Moreover, the whole cycle must be closed, so that En must catalyse I1 formation for some integer n > 1. Alternative concepts 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. 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. 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. Mathematical model Elementary hypercycle The dynamics of the elementary hypercycle can be modelled using the following differential equation: where In the equation above, xi is the concentration of template Ii; x is the total concentration of all templates; ki is the excess production rate of template Ii, which is a difference between formation fi by self-replication of the template and its degradation di, usually by hydrolysis; ki,j is the production rate of template Ii catalysed by Ij; 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 (xi = 1). Assuming that at high concentration x the term ki 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: where according to the cyclic properties, it can be assumed