1 Why are equally-sized gametes so rare? The instability of isogamy and the cost of anisogamy Matsuda, H. & Abrams, P.A. (1999) Evolutionary Ecology Research 1: unpublished study Hiroyuki Matsuda (Department of Environmental Management, Yokohama National University (YNU))

2 Brief self-introduction Editor-in-Chief of Japanese Journal of Conservation Ecology (by ESJ) IUCN/SSC Japan vascular plants, extinction risk assessment of Japanese Red Data Book Founder of adaptive management in Japan (for sika deer in Hokkaido Prefecture) 1 st author of “Guideline for Nature Restoration Projects” by ESJ committee.

3 Overview of this talk Parker, Baker & Smith (PBS 1972) gave a solution for isogamy as an evolutionary game, and many authors consider factors for anisogamy (e.g., effects for cytoplasm and parasitism). But a Nash solution is not the solution of evolutionary dynamics. Matsuda & Abrams (M&A 1999): isogamy is theoretically unlikely, and consider the condition of isogamy and the evolution of slightly different gamete sizes (“slight anisogamy”).

4 Tatsuya Togashi told me: Isogamy and “slight anisogamy” exists in green algae. In some green algae, the relative size difference between female and male gametes is ca is 3  2 (the cubic root of 2) Male gametes is a product of one more fission than females. A male gamete is half of a female in volume. I define size as volume in this talk.

5 Let x and y be the gamete sizes of female and male Uninvadability is a state that any mutant of “either sex” does not have a higher fitness than the population mean. Fitness is at a local maximum, i.e.,  2 F x /  x 2 < 0 and  2 F y /  y 2 < 0. However, convergence stability depends on Jacobian of evolutionary game dynamics… Evolutionary Stability (ES): uninvadability vs. convergence stability

6 Jacobian for Evolutionary Stability: uninvadability vs. convergence stability Uninvadability (Maynard Smith & Price 1973) Nash(1950) solution Convergence stability (~Eshel & Motro 1981) a+d<0 ad – bc > 0 a < 0, d < 0 b: , c : 

7 The origin and evolution of gamete dimorphism and the male-female phenomenon. (Parker et al. 1972) male’s mating success ~ #eggs/#sperms zygote’s fitness ~ zygote size F y = (K/y) m y (y, x*, y*) s(x*+y) F x = (K/x) m x (x, x*, y*) s(x+y*) e.g., m y (y, x*, y*) = cx*/y*, m x = c Parent’s fitness No. of gametes mating success zygote’s fitness ××=

8 Optimal egg size of parthenogenesis Parthenogenesis: F(x) = (K/x)s(x)↑, dF/dx = (K/x)s’(x) – (K/x 2 )s(x) = 0 xs’(x) = s(x) x par *=s(x)/s’(x) A smaller or larger egg is uninvadible. s(x)s(x) x par * egg size x O

9 Evolutionary game between male and female If  m y /  y =  m x /  x = 0 (mating success of a gamete is independent of its own gamete size but depends on x*/y*), Nash solution for non-cooperative game  F x /  y = m x (K/y)[s’(x*+y) – m x s(x*+y)/y] = 0  F y /  x = m y (K/y)[s’(x+y*) – m y s(x+y*)/x] = 0 Isogamic (symmetrical) solution (x*=y*): x iso *=y iso *=s(2x*)/s’(2x*) F y = (K/y) m x s(x*+y)

10 Uninvadable zygote size (Parker et al. 1972) Isogamy: x iso * = y iso * = s(2x*)/s’(2x*) A smaller or larger mutant is always uninvadable. zygote size y+x* Survival rate of zygote s 2x iso * x par * x iso *

11 Co-Evolution Dynamics dx*/dt = g(  F x /  x) dy*/dt = g(  F y /  y) (  F x /  x)= –(K/x* 2 ) m x s(x*+y*) + (K/x*) m x s’(x*+y*) + (K/x*) (  m x /  x) s(x*+y*) g: Additive genetic variance small phenotypic variance, no covariance Nash solution is an equilibrium, but it is neither necessary nor sufficient condition for convergence stability.

12 Isogamy is usually convergence unstable! If (  m x /  x)= (  m y /  y) = 0, and if s’’ < 0, then at x*=y*= s(2x*)/s’(2x*), Condition for convergence stability (s)(–s + 2s’’ x** 2 ) > 0. It does not hold at all (s>0 & s’’<0)

13 Dynamics of gamete sizes Gamete size x Irrespective of existence of a Nash solution, anisogamy evolves if the mating success of a gamete is independent of its own gamete size. Gamete size y s = log zs = z 6 /(1+z 6 )s = 1-exp(z 2 )

14 Why is isogamy possible? or (2) 3 or more mating types, or (3) isogamy is possible if Continuous change of ovary size is difficult. Mutation of gamete size may change with the number of fissions (discrete event) And one more fission may lose some functions of gamete (mobility, fertility, etc.) Gamete size x Gamete size y (1) Mating success depends on the gamete size, e.g., m(y, x*, y*) = (x*/y*)(y-y 0 ) 2,…(M & A 1999),

15 Why is “slight anisogamy” possible? Two more fission will lose some functions of gametes in mobility. Visibility (eye spots) Chemotaxis (chemical sensors) ? Are male gametes with two more fission really absent in all green algae? I do not know.

16 Cost of anisogamy vs. isogamy (Matsuda & Abrams 1999) If zygote size of isogamy equals egg size, the cost of anisogamy is two-fold, But zygote size of isogamy < egg size of anisogamy zygote or egg size Survival rate of zygote s 2x*2x* x par * s = z n /(1+z n ) Cost of anisogamy n n=6

17 Summary The evolution of anisogamy is a good theoretical example of convergence instability that differs from uninvadability. I think this is not a side-effect of prevention of cytoplasmic competition. Isogamy and slight anisogamy is possible if mating success depends on its own gamete size. After sexual reproduction established, the cost of anisogamy is << two-fold.

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19 The biological origin of gender … is difference between gamete sizes. Male fitness usually has a bigger variance, (speculations) Males like a risky gamble, Monogamy has decreased this difference,

20 Fallacy of uninvadability & Nash solution One of the most common phenomena in multicellular organisms, anisogamy, is not explained by uninvadability. Possibility of difference between uninvadability and convergence stability is universal. Convergence stability is a better criterion of evolutionary stability.

21 Why have anisogamy evolved? (Parker et al. 1972) If a very small mutant (sperm) appears, it is invadable. zygote size y+x* Survival rate of zygote s 2x*2x* x par * x iso *