hyper

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 * TODO List**
 * REF Eigen Schuster -> 1979 or 1989?
 * REF Dobzanksky

=[|Hypercycles]=

A first attempt to cross the information threshold was also 'invented' by Eigen and Schuster (1979) who considered an ecological solution: i.e. an ecology of interacting molecules that form stable networks and so together maintain more sequence length than each on their own. Their ecological modelling approach focussed on ecological stability (and thus neglected mutations). A very simple form of this idea is depicted in the first picture (left), with the accompanying ODE:

//dA/dt = a1*A + b1*A*B - O// //dB/dt = a2*B - O//

where //a1// and //a2// are the catalysis-independent replication rates, //b1// is a catalysis-dependent replication rate and O is a chemostat-assumption term (see the quasispecies-equation).

If we generalise this two species model to any number of species (second picture), we get

//dX i /dt = a i *X i + b i *X i *X j - O i //

with the second term signifying that //X i // receives catalysis from //X j //.

If we look at the simplified ODE, //a1 + b1 >> a2// would imply that //A// outcompetes //B//. The other way around (//A// catalysing //B//) will not work either, then B will outcompete A. We will always expect the ones that are not catalyzed to be outcompeted, and this holds for any number of species that catalyse each other. Hence we need feedback: a cycle.

Hypercycles in ODEs
The resulting dynamics depend on the cycle length. There is always one non-trivial equilibrium.
 * **Cycle length** || **Stability** ||
 * <= 3 || fixed point ||
 * 4 || stable spiral ||
 * >= 5 || limit cycle (increasing amplitude) ||
 * || attofoxes! ||

In this model the only stable topology is a circle, because non-catalysing branches will always outcompete other molecules (i.e. parasites that do not give catalysis but do receive it will quickly outcompete the mutualists). When mutations are added to the ecological system it is clear that even the stable topologies (cycles) are quickly destroyed by parasites which destroy its cyclical nature. (Note that this is different from continually adding lots of mutants like in the quasi-species equation.)

Moreover, new hypercycles without parasites cannot invade the system since the X 2 term dominates the equations in the model, i.e. concentration is more important than growth rate. Hence, cycles with a better growth rate but low concentration could never invade an already established cycle. This in itself is an interesting consideration for Darwinian selection. Darwinian selection only works if replication rates are more of less linear, i.e. if in a*X n :
 * **n>1**, then **survival of the first** (very strong founder control)
 * **n<1,** then **survival of everyone** (see question on complex formation)

This consideration of the stability of ecological systems in terms of evolution (mutants invading) is illustrative of the importance of a famous essay by [|Dobzhanksky] (1973) (REF): [|Nothing in biology makes sense except in light of evolution.]

We have added at this stage: //in light of CAs (local interactions, micro-macro transition, non-linear dynamics, simple rules to complex behaviour)//

The bottom line of this course is that: //Nothing in biology makes sense except in light of both!//

We will now consider removing the **infinite** and the **well-mixed** assumptions of the ODE hypercycle model. (Note that these are assumptions that are implicit in the ODE model formalism and therefore came in through the back door!)

Next: Hypercycles in space