Continuous growth of autocatalytic sets

BioSystems 59 (2001) 61 – 69 www.elsevier.com/locate/biosystems

Continuous growth of autocatalytic sets Shigenobu Mitsuzawa a,b,*, Sei-ichiro Watanabe a a

Department of Earth and Planetary Sciences, Graduate School of Science, Nagoya Uni6ersity, Furo-cho, Chigusa, Nagoya, Aichi 464 -8602, Japan b Theoretical/Computational Physics Group, National Laboratory for High Energy Physics (KEK), 1 -1 Oho, Tsukuba, Ibaraki 305 -0801, Japan Received 30 August 2000; received in revised form 4 December 2000; accepted 8 December 2000

Abstract We investigated a topology of the reaction network for polymerization of amino acids as the emergence and the continuous growth of autocatalytic sets. Reaction networks including transpeptidation reaction can yield newer and longer peptides with time having higher concentrations. Conditions necessary for the occurrence of transpeptidation can be met in actual amino acid/peptide systems. © 2001 Elsevier Science Ireland Ltd. All rights reserved. Keywords: Origin of life; Polymerization of amino acids; Transpeptidation

1. Introduction One of the fundamental questions regarding the origin of life is about which came first, either proteins or nucleic acids. The view that nucleic acids, probably RNA, were the primordial molecules of life (Eigen et al., 1981) has been favored by many researchers, but has not been substantiated as yet (Miller, 1992). On the other hand, some researchers have suggested the view that metabolisms of peptides could have originated prior to or independently of the replication of nucleic acids (Dyson, 1985; Kauffman, 1993). This view is based upon the vantage point that * Corresponding author. Tel.: +81-298-796102; fax: + 81298-645755. E-mail address: [email protected] (S. Mitsuzawa).

peptides could be synthesized more easily than nucleic acids in abiotic conditions. Ito et al. (1990) suggested that polypeptides carrying molecular weights of 1000 –4000 Da had secondary structures and catalytic functions for many metabolic reactions. Seffens (1995) investigated the possibility that peptide systems might have exhibited metabolic and replicative activities. Recently, reaction networks of self- and cross-replicating peptides consisting of about 30 residues have been constructed in the laboratory by Lee et al. (1996) and other researchers(Lee et al., 1997; Yao et al., 1998). These observations strongly support the possibility of peptide systems of abiotic origin. In order to have functional peptide systems, a set of long peptide chains must be prepared abiotically. In this study, we investigate how longer peptide chains could be synthesized from amino

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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 polymers with high concentrations unless the equilibrium 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 polymerization reaction, based on Bagley and Farmer (1991), though with some modifications and simplifications 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

Fig. 1. Graph of a pair of condensation and hydrolysis reactions. For example, hi (A) and ih (B) join together to form hiih (C). At the same time, hiih (C) hydrolyzes into hi (A) and ih (B). hihii (E) catalyzes both reactions.

dxC dxA dxB = − =− dt dt dt



(1)

= B AF EC

(2)

= (1+ wxE )(sxAxB − xC )

(3)

dxE =0 dt

(4)

where s is the equilibrium constant of the reaction, xN (N is A, B, C or E) is the concentration of the respective polymers, and w is the catalytic efficiency. For simplicity, we assume that all catalyzed reactions have the same catalytic efficiency w. (1+ wxE ) is the enhancement rate with a catalyst. When the net flow, B AF EC, 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 dxC dx dx = − A = − B = B AFC = (sxAxB − xC ) dt dt dt (5) For simplicity, we assume that all reactions, whether catalyzed or not, have the same equilibrium constant s. With the use of the net flows of reaction, the kinetic equation of each polymer species i in the reaction system is expressed as dxi A (E) A (E) = − % B iF (E) C − % i FC + % B Fi dt {B,(E)} {A,(E)} {A,(E)} − kdxi + 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 kd the dissipation rate constant. In this study, we set the same dissipation rate constant for all polymers. However, if we suppose that polymerization 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

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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 calculate the concentration distributions of reactants at steady state. At steady state, the total concentration of monomers in the system, m0 =%Li xi i

where Li is the length of each polymer species i, is derived from Eq. (6) as Sr m0 = (7) kd where S is the number of kinds of monomers, i.e. i Li (i runs over monomeric species). We shall normalize the concentration by the quantity m0. For expressing the actual concentration, we use a hatted symbol of the parameter such as m ˆ 0. 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 kcal/mol (Voet and Voet, 1995) and condensation of two Gly into Gly –Gly needs 3.6 kcal/mol (Borsook, 1953). Then the equilibrium constant of the following reaction, Gly – AMP+Gly – AMP “ Gly – Gly –AMP + AMP, is s= m ˆ 0exp(− G 0/RT) =0.25, assuming m ˆ 0= 0.001 M (Borsook, 1953), G 0 =( − 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,

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Fig. 2. Graph of a transpeptidation reaction.

1. Pick up those polymers whose concentrations newly exceed a given threshold T(= 1/V) corresponding 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 concentration 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 transpeptidation is in net cleavage reaction, which was not considered by Bagley and Farmer’s model.

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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 simulation (Fig. 3) for the reaction network shown in Fig. 2 with parameters S = 2, s = 0.25, w = 105, T = 10 − 5, kd = 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 concentration profile of a polymer with L amino acids in equilibrium, i.e. kd =0, which is expressed as (Bagley and Farmer, 1991) z L S

1 + 4s −1 xL = , z=1− (8) s 2s 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 transpeptidation, 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.

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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 reaction 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 transpeptidation 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. Theoretically, 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.

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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 limited to ours.

