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Search for a small egg by spermatozoa in restricted geometries.

Yang J, Kupka I, Schuss Z, Holcman D - J Math Biol (2015)

Bottom Line: In the proposed model the swimmers' trajectories are rectilinear and the speed is constant.Because hitting a small target by a trajectory is a rare event, asymptotic approximations and stochastic simulations are needed to estimate the mean search time in various geometries.We consider searches in a disk, in convex planar domains, and in domains with cusps.

View Article: PubMed Central - PubMed

Affiliation: Applied Mathematics and Computational Biology, Ecole Normale Supérieure, IBENS, 46 rue d'Ulm, 75005, Paris, France.

ABSTRACT
The search by swimmers for a small target in a bounded domain is ubiquitous in cellular biology, where a prominent case is that of the search by spermatozoa for an egg in the uterus. This is one of the severest selection processes in animal reproduction. We present here a mathematical model of the search, its analysis, and numerical simulations. In the proposed model the swimmers' trajectories are rectilinear and the speed is constant. When a trajectory hits an obstacle or the boundary, it is reflected at a random angle and continues the search with the same speed. Because hitting a small target by a trajectory is a rare event, asymptotic approximations and stochastic simulations are needed to estimate the mean search time in various geometries. We consider searches in a disk, in convex planar domains, and in domains with cusps. The exploration of the parameter space for spermatozoa motion in different uterus geometries leads to scaling laws for the search process.

No MeSH data available.


Related in: MedlinePlus

Search for a small target in a disk and a ball. a Schematic representation of the dynamics in a disk: the motion is reflected on the boundary at a random angle. The target size  is on the boundary. b A path with many random reflection escapes ultimately at the target (green dot). c Probability density function of the arrival time obtained from stochastic simulations. d The expected search time is the MFPT of the trajectory to the target, obtained from stochastic simulations (blue circles) and compared to the analytical (9) (red). e Search in a ball. f Comparison of the analytical curve (red) with the stochastic simulations (blue circles). Non-dimensional parameters: ,  (color figure online)
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Fig2: Search for a small target in a disk and a ball. a Schematic representation of the dynamics in a disk: the motion is reflected on the boundary at a random angle. The target size is on the boundary. b A path with many random reflection escapes ultimately at the target (green dot). c Probability density function of the arrival time obtained from stochastic simulations. d The expected search time is the MFPT of the trajectory to the target, obtained from stochastic simulations (blue circles) and compared to the analytical (9) (red). e Search in a ball. f Comparison of the analytical curve (red) with the stochastic simulations (blue circles). Non-dimensional parameters: , (color figure online)

