Chalmers Conferences, 9th European Conference on Mathematical and Theoretical Biology

Effect of Neutral Spaces in a Markov Chain Model of Bacterial Evolution
Kush Chandra Khosla, Dan Nichol, Jacob Scott

Last modified: 2014-04-01


Background: In response to the growing problem of drug resistance in bacteria, scientists have built computational models to more rigorously understand the evolutionary process. We previously encoded the Strong Selection Weak Mutation (SSWM) model of evolution in a Markov Chain which allowed for efficient and analytically tractable simulation of the evolutionary process. Our initial work showed that different orderings of drug treatments, abstracted as fitness landscapes, affected the final populations which emerged. The process of using two fitness landscapes to direct population evolution is known as steering. Limitations on our original model included the omission of neutral spaces, areas in the genotype space in which each genotype is mapped to the same fitness; as well as the SSWM assumption, which we modeled as a low mutation rate so that the population remains homogenous. In this extension of our model we incorporate neutral spaces and obtain results to understand their effect. As a next step, we relax the SSWM assumptions so we that can model another biological process, known as a quasispecies phenomenon. A quasispecies assumes a high mutation rate, and, as a result, mutations occur and populations to not converge to a single peak on the fitness landscape. We further introduce an explicit stochastic model to predict quasispecies evolution with varying mutation rates on different genotype spaces.

Methods: As initial work showed that order of application can affect steering, we next test the addition of neutral spaces in a similar manner. To understand how size and shape of neutral spaces affect end populations, we incorporate randomly generated neutral spaces into fitness landscapes and use our Markov Chain model, written in the python programming language, to simulate evolution.  Each in silico experiment involves the application of two selective pressures (fitness landscapes), where, by varying the parameters of the neutral space, we can determine its effect on time to convergence and end population. We further enhance the model to understand the quasispecies phenomenon by testing populations of varying mutation rates. Throughout our studies we undo one part of the original model at a time so that we can compare results to determine its role in evolution.

Results: The inclusion of neutral spaces does not greatly affect end populations. We find that neutral spaces have the biggest effect when they contain many genotypes, have a large fitness value and are applied second. While average time to convergence stays relatively constant, in certain worst case scenarios convergence takes exponential time. As we relax the SSWM assumption, we find that our quasispecies model leads to populations concentrated less at the peak.

Conclusions/future work: We have extended a previous model of evolution by incorporating neutral spaces and high mutation rates to study steering. We have found that higher mutation rates create more diverse populations and that neural spaces, in some cases, can cause an exponential slow down in convergence time. We plan to test this model on realistic fitness landscapes in the future.


bacteria; neutral spaces; markov chain; population; dynamics; quasispecies; mathematical; mode; in silico;