Volume 94, No. 2 THE QUARTERLY REVIEW OF BIOLOGY June 2019 MATHEMATICAL MODELS OF FERTILIZATION—AN ECO-EVOLUTIONARY PERSPECTIVE Jussi Lehtonen School of Life and Environmental Sciences, University of Sydney Sydney, New South Wales 2006 Australia Evolution and Ecology Research Centre, School of Biological, Earth and Environmental Sciences, University of New South Wales Sydney, New South Wales 2052 Australia e-mail: jussi.lehtonen@iki.fi Louis Dardare Evolution and Ecology Research Centre, School of Biological, Earth and Environmental Sciences, University of New South Wales Sydney, New South Wales 2052 Australia e-mail: [email protected] keywords fertilization functions, egg, sperm, gamete, fertilization models, sperm limitation abstract Mathematical models of fertilization have been developed for many taxa, for both external and inter- nal fertilizers. They estimate the proportion or number of fertilized gametes based on gamete concentrations and parameters relating to the biology of the model organism, as well as serve multiple purposes: a pre- dictive purpose, with applications in, for example, artificial insemination; they clarify causal components of fertilization success such as concentration, size, collision rates and swimming speed of gametes, and polyspermy block times; and they function as components of models in evolutionary ecology, which often require understanding of fitness consequences of resource allocation between gametes and other traits. We pay particular attention to this last category, which has received less attention than other uses. Many evolutionary models assume the simplest relationship between fertilization success and gamete numbers: all eggs are fertilized. In nature, however, it is not uncommon for eggs to be sperm limited, and fertiliza- tion success must decrease as sperm density approaches zero. Fertilization functions become important in the range between these two extremes. We focus on models in evolutionary ecology, but aim for a resource that is useful regardless of topic or taxon by reviewing models developed for different purposes in a com- mon mathematical framework. The Quarterly Review of Biology, June 2019, Vol. 94, No. 2 Copyright © 2019 by The University of Chicago Press. All rights reserved. 0033-5770/2019/9402-0003$15.00 177 This content downloaded from 129.078.056.178 on May 20, 2019 22:47:05 PM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 178 THE QUARTERLY REVIEW OF BIOLOGY Volume 94 Introduction clear thatif the number ofsperm is zero, then ERTILIZATION is a fundamental pro- thenumberof fertilized eggsmustalso be zero. F cess and crucial component of fitness Whathappensinbetween,whenspermispre- for all sexually reproducing organisms, and sent but not necessarily sufficient to fertilize the only way for obligately sexual organisms all eggs? This intermediate scenario, not un- to reproduce and to pass on their genetic common in nature, is often known as sperm information to the next generation. Many limitation (or, more generally, gamete limita- aspects of this process have been modeled us- tion), and it is considered to be a particularly ing mathematical models (henceforth called potent selective force in marine external fer- fertilization functions—see Appendix 1), which tilizers (Levitan and Petersen 1995; Yund describe the proportion or number of fertil- 2000; Levitan 2010), which was likely the ized gametes as a function of total gamete mode of reproduction of our ancestors (Le- numbers. These functions have potential ap- vitan2010). Althoughseveralpublished mod- plications in several fields. They have signif- els do account for sperm limitation in some icantly added to our understanding of the form (Cox and Sethian 1985; Ball and Parker ecology of marine broadcast spawners (Vogel 1997; Mesterton-Gibbons 1999; Levitan 2000; et al. 1982; Styan 1998; Millar and Anderson Bode and Marshall 2007; Iyer and Rough- 2003) where empirical and theoretical devel- garden 2008; Lehtonen and Kokko 2011; Hen- opments have reinforced each other (Styan shaw et al. 2014), many (but not all) of these and Butler 2000; Franke et al. 2002; Levitan models have derived the fertilization pro- et al. 2007; Okamoto 2016; Levitan 2018). cess anew from the ground up, without refer- An obvious application, with potential eco- ence to existing models of fertilization. The nomic impact, is artificial insemination of use of fertilization functions already available livestock, where a good understanding of fer- in the literature may streamline development tilization is necessary for optimal use of sperm of such models, while making it easier to ac- when the aim is to fertilize as many females count for various features of fertilization, such as possible using a limited supply of high- as polyspermy (Styan 1998; Millar and Ander- quality sperm. In other words, the aim is to son 2003; Bode and Marshall 2007) or isog- use enough sperm per insemination, but no amy (Togashi et al. 2007; Lehtonen 2015). more than necessary (Salisbury and Van- The aim of this