Clas Blomberg - Physics of life-Elsevier Science (2007)

206 Part V. Stochastic dynamics
If x here is relatively large (at least equal to 10), then the integrand is largest close to the upper
limit. A good approximation of the integral is equal to e x /x(1 – 1/x). Thus, we get a relation:
⎛ b ⎞
aP1( 0) ∑
P1( n) exp( a/ b) ⋅ g
0 1
a P 1( 1)
⎝⎜
⎠⎟
n0
Use this for (20.16):
rP 1 (0) bP 1 (1)
This now gives an expression for the rate r (the probability of extinction per time):
⎛ a ⎞⎛
b
r aexp 1 ⎞⎛
⎞
⎝⎜
b⎠⎟
1
⎝⎜
g ⎠⎟
⎝⎜
a ⎠⎟
0
(20.18)
If a/b is 10 or larger, this is a small number, meaning that the extinction rate is small and a
probability distribution represented by P 1 (n) will survive for a considerable time. If a 10,
b 1, r a exp(a/b) 0.000045, 1/r 22000 time units. Larger values of a provide
more extreme values. a 20, b 1, yields r 4 10 8 , 1/r 2.4 10 7 time units.
Still more, if a 100, b 1, then r 4 10 44 ; 1/r 2.7 10 43 time units. Our results
show that if the quotient a/b is large, which corresponds to a large “stationary population”, then
the probability of extinction is very small even after long times. The population remains and,
not unexpected, the probability distribution, represented by P 1 (n) has a maximum close to a/b.
(If a/b is very large, the maximum is at n a/b, while for a/b 10, the maximum is at n 9.)
What we see here is that even if there is only one stationary possibility, meaning that
everything gets extinct, there can be a time-dependant contribution which survives for very
long times and thus can be interpreted as a “metastable” distribution that corresponds to the
actual “deterministic differential equation description”. There need not be any contradiction
here, although it is true that for any stochastic formulation of population dynamics, where
the system can get extinct, the only true stationary probability distribution is the extinct one.
20F
Reaction kinetics as step processes
Chemical reactions are also driven by fluctuations. Molecules move by diffusion and randomly
encounter each other. If positions and internal states are appropriate, molecules can
bind to each other or re-distribute bonds—reactions occur. This also concerns splitting of
bonds, which can be considered similar to our previous discussion of barrier passage and
macromolecule transitions.
Chemical reactions, in biological contexts often considered in networks are generally calculated
by deterministic equations, representing average numbers. This is mostly appropriate