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

Chapter 33. Origin of life 373
loops, where, for instance, polymer ‘1’ catalyses the production of polymer ‘2’, which in
turn catalyses the production of polymer ‘3’. The last one can then catalyse the production
of a fourth and so on, eventually leading to one that catalyses the production of the first one.
Such schemes have been much studied in different frameworks, primarily (Eigen et al.
(1980). There is a scheme:
X ( cat) X ( temp) →X 2X
;
X ( cat) X ( temp)
→X 2X
⋮
X
1 2 1 2
2 3 2 3
n
( cat)
X1( temp)
→Xn 2 X1
We distinguish the components in the reactions that serve as catalysts and those that serve
as templates. We assume the same rate constants for all of the different catalytic reactions,
and we get equations:
dX
dt
dX
dt
dX
dt
⋮
dX
dt
1
2
3
n
kMX X
n
kMX X
kMX X
kMX
gX
1 1
gX
1 2 2
gX
2 3 3
X
gX
n1
n n
(33.21)
To this, there is an equation for M of the same type as previously. There is a stationary solution
with all X-concentrations equal, which is stable only up to three components (n 3).
With n larger than 3, the components will oscillate, and this can be rather drastic for a large
number of components. The oscillations appear in the following manner.
Start with a situation where component ‘n’ is relatively large but decreasing. As long as
it is above a certain threshold value, component ‘1’ increases. As ‘1’ then becomes large,
component ‘2’ increases, although it at first is not large enough for ‘3’ to increase. As component
‘n’decreases further, ‘1’will reach a maximum and then decrease. Component ‘2’will
still increase at that stage, and when it passes a threshold, also ‘3’ increases. As ‘1’ no longer
gets sufficient catalytic support from ‘n’, it goes down. ‘2’ will turn while ‘3’ still increases
and the next component ‘4’ is on its way up. And so on. In the decreasing intervals, the
components may go down very far when there are many components. The minima are
roughly 2 orders of magnitude lower for each further component in the scheme. (Although
the general features are the same in alternative schemes, details may vary.)