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Why are equally-sized gametes so rare? The instability of isogamy and the cost of anisogamy

http://risk.kan.ynu.ac.jp/matsuda/2005/050720Wien.ppt. 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 Matsuda, H. & Abrams, P.A. (1999) Evolutionary Ecology Research 1:769-784 + unpublished study Hiroyuki Matsuda

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Why are equally-sized gametes so rare? The instability of isogamy and the cost of anisogamy

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  1. http://risk.kan.ynu.ac.jp/matsuda/2005/050720Wien.ppt 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of isogamy and the cost of anisogamy Matsuda, H. & Abrams, P.A. (1999) Evolutionary Ecology Research 1:769-784 + unpublished study Hiroyuki Matsuda (Department of Environmental Management, Yokohama National University (YNU))

  2. http://risk.kan.ynu.ac.jp/matsuda/2005/050720Wien.ppt 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) • 1st author of “Guideline for Nature Restoration Projects” by ESJ committee.

  3. http://risk.kan.ynu.ac.jp/matsuda/2005/050720Wien.ppt 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. http://risk.kan.ynu.ac.jp/matsuda/2005/050720Wien.ppt 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. 1.26. • 1.26 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. Evolutionary Stability (ES):uninvadability vs. convergence stability • Let x and y be the gamete sizes of femaleand 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., 2Fx/x2< 0 and 2Fy/y2< 0. • However, convergence stability depends on Jacobian of evolutionary game dynamics…

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

  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 • Fy= (K/y) my(y, x*, y*)s(x*+y) • Fx= (K/x) mx(x, x*, y*)s(x+y*) • e.g., my(y, x*, y*) = cx*/y*, mx = 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/x2)s(x) = 0 • xs’(x) = s(x) • xpar*=s(x)/s’(x) • A smaller or larger egg is uninvadible. s(x) xpar* O egg size x

  9. Evolutionary game between male and female Fy= (K/y) mxs(x*+y) • If my/y= mx/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 • Fx/y = mx(K/y)[s’(x*+y) – mxs(x*+y)/y]= 0 • Fy/x = my(K/y)[s’(x+y*) – mys(x+y*)/x]= 0 • Isogamic (symmetrical) solution (x*=y*): • xiso*=yiso*=s(2x*)/s’(2x*)

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

  11. Co-Evolution Dynamics • dx*/dt = g(Fx/x) • dy*/dt = g(Fy/y) • (Fx/x)= –(K/x*2) mxs(x*+y*) + (K/x*) mxs’(x*+y*) + (K/x*) (mx/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 (mx/x)= (my/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 • Irrespective of existence of a Nash solution, anisogamy evolves if the mating success of a gamete is independent of its own gamete size. s = log z s = 1-exp(z2) s = z6/(1+z6) Gamete size y Gamete size x

  14. Gamete size y Gamete size x 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.) (1) Mating success depends on the gamete size, e.g., m(y, x*, y*) = (x*/y*)(y-y0)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. xpar* Survival rate of zygote s 2x* zygote or egg size 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 n=6 Cost of anisogamy s = zn/(1+zn) n

  17. http://risk.kan.ynu.ac.jp/matsuda/2005/050720Wien.ppt 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.

  18. 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,

  19. 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.

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

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