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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

Exploring various parameters and the geometry of uterus-like domain. a Pdf of the search time for  for various values of the cervix size . b Changing the aspect ratio  affects the pdf of arrival time. The pdfs of the arrival time, normalized by the maximum are identical (inset). c Normalized pdfs for different parameters  and . d Expected search time for various aspect ratios . All fixed parameters are the same as in Fig. 4
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Fig5: Exploring various parameters and the geometry of uterus-like domain. a Pdf of the search time for for various values of the cervix size . b Changing the aspect ratio affects the pdf of arrival time. The pdfs of the arrival time, normalized by the maximum are identical (inset). c Normalized pdfs for different parameters and . d Expected search time for various aspect ratios . All fixed parameters are the same as in Fig. 4

Mentions: We now explore the consequences of changing various parameters on the arrival time, this includes looking at the geometry of the uterus-like domain. We vary here several parameters such as the target size , the length L and the width W and the entrance size , which represents the cervix outside radius. First, by varying the size (in range ), while keeping the other parameters constant (Fig. 5a), we obtain that the pdfs of arrival time show only a slight difference (less than 20 %). Thus, the numerical simulations show that the cervix size has little influence on the search time.Fig. 5


Search for a small egg by spermatozoa in restricted geometries.

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

Exploring various parameters and the geometry of uterus-like domain. a Pdf of the search time for  for various values of the cervix size . b Changing the aspect ratio  affects the pdf of arrival time. The pdfs of the arrival time, normalized by the maximum are identical (inset). c Normalized pdfs for different parameters  and . d Expected search time for various aspect ratios . All fixed parameters are the same as in Fig. 4
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

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

Fig5: Exploring various parameters and the geometry of uterus-like domain. a Pdf of the search time for for various values of the cervix size . b Changing the aspect ratio affects the pdf of arrival time. The pdfs of the arrival time, normalized by the maximum are identical (inset). c Normalized pdfs for different parameters and . d Expected search time for various aspect ratios . All fixed parameters are the same as in Fig. 4
Mentions: We now explore the consequences of changing various parameters on the arrival time, this includes looking at the geometry of the uterus-like domain. We vary here several parameters such as the target size , the length L and the width W and the entrance size , which represents the cervix outside radius. First, by varying the size (in range ), while keeping the other parameters constant (Fig. 5a), we obtain that the pdfs of arrival time show only a slight difference (less than 20 %). Thus, the numerical simulations show that the cervix size has little influence on the search time.Fig. 5

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