The effects of stochasticity and population structure on mathematical descriptions of evolutionary dynamics
Lead Research Organisation:
University College London
Department Name: Mathematics
Abstract
Research Area: Mathematical Biology
Darwinian evolution is based on three fundamental principles: reproduction, mutation and selection. These determine how traits change in time within a population. There are numerous mathematical descriptions of the resulting evolutionary dynamics. In previous work, we showed how several of these frameworks could be unified by two equivalent descriptions of the dynamics: the Price equation and the replicator-mutator equation, Page and Nowak, 2002. These descriptions are, however, deterministic. In small populations and/or when selective coefficients have small magnitude, random effects are important. There are mathematical descriptions of the stochastic dynamics that occur in neutral evolution, e.g. Kimura, 1968. We aim to develop a unifying framework, which can describe both stochastic effects, like genetic drift, and also incorporate the Price/ replicator-mutator framework.
We intend to begin with a stochastic description of discrete individuals. From these we intend to derive a macroscopic description, using e.g. equations for the probabilities/ probability densities of different phenotypes. This will build on work by Champagnat et al., 2006, and aim to explain how their models relate to previous mathematical descriptions.
We will begin with stochastic process models and derive from them Fokker-Planck equations/ other macroscopic descriptions. We aim to describe what is missing in the framework, such as effects of population structure, resulting e.g. from spatial location.
Incorporating the effects of population structure into models of evolutionary dynamics is an area of great current interest, where recent advances have been made in special cases, see for example Bolker and Pacala, 1996, Allen and Tarnita, 2014, Allen, Nowak and Dieckmann, 2013, Barton, Depaulis and Etheridge, 2002, Chalub and Souza, 2009. We intend to build on these works.
References:
Page & Nowak (2002) Unifying evolutionary dynamics. Journal of Theoretical Biology 219, 93-98.
Kimura (1968) Evolutionary rate at the molecular level. Nature 217, 624-626.
Champagnat, Ferriere, Meleard (2006) Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models. Theoretical Population Biology 69, 297-321.
Bolker and Pacala (1996) Using moment equations to understand stochastically driven spatial pattern formation in ecological systems. Theoretical Population Biology 52,179-197.
Allen and Tarnita (2014) Measures of success in a class of evolutionary models with fixed population size and structure, Journal of Mathematical Biology 68,109-143.
Allen, Nowak and Dieckmann (2013) Adaptive dynamics with interaction structure. American Naturalist 181, E139-163.
Barton, Depaulis and Etheridge (2002) Neutral Evolution in Spatially Continuous Populations. Theoretical Population Biology 61, 31-48.
Chalub and Souza (2009) From discrete to continuous evolution models: a unifying approach to drift-diffusion and replicator dynamics, Theoretical Population Biology 76, 268-277.
Darwinian evolution is based on three fundamental principles: reproduction, mutation and selection. These determine how traits change in time within a population. There are numerous mathematical descriptions of the resulting evolutionary dynamics. In previous work, we showed how several of these frameworks could be unified by two equivalent descriptions of the dynamics: the Price equation and the replicator-mutator equation, Page and Nowak, 2002. These descriptions are, however, deterministic. In small populations and/or when selective coefficients have small magnitude, random effects are important. There are mathematical descriptions of the stochastic dynamics that occur in neutral evolution, e.g. Kimura, 1968. We aim to develop a unifying framework, which can describe both stochastic effects, like genetic drift, and also incorporate the Price/ replicator-mutator framework.
We intend to begin with a stochastic description of discrete individuals. From these we intend to derive a macroscopic description, using e.g. equations for the probabilities/ probability densities of different phenotypes. This will build on work by Champagnat et al., 2006, and aim to explain how their models relate to previous mathematical descriptions.
We will begin with stochastic process models and derive from them Fokker-Planck equations/ other macroscopic descriptions. We aim to describe what is missing in the framework, such as effects of population structure, resulting e.g. from spatial location.
