A simple model of the thermal prebiotic oligomerization of
amino acids
F.G. Mosqueira
a,*, S. Ramos-Bernal
b, A. Negro´n-Mendoza
baDireccio´n General de Di
6ulgacio´n de la Ciencia,UNAM.Cd.Uni6ersitaria,A.p.70-487,04510Me´xico,D.F., Mexico bInstituto de Ciencias Nucleares,UNAM. Cd.Uni
6ersitaria,A.p.70-543,04510Me´xico,D.F., Mexico Received 1 September 1999; accepted 7 April 2000
Abstract
We construct a probabilistic model with the aid of the Markov chain formalism to describe and give a physico-chemical justification to an oligomerization process of a set of amino acids under certain prebiotic conditions. Such chemical process shows a remarkable bias in the polymer products that our model can explain. Some predictions and limitations are also discussed. © 2000 Elsevier Science Ireland Ltd. All rights reserved.
Keywords:Markov chain; Oligomerization of amino acids; Self ordering principle of amino acids; Thermal proteins
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1. Introduction
The principle of self ordering of amino acids, concerning the origins of life research, has been extensively investigated and interpreted by Fox et al. (1977) (see for an overview Fox and Dose, 1977). Such principle establishes that the reactiv-ity between different amino acids is not even. A remarkable feature in these experiments is the finding of a relatively reduced number of thermal proteins (or proteinoids in former Fox’s terminol-ogy) as compared with a much larger number of such polymers, which would be expected assum-ing an even probability of reaction between amino acids. Such finding is of fundamental importance in the origin of life research, because it is possible to envisage feasible self-reproducing phenomena.
The study of the structure and sequencing of such polymers is a difficult task. We only have knowledge of a few studies done on small tripep-tides (Fox et al., 1977; Nakashima et al., 1977). They investigated the thermal synthesis of tripep-tides involving only glutamic acid, glycine, and tyrosine. Their experiments show the production of only two tripeptides among 36 possible tripep-tides containing tyrosine that could be formed under the assumption of an even probability of reaction between the different amino acids. Fur-thermore, Hartmann et al. (1981) performed a mechanistic study of this reaction.
The relevance of the self-ordering principle to the emergence of a ‘minimum living chemical system’ has been already estimated by Mosqueira (1988), using simple probability theory. This study considered the emergence of a set of oligomers,
Dedicated to the memory of Sidney W. Fox * Corresponding author.
{n}, under the following premises. (1) Plurality, each single oligomer is assigned a single function; thus we need a set {n} of oligomers. (2) Simul-taneity, it is assumed that the set {n} is located together at some physical space in order to be kinetically connected. (3) Number of participating oligomers, it falls into the range 8+n+14. (4) Degree of polymerization x, it is assumed x around 40 (monomers). (5) Alphabet a, it is adopted a two letter alphabet, i.e. a=2. Finally, it is assumed, of course, that the probability of reaction between different amino acids is even. These considerations concluded that the polymer-ization phenomena associated with the origin of life (that is, to the origin of {n}) had to be strongly biased, otherwise the probability of nu-cleation of such ‘minimum of life’ would be excluded.
In the present study we intend to delve into these questions and give a theoretical justification to the implicit bias of the self-ordering principle. This is done by assuming different electromag-netic interactions among the reacting amino acids. To that end, in accordance with Dickerson and Geis (1969), we classify amino acids into four groups, polar positive (p+), polar negative (p−),
neutral (n), and non polar (np). Further, we use the Markov chains formalism to model such pre-biotic polymerization process.
2. The model
Let us define a finite Markov chain (Moran, 1984). Consider events that can occur at succes-sive discrete stages and denote them by a variable k, which can take the values 0, 1, …, n, …. At each stage a finite number of eventsE1,E2, …,En,
… can occur. These are the possible states of the system.
At each stage of k+1 we suppose that the eventsE1, …,En occur with certain probabilities,
which depend only on the events that occurred at stagekand not on anything which had happened previously. We expresspijfor the probability ofEj
to occur at stage k+1 conditional on Ei having
occurred at stage k.
The set of quantities, pij, i=1, …, n, j=1, …,
n are known as the transition probabilities, are non-negative, and satisfy the conditions
%
j
pij=1, i=1, …,n. (1)
Besides, P=(pij) is a (n×n) matrix and it is
known as the transition probability (or reactivity) matrix of the system (or stochastic matrix of the system).
If the probabilities of the events E1, …, En at
any stage ofkare denoted byp1(k), …,pn(k), and
call it the state matrix after k stages, we have
pj(k+1)=% i
pi(k)pij, (2)
and these equations can be written in the matrix form
p(k+1)=p(k)P (3)
where, p(k) is a row vector (or 1×n matrix) whose elements are p1(k), …, pn(k). Let us define
a 1×ninitial state matrix (or an initial state row vector) p(0).
