Hypercycle (chemistry)

• 1971 (1971): Eigen introduces the hypercycle concept[1]
• 1977 (1977): Eigen and Schuster extend the hypercycle concept, propose a hypercycle theory and introduce the concept of quasispecies[2]
• 1982 (1982): Discovery of ribozyme catalytic properties[16][17]
• 2001 (2001): Partial RNA polymerase ribozyme is designed via directed evolution[18]
• 2012 (2012): Experimental demonstration that ribozymes can form collectively autocatalytic sets[15]

A presentation of the error threshold problem

When a model of replicating molecules was created,[1][2] it was found that, for effective storage of information, macromolecules on prebiotic Earth could not exceed a certain threshold length. This problem is known as the error threshold problem. It arises because replication is an imperfect process, and during each replication event, there is a risk of incorporating errors into a new sequence, leading to the creation of a quasispecies. In a system that is deprived of high-fidelity replicases and error-correction mechanisms, mutations occur with a high probability. As a consequence, the information stored in a sequence can be lost due to the rapid accumulation of errors, a so-called error catastrophe. Moreover, it was shown that the genome size of any organism is roughly equal to the inverse of mutation rate per site per replication.[19][20][21] Therefore, a high mutation rate imposes a serious limitation on the length of the genome. To overcome this problem, a more specialized replication machinery that is able to copy genetic information with higher fidelity is needed. Manfred Eigen suggested that proteins are necessary to accomplish this task.[1] However, to encode a system as complex as a protein, longer nucleotide sequences are needed, which increases the probability of a mutation even more and requires even more complex replication machinery. John Maynard Smith and Eörs Szathmáry named this vicious circle Eigen's Paradox.[22]

According to current estimations, the maximum length of a replicated chain that can be correctly reproduced and maintained in enzyme-free systems is about 100 bases, which is assumed to be insufficient to encode replication machinery. This observation was the motivation for the formulation of the hypercycle theory.[23]

It was suggested that the problem with building and maintaining larger, more complex, and more accurately replicated molecules can be circumvented if several information carriers, each of them storing a small piece of information, are connected such that they only control their own concentration.[1][2] Studies of the mathematical model describing replicating molecules revealed that to observe a cooperative behaviour among self-replicating molecules, they have to be connected by a positive feedback loop of catalytic actions.[24][25][26] This kind of closed network consisting of self-replicating entities connected by a catalytic positive-feedback loop was named an elementary hypercycle. Such a concept, apart from an increased information capacity, has another advantage. Linking self-replication with mutual catalysis can produce nonlinear growth of the system. This, first, makes the system resistant to so-called parasitic branches. Parasitic branches are species coupled to a cycle that do not provide any advantage to the reproduction of a cycle, which, in turn, makes them useless and decreases the selective value of the system. Secondly, it reinforces the self-organization of molecules into the hypercycle, allowing the system to evolve without losing information, which solves the error threshold problem.[2]

A hypercycle with translation.

Analysis of potential molecules that could form the first hypercycles in nature prompted the idea of coupling an information carrier function with enzymatic properties. At the time of the hypercycle theory formulation, enzymatic properties were attributed only to proteins, while nucleic acids were recognized only as carriers of information. This led to the formulation of a more complex model of a hypercycle with translation. The proposed model consists of a number of nucleotide sequences I (I stands for intermediate) and the same number of polypeptide chains E (E stands for enzyme). Sequences I have a limited chain length and carry the information necessary to build catalytic chains E. The sequence I_i provides the matrix to reproduce itself and a matrix to build the protein E_i. The protein E_i gives the catalytic support to build the next sequence in the cycle, I_i+1. The self-replicating sequences I form a cycle consisting of positive and negative strands that periodically reproduce themselves. Therefore, many cycles of the +/− nucleotide collectives are linked together by the second-order cycle of enzymatic properties of E, forming a catalytic hypercycle. Without the secondary loop provided by catalysis, I chains would compete and select against each other instead of cooperating. The reproduction is possible thanks to translation and polymerization functions encoded in I chains. In his principal work, Manfred Eigen stated that the E coded by the I chain can be a specific polymerase or an enhancer (or a silencer) of a more general polymerase acting in favour of formation of the successor of nucleotide chain I. Later, he indicated that a general polymerase leads to the death of the system.[2] Moreover, the whole cycle must be closed, so that E_n must catalyse I₁ formation for some integer n > 1.[1][2]

During their research, Eigen and Schuster also considered types of protein and nucleotide coupling other than hypercycles. One such alternative was a model with one replicase that performed polymerase functionality and that was a translational product of one of the RNA matrices existing among the quasispecies. This RNA-dependent RNA polymerase catalysed the replication of sequences that had specific motifs recognized by this replicase. The other RNA matrices, or just one of their strands, provided translational products which had specific anticodons and were responsible for unique assignment and transportation of amino acids.[2]