Mentions: The probability to hit the target in exactly k steps is a geometric distribution . Thus the expected number of steps to the target is4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathbb {E}}[N_{hit}] = \sum \limits _{k=0}^{\infty } kq_k=\frac{1}{p(\varepsilon )}=\frac{2\pi R}{\varepsilon }. \end{aligned}$$\end{document}E[Nhit]=∑k=0∞kqk=1p(ε)=2πRε.To determine the expected search time, we assume that the target is centered at and that k reflections occur at the uniformly distributed angles (Fig. 2), where the angle of reflection is . Each ray is represented by its projection on the circle of radius R. The distance between the reflection points is5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} d_k=2R\left/ \sin \frac{\theta _k-\theta _{k-1}}{2}\right/ . \end{aligned}$$\end{document}dk=2Rsinθk-θk-12.The expectation of the right hand side is given by6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathbb {E}}_{\theta _k,\theta _{k-1}}\left[ \left/ \sin \frac{\theta _k-\theta _{k-1}}{2}\right/ \right]= & {} \int _{0}^{2\pi } \int _{0}^{2\pi }\left/ \sin \frac{\theta _k-\theta _{k-1}}{2}\right/ \frac{d\theta _k}{2\pi }\frac{d\theta _{k-1}}{2\pi }\end{aligned}$$\end{document}Eθk,θk-1sinθk-θk-12=∫02π∫02πsinθk-θk-12dθk2πdθk-12π7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}= & {} 2\int _{0}^{\pi /2}\left/ \sin \frac{\theta }{2} \right/ \frac{d\theta }{\pi }=\frac{2}{\pi }. \end{aligned}$$\end{document}=2∫0π/2sinθ2dθπ=2π.The time spent on a single ray is , therefore the mean time to exit in N steps is8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathbb {E}[\tau \,/\,N]=\frac{1}{v_0}\mathbb {E}\sum _{k=1}^{N}2R \left/ \sin \frac{\theta _k-\theta _{k-1}}{2}\right/ =\frac{4RN}{\pi v_0}. \end{aligned}$$\end{document}E[τ/N]=1v0E∑k=1N2Rsinθk-θk-12=4RNπv0.Thus the mean first passage time to the target is9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathbb {E}[\tau ]=\sum _{N=1}^\infty \mathbb {E}[\tau \,/\,N](1-p(\varepsilon ))^{N-1}p(\varepsilon )=\frac{4R}{\pi v_0 p(\varepsilon )}=\frac{8R^2}{ v_0 \varepsilon }=\frac{8S}{\pi v_0 \varepsilon }, \end{aligned}$$\end{document}E[τ]=∑N=1∞E[τ/N](1-p(ε))N-1p(ε)=4Rπv0p(ε)=8R2v0ε=8Sπv0ε,where . Formula (9) shows that the present search process is asymptotically of the order , as goes to zero, much longer than the one for a free Brownian particle with diffusion coefficient D, searching for a target of size , in a two-dimensional domain of surface S, for which the Narrow escape theory (Cheviakov et al. 2012; Schuss et al. 2007) gives10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathbb {E}[\tau ]\approx \frac{S}{\pi D} \log \frac{1}{\varepsilon } +O(1). \end{aligned}$$\end{document}E[τ]≈SπDlog1ε+O(1).The results of numerical simulations of a search for a small arc (Fig. 2a) in a circular disk are shown in Fig. 2b, where the search starts at a point uniformly distributed on the boundary (red point). The tail of the exit time density decays exponentially (Fig. 2c). The increase at short times corresponds to a fraction of trajectories that shoot-up directly to the target. Finally, the numerical approximation for the expected search time is well matched by the asymptotic formula (9), where\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathbb {E}[ \tau ] \sim \frac{1}{\varepsilon } \end{aligned}$$\end{document}E[τ]∼1ε(Fig. 2d, where for the simulations, we took non dimensional parameters , ).


Search for a small egg by spermatozoa in restricted geometries.

Yang J, Kupka I, Schuss Z, Holcman D - J Math Biol (2015)

