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

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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

• Matsuda, H. Abrams, P.A. (1999) Evolutionary
  Ecology Research 1769-784
• unpublished study
• Hiroyuki Matsuda
• (Department of Environmental Management, Yokohama
  National University (YNU))

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Brief self-introduction
http//risk.kan.ynu.ac.jp/matsuda/2005/050720Wien.
ppt

• 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.

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Overview of this talk
http//risk.kan.ynu.ac.jp/matsuda/2005/050720Wien.
ppt

• 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 (MA 1999) isogamy is
  theoretically unlikely, and consider the
  condition of isogamy and the evolution of
  slightly different gamete sizes (slight
  anisogamy).

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Tatsuya Togashi told me
http//risk.kan.ynu.ac.jp/matsuda/2005/050720Wien.
ppt

• 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.

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Evolutionary Stability (ES)uninvadability vs.
convergence stability

• 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 lt 0
  and ?2Fy/?y2 lt 0.
• However, convergence stability depends on
  Jacobian of evolutionary game dynamics

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Jacobian for Evolutionary Stabilityuninvadabilit
y vs. convergence stability
Uninvadability (Maynard Smith Price 1973)
Nash(1950) solution
Convergence stability (Eshel Motro 1981)
adlt0
a lt 0, d lt 0
ad bc gt 0
b? , c ?
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The origin and evolution of gamete dimorphism and
the male-female phenomenon. (Parker et al. 1972)
Parents fitness
No. of gametes
mating success
zygotes fitness

• males mating success eggs/sperms
• zygotes fitness zygote size
• Fy (K/y) my(y, x, y) s(xy)
• Fx (K/x) mx(x, x, y) s(xy)
• e.g., my(y, x, y) cx/y, mx c

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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)
• xpars(x)/s(x)
• A smaller or larger
  egg is uninvadible.

s(x)
xpar
O
egg size x
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Evolutionary game between male and female
Fy (K/y) mxs(xy)

• 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(xy) mxs(xy)/y 0
• ?Fy/?x my(K/y)s(xy) mys(xy)/x 0
• Isogamic (symmetrical) solution (xy)
• xisoyisos(2x)/s(2x)

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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.

xpar
xiso
Survival rate of zygote s
2xiso
zygote size yx
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Co-Evolution Dynamics

• dx/dt g(?Fx/?x)
• dy/dt g(?Fy/?y)
• (?Fx/?x) (K/x2) mxs(xy) (K/x)
  mxs(xy) (K/x) (?mx/?x) s(xy)
• 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.

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Isogamy is usually convergence unstable!

• If (?mx/?x) (?my/?y) 0, and
• if s lt 0, then
• at xy s(2x)/s(2x),
• Condition for convergence stability
• (s)(s 2s x2) gt 0.
• It does not hold at all (sgt0 slt0)

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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 z6/(1z6)
s 1-exp(z2)
Gamete size y
Gamete size x
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Why is isogamy possible?
(1) Mating success depends on the gamete size,
e.g., m(y, x, y) (x/y)(y-y0)2,(M A 1999),

• 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.)

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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 lt egg size of
  anisogamy

n6
Cost of anisogamy
s zn/(1zn)
n
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Summary
http//risk.kan.ynu.ac.jp/matsuda/2005/050720Wien.
ppt

• 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 ltlt two-fold.

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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.

xpar
xiso
Survival rate of zygote s
2x
zygote size yx