Proc. Natl. Acad. Sci. USA Vol. 81, pp. 6078-6079, October 1984 Evolution Search, encounter rates, and the evolution of anisogamy (isogamy/gamete dimorphism/sexual dimorphism/gamete motility) PAUL ALAN COX*t AND JAMES A. SETHIAN*§ *Department of Botany and tDepartment of Mathematics and Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720 Communicated by Edward 0. Wilson, June 15, 1984

ABSTRACT We describe analytical and numerical models 12-14); we further assume that the gametes jerk in unison. to study gamete encounters in two and three dimensions be- This will be the model for our numerical simulation. For our tween differently sized gametes without assuming pre-existing mathematical model, we further simplify as follows: (i) we mating types. Our results indicate that an isogamous popula- evaluate the number ofcollisions during a time t less than the tion can be successfully invaded by a gametangium if it pro- time between jerks; hence, during that time each gamete duces gametes of a different size. The existence of a low adap- moves in a straight line; and (ii) we ignore the fact that once tive peak for isogamy and a much higher adaptive peak for fusion occurs, a pair is no longer available for fertilization anisogamy suggests that stochastic forces may be initially im- and instead concern ourselves with calculating the total portant in driving isogamy through the fitness saddle to ani- number of encounters. sogamy. We begin by considering motion in a plane. Let VMAX be the total reproductive mass available for each type A and Previous models (1-7) for the evolution of anisogamy from type B gametangium, and let nA (nB) be the number oftype A isogamy have ignored the effect ofgamete size on encounter (type B) gametes produced by each type A (the type B) gam- rates.¶ These models predict anisogamy to be favored only etangium. Then, when zygote fitness increases exponentially with respect to VMAX VMAX zygote size. nA - Cl nB - C1 3 We here report analytic and numerical results that show rA rB anisogamy to be evolutionarily stable except for a small re- gion of local stability for isogamy for small gamete sizes. Our We have results hold for a wide range of fitness functions and, in par- uArA = T = uBrB, ticular, for both a simple linear relation between fitness and zygote size and a fitness function based on a normal distribu- where C1, C2, and T are constants and UA (UB) is the velocity tion around an optimal zygote size. of the type A (B) gamete. We assume that both A and B ga- Consider a collection of type A gametangia with equal re- metes are uniformly distributed and that the direction each productive masses capable of producing type A gametes, all gamete moves is chosen at random. We choose units so that ofconstant and equal volume. We assume that the number of there are nA type A gametes per unit area (not, of course, all A gametangia is so large that the gametes from a single A from the same parent). The problem of determining the num- gametangium will not collide with each other but instead will ber of encounters is similar to problems in search theory fuse with type A gametes from other gametangia. Suppose (15); following those investigations, we can determine RAB. that this population is invaded by an additional gametangi- Given a type A (B) gamete moving with vector velocity UA um, which we label B, with the same available reproductive (UB), let (p be the angle between the two, measured in a coun- mass, which can be divided up into N type B gametes. Let terclockwise direction. The vector velocity w' corresponding I" be the effectiveness of this invasion-that is, to the motion of the A gamete relative to the B gamete has magnitude I" = K" R" W", w = (uA + UB - 2uAuBcosp)1/2. where KA is the fraction of encounters between A and B We first consider collisions with reference to a single B ga- gametes that result in fusion, RA is the number of encoun- mete. For a particular angle q', only those gametes located in ters, and WA is the fitness of the zygotes parented by A and the region corresponding to the circle of radius rA + rB trans- B gametes. KA is related to the species-specific reproduc- lated in the direction -w' a distance wt can collide with the B tive physiology; we shall ignore this factor and assume KA gamete and there are nA(rA + rB)wt such gametes. Integrat- to be unity. We study the effect of varying N on the invasion all we the effectiveness I. ing over possible angles 'p find total number of We idealize type A (B) gametes as spheres of radius rA collisions RAB (rB), each with a flagellum capable of producing a constant, (rA + rB)2 given thrust that is the same regardless of the gamete size. RAB =C 4 4 () Thus, we assume an inverse relation between size and veloc- ity [this relation is verified by published data (10) and in the where C is a constant, sin(O = 2(uAuB)½/(UA + UB), and E is a idealized model situation of steady-state flow past a sphere complete elliptic integral of the second kind. at low Reynolds number is provided by Stokes law (11)]. We assume that gamete motion consists of short, smooth swim- tPresent address: Department of Botany and Range Science, ming paths, followed by a sudden reorientation in a new di- Brigham Young University, Provo, UT 84602. rection, after which the swimming motion is resumed (10, 1Present address: Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012. lExcept for a comment concerning locomotion based purely on mo- The publication costs of this article were defrayed in part by page charge lecular bombardment (8) and a study using statistical mechanics to payment. This article must therefore be hereby marked "advertisement" analyze anisogamy on the basis of encounter rates and life expec- in accordance with 18 U.S.C. §1734 solely to indicate this fact. tancy (9).

