acids that may be long enough to carry primitive metabolic capabilities.
Kauffman 1986 proposed an autocatalytic set of polymers. Simulation studies of autocatalytic
sets Bagley and Farmer, 1991, however, revealed that it is not possible to synthesize longer poly-
mers with high concentrations unless the equi- librium constant of condensation is large enough.
In the present study, we performed numerical experiments by varying the topology of reaction
networks. Our major finding is that even if the equilibrium constant of condensation is small,
certain reaction networks can synthesize many kinds
of longer
polymers with
higher concentrations.
2. Numerical model
2
.
1
. Basic equations We start with the basic equations of polymer-
ization reaction, based on Bagley and Farmer 1991, though with some modifications and sim-
plifications as shown in Appendix A. Each polymerization reaction is reversible Fig.
1. Two polymers condense into a single longer polymer giving off a water molecule, and at the
same time, the longer polymer cleaves itself into two shorter polymers. In this study, we denote the
two reactants for condensation as ‘A’ and ‘B’, and the product as ‘C’. Note that A + B B + A
in the respect of the sequence synthesized. The reaction is catalyzed by another polymer ‘E’. The
rate equations for A, B, C and E of the catalyzed reaction are
dx
C
dt =
− dx
A
dt = −
dx
B
dt 1
=
B A
F
C E
2 =
1 + nx
E
kx
A
x
B
− x
C
3 dx
E
dt =
4 where k is the equilibrium constant of the reac-
tion, x
N
N is A, B, C or E is the concentration of the respective polymers, and n is the catalytic
efficiency. For simplicity, we assume that all cata- lyzed reactions have the same catalytic efficiency
n . 1 + nx
E
is the enhancement rate with a cata- lyst. When the net flow,
B A
F
C E
, is positive, the reaction is a net condensation reaction. When it is
negative, the reaction is a net cleavage reaction. When the net flow vanishes, the reaction is in
equilibrium. In the absence of catalytic reaction, the rate equations for A, B and C are
dx
C
dt = −
dx
A
dt = −
dx
B
dt =
B A
F
C
= kx
A
x
B
− x
C
5 For simplicity, we assume that all reactions,
whether catalyzed or not, have the same equi- librium constant k.
With the use of the net flows of reaction, the kinetic equation of each polymer species i in the
reaction system is expressed as dx
i
dt = −
{B,E} B
i
F
C E
−
{A,E} i
A
F
C E
+
{A,E} B
A
F
i E
− k
d
x
i
+ r if i is a monomer
6 The first three terms on the right-hand side are
the sums over the net flows of reactions in which polymer i denotes A, B, and C. The fourth term
represents the flows of the polymers leaving the system into the outside. The term is proportional
to the concentrations of polymers, in which we call k
d
the dissipation rate constant. In this study, we set the same dissipation rate constant for all
polymers. However, if we suppose that polymer- ization takes place within enclosures Oparin,
1957, the longer polymers should have the smaller values of dissipation rate constant because
it is difficult for longer polymers entrapped by enclosures to escape into the outside. Thus the
Fig. 1. Graph of a pair of condensation and hydrolysis reac- tions. For example, ab A and ba B join together to form
abba C. At the same time, abba C hydrolyzes into ab A
and ba B. ababb E catalyzes both reactions.
assumption of the same dissipation rate constant for all polymers makes our model least vulnerable
to polymer elongation of artifact origin. If i is a monomeric species, it is supplied from the outside.
The fifth term represent the supply of the monomeric species, where r is the supply rate.
In the following numerical experiments, we cal- culate the concentration distributions of reactants
at steady state. At steady state, the total concen- tration of monomers in the system,
m =
i
L
i
x
i
where L
i
is the length of each polymer species i, is derived from Eq. 6 as
m =
Sr k
d
7 where S is the number of kinds of monomers, i.e.
i
L
i
i runs over monomeric species. We shall normalize the concentration by the quantity m
. For expressing the actual concentration, we use a
hatted symbol of the parameter such as m ˆ
. We express the magnitude of the flow between the
system and the outside with use of the dissipation rate constant.
Now we shall estimate the actual value of the equilibrium constant in a real peptide system. For
example, we consider aminoacyl-adenylate, e.g. Gly – AMP, for the monomer Paecht-Horowitz et
al., 1970. Cleavage of Gly – AMP into Gly and AMP gives off free energy of about 7 kcalmol
Voet and Voet, 1995 and condensation of two Gly into Gly – Gly needs 3.6 kcalmol Borsook,
1953. Then the equilibrium constant of the fol- lowing reaction, Gly – AMP + Gly – AMP Gly –
Gly – AMP + AMP,
is k =
m ˆ
exp − G RT = 0.25,
assuming m
ˆ =
0.001 M Borsook, 1953, G =
− 7 + 3.6 kcal mol, and T = 310 K.
2
.
2
. Algorithms for setting up catalyzed reactions In order to simulate behaviors of the reaction
networks, we elaborate a list of the catalyzed reactions to be expected.
Procedure of the simulation is as follows: The initial condition is such that only monomers are
present. Then,
Fig. 2. Graph of a transpeptidation reaction.
1. Pick up those polymers whose concentrations newly exceed a given threshold T = 1V cor-
responding to the presence of a single molecule in the system volume V.
2. Enumerate all possible catalyzed reactions in which selected polymers can participate. Then
determine whether or not each reaction is physically realizable as referring to the list of
the catalyzed reactions.
3. Solve Eq. 6 at steady state conditions. If some polymers newly exceed the threshold,
go back to Step 1, or else stop the whole operation.
3. Numerical simulations
