Why are equally-sized gametes so rare? The instability of isogamy and the cost of anisogamy
-
Upload
jaquelyn-dickerson -
Category
Documents
-
view
19 -
download
0
description
Transcript of Why are equally-sized gametes so rare? The instability of isogamy and the cost of anisogamy
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 1:769-784
+ unpublished study
Hiroyuki Matsuda (Department of Environmental Management,
Yokohama National University (YNU))
http://risk.kan.ynu.ac.jp/matsuda/2005/050720Wien.ppt
2
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
3
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
4
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
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., 2Fx/x2 < 0 and 2Fy/y2 < 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 (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
//
//
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 e
gg is uninvadible.s(x)
xpar*egg size xO
9
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)
10
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*
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 neit
her 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
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)
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 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),
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
Sur
viva
l rat
e of
zyg
ote
s
2x*
xpar*
s = zn/(1+zn)
Cos
t of
anis
ogam
y
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.
http://risk.kan.ynu.ac.jp/matsuda/2005/050720Wien.ppt
18
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*
Surv
ival
rat
e of
zyg
ote
s
2x*
xpar*xiso*