Evolution of self-replicators and hypercycles
One of the very basic scientific questions that is coming to the forefront of biology and biochemistry is how did life evolve in the pre-biotic conditions ? As part of the effort to provide hypotheses that might address this question there is an even more basic question that is of concern today ñ how can cooperation and hierarchical organization spontaneously evolve?
As an example of hierarchical organization we can consider multi-cellular organisms. These consist of organs, which consist of cells, which contain genomes, which contain chromosomes and then genes. Many multicellular organisms are then themselves hierarchically organized into societies. A theoretical framework that presents a case for why hierarchical organizations make sense, and why they should evolve has been presented in a recent book by Richard E. Michod, ìDarwinian Dynamicsî, Princeton University Press, 1999.
In the opening chapters of Michodís book several models of how the very basic units of life, self-replicating molecules and hypercycles are presented. I thought it would be interesting to translate these models into computer code and examine their behavior. Below I describe a model of self-replication. On two connected pages I have started a description of Hypercycles and quasi-species. I would suggest we put open questions on a page and link from that Self-Replicator Open Questions.
Self Replication
Michod considers two ways of representing the reactions of self-replicating system of molecules.
In the first model, the kinetics are those of a reversible reaction with constants, bi , and di , in the forward and reverse direction. The subtle point is that these two reactions donít really represent birth and death, as separate capacities of the molecule AATG (this is a single strand RNA). An equation that describes the kinetics of the first set of reactions might read:
The bi constant is ìspontaneous creationî (maybe Pasteur was wrong!), and the di represents the classical first order reaction.
In the second scheme the kinetics can be represented by a Michaelis-Menton relationship that includes a dependence on the availability of the resources.
Where in this case the parameters bi,di,Ri are interpretable are hereditable traits. Actual values for bi,di, Ri are thought to be deriviable from free energies of base-pair formation and the shape and folding of the RNA molecule, the death rate is dependent on the rate of hydrolysis of the molecule which again comes down to shape. To speak in generalities, an open structure with many points at which bases can be added will likely have a high replication rate, but also a high death rate, whereas a densely folded molecular that protects much of its length from the outside environment will be slow to grow but also slow to die. Michod refers to the work of M. Eigen for more detail. (I am sure there is a lot to be said here).
Note, however, that if Xi is initialized to zero there is no way for the ìself-replicatorî to get going in the environment, the derivative will be zero. Thus I hypothesize (and I am sure this is not original) that both sets of reactions occured, but that the ìspontaneous creationî reaction has a far lower rate than the self-replicating reaction and that we can write an overall kinetic equation for the ith replication as:
A key feature of this equation is that ìbirthî is a linear function of the density of the replicator, Michod has termed this ìMalthusian.î
In addition to the dynamics of each of the self-replicators the dynamics of the environment itself must be considered. In the simplest case, the environment can be represented by a fixed total set of resources RT some of which are bound up in the self-replicators. This leads to the balance
Once the basics of self-replication have been considered it is interesting to try to take the next step and consider the evolution of mutual-replication which at this basic molecular level can be captured by a model of Hypercycles.
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