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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+ unpublished study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))

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

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Brief self-introduction Editor-in-Chief of Japanese Journal of Cons

ervation 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 Restorati

on Projects” by ESJ committee.

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

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Overview of this talk Parker, Baker & Smith (PBS 1972) gave a solutio

n 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”).

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

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Tatsuya Togashi told me: Isogamy and “slight anisogamy” exists in green al

gae. In some green algae, the relative size difference be

tween female and male gametes is ca. 1.26. 1.26 is 32 (the cubic root of 2) Male gametes is a product of one more fission tha

n females. A male gamete is half of a female in volume.

I define size as volume in this talk.

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

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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., 2Fx/x2 < 0 and 2Fy/y2 < 0.

However, convergence stability depends on Jacobian of evolutionary game dynamics…

Evolutionary Stability (ES):uninvadability vs. convergence stability

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Jacobian for Evolutionary Stability:uninvadability vs. convergence stability

Uninvadability (Ma

ynard Smith & Price 1973) Nash(1950) solution

Convergence stability (~Eshel & Motro 1981)

a+d<0

ad – bc > 0

a < 0, d < 0

b: , c :

dc

ba

yFyxF

yxFxF

yy

xx222

222

//

//

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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

××=

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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 e

gg is uninvadible.s(x)

xpar*egg size xO

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Evolutionary game between male and female 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*)

Fy = (K/y) mxs(x*+y)

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Uninvadable zygote size (Parker et al. 1972)

Isogamy: xiso* = yiso* = s(2x*)/s’(2x*)

A smaller or larger mutant is always uninvadable.

zygote size y+x*

Surv

ival

rat

e of

zyg

ote

s

2xiso*

xpar*xiso*

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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 neit

her necessary nor sufficient condition for convergence stability.

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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)

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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.

Gam

ete

size

y

s = log z s = z6/(1+z6)s = 1-exp(z2)

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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 num

ber of fissions (discrete event) And one more fission may lose some functions of

gamete (mobility, fertility, etc.) Gamete size x

Gam

ete

size

y

(1) Mating success depends on the gamete size, e.g., m(y, x*, y*) = (x*/y*)(y-y0)2,…(M & A 1999),

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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.

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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

Sur

viva

l rat

e of

zyg

ote

s

2x*

xpar*

s = zn/(1+zn)

Cos

t of

anis

ogam

y

n

n=6

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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.

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

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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,

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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.

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Why have anisogamy evolved?(Parker et al. 1972)

If a very small mutant (sperm) appears, it is invadable.

zygote size y+x*

Surv

ival

rat

e of

zyg

ote

s

2x*

xpar*xiso*