Quasi-species

The model, called quasi-species , was proposed by Manfred Eigen and Peter Schuster , based on an earlier work by Manfred Eigen. It qualitatively describes the evolution of a closed system of self-reproducing molecules, e.g. B. RNA or DNA . It was conceived as a contribution to the search for the origin of life and transfers Darwin's theory of evolution with mutation and selection to the molecular level. Quantitative statements are difficult to make with this model because the initial conditions cannot be established or measured in practice. But there are also models that explain the evolution of an open system.

Assumptions

This theory makes four assumptions:

The molecules to be replicated are composed of a small number of building blocks.
New molecules are only created through (possibly faulty) replications of the existing molecules.
Substrates - substances necessary for the structure of the molecules - are available in large quantities, degradation products leave the container.
Molecules can break down again into their building blocks according to their stability. The age of a molecule doesn't matter.

experiment

Evolution occurs in the reaction vessel: the concentration of a type of molecule depends heavily on its stability (rate of decay) and its rate of reproduction. Errors ( mutation ) can occur during replication , which may offer the molecule a better chance of survival - the original molecule is then displaced ( selection ). Since more or less strong mutations always occur, closely related (similar) molecules are called quasi-species. These have been observed in RNA viruses. Since mutations in similar molecules regularly occur, the success of reproduction does not only depend on one species, but on a whole "cloud" of similar species, which transform into each other again and again and reproduce equally well: optimally in the "middle", towards the "edge" getting worse and worse. A species can thus also arise through mutation from another species, not just through its own replication. It should therefore be noted that the success of a quasi-species depends on its reproduction rate, its death rate, but also on its mutation rate: Without errors in replication, only one species exists - an improvement is impossible; if the mutation rate is too high, the quasi-species spreads over the entire population. Slowly growing species are not necessarily displaced by faster growing ones: a faster growing species can mutate into the slower growing one, thus compensating for the extinction caused by decay and runoff on a small scale.

Hypercycles can even occur: A molecule A reproduces a molecule B, which in turn replicates A. The molecules A and B multiply independently of each other and also form a quasi-species.

Mathematical description

A simple mathematical model for the quasi-species: there are possible sequences and organisms with sequence i . Let's say that every individual reproduces at the rate of reproduction . Some are clones of their "parents" and have the sequence i , but some are mutated and have a different sequence. Let us say that the fraction of the j types that descend from an i type is what we call the mutation rate. Define as the total number of i- type organisms after the first round of reproduction. Then applies ${\ displaystyle S}$ ${\ displaystyle n_ {i}}$ ${\ displaystyle A_ {i}}$ ${\ displaystyle q_ {ji}}$ ${\ displaystyle n '_ {i}}$

{\ displaystyle n '_ {i} = \ sum _ {j} w_ {ij} n_ {j} \,}

where and is. Sometimes a death rate or a decay rate is introduced such that: ${\ displaystyle w_ {ij} \, = \, A_ {j} q_ {ij}}$ ${\ displaystyle \ sum _ {j} q_ {ij} = 1 \,}$ ${\ displaystyle D_ {i}}$

{\ displaystyle w_ {ij} = A_ {j} q_ {ij} -D_ {i} \ delta _ {ij} \,}

wherein is 1 when i = j, otherwise 0. The n-th generation was obtained by us in the above formula W through the n-th power of W replace. ${\ displaystyle \ delta _ {ij}}$

That is a system of linear equations . The usual solution is to diagonalize the W matrix first . Their diagonal entries will be eigenvalues for different mixtures ( eigenvectors ) of the W matrix, which are called the quasi-species. After many generations, only the eigenvector with the highest eigenvalue will prevail and this quasi-species will dominate. The eigenvectors indicate the relative ratio of each sequence in equilibrium.

A simple example

The concept of quasi-species can be illustrated by a simple system consisting of 4 sequences: Sequence 1 is [0,0] and the sequences [0,1], [1,0] and [1,1] are assumed to be 2, 3 and numbered 4. Suppose sequence [0,0] never mutates and always produces offspring. The other 3 sequences produce offspring on average , i.e. less than 1, but the other two types, whereby applies. The matrix W then looks like this: ${\ displaystyle 1-k}$ ${\ displaystyle k}$ ${\ displaystyle 0 \ leq k \ leq 1}$

{\ displaystyle \ mathbf {W} = {\ begin {bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1-k & k & k \\ 0 & k & 1-k & k \\ 0 & k & k & 1-k \ end {bmatrix}}}

The diagonalized matrix is:

{\ displaystyle \ mathbf {W '} = {\ begin {bmatrix} 1-2k & 0 & 0 & 0 \\ 0 & 1-2k & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 + k \ end {bmatrix}}}

and the eigenvectors for these eigenvalues are:

Eigenvalue	Eigenvector
1−2k	[0, −1,0,1]
1−2k	[0, −1,1,0]
1	[1,0,0,0]
1 + k	[0,1,1,1]

Only the eigenvalue is greater than one. For the nth generation, the associated eigenvalue will be and so grow over time over all limits. This eigenvalue belongs to 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 quasi-species are not allowed. The third consists only of the non-mutating sequence 1. It can be seen that species 1 appears to be the fittest, since it reproduces itself at the highest rate - but in the long run it can succumb to the quasi-species from the other three sequences. Cooperation can pay off in evolution! ${\ displaystyle 1 + k}$ ${\ displaystyle (1 + k) ^ {n}}$

Individual evidence

↑ Manfred Eigen, Peter Schuster: The Hypercycle: A Principle of Natural Self-Organization. In: Die Naturwissenschaften, Volume 64, No. 11, 1977, pp. 541-565, doi: 10.1007 / BF00450633 .
↑ Manfred Eigen: Selforganization of Matter and the Evolution of Biological Macromolecules. In: The natural sciences. Volume 58, No. 10, 1971, pp. 465-523, doi: 10.1007 / BF00623322 .
↑ Luis P. Villarreal, Günther Witzany: Rethinking quasispecies theory: From fittest type to cooperative consortia. In: World Journal of Biological Chemistry. Volume 4, No. 4, 2013, pp. 79-90. doi: 10.4331 / wjbc.v4.i4.79

[1] Manfred Eigen, Peter Schuster: The Hypercycle: A Principle of Natural Self-Organization. In: Die Naturwissenschaften, Volume 64, No. 11, 1977, pp. 541-565, doi: 10.1007 / BF00450633 .

[2] Manfred Eigen: Selforganization of Matter and the Evolution of Biological Macromolecules. In: The natural sciences. Volume 58, No. 10, 1971, pp. 465-523, doi: 10.1007 / BF00623322 .

[3] Luis P. Villarreal, Günther Witzany: Rethinking quasispecies theory: From fittest type to cooperative consortia. In: World Journal of Biological Chemistry. Volume 4, No. 4, 2013, pp. 79-90. doi: 10.4331 / wjbc.v4.i4.79