Another concept devised by Eigen and Schuster was a model in which each RNA template's replication was catalysed by its own translational product; at the same time, this RNA template performed a transport function for one amino acid type. Existence of more than one such RNA template could make translation possible.[2]

Nevertheless, in both alternative concepts, the system will not survive due to the internal competition among its constituents. Even if none of the constituents of such a system is selectively favoured, which potentially allows coexistence of all of the coupled molecules, they are not able to coevolve and optimize their properties. In consequence, the system loses its internal stability and cannot live on. The reason for inability to survive is the lack of mutual control of constituent abundances.[2]

The dynamics of the elementary hypercycle can be modelled using the following differential equation:[3]

${\displaystyle {\dot {x_{i}}}=x_{i}\left(k_{i}+\sum _{j}k_{i,j}x_{j}-{\frac {1}{x}}\phi \right)}$

where

${\displaystyle {\begin{aligned}x&=\sum _{i}x_{i},\\k_{i}&=f_{i}-d_{i}.\end{aligned}}}$

In the equation above, x_i is the concentration of template I_i; x is the total concentration of all templates; k_i is the excess production rate of template I_i, which is a difference between formation f_i by self-replication of the template and its degradation d_i, usually by hydrolysis; k_i,j is the production rate of template I_i catalysed by I_j; and φ is a dilution flux; which guarantees that the total concentration is constant. Production and degradation rates are expressed in numbers of molecules per time unit at unit concentration (x_i = 1). Assuming that at high concentration x the term k_i can be neglected, and, moreover, in the hypercycle, a template can be replicated only by itself and the previous member of the cycle, the equation can be simplified to:[3]

${\displaystyle {\dot {x_{i}}}=x_{i}\left(k_{i,i-1}x_{i-1}-{\frac {1}{x}}\phi \right)}$

where according to the cyclic properties, it can be assumed that

${\displaystyle {\begin{aligned}k_{i,0}&=k_{i,n},\\x_{0}&=x_{n}.\end{aligned}}}$

A hypercycle with translation consists of polynucleotides I_i (with concentration x_i) and polypeptides E_i (with concentration y_i). It is assumed that the kinetics of nucleotide synthesis follows a Michaelis–Menten-type reaction scheme in which the concentration of complexes cannot be neglected. During replication, molecules form complexes I_iE_i-1 (occurring with concentration z_i). Thus, the total concentration of molecules (x_i⁰ and y_i⁰) will be the sum of free molecules and molecules involved in a complex:

${\displaystyle {\begin{aligned}x_{i}^{0}&=x_{i}+z_{i},\\y_{i}^{0}&=y_{i}+z_{i+1}.\end{aligned}}}$

The dynamics of the hypercycle with translation can be described using a system of differential equations modelling the total number of molecules:

${\displaystyle {\begin{aligned}{\dot {x}}_{i}^{0}&=f_{i}z_{i}-{\frac {x_{i}^{0}}{c_{I}}}\phi _{x},\\{\dot {y}}_{i}^{0}&=k_{i}x_{i}-{\frac {y_{i}^{0}}{c_{E}}}\phi _{y}\end{aligned}}}$

where

${\displaystyle {\begin{aligned}c_{I}&=\sum _{i}x_{i}^{0},\\c_{E}&=\sum _{i}y_{i}^{0}.\end{aligned}}}$

In the above equations, c_E and c_I are total concentrations of all polypeptides and all polynucleotides, φ_x and φ_y are dilution fluxes, k_i is the production rate of polypeptide E_i translated from the polynucleotide I_i, and f_i is the production rate of polynucleotide I_i synthesised by the complex I_iE_i-1 (through replication and polymerization).[3]

Coupling nucleic acids with proteins in such a model of hypercycle with translation demanded the proper model for the origin of translation code as a necessary condition for the origin of hypercycle organization. At the time of hypercycle theory formulation, two models for the origin of translation code were proposed by Crick and his collaborators. These were models stating that the first codons were constructed according to either an RRY or an RNY scheme, in which R stands for the purine base, Y for pyrimidine, and N for any base, with the latter assumed to be more reliable. Nowadays, it is assumed that the hypercycle model could be realized by utilization of ribozymes without the need for a hypercycle with translation, and there are many more theories about the origin of the genetic code.[27]

