Biological Systems, University of California, Irvine, Irvine, United States
| | - Jun Allard
- dDepartment of Physics and Astronomy, University of California, Irvine, Irvine, United States
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Srivastava V, Cheviakov A. Brownian dynamics simulations for the narrow escape problem in the unit sphere. Phys Rev E 2021; 104:064113. [PMID: 35030881 DOI: 10.1103/physreve.104.064113] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/09/2021] [Accepted: 11/15/2021] [Indexed: 06/14/2023]
Abstract
The narrow escape problem is a first-passage problem that concerns the calculation of the time needed for a Brownian particle to leave a domain with localized absorbing boundary traps, such that the measure of these traps is asymptotically small compared to the domain size. A common model for the mean first-passage time (MFPT) as a function of particle's starting location in a given domain with constant diffusivity is given by a Poisson partial differential equation subject to mixed Dirichlet-Neumann boundary conditions. The primary objective of this work is to perform direct numerical simulations of multiple particles undergoing Brownian motion in a three-dimensional spherical domain with boundary traps, compute MFPT values by averaging Brownian escape times, and compare these with explicit asymptotic results obtained previously by approximate solution of the Poisson problem. A close agreement of MFPT values is observed already at 10^{4} particle runs from a single starting point, providing a computational validation of the Poisson equation-based continuum model. Direct Brownian dynamics simulations are also used to study additional features of particle dynamics in narrow escape problems that cannot be captured in a continuum approach, such as average times spent by particles in a thin layer near the domain boundary, and effects of isotropic vs anisotropic near-boundary diffusion.
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Affiliation(s)
| | - Alexei Cheviakov
- Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan S7N 5E6, Canada
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Handy G, Lawley SD. Revising Berg-Purcell for finite receptor kinetics. Biophys J 2021; 120:2237-2248. [PMID: 33794148 DOI: 10.1016/j.bpj.2021.03.021] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/15/2020] [Revised: 02/18/2021] [Accepted: 03/25/2021] [Indexed: 11/29/2022] Open
Abstract
From nutrient uptake to chemoreception to synaptic transmission, many systems in cell biology depend on molecules diffusing and binding to membrane receptors. Mathematical analysis of such systems often neglects the fact that receptors process molecules at finite kinetic rates. A key example is the celebrated formula of Berg and Purcell for the rate that cell surface receptors capture extracellular molecules. Indeed, this influential result is only valid if receptors transport molecules through the cell wall at a rate much faster than molecules arrive at receptors. From a mathematical perspective, ignoring receptor kinetics is convenient because it makes the diffusing molecules independent. In contrast, including receptor kinetics introduces correlations between the diffusing molecules because, for example, bound receptors may be temporarily blocked from binding additional molecules. In this work, we present a modeling framework for coupling bulk diffusion to surface receptors with finite kinetic rates. The framework uses boundary homogenization to couple the diffusion equation to nonlinear ordinary differential equations on the boundary. We use this framework to derive an explicit formula for the cellular uptake rate and show that the analysis of Berg and Purcell significantly overestimates uptake in some typical biophysical scenarios. We confirm our analysis by numerical simulations of a many-particle stochastic system.
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Affiliation(s)
- Gregory Handy
- Departments of Neurobiology and Statistics, Chicago, Illinois; Grossman Center for Quantitative Biology and Human Behavior, University of Chicago, Chicago, Illinois
| | - Sean D Lawley
- Department of Mathematics, University of Utah, Salt Lake City, Utah.
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Dobramysl U, Holcman D. Triangulation Sensing to Determine the Gradient Source from Diffusing Particles to Small Cell Receptors. PHYSICAL REVIEW LETTERS 2020; 125:148102. [PMID: 33064548 DOI: 10.1103/physrevlett.125.148102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/11/2019] [Accepted: 09/03/2020] [Indexed: 06/11/2023]
Abstract
How does a cell locate the source of molecular guidance cues from within a concentration gradient? We present a computational approach to recover the source from the absorbed fluxes at narrow receptor windows located on the surface of the cell. In the limit of fast binding, we solve the steady-state diffusion equation using an asymptotic approach and hybrid stochastic-analytical simulations. We show that the sensitivity to the gradient direction decays too rapidly to enable long-distance sensing. We illustrate how this constraint can be alleviated when triangulating the source with an increasing number of receptor windows and quantify the susceptibility of this process to flux perturbations.
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Affiliation(s)
- Ulrich Dobramysl
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, United Kingdom
- Wellcome Trust / CRUK Gurdon Institute, University of Cambridge, Cambridge CB2 1QN, United Kingdom
| | - David Holcman
- Group Computational Biology and Data modeling, IBENS, Ecole Normale Superieure - PSL, Paris, France
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