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Suppose we assume that Vp, is the optimal size of a zy- In three dimensions, a similar derivation produces an en- gote and that the fitness function is a bivariate normal distri- counter rate RAB of the form bution with mean and variance a2. The invasion effec- V0p, (rA +rB)2[1 i 3 1 131 tiveness becomes - RAB = C 2 2 + rrB rA rE rA rB =C(rA + rB)2 4(- 7r(r3+r3,,) 2 r IAB = 4 4 E((W rr~B-O)/v By using this encounter rate, invasion effectiveness for vari- ous fitness functions can be examined. The qualitative re- zygote size corresponding to a radius sults remain unchanged; anisogamy is favored except for a In Fig. 1, with optimal for isogamy for relatively small- of (2)4A 3 units and a2 = 12.5, we plot IAB for rA and rB be- small adaptive fitness peak tween 1 and 5 units, where the unit length is the radius of the sized gametes. smallest possible gamete, the axes on the base of the plot To test the applicability of these conclusions to our origi- corresponding to rA and rB are increasing in the direction out nal model for gamete motion, we developed a numerical so computer simulation of gamete encounters. We studied mo- of the page, and the invasion effectiveness is normalized A gametes. that the maximal value is 1. The dark bands are curves of tion in a plane and began with a uniform grid of fol- We injected into the center of the domain a type B gametan- constant zygote volume. We interpret these results as The position lows. Assume that each of the type A gametangium produces gium capable of dividing into N type B gametes. that the invasion ofboth types ofgametes was advanced a time step by motion type A gametes of unit size. Fig. 1 shows a Gaussian dis- effectiveness of the invading B gametangium depends on the in a random direction with speed drawn from of unit tribution with mean inversely proportional to the radius and size of the B gametes produced. If the B gametes are type B gamete size also, the invasion is more successful than that associat- variance prescribed. If a type A gamete and collided during a time step, they were assumed to have fertil- ed with slightly larger (and hence fewer) B gametes. Howev- The results of er, ifthe B gametangium produces B gametes above a certain ized and were removed from the calculation. result- computing the invasion effectiveness as a function of N for size, even though there are relatively few ofthem, the densities (10-13) agree ing invasion can be far more effective than that associated known gamete sizes, velocities, and we have assumed that the well with the theoretical conclusions, once again indicating with unit-sized B gametes. Since a of isogamy of A gametes from a single A gametangium is small anisogamy to be favored except for local peak number at small gamete sizes. relative to the total number of A gametes produced by all A (16) charac- gametangia, we can ignore collisions among gametes parent- Our results indicate an adaptive topography a B gametangi- terized by high fitness peaks corresponding to anisogamy ed by the same A gametangium. Invasion by fitness corresponding to isogamy. um producing unit-sized gametes is thus comparable to inva- and a much smaller peak effec- Isogamy is predicted to be locally stable at relatively small sion by an A gametangium, and the resulting invasion with results that show the B is nearly equal to that for an gamete sizes; this is in agreement tiveness for gametangium isogamous algae to have significantly smaller zygote sizes A gametangium. Hence, our results show that a B gametangi- numerical ex- and larger B gametes will have a higher than anisogamous taxa (17). The details of the um producing fewer periments, full derivations, and a study of the related issues invasion effectiveness than any of the A gametangia produc- elsewhere (18). ing unit-sized gametes. In addition, our results indicate that a of oogamy and chemotaxis will be presented population of large A gametes produced by type A gametan- During this study P.A.C. was supported by a Miller Fellowship by gia could always be successfully invaded by a type B gam- the Miller Institute for Basic Research in Science and J.A.S. was etangium producing smaller type B gametes. This result sug- supported by a National Science Foundation Mathematical Sciences gests that relative sizes of isogametes depend purely on fit- Post-doctoral Fellowship. This work was supported in part by the ness considerations. This qualitative character of the fitness Director, Office of Energy Research, Office of Basic Energy Sci- if instead one considers fitness ences, Engineering, Mathematical and Geosciences Division of the surface remains unchanged under Contract DE-AC03-76SF00098. as a linear function of volume. U.S. Department of Energy 1. Parker, G. A., Baker, R. R. & Smith, V. G. F. (1971) J. 14 Theor. Biol. 96, 529-553. 0.9 2. Parker, G. A. (1978) J. Theor. Biol. 73, 247-270. 1.0 3. Bell, G. J. (1978) J. Theor. Biol. 96, 281-294. 0.9 4. Parker, G. A. (1982) J. Theor. Biol. 96, 281-294. Smith, J. (1978) The Evolution of Sex (Cambridge . 0.8 cn 5. Maynard 0 Univ. Press, Cambridge, MA). o 04 0.7 6. Williams, G. C. (1975) Sex and Evolution (Princeton Univ. 0.5 Press, Princeton, NJ). o.6 c 7. Ghiselin, M. T. (1974) The Economy ofNature and the Evolu- 0.5 .> tion of Sex (Univ. of California Press, Berkeley). 8. Schuster, P. & Sigmund, K. (1981) J. Theor. Biol. 94, 107-110. 16 4 0.4 9. Scudo, F. M. (1965) Evolution 21, 285-291. 0.3 f 10. Pommerville, J. (1978) Exp. Cell Res. 113, 161-172. 0.2 0 11. Landau, L. D. & Lifshitz, E. M. (1959) Fluid Mechanics (Per- gamon, Oxford). 0.1 12. Katz, D. F. & Yanagimachi, R. (1980) Biol. Reprod. 22, 759- 0.0 764. 13. Dresdner, R. D. & Katz, D. F. (1981) Biol. Reprod. 25, 920- 930. 14. David, G., Serres, S. & Jouannet, P. (1981) Gamete Res. 4, 83- 95. 15. Koopman, B. 0. (1956) Oper. Res. 4, 324-346. 16. Wright, S. (1977) Evolution ofthe Genetics ofPopulation: Ex- perimental Results and Evolutionary Deductions (Univ. of Chicago Press, Chicago), Vol. 3. 17. Madsen, J. D. & Waller, D. M. (1983) Am. Nat. 212, 443-447. FIG. 1. Invasion effectiveness IAB. 18. Cox, P. A. & Sethian, J. A. (1984) Am. Nat., in press. Downloaded by guest on September 30, 2021