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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 process in uterus-like geometry explored by simulations. a Schematic representation of a two-dimensional uterus-like domain containing a typical trajectory before reaching a small target. b Probability density distribution of the search time, when the target size  non dimensional unit (, ). c Probability density distribution of arrival time for different target size  and the inset is the normalized pdf of arrival time normalized by the mean for various opening sizes . d The mean arrival time in a uterus-like geometry decays approximately with , where the target size is  and
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Fig4: Search process in uterus-like geometry explored by simulations. a Schematic representation of a two-dimensional uterus-like domain containing a typical trajectory before reaching a small target. b Probability density distribution of the search time, when the target size non dimensional unit (, ). c Probability density distribution of arrival time for different target size and the inset is the normalized pdf of arrival time normalized by the mean for various opening sizes . d The mean arrival time in a uterus-like geometry decays approximately with , where the target size is and

Mentions: Classical models of spermatozoa motion include the beating of flagella in viscoelastic fluids (Fu et al. 2008) and attraction to a flat wall due to hydrodynamic interactions of the swimmer with the surface (Elgeti et al. 2010; Berke et al. 2008; Smith et al. 2011; Gaffney et al. 2011; Kantsler et al. 2014, 2013). For high spermatozoa concentration, collective modes of locomotion, different from those displayed by isolated cells, have been described by long-time kinematics of their relative locomotion (Michelin and Lauga 2010). The study of asymmetric flagellar bending was based on cytoplasmic calcium dynamics in the flagellum (Olson et al. 2011). The trajectories of spermatozoa can also be influenced by fluid motion (Marcos et al. 2012). However none of the present studies have addressed the generic question of the search of a small egg in the context of the uterus. Indeed, we shall explore here the role of the uterus geometry, quantified by various parameters such as the height, width, the radius of the cervix , local curvature near the fallopian tubes or the aspect (as shown in Fig. 4) that measures the non-convexity, all should be key parameters in directing the spermatozoa toward the egg, yet this possibility has not been explored so far. However, contrary to the mentioned references, in the present work, we will use a very crude model of the spermatozoa motion, approximated as a ballistic directed motion, to derive asymptotic formula for the search process in convex geometries and use numerical simulations for non-convex ones. The novelty and difficulty here are in geometry of the uterus-like domain, where the motion occurs and the search for the small egg target.


Search for a small egg by spermatozoa in restricted geometries.

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

Search process in uterus-like geometry explored by simulations. a Schematic representation of a two-dimensional uterus-like domain containing a typical trajectory before reaching a small target. b Probability density distribution of the search time, when the target size  non dimensional unit (, ). c Probability density distribution of arrival time for different target size  and the inset is the normalized pdf of arrival time normalized by the mean for various opening sizes . d The mean arrival time in a uterus-like geometry decays approximately with , where the target size is  and
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4940446&req=5

Fig4: Search process in uterus-like geometry explored by simulations. a Schematic representation of a two-dimensional uterus-like domain containing a typical trajectory before reaching a small target. b Probability density distribution of the search time, when the target size non dimensional unit (, ). c Probability density distribution of arrival time for different target size and the inset is the normalized pdf of arrival time normalized by the mean for various opening sizes . d The mean arrival time in a uterus-like geometry decays approximately with , where the target size is and
Mentions: Classical models of spermatozoa motion include the beating of flagella in viscoelastic fluids (Fu et al. 2008) and attraction to a flat wall due to hydrodynamic interactions of the swimmer with the surface (Elgeti et al. 2010; Berke et al. 2008; Smith et al. 2011; Gaffney et al. 2011; Kantsler et al. 2014, 2013). For high spermatozoa concentration, collective modes of locomotion, different from those displayed by isolated cells, have been described by long-time kinematics of their relative locomotion (Michelin and Lauga 2010). The study of asymmetric flagellar bending was based on cytoplasmic calcium dynamics in the flagellum (Olson et al. 2011). The trajectories of spermatozoa can also be influenced by fluid motion (Marcos et al. 2012). However none of the present studies have addressed the generic question of the search of a small egg in the context of the uterus. Indeed, we shall explore here the role of the uterus geometry, quantified by various parameters such as the height, width, the radius of the cervix , local curvature near the fallopian tubes or the aspect (as shown in Fig. 4) that measures the non-convexity, all should be key parameters in directing the spermatozoa toward the egg, yet this possibility has not been explored so far. However, contrary to the mentioned references, in the present work, we will use a very crude model of the spermatozoa motion, approximated as a ballistic directed motion, to derive asymptotic formula for the search process in convex geometries and use numerical simulations for non-convex ones. The novelty and difficulty here are in geometry of the uterus-like domain, where the motion occurs and the search for the small egg target.

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