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

372 Part VIII. Applications
developed in later work (Eigen and Schuster, 1979), for a system where the polymer replication
is catalysed by other polymers in the same group, or formed (as proteins) by polymers
in the group. We will here consider the first possibility which also represents the idea
of a RNA world, a state on the path to the origin of life where self-replicating polymers
appear which also could catalyse the production of themselves or other polymers (Gilbert,
1986; Joyce, 1989; Fontana et al., 1989).
We start with a situation of a polymer, X, which also catalyses its own production, leading
to a growth, proportional to X 2 . We in this part stick consistently to the formalism of the
use of monomers, M. Equations for one such species are:
dX
dt
dM
dt
KMX
2
gX
abMNkMX
2
(33.19)
The equation for M is of the same type as previously. Relations are more complicated than in
the previous (linear) case, but can be readily written down. There is a stable stationary state:
g 1
X
K M M a2
;
2
ab 4b2
Ng2
bK
(33.20)
The condition for a solution, similar to eq. (33.5) is now: (a 2 K)/(4Nbg 2 ) 1. (Note that K
is of a different kind than k in the preceding case.)
We also see that the growth condition is more complex than previously. It follows from
the equations that in order for X to grow, the product MX must be larger than a certain value
(g/K). Thus, it cannot grow from very small values. This can be changed by keeping the linear
term of the previous case, i.e. to put the right hand side of the X-derivative equal to:
KMX 2 kMX gX
We do not go further with that question now. With two competing species, it is possible to
get stable, stationary states of both the species, and initial states will determine what happens
after long times. In some regions in a X 1 , X 2 , M-space, solutions go to a stationary
state with only X 1 , in other regions, it goes to a state with only X 2 . If we also take fluctuations
explicitly into account (and not only consider mean values), it is a still more open
question which species of two competing ones will survive.
The hypercycles and the RNA scenario also show possibilities of co-operativity. A catalytic
polymer can facilitate the production of other polymers. There can be networks of
various polymers catalysing the production of others, and there can be closed catalytic