review is to combine exist- Demark 1961; Foote and Kaproth 1997; Den ing fertilization functions, currently scattered Daas et al. 1998). Our goal is to provide a throughout different (often quite specialized) resource that is of value regardless of topic, research topics, into a clear and easily acces- but we will pay particular attention to an ap- sible source. Fertilization functions may have plication that has received less attention than been initially published using a variety of no- the two mentioned above: the use of fertil- tations and conventions; here we will present ization functions as components of mathe- them in a common mathematical framework matical models in evolutionary ecology. Many that facilitates comparison and selection of models, including the majority of sperm com- fertilization functions, as well as their inte- petition models (Parker 1998; Parker and gration into further evolutionary models. We Pizzari 2010), assume the simplest possible will examine features that are common to relationship between the proportion of fer- most fertilization functions regardless of taxon, tilized eggs and the total number of sperm: while pointing out their unique features. In that is, that there are always sufficient sperm doing so, we also discuss extensions of exist- to fertilize all eggs. This is often well justified ing models, and suggest future directions in the biological scenarios that these models for research on mathematical models of fer- investigate, and we do not criticize this ap- tilization. proach—instead, we aim to complement it. We show that comprehensive mathematical Scope, Purpose, Structure, and machinery exists that can be integrated into Notation of Fertilization Functions such models with relative ease whenever the biological question of interest necessitates this. Fertilization functions can be applied to a And such cases clearly exist: it is intuitively great taxonomic range of sexually reproduc- This content downloaded from 129.078.056.178 on May 20, 2019 22:47:05 PM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). June 2019 MATHEMATICAL MODELS OF FERTILIZATION 179 ing organisms, from marine broadcast spawn- very practical approach; in a typical study spe- ers (Rothschild and Swann 1951; Vogel et al. cies, the relative size difference between eggs 1982; Styan 1998; Millar and Anderson 2003) and sperm is enormous (Parker 1982), and it to cattle (van Duijn 1964, 1965; Schwartz is intuitive to think of the proportion of fer- et al. 1981; Fearon and Wegener 2000), hu- tilized eggs varying as a function of sperm mans (at least on a qualitative and illustrative density. In other words, the fertilization func- level; Amann 1989), and even to isogamous tion is considered from an egg’s perspective, organisms that reproduce sexually via syn- in line with how we typically envision an egg gamy, but do not have separate male and fe- being fertilized by a spermatozoon, rather male gametes (Togashi et al. 2007; Lehtonen than vice versa. However, things are not as 2015). Although some models have been tai- clear-cut once we broaden our taxonomic lored to a specific study organism or taxon, focus to isogamous or near-isogamous spe- most fertilization functions have several fea- cies, which are common and can be found in tures in common. Case-specific details may all eukaryotic supergroups (Lehtonen et al. be vital when studying the ecology of par- 2016a). In such cases, where gametes are ticular species or taxa, but broader, species- morphologically similar, there is no obvious independent features of fertilization can preferred perspective, and it becomes am- become more important in evolutionary mod- biguous to speak of proportions or probabil- els that aim to address questions that apply to ities of fertilization. a broad range of organisms. The fact that such In this article we therefore adopt two nota- near-universal properties of fertilization func- tions for fertilization functions. We use the tions exist in the first place is not surprising, letter p for the proportion of fertilized gam- given that regardless of case-specific details, etes (or, equivalently, probability of fertiliza- all fertilization functions aim to describe the tion). In the standard, anisogamous case we same fundamental process: the coming to- denote the number of eggs with x and the gether of gametes for successful fertilization number of sperm with y. In situations where as a function of gamete numbers or concen- there are no morphologically diverged gam- trations. The fertilization functions we discuss etes (isogamy), these letters arbitrarily de- in this article generally refer to total fertiliza- note the two mating types (Lehtonen et al. tions as a function of total gamete numbers: 2016a; see Appendix 1). x and y can also be they describe the overall outcome of a fertil- used as subscripts (as in px or py) to clarify ization event, but not how that outcome is di- whether we mean the fertilization probabil- videdbetween,say,competingmales.Wealso ity of an x or y gamete.