4. Discussion

4.1. Necessary conditions for transpeptidation

6cat =

In Fig. 8, the concentrations of the elite, B1 are displayed and also the longer product of transpeptidation, B2 of Fig. 2 with parameters kd and (1 +wxE ). Here we investigate the condition of the system for satisfying that the concentration of B2 is higher than that of B1, which manifests the occurrence of transpeptidation to be effective. From Fig. 8, we read the condition (1 + wxE )\4 103.5

is necessary for transpeptidation to be effective. We now examine whether shorter oligopeptides can bear the high catalytic efficiencies satisfying Eq. (9), by adopting a Michaelis –Menten scheme. The Michaelis –Menten equation expresses the rate of a catalyzed hydrolysis reaction of a substrate as (Voet and Voet, 1995)

(9)

k. catxˆE TxˆS K. M + xˆS

(10)

where xˆS denotes the concentration of the substrate, and xˆE T denotes the total concentration of the enzyme, that is, the sum of the concentrations of the free enzyme and of the enzyme –substrate complex. The values of rate constants for carboxypeptidase A are k. cat = 578 s − 1 and k. M = 8.8× 10 − 5 M at 25°C (Radzicka and Wolfenden, 1995). Here we assume that these values represent contemporary peptidase activity. The reaction

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constant of hydrolysis of peptide without catalyst is k. non =3.0× 10 − 9 s − 1 at 25°C (Radzicka and Wolfenden, 1995). Shimizu (1996) reported that a dipeptide Ser –His, which is relevant to the catalytic site of chimotrypsin, has a peptidase activity. According to his experimental data, the strength of bonding interpreted as 1/K. M between Ser – His and a substrate is equal to that of chimotrypsin. The reaction rate interpreted as k. cat of Ser – His is smaller by a factor of 0.001 than that of chimotrypsin. These facts and the above values of parameters give k. cat =0.578 s − 1 and K. M = 8.8× 10 − 5 M for Ser – His. We assume that some other oligopeptides have weak peptidase activities as with Ser – His. The enhancement rate with a catalyst in our model, (1+ wxE ), is expressed as (1 +wxE )=

6cat k. nonxˆS

(11)

Inserting Eq. (10) into Eq. (11) with k. cat = 0.578 s − 1, K. M = 8.8 × 10 − 5 M, k. non =3.0 × 10 − 9 s − 1, m ˆ 0 =0.001 M, and the assumption, xˆS  K. M, gives

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ˆ0 k. catm K. Mk. non

(12)

= 2.2× 109

(13)

w=

Based upon the results demonstrated in Fig. 3, it turns out that Eq. (9) is met if the catalysts are either any polymers with the length of under 5 or a part of polymers with the length of 5 for the case of w= 2.2×109. Thus we conclude that short peptides can be the catalysts whose catalytic efficiencies are high enough for transpeptidation to be effective. In fact, the crucial parameter is the equilibrium constant which determines the gradient of the concentration against the length of a polymer (Eq. (8)). If the equilibrium constant is too small, the enhancement rate with a catalyst cannot be high enough for transpeptidation to become effective because the concentration of a catalyst is too small. Recently, the high temperature and high pressure conditions like those expected near submarine hydrothermal systems have been considered as a suitable site for polymerization of amino acids (Shock, 1992; Qian et al., 1993; Imai et al., 1999). Thus autocatalytic sets including transpep-

Fig. 8. Concentrations of B1 and B2 in Fig. 2 varying kd and (1 +wxE ).

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tidation might have emerged in such a condition that realizes a high value for the equilibrium constant with non-activated free amino acids.

4.2. Could the transpeptidation reaction system emerge? Silver and James (1981a,b) reported that pepsin catalyzes the transpeptidation of oligopeptides. We expect that some short peptides which might have relevance to contemporary peptidase activities could exhibit similar catalytic function to pepsin. Our model assumes a constant flux of amino acids and peptides. It might be implausible to expect a constant flux of appropriate materials for a sustained period of time in a natural setting. One possible strategy for the reaction system to avoid attainment of equilibrium conditions could be the formation of an enclosure dispelling water molecules which persisted during some periods in equilibrium conditions (Bagley and Farmer, 1991). Another strategy is to put reaction networks within submarine hydrothermal systems. The hydrothermal systems can be regarded as continuous-flow reactors guaranteeing the persistence of mass flux (Corliss, 1986). In our simulations, we studied only two different kinds of monomers. If we consider more than two different kinds, much wider varieties of polymers and reactions would be expected, with a more enhanced likelihood of having transpeptidations. Acknowledgements We are grateful to Katsuro Ogawa and Shigeo Yoshida for valuable discussions. We also appreciate Tetsuyuki Yukawa for reading the manuscript and for useful comments. Appendix A In this work, we use a numerical model, based on Bagley and Farmer (1991) adding some modifications and simplifications, which we summarize here.

A.1. Rate equation Bagley and Farmer expressed the rate equations of a catalyzed reaction as dxC = (1+ wxE )(kfxAxB − krHxC ) dt

(14)

where kf is the rate constant of the condensation reaction, kr is the rate constant of cleavage reaction, and H is the concentration of water. The equilibrium constant is s=

kf krH

As with Bagley and Farmer, we assume for simplicity that all reactions have the same values of kf and kr, respectively, and we normalize the time parameter by krH. Then Eq. (14) is rewritten as dxC = (1+ wxE )(sxAxB − xC ) dt

(15)

A.2. Complexes in the catalyzed reactions Bagley and Farmer’s model considers the complexes into which the catalyst and the reactants are bound together. For simplicity, our model did not consider this effect. We have confirmed that this assumption is appropriate in actual oligopeptide syntheses.

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