by Olivia Jack , Dirk Brockmann

This explorable illustrates the dynamics of the famous hypercycle model. It was originally conceived by Peter Schuster and Nobel laureate Manfred Eigen (1927-2019) in 1979 to investigate the chemical basis of the origin of life. Because living things make copies of themselves, in the beginning complex chemicals like polymer chains, including small RNA molecules (see e.g. RNA-world), had to acquire the ability to catalyse their own synthesis from smaller parts, e.g. single nucleotides.

Before the hypercycle, Eigen had developed the quasispecies model for self-replicating entities in 1971. One of the key insights of the quasispecies model was the error threshold. On one hand, self-replicating molecules had to have a significant size and complexity to be able to catalyse their own replication; on the other hand, mutations and errors in the replication processes changed the information encoded in the molecule potentially putting an end to their ability to replicate.

The hypercycle model solves this problem. In essence it consists of a small number of different self-replicating entities, $A_1$, $A_2$,…,$A_n$, each of which facilitates its own replication but also receives cooperative catalytic support from other entities, such that $A_1$ helps $A_2$ which in turn helps $A_3$, etc. in a cyclic fashion. This way the entire hypercycle can be interpreted as a cooperative entity that replicates as a whole.

First press Play and keep on reading….

This is how it works

This model evolves on a square lattice. A lattice site can be occupied by an individual of one of the $N$ species (you can select between 3, 6 and 9 different species) in the system or it can be empty. Colors depict different species, while empty sites are grey. The dynamics is governed by four basic events that can happen:

  1. An individual can die, so its site becomes vacant. This is governed by the decay rate slider.
  2. An individual can move to a vacant adjacent site, this is controlled with the diffusion slider.
  3. An individual can replicate into an empty adjacent site at a baseline replication rate
  4. Finally, a species replication rate can be increased if an individual is spatially adjacent to individuals of another species of a particular type. This is the catalytic, cooperative support a species receives from another species as depicted in the diagram in the control box.

The last part is the important element. The structure in which species support each other species is cyclic, so species $n$ receives support from species $n-1$ (species $1$ receives support from species $N$ to close the circle). The entire system is called a hypercycle.

Try this

For the default set of parameters, 9 species are elements of the hypercycle. A self-sustained pattern of spiral waves should emerge in which the wave fronts of one species expand into regions of the supporting species. The smooth-switch makes the emergent pattern visually smoother by coloring each pixel according to the majority of the type of species type around it. It changes no part of the model. With the speed slider you can change the update rate of the model.

Keeping the parameters fixed, you can change the number of species to 3 or 6 and wait for the pattern to display stable, sustained dynamic patterns.

No catalytic support

First try to explore what happens when a system has no cooperative support among the species, by turning the catalytic support slider to the minimum. You may need to increase the diffusion slider a bit to observe a stable dynamic pattern. Increasing the diffusion may be necessary in order to increase the availability of empty site of species to replicate into.

The impact of cooperative support

Now slowly increase the catalytic support. You should see that the quality of the pattern changes into something more cyclic and less patchy. You may even decrease the baseline replication rate to its minimum and see sustained spiral wave patterns.

Parasitic Invasion

The hypercycle model was criticized because it is susceptible to “parasitic molecules”. Those are entities that receive catalytic support from one of the species of the hypercycle but do not contribute to the whole by providing catalytic support to others. When such a parasite is introduced, it can compromise the entire system. This always happens if the parasitic entity receives more support than the other elements in a well mixed situation.

In a 1990 paper, Maarten Boerlijst and Paulien Hogeweg analyzed the spatial hypercycle model explored here with a special focus on the impact of parasitic entities.

The surprising insight: In the spatial hypercycle model, parasitic invasion can be avoided. When parasites are introduced they can be forced out of the system even when they received more catalytic support than all other species.

Parasites can compromise the hypercycle

First, rewind and reset the parameters by pressing counterclockwise arrow button and back button.

Now, select 3 species in the control box and let the system evolve. You should see an expanding pattern of patches of all three species that is self-sustained.

Finally, press the add parasites button. This injects individual pixels of the parasitic species into the system.

You should see growing black patches that “eat” into the existing pattern leaving a trace of white (empty) region. This is because the parasite requires regions of the species that supports it, but cannot replicate sufficiently by itself.

At the end, everything is dead.

Robustness with respect to invasive parasites

While keeping everything else fixed, you can repeat the simulation with 6 or with 9 species, and you will see that on average the parasites cannot invade and overtake the system. This is because its hard for them to encounter regions where the one species they need is abundant.

The hypercycle is therefore robust with respect to invasion of parasites.

Further Information

Related Explorables:

The Prisoner's Kaleidoscope

The prisoner's dilemma game on a lattice

The Prisoner's Kaleidoscope

The prisoner's dilemma game on a lattice


The spatial rock-paper-scissors game

Hopfed Turingles

Pattern Formation in a simple reaction-diffusion system

A Patchwork Darwinge

Evolution: Variation and Selection