Search for a small egg by spermatozoa in restricted geometries.
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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.
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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 |
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Mentions: Another consequence of the present analysis is that the number of swimmers matters when we consider the first one that finds a neighborhood of the egg. By increasing the number of trajectories (Fig. 6), when computing the arrival time of the first one, the trajectories that are almost optimizing the search time are located in a neighborhood around the optimal ones (white dashed lines in Fig. 6). The optimal trajectory is indeed the geodesic that joins the initial point to the final egg location. The shortest search time is achieved by the geodesic that minimizes the functional53\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Lambda _{min} = \inf _{A_T} \int _0^T d(s)\,ds, \end{aligned}$$\end{document}Λmin=infAT∫0Td(s)ds,where d(s) is the Euclidean distance at time s and54\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A_T =\{x: x(0)=x, x(T)=y, \quad \hbox {where } x \hbox { are piecewise constant trajectories} \},\qquad \end{aligned}$$\end{document}AT={x:x(0)=x,x(T)=y,wherexare piecewise constant trajectories},as indicated by the results of our simulations (Fig. 6). Possible future directions include the exploration of other parameters of the search process, such as the fitness of spermatozoa, the time window when they can sense the presence of the egg and chemotaxis processes, and so on. These elements should certainly be included in future models, as well as a possible killing field, which should account for sperm degradation during their sojourn time. |
View Article: PubMed Central - PubMed
Affiliation: Applied Mathematics and Computational Biology, Ecole Normale Supérieure, IBENS, 46 rue d'Ulm, 75005, Paris, France.
No MeSH data available.