Search for a small target in a disk and a ball. a Schematic representation of the dynamics in a disk: the motion is reflected on the boundary at a random angle. The target size  is on the boundary. b A path with many random reflection escapes ultimately at the target (green dot). c Probability density function of the arrival time obtained from stochastic simulations. d The expected search time is the MFPT of the trajectory to the target, obtained from stochastic simulations (blue circles) and compared to the analytical (9) (red). e Search in a ball. f Comparison of the analytical curve (red) with the stochastic simulations (blue circles). Non-dimensional parameters: ,  (color figure online)
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Fig2: Search for a small target in a disk and a ball. a Schematic representation of the dynamics in a disk: the motion is reflected on the boundary at a random angle. The target size is on the boundary. b A path with many random reflection escapes ultimately at the target (green dot). c Probability density function of the arrival time obtained from stochastic simulations. d The expected search time is the MFPT of the trajectory to the target, obtained from stochastic simulations (blue circles) and compared to the analytical (9) (red). e Search in a ball. f Comparison of the analytical curve (red) with the stochastic simulations (blue circles). Non-dimensional parameters: , (color figure online)
Mentions: The probability to hit the target in exactly k steps is a geometric distribution . Thus the expected number of steps to the target is4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathbb {E}}[N_{hit}] = \sum \limits _{k=0}^{\infty } kq_k=\frac{1}{p(\varepsilon )}=\frac{2\pi R}{\varepsilon }. \end{aligned}$$\end{document}E[Nhit]=∑k=0∞kqk=1p(ε)=2πRε.To determine the expected search time, we assume that the target is centered at and that k reflections occur at the uniformly distributed angles (Fig. 2), where the angle of reflection is . Each ray is represented by its projection on the circle of radius R. The distance between the reflection points is5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} d_k=2R\left/ \sin \frac{\theta _k-\theta _{k-1}}{2}\right/ . \end{aligned}$$\end{document}dk=2Rsinθk-θk-12.The expectation of the right hand side is given by6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathbb {E}}_{\theta _k,\theta _{k-1}}\left[ \left/ \sin \frac{\theta _k-\theta _{k-1}}{2}\right/ \right]= & {} \int _{0}^{2\pi } \int _{0}^{2\pi }\left/ \sin \frac{\theta _k-\theta _{k-1}}{2}\right/ \frac{d\theta _k}{2\pi }\frac{d\theta _{k-1}}{2\pi }\end{aligned}$$\end{document}Eθk,θk-1sinθk-θk-12=∫02π∫02πsinθk-θk-12dθk2πdθk-12π7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}= & {} 2\int _{0}^{\pi /2}\left/ \sin \frac{\theta }{2} \right/ \frac{d\theta }{\pi }=\frac{2}{\pi }. \end{aligned}$$\end{document}=2∫0π/2sinθ2dθπ=2π.The time spent on a single ray is , therefore the mean time to exit in N steps is8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathbb {E}[\tau \,/\,N]=\frac{1}{v_0}\mathbb {E}\sum _{k=1}^{N}2R \left/ \sin \frac{\theta _k-\theta _{k-1}}{2}\right/ =\frac{4RN}{\pi v_0}. \end{aligned}$$\end{document}E[τ/N]=1v0E∑k=1N2Rsinθk-θk-12=4RNπv0.Thus the mean first passage time to the target is9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathbb {E}[\tau ]=\sum _{N=1}^\infty \mathbb {E}[\tau \,/\,N](1-p(\varepsilon ))^{N-1}p(\varepsilon )=\frac{4R}{\pi v_0 p(\varepsilon )}=\frac{8R^2}{ v_0 \varepsilon }=\frac{8S}{\pi v_0 \varepsilon }, \end{aligned}$$\end{document}E[τ]=∑N=1∞E[τ/N](1-p(ε))N-1p(ε)=4Rπv0p(ε)=8R2v0ε=8Sπv0ε,where . Formula (9) shows that the present search process is asymptotically of the order , as goes to zero, much longer than the one for a free Brownian particle with diffusion coefficient D, searching for a target of size , in a two-dimensional domain of surface S, for which the Narrow escape theory (Cheviakov et al. 2012; Schuss et al. 2007) gives10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathbb {E}[\tau ]\approx \frac{S}{\pi D} \log \frac{1}{\varepsilon } +O(1). \end{aligned}$$\end{document}E[τ]≈SπDlog1ε+O(1).The results of numerical simulations of a search for a small arc (Fig. 2a) in a circular disk are shown in Fig. 2b, where the search starts at a point uniformly distributed on the boundary (red point). The tail of the exit time density decays exponentially (Fig. 2c). The increase at short times corresponds to a fraction of trajectories that shoot-up directly to the target. Finally, the numerical approximation for the expected search time is well matched by the asymptotic formula (9), where\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathbb {E}[ \tau ] \sim \frac{1}{\varepsilon } \end{aligned}$$\end{document}E[τ]∼1ε(Fig. 2d, where for the simulations, we took non dimensional parameters , ).

Bottom Line: In the proposed model the swimmers' trajectories are rectilinear and the speed is constant.Because hitting a small target by a trajectory is a rare event, asymptotic approximations and stochastic simulations are needed to estimate the mean search time in various geometries.We consider searches in a disk, in convex planar domains, and in domains with cusps.

View Article: PubMed Central - PubMed

Affiliation: Applied Mathematics and Computational Biology, Ecole Normale Supérieure, IBENS, 46 rue d'Ulm, 75005, Paris, France.

ABSTRACT
The search by swimmers for a small target in a bounded domain is ubiquitous in cellular biology, where a prominent case is that of the search by spermatozoa for an egg in the uterus. This is one of the severest selection processes in animal reproduction. We present here a mathematical model of the search, its analysis, and numerical simulations. In the proposed model the swimmers' trajectories are rectilinear and the speed is constant. When a trajectory hits an obstacle or the boundary, it is reflected at a random angle and continues the search with the same speed. Because hitting a small target by a trajectory is a rare event, asymptotic approximations and stochastic simulations are needed to estimate the mean search time in various geometries. We consider searches in a disk, in convex planar domains, and in domains with cusps. The exploration of the parameter space for spermatozoa motion in different uterus geometries leads to scaling laws for the search process.

No MeSH data available.


Related in: MedlinePlus