By applying Eq. (3) repeatedly we see that
p(k)=p(0)Pk (4)
where k is an integer.
Now, we arrange the four possible electromag-netic interactions between amino acids into a 4×
4 matrix as follows:
Á
Thus, for example, the element p13 is equal to
p+n. Besides, the state of the system is
repre-sented at any stage k by a matrix of the state of the system that is a row matrix with four elements:
(p+ p− n np) (6)
Finally, we should make a succinct comment on the interpretation that we give to (pij) in Eq. (5),
Fig. 1.
focused on the sequencing and identification of synthesized trimers. Experimental restriction al-lowed only to report those trimers containing tyrosine. It should be underlined that one would expect 36 possible tripeptides from such thermal polycondensation of three amino acids. However, Nakashima et al. (1977) obtained only two trimers, pyroglutamyl – glycyl – tyrosine and pyroglutamil – tyrosyl – glycine.
The mechanism of reaction to form the trimers is known (Hartmann et al., 1981). In the first stage it is formed pyroglutamic acid 1 from glu-tamic acid (see Fig. 1). In the second stage, slower than the first one, it is formed a diketopiperazine 2 from the other two amino acids tyrosine and glycine. Finally, in the third stage, pyroglutamic acid reacts with the diketopiperazine to form equimolecular amounts of the two tripeptides, pyroGlu – Gly – Tyr 3 and pyroGlu – Tyr – Gly 4 (see Fig. 2).
4. Application of the model
The mathematical modeling of this chemical reaction should describe its different kinetic stages. Thus, we will consider in turn each of the three main steps we have just reviewed.
4.1. Internal cyclization
A glutamic acid molecule has three centers of charge (two negatives and one positive) with no predominance of either of them. We conceive the Markov chain, a matrix element pij signifies the
probability that an entityi becomesan entityj. In our approach, we interpret it as the probability of chemical reaction between entities i and j. As we assume that this probability is the same for reac-tion betweeniand jas with the reaction ofj and i, by construction matrix (Eq. (5)) is symmetrical. With such interpretation we obtain interesting results from the Markov formalism.
3. Chemical aspects of the thermal prebiotic production of trimers
Dry heating of a mixture of the amino acids glutamic acid, glycine, and tyrosine yields, among other products, only two tyrosine-containing trimers. Glutamic acid was the necessary trifunc-tional amino acid since it is known to produce smaller peptides than other trifunctional amino acids (Nakashima et al., 1977). Their analysis was
first stage (the formation of pyrGlu) as aninternal cyclization process that proceeds readily, because in the same molecule we have p+ and p− nearby
that by internal rotation react rapidly to get pyrGlu. The product of this intra-reaction has a concentrated negative charge on it (see 1 in Fig. 1), giving rise to a powerful initiator for the polymerization reaction. Such characteristic has been pointed out previously on the basis of chemical analysis (Fox et al., 1977; Melius and Hubbard, 1987).
As applied to our model, we have the following initial state matrix (Eq. (6))
(0.5 0.5 0 0). (7)
The transition matrixP is formed from Eq. (5) using only the relevant electromagnetic interactions involved within this particular reaction
Thus, substitution of Eqs. (7) and (8) into Eq. (4) for k=1 yields
(0.5 0.5 0 0). (9)
that is the same as the initial state matrix (Eq. (7)).
4.2. External cyclization
Now, let us consider the formation of the dike-topiperazine produced by the cyclization reaction between glycine and tyrosine. For this reason it will be conceived as an external cyclization reac-tion. We assume glycine is np and tyrosine is n (Dickerson and Geis, 1969). We will use equal concentrations for these amino acids, to stick on approximately to the proportions used in the ex-periments (Fox et al., 1977; Nakashima et al., 1977). Thus, the initial state matrix is
(0 0 0.5 0.5) (10)
Again, the transition probability matrix P is formed only with the relevant electromagnetic
interactions for this reaction. Eq. (5) yields
P=
Substitution of Eqs. (10) and (11) into Eq. (4) for k=1 yields
(0 0 0.5 0.5) (12)
that is the same matrix as the initial state matrix (Eq. (10)).
It is of interest to use other initial concentration reflecting more closely actual results in prebiotic synthesis, such as a ratio of molar concentrations Gly:Tyr equal to 99:1. Under these conditions the initial state matrix (Eq. (10)) becomes (0 0 0.99 0.01) and, its application as above yields (Eq. (12)) as well. We will comment later on this result in Section 5.