Incorporating the effects of population structure into models of evolutionary dynamics is an area of great current interest, where recent advances have been made in special cases, see for example Bolker and Pacala, 1996, Allen and Tarnita, 2014, Allen, Nowak and Dieckmann, 2013, Barton, Depaulis and Etheridge, 2002, Chalub and Souza, 2009. We intend to build on these works.
References:
Page & Nowak (2002) Unifying evolutionary dynamics. Journal of Theoretical Biology 219, 93-98.
Kimura (1968) Evolutionary rate at the molecular level. Nature 217, 624-626.
Champagnat, Ferriere, Meleard (2006) Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models. Theoretical Population Biology 69, 297-321.
Bolker and Pacala (1996) Using moment equations to understand stochastically driven spatial pattern formation in ecological systems. Theoretical Population Biology 52,179-197.
Allen and Tarnita (2014) Measures of success in a class of evolutionary models with fixed population size and structure, Journal of Mathematical Biology 68,109-143.
Allen, Nowak and Dieckmann (2013) Adaptive dynamics with interaction structure. American Naturalist 181, E139-163.
Barton, Depaulis and Etheridge (2002) Neutral Evolution in Spatially Continuous Populations. Theoretical Population Biology 61, 31-48.
Chalub and Souza (2009) From discrete to continuous evolution models: a unifying approach to drift-diffusion and replicator dynamics, Theoretical Population Biology 76, 268-277.
Organisations
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509577/1 | 01/10/2016 | 24/03/2022 | |||
| 1777631 | Studentship | EP/N509577/1 | 01/10/2016 | 13/05/2021 | Jessica Renton |
| Description | Cancer occurs when cells in the body mutate and invade healthy tissues forming tumours. Our work looks at the early stage of this process, considering the likelihood of invasion when a single cell acquires a mutation. This is dependent on several factors including the type of mutation, the structure of the tissue and the specific mechanisms for cell division and death. We have focussed on epithelia, the tissues which form surfaces in the body such as skin, which are the origin point for most cancers. Previous work has used evolutionary graph theory to model evolution in these systems. We have instead used an explicit model of an epithelium called the Voronoi tessellation model, which has the advantage that the population structure is not fixed and there is more flexibility in how divisions and deaths can be implemented. We have looked at cooperative mutations, whereby mutated cells do not boost their own fitness directly, but provide a benefit to their neighbours. We have found that the spatial coupling of division and death events plays an important role in the success of cooperative mutations. When there is no interdependence of the locations of divisions and deaths, cooperation is more successful. Conversely, when divisions are constrained to occur only in cells neighbouring a recent death cooperation is less successful. Furthermore, if we implement density-dependence of proliferation, whereby cells can only divide if they exceed a certain size threshold, there is a spectrum of behaviour between these extremes. We find that the success of cooperation depends not only on the spatial coupling of death and division, but additionally, on mechanical properties of the epithelium. In healthy epithelia there is a strict balance maintained between divisions and deaths in order that population size remains constant. This process is still being understood, however density-dependent death and division play a role. Therefore it is likely that division and death are spatially coupled. Understanding these processes and how they affect evolutionary outcomes could therefore be important for modelling oncogenesis. |
| Exploitation Route | Our results show the importance of taking into account population structure and spatial coupling of birth and death when modelling evolution in epithelia. It is therefore important that models which seek to predict cancer evolution and outcomes should factor in these results. Furthermore our results should be taken into account when using modelling to evaluate potential efficacy of cancer therapies. Our results are also relevant to studies of cooperation in cell populations beyond cancer, and to microbial populations that produce diffusible public goods. |
| Sectors | Healthcare,Pharmaceuticals and Medical Biotechnology |
| URL | https://github.com/jessierenton |
| Description | Sir George Jessel Studentship |
| Amount | £1,800 (GBP) |
| Organisation | University College London |
| Sector | Academic/University |
| Country | United Kingdom |
| Start | 02/2019 |
| End | 02/2019 |