Sperm numbers and fertility: a kinetic approach.

Abstract

A kinetic theory is derived for the relationship between numbers of spermatozoa, sperm movement, and the probability of fertilization. The theory shows good agreement with experimental results. Cf. A.B.A., 33, Nos. 1872 and 1873. F. S. (Abstract retrieved from CAB Abstracts by CABI’s permission)

Mathematical Models of Fertilization—An Eco-Evolutionary Perspective

“…The case of a threshold of sperm concentration seems inconclusive. Van Duijn (1965) and Pace et al (1981) both found that the "threshold versions" fit their empirical data best. On the other hand, fertilization is possible when sperm concentration in the immediate vicinity of eggs is low, such as in marine invertebrates (Franke et al 2002) as well as mammals (Hunter 1996).…”

“…Fertilization functions can be applied to a great taxonomic range of sexually reproduc-ing organisms, from marine broadcast spawners (Rothschild and Swann 1951;Vogel et al 1982;Styan 1998;Millar and Anderson 2003) to cattle (van Duijn 1964(van Duijn , 1965Schwartz et al 1981;Fearon and Wegener 2000), humans (at least on a qualitative and illustrative level ;Amann 1989), and even to isogamous organisms that reproduce sexually via syngamy, but do not have separate male and female gametes (Togashi et al 2007;Lehtonen 2015). Although some models have been tailored to a specific study organism or taxon, most fertilization functions have several features in common.…”

“…Here γ is a parameter relating to sperm mortality, as well as other factors that affect mortality outside of sperm numbers (see van Duijn 1964van Duijn , 1965Fearon and Wegener 2000). Found to fit empirical data on bovine artificial insemination better than other exponential forms by Pace et al (1981).…”

“…Whether such an assumption is realistic is debatable, and the answer may be taxonspecific. Schwartz et al (1981) question the derivation of equation F 5 originally proposed by van Duijn (1964Duijn ( , 1965 for artificial insemination of livestock, stating that the assumptions about aspects of sperm kinetics introduced by the formula were complicated and not well understood. The equation is a good fit to the data presented by van Duijn, as well as that presented by Schwartz et al (1981), but in both cases there are no data for very low sperm concentrations and low fertilization probability, which is where the threshold or acceleration effect would be observable.…”

“…Although both sperm and eggs senesce (Salisbury and Hart 1970), on the timescale of a single fertilization event sperm aging is dominant and, hence, most fertilization functions only account for sperm aging. Many models account for gamete aging explicitly (van Duijn 1964(van Duijn , 1965Vogel et al 1982;Lehtonen 2015;Okamoto 2016), although the mathematical details may differ significantly between models. For example, in function F 7 (Table 1; Togashi et al 2007;Lehtonen 2015), t can denote fixed gamete life span or a fixed experimental time, which results in an overall fertilization efficiency parameter that can be decomposed into multiplicative components: gamete encounter rate and gamete life span.…”