4.3. Oligomerization
Finally, at this stage we should consider the reaction between the initiator pyroGlu (p−) and
the diketopiperazine Gly – Tyr that we assume is a neutral molecule (n) (which is undistinguishable from Tyr – Gly because it is a cyclic molecule). Let us start this reaction with equal concentrations of species. The initial state matrix is
(0 0.5 0.5 0), (13)
and the transition matrix P is formed using only the electromagnetic interactions among p− and n
molecules in Eq. (5).
P=
Once more, substitution of Eqs. (13) and (14) into Eq. (4) for k=1 yields
(0 0.5 0.5 0), (15)
5. General remarks
Our proposed model intends primarily to de-scribe and give a theoretical explanation of the limited number of tyrosine-containing trimers ob-tained experimentally (Fox et al., 1977). With such knowledge, it may be possible to make some predictions concerning other trimers and larger oligomers. There are several aspects that should be discussed separately.
5.1. The initiator is pyroglutamic acid
The first question to be explained is why pyrog-lutamic acid acts as an initiator. We have already referred to its formation process in Section 4.1 and found that it goes from a three charge species (glutamic acid) to a one charge species (pyroglu-tamic acid). The latter has a concentrated negative charge that may induce electronic displacements more easily than other participating chemical spe-cies (i.e. those classified as n and np) and thus play the role of the initiator in this polymerization process. However, there are six trimers that could initiate with pyroGlu:
1. pyroGlu –a– Glu – Tyr;
2. pyroGlu –g– Glu – Tyr;
3. pyroGlu – Gly – Tyr; 4. pyroGlu – Tyr – Glu; 5. pyroGlu – Tyr – Gly; 6. pyroGlu – Tyr – Tyr.
We now give some justification that eliminates four of them. We may discard trimers 1, 2, and 4 which have Glu in an internal position because it is known, under kinetic basis, that Glu consumes rapidly to become pyroGlu (Hartmann et al., 1981). That is, internal cyclization goes faster than external cyclization. So, there would be no Glu left to participate in an internal position in the trimers and we do not see such trimers. Trimer 6 may also be eliminated because it would be much more difficult to form a diketopiperazine Tyr – Tyr, due to its bulky residues, than Gly – Tyr (or Tyr – Gly). In fact, it has been reported the synthesis of only trace amounts of Tyr – Tyr dike-topiperazine (Hartmann et al., 1981). Thus,
finally, the only trimers that remain are 3 and 5 that are the ones that are observed ex-perimentally.
In this connection, the same research group identified also trace amounts of Gly – Gly dike-topiperazine. From the size of the glycine molecule we would expect larger quantities of such cyclic molecule. However, glycine is classified as an n and np species at a time, which in turn do not generate strong electromagnetic interactions to induce chemical changes. So, this reaction does not proceed to a larger extent.
A comment should be made with respect to another possible initiator, from the set of amino acids considered. Aspartic acid is also a three charge species susceptible of internal cyclization. However, this amino acid has one methyl group less than glutamic acid and therefore the cyclic molecule to be formed (with four atoms) is less stable than pyroGlu with five atoms in the cycle. For this reason, we predict a more extensive use as initiator of Glu as compared with Asp.
5.2. Elongation and termination
We are assuming that elongation in this poly-merization process is carried out by the reaction of products of both internal and external cycliza-tions (see Section 4.3). That is, we envisage that cyclic Gly – Tyr 2 is cleaved by pyroGlu 1 and yields the two possible linear trimers 3 and 4 (see Fig. 2). As a result, in both products (pyroGlu – Gly – Tyr and pyroGlu – Tyr – Gly) we reconstitute a free carboxylic group at the growing end of the trimers that will be equivalent to that of pyroGlu. So, in a similar way as pyroGlu acted as an initiator, both trimers may also play the role of initiators and react with another cyclic dike-topiperazine to produce pentamers, i.e. four py-roGlu – tetrapeptides. Thus, we would expect the synthesis of oligomers containing an odd number of residues from this mechanism. Furthermore, Hartmann et al. (1981) pointed out that at the temperature of this reaction (180°C), there is no stereoselective residue effect between Gly and Tyr and this is why the trimers are found in equimolar amounts.
above that elongation is an autocatalytic process, because two one-charge molecules are obtained from one-charge molecule. Further, two initiators cannot interact and cancel each other (both have a negative charge). Altogether, the termination of the polymerization process comes only with the exhaustion of the initiator species and/or the diketopiperazines.