Eigen made several assumptions about conditions that led to the formation of the first hypercycles.[2] Some of them were the consequence of the lack of knowledge about ribozymes, which were discovered a few years after the introduction of the hypercycle concept[16][17] and negated Eigen's assumptions in the strict sense. The primary of them was that the formation of hypercycles had required the availability of both types of chains: nucleic acids forming a quasispecies population and proteins with enzymatic functions. Nowadays, taking into account the knowledge about ribozymes, it may be possible that a hypercycle's members were selected from the quasispecies population and the enzymatic function was performed by RNA. According to the hypercycle theory, the first primitive polymerase emerged precisely from this population. As a consequence, the catalysed replication could exceed the uncatalysed reactions, and the system could grow faster. However, this rapid growth was a threat to the emerging system, as the whole system could lose control over the relative amount of the RNAs with enzymatic function. The system required more reliable control of its constituents—for example, by incorporating the coupling of essential RNAs into a positive feedback loop. Without this feedback loop, the replicating system would be lost. These positive feedback loops formed the first hypercycles.[2]

In the process described above, the fact that the first hypercycles originated from the quasispecies population (a population of similar sequences) created a significant advantage. One possibility of linking different chains I—which is relatively easy to achieve taking into account the quasispecies properties—is that the one chain I improves the synthesis of the similar chain I’. In this way, the existence of similar sequences I originating from the same quasispecies population promotes the creation of the linkage between molecules I and I’.[2]

After formation, a hypercycle reaches either an internal equilibrium or a state with oscillating concentrations of each type of chain I, but with the total concentration of all chains remaining constant. In this way, the system consisting of all chains can be expressed as a single, integrated entity. During the formation of hypercycles, several of them could be present in comparable concentrations, but very soon, a selection of the hypercycle with the highest fitness value will take place.[1] Here, the fitness value expresses the adaptation of the hypercycle to the environment, and the selection based on it is very sharp. After one hypercycle wins the competition, it is very unlikely that another one could take its place, even if the new hypercycle would be more efficient than the winner. Usually, even large fluctuations in the numbers of internal species cannot weaken the hypercycle enough to destroy it. In the case of a hypercycle, we can speak of one-for-ever selection, which is responsible for the existence of a unique translation code and a particular chirality.[2]

The above-described idea of a hypercycle's robustness results from an exponential growth of its constituents caused by the catalytic support. However, Eörs Szathmáry and Irina Gladkih showed that an unconditional coexistence can be obtained even in the case of a non-enzymatic template replication that leads to a subexponential or a parabolic growth. This could be observed during the stages preceding a catalytic replication that are necessary for the formation of hypercycles. The coexistence of various non-enzymatically replicating sequences could help to maintain a sufficient diversity of RNA modules used later to build molecules with catalytic functions.[28]

From the mathematical point of view, it is possible to find conditions required for cooperation of several hypercycles. However, in reality, the cooperation of hypercycles would be extremely difficult, because it requires the existence of a complicated multi-step biochemical mechanism or an incorporation of more than two types of molecules. Both conditions seem very improbable; therefore, the existence of coupled hypercycles is assumed impossible in practice.[2]

Evolution of a hypercycle ensues from the creation of new components by the mutation of its internal species. Mutations can be incorporated into the hypercycle, enlarging it if, and only if, two requirements are satisfied. First, a new information carrier I_new created by the mutation must be better recognized by one of the hypercycle's members I_i than the chain I_i+1 that was previously recognized by it. Secondly, the new member I_new of the cycle has to better catalyse the formation of the polynucleotide I_i+1 that was previously catalysed by the product of its predecessor I_i. In theory, it is possible to incorporate into the hypercycle mutations that do not satisfy the second condition. They would form parasitic branches that use the system for their own replication but do not contribute to the system as a whole. However, it was noticed that such mutants do not pose a threat to the hypercycle, because other constituents of the hypercycle grow nonlinearly, which prevents the parasitic branches from growing.[2]

According to the definition of a hypercycle, it is a nonlinear, dynamic system, and, in the simplest case, it can be assumed that it grows at a rate determined by a system of quadratic differential equations. Then, the competition between evolving hypercycles can be modelled using the differential equation:[3]

${\displaystyle {\dot {C_{l}}}=q_{l}C_{l}^{2}-C_{l}{\frac {\phi }{C}}}$

where

${\displaystyle C=\sum _{l}C_{l}.}$

Here, C_l is the total concentration of all polynucleotide chains belonging to a hypercycle H_l, C is the total concentration of polynucleotide chains belonging to all hypercycles, q_l is the rate of growth, and φ is a dilution flux that guarantees that the total concentration is constant. According to the above model, in the initial phase, when several hypercycles exist, the selection of the hypercycle with the largest q_l value takes place. When one hypercycle wins the selection and dominates the population, it is very difficult to replace it, even with a hypercycle with a much higher growth rate q.[3]