Quasispecies model

Referred to as quasispecies model was proposed by Manfred Eigen and Peter Schuster, based on an earlier work of Manfred Eigen. It qualitatively describes the evolution of a closed system of self-replicating molecules, such as RNA or DNA. It was designed as a contribution to the search for the origin of life and transmits the Darwinian theory of evolution by mutation and selection at the molecular level. Quantitative statements are difficult to make with this model, since the initial conditions are not to manufacture or remeasure in practice. There are, however explain the evolution of an open system also models.

Assumptions

Four assumptions makes this theory:

Experiment

In the reaction vessel there is evolution: The concentration of a molecule type is highly dependent on the stability ( decomposition rate) and its reproduction rate. In replication, there may be errors occur (mutation), which may provide the molecular improved chances of survival - the original molecule is then displaced (selection). Since it is always more or less severe mutations occur, it is called closely related ( similar ) molecules a quasispecies. These were observed in RNA viruses. Since mutation regularly found in similar molecules, the reproductive success depends not only from one species, but of a whole "cloud" of similar species which are converted again into each other and similarly well reproduce: In the "middle" optimal, to the "Edge " go from bad to worse. One species may occur as by mutation of a different species, not only through their own replication. So It should be noted that the success of a quasi-species depends on their rate of reproduction, their death rate, but also on their mutation rate: without replication errors exists only a single species - an improvement is impossible, at too high a mutation rate of the quasispecies widened over the entire population. Slow-growing species are not necessarily displaced by faster growing: A faster- growing species can compensate for the slower growing into mutate and so the extinction by disintegration and drain on a small level.

It can even hypercycles occur: a molecule A reproduces a molecule B, which in turn replicates A. The molecules A and B so multiply interdependent and also form a quasi-species.

Mathematical Description

A simple mathematical model for the quasi-species: there is possible sequences and organisms with sequence i say we, each individual reproduce with the reproduction rate. Some are clones of their " parents " and have the sequence i, but some are mutated, and have a different sequence. Let's say that the fraction of the j- types, which is derived from an i- type, which we call mutation rate. If we define as the total number of i- type organisms after the first round of reproduction. Then we have

And where is. Sometimes one introduces a mortality rate or decay rate, so that applies:

Wherein is 1 if i = j 0 otherwise The n-th generation, we obtained by replacing in the above formula, W through the n -th power of W.

Which is a linear system of equations. The usual approach is to first diagonalize the W matrix. Your diagonal entries are eigenvalues ​​to be various mixtures ( eigenvectors ) of the matrix W, called the quasi-species. After many generations, only the eigenvector will prevail with the highest eigenvalue and these quasi-species will dominate. The eigenvectors indicate the relative proportion of each sequence at equilibrium.

A simple example

The concept of quasispecies can be illustrated by a simple system composed of 4 sequences: Sequence 1 is [0,0] and the sequences [0,1], [1,0] and [1,1] are 2, 3 and 4 numbered. Suppose sequence [ 0,0] never mutate and always produces an offspring. The other three sequences produce progeny average, less than 1, but of the other two types, with the proviso. The matrix W then looks like this:

The diagonalized matrix is:

And the eigenvectors for these eigenvalues ​​are:

Only the eigenvalue is greater than one. For the n - th generation of the corresponding eigenvalue will be and so grow over time across all borders. The eigenvalue associated with the eigenvector [0, 1, 1, 1], which represents the quasi-species, consisting of species 2, 3 and 4 - which will be present in the same concentration after a long time. Since all population numbers must be positive, the first two quasispecies are not allowed. The third consists only of the non- mutated sequence 1 can be seen that species 1 seems best shape since it reproduces itself at the highest rate - yet they may be subject to duration of the quasispecies from the other three sequences. Cooperation can pay evolutionarily!