5.3. Mathematical modeling
5.3.1. Steady state
1. There is a characteristic feature in the applica-tion of the mathematical model to the three stages of the polymerization process (see Sec-tion 4). In every stage we assign a given initial state matrix and substitute in Eq. (4). As a result of this, we recover precisely the initial state matrix. That is, we ‘instantly’ obtain the steady state of the Markov chain that is defined as follows
p(k)=p(k)P (16)
corresponding to every stage in the polymer-ization process.In Section 4.2, we provide an additional proof of arriving to such steady state as we tried two different initial conditions and get, in both cases, the same result, i.e. the steady state for such stage of the chemical reaction. This is a result of great relevance in prebiotic synthesis, since it shows some sort of initial concentration independence for that part of the synthesis process.
2. If we establish an analogy between matrix multiplication and chemical reaction among reactants, we also might interpret the product of such matrix multiplication as the product of the chemical reaction. In such case, we do not obtain a match between the state matrix at stage k and the electromagnetic properties of the products for the same stage (although it truly represents the electromagnetic properties for the reactants for the stage k+1). For example, (see Section 4.1) the new state matrix generated in Eq. (9) does not represent the electromagnetic properties of the chemical
product of such reaction (pyroGlu), with only one charge on it. Instead, it denotes two charges (Eq. (9)) characteristics of the reactant (Glu).
5.3.2. Predictions
We have already discussed and justified the synthesis of the two sole trimers obtained (see Sections 5.1 and 5.2). We are now able to make predictions with respect to other oligomers that might be presented in the reaction mixture. Let us confine ourselves to make comments on possible pentamers (pyroGlu – tetrapeptides), and hep-tamers (pyroGlu – hexapeptides) synthesized under the same constraints of the experiment, i.e. dry heating of an initial mixture of glutamic acid, glycine, and tyrosine, with trace amounts of syn-thesized Gly – Gly and Tyr – Tyr diketopiperazines. Let us consider the trimer pyroGlu – Gly – Tyr as the initiator (see Fig. 2). Then we would obtain two possible pyroGlu – tetrapeptides, (a) py-roGlu – Gly – Tyr – Gly – Tyr, and (b) pypy-roGlu – Gly – Tyr – Tyr – Gly. It would be interesting to check experimentally if again we obtain an equimolar mixture of such pentamers, as we would predict from the used temperature and previous results. If not, then stereo-selective ef-fects might have some effect on the polymeriza-tion process that could be introduced into the model.
From the trimer pyroGlu – Tyr – Gly as initiator we would obtain, (c) pyroGlu – Tyr – Gly – Gly – Tyr and (d) pyroGlu – Tyr – Gly – Tyr – Gly in an equimolar mixture, which could be the same as in the previous pentamer.
Each pentamer (a), (b), (c), and (d) may play de role of an initiator (see Section 5.2) and react with cyclic Gly – Tyr to produce eight heptamers (i.e. eight pyroGlu – hexamers), presumably in equimo-lar quantities. A simiequimo-lar exercise as with the syn-thesis of pentamers above would yield the sequences of these eight pyroGlu – hexamers.
5.3.3. Other polymerization mechanisms
alter-native mechanisms of polymerization that are known to proceed as well. For example, Rohlfing (1976) used other experimental conditions (lower temperature and longer time of heating) and obtained polymers containing chromophores and no simple peptides. Besides, Harada and Fox (1958) have also observed polymerization via monomer addition. Our model could be applied to the latter situation very easily, although we have not considered this problem in the present study.
We developed the model according to known experimental results. However, such experiments involved a limited number of amino acids. This situation originates in turn that most of the matrix elements in P(Eq. (5)) are null. However, in more complex situations with a larger set of different participating amino acids, the majority of the matrix elements would be different from 0 to consider the pair-wise interaction among them, denoting various degrees of electromagnetic interaction among four categories of amino acids, according to probabilistic rules (Eq. (1)). This remains to be done in the future.
5.3.4. Early stages of the polymerization process The probabilistic method we have employed is based in a one-nearest-neighbor influence. We might expect this to be the case as long as the oligomer maintains a fairly linear and rigid configuration, as would occur in the early stages of the polymerization. However, as the degree of polymerization increases, the oligomer starts to recoil on itself and eventually two or more nearest-neighbor influence the reactive end of the oligomer. In that case, the Markov chain formal-ism of order one that we have applied, is no more a suitable procedure to model this system. It might become a higher order Markovian chain, as we should now take into account additional
elec-tromagnetic influences of distant amino acids residues of the polycondensed oligomer.
From a physical point of view, a higher order-Markovian situation coincides with the consider-ation of a n-body-problem (n=3, 4, …) that has at most an approximate solution. Instead, when n=2 we have a Markov chain of order one and we might expect an exact physical solution.
Acknowledgements
We would like to thank Rene´ Cantu´ Garza for drawings.
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