Limits...
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

Schematic representation of the human uterus, showing the fallopian tubes, ovaries, cervix, vagina, and the ovum. We emphasize various location of the target (round dashed circle)
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC4940446&req=5

Fig1: Schematic representation of the human uterus, showing the fallopian tubes, ovaries, cervix, vagina, and the ovum. We emphasize various location of the target (round dashed circle)

Mentions: We recall that the adult female uterus (Gray et al. 1974) is pear-shaped and is about 7.5 cm long, has a maximum diameter of 5 cm and a height of 3.4 cm, with a mean volume of tens of . It is a hollow thick-walled non-convex muscular organ. On its upper part, the uterine tubes open, one on either side, while below, its cavity connects to that of the vagina. After an egg is released from the ovaries and moves inside the uterine cavity through the uterine tubes, it waits for fertilization (see Fig. 1). In summary, fertilization occurs most likely between the junction at the end of the uterus and somewhere in the fallopian tube, but not inside the uterus. We thus define the position of the target for this search process as the entrance of the fallopian tube, modeled here as a small gap between straight line and quarter-ellipses (see Fig. 1).Fig. 1


Search for a small egg by spermatozoa in restricted geometries.

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

Schematic representation of the human uterus, showing the fallopian tubes, ovaries, cervix, vagina, and the ovum. We emphasize various location of the target (round dashed circle)
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

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

Fig1: Schematic representation of the human uterus, showing the fallopian tubes, ovaries, cervix, vagina, and the ovum. We emphasize various location of the target (round dashed circle)
Mentions: We recall that the adult female uterus (Gray et al. 1974) is pear-shaped and is about 7.5 cm long, has a maximum diameter of 5 cm and a height of 3.4 cm, with a mean volume of tens of . It is a hollow thick-walled non-convex muscular organ. On its upper part, the uterine tubes open, one on either side, while below, its cavity connects to that of the vagina. After an egg is released from the ovaries and moves inside the uterine cavity through the uterine tubes, it waits for fertilization (see Fig. 1). In summary, fertilization occurs most likely between the junction at the end of the uterus and somewhere in the fallopian tube, but not inside the uterus. We thus define the position of the target for this search process as the entrance of the fallopian tube, modeled here as a small gap between straight line and quarter-ellipses (see Fig. 1).Fig. 1

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