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
3
.
1
. Topology of reaction network Bagley and Farmer 1991 estimated the con-
centration distribution of an autocatalytic set. The topology of the catalyzed reaction network is such
that each polymer in the network is formed by net condensation reactions starting from monomers.
Now we call the polymers which are formed in this way as elites. We introduce another type of
topology, a scheme of transpeptidation Silver and James, 1981a,b; Kauffman, 1986 into the
reaction network. Fig. 2 illustrates this reaction network. Catalysts and uncatalyzed reactions are
not shown in the figure. C1 as well as A1 and B1 is the elite. Reaction II cutting a peptide bond in
C1 chain is a net cleavage reaction. Then the sequence of Reactions I and II can be considered
a reaction in which a part of A1 is transferred and connected to B1. The significance of transpeptida-
tion is in net cleavage reaction, which was not considered by Bagley and Farmer’s model.
Fig. 3. Distribution of the concentrations of polymers in the reaction network including a transpeptidation reaction. Filled diamonds are the elites. Empty squares are the products of transpeptidation. Crosses are the polymers which are synthesized by uncatalyzed
reactions. The polymers are sorted with their lengths in the horizontal axis.
3
.
2
. Transpeptidation Here we show the result of numerical simula-
tion Fig. 3 for the reaction network shown in Fig. 2 with parameters S = 2, k = 0.25, n =
10
5
, T = 10
− 5
, k
d
= 10. We assume that all of the
catalysts are
substituted by
one identical
monomeric species, since our result will turn out to be indifferent to the variety of catalytic species.
The dotted line in Fig. 3 denotes the concentra- tion profile of a polymer with L amino acids in
equilibrium, i.e. k
d
= 0, which is expressed as
Bagley and Farmer, 1991 x
L
= r
S
L
k , r = 1 −
1+4k−1 2k
8 We find that the concentrations of the elites which
are denoted as filled diamonds show an exponen- tial decrease of their concentrations with length
and that the longer species due to the transpepti- dation,
denoted as
B2, has
the higher
concentration. Next we consider the case where the product of
the transpeptidation reaction is an elite Fig. 4.
Fig. 4. Graph of a transpeptidation reaction whose product is also synthesized by a net condensation reaction.
Fig. 5. Distribution of the concentrations of polymers in the reaction network of Fig. 4.
This catalyzed reaction network is the one shown in Fig. 2 supplemented by an additional catalyzed
reaction, which exhibits a net condensation reac- tion yielding B2 as an elite synthesized from a
monomer and a dimer elite. In this case, the concentration of B2 decreases down to the level of
the elites Fig. 5.
3
.
3
. Continuous growth Next we consider the case that some transpepti-
dation reactions are connected. That is to say, a product of a transpeptidation reaction becomes a
substrate of another transpeptidation reaction. Fig. 6 demonstrates the reaction network. B2 is
the longer product of the first transpeptidation reaction which is depicted in Fig. 2. An additional
transpeptidation taking two of the B2 molecules as substrates yields A3 and B3. In the same
manner, the third one taking two of the B3 molecules as substrates yields A4 and B4. Theo-
retically, the chain of transpeptidation reactions can extend itself to an infinite length. As shown in
Fig. 7, the longer products of transpeptidations have higher concentrations than those of the
elites.
The reaction network can continuously grow by repeating the cycle of incorporating a newer
transpeptidation. Although the reaction networks
Fig. 6. Graph of a chain of transpeptidation reactions.
Fig. 7. Distribution of the concentrations of polymers in the reaction network of Fig. 6.
we studied are limited, the possibility of having transpeptidation reactions do not have to be lim-
ited to ours.
4. Discussion
