| TAZ-TFG-2024-4379 |
El modelo de cuasiespecies
Embid López, Miguel
Gómez Ambrosi, Carlos (dir.)
Universidad de Zaragoza,
CIEN,
2024
Graduado en Matemáticas
Resumen: This thesis dives into the quasispecies model, a mathematical framework that offers a structured approach to understanding viral evolution, focusing on mutation and natural selection. The analysis centers around RNA viruses, which is a proper subject for such study due to their high mutation rates and simpler replication processes compared to DNA viruses.
The study begins by introducing the Eigen model, a foundational model in evolutionary biology that describes the behavior of populations over time when treating self replicating molecules. The Eigen model plays a central role in understanding how populations evolve when errors in replication (mutations) occur. The thesis presents two versions of this model: one in continuous time and one in discrete time. Both variations are used to explain how mutation and natural selection jointly influence the distribution of different species or genomes within a population, providing insights into the underlying evolutionary dynamics.
An important part of this analysis is the application of the Perron-Frobenius theorem, a mathematical tool that ensures the existence and uniqueness of equilibrium solutions under certain conditions. This theorem is crucial in determining the steady-state or equilibrium distribution of species within a population subjected to both mutation and selection processes. In this steady state, which is called the “quasispecies”, a specific distribution of the organisms tends to dominate the population. This concept contrasts with the traditional view where evolution leads to the survival of the single “fittest” organism; instead, the quasispecies framework suggests that evolution results in a population made up of a spectrum of organisms.
The thesis also explores an alternative model of viral evolution, the Crow-Kimura model. This model takes a different approach by decoupling the processes of mutation and replication, treating them as independent events. Unlike the Eigen model, where mutations occur during the replication process, the Crow-Kimura model allows mutations to happen at any point in the life cycle of an organism. This alternative perspective provides a broader understanding of how these processes might influence the evolution of populations in a different context.
The thesis then explores the two major results derived from the quasispecies theory. These results are key to understand the complexity of evolutionary dynamics in systems where mutation plays a significant role:
- Absence of an Evolutionary Optimization Principle: In classical evolutionary theory, particularly in Fisher’s fundamental theorem of natural selection, it is said that a population will tend to increase its average fitness over time. This idea suggests that natural selection alone drives a population toward optimal fitness. However, the quasispecies model challenges this principle by showing that the presence of mutations can avoid that a population ever achieves an optimal fitness state. The continuous introduction of mutations into the population prevents the dominance of the fittest genome, leading instead to a more diverse distribution of genomes.
- Error Threshold Phenomenon: This is a crucial aspect of the quasispecies model. The error threshold refers to the maximum mutation rate that a population can sustain before losing genetic information. If the mutation rate goes above this limit, the replication process makes too many mistakes to reliably pass on working genetic material. This leads to a breakdown in the population’s ability to maintain its evolutionary advantage. This phenomenon is particularly relevant to RNA viruses, which are known for their high mutation rates.
The mathematical analysis of these models is complemented by various simulations that help illustrate the behavior of populations under different initial states. Some of these simulations demonstrate how, under certain conditions, small variations in the initial distribution of genomes can lead to drastically different outcomes in the population’s long-term equilibrium state. Meanwhile, other simulations show how, under the complementary conditions, the steady state only depends on the underlying system and all the possible initial states lead to the same stationary distribution.
Furthermore, the thesis highlights the importance of the balance between mutation and selection. While natural selection tends to favor organisms with higher fitness, mutations introduce variability that can prevent a population from being solely composed of the most fit organisms. This balance is essential for maintaining genetic diversity within the population, which in turn allows the population to adapt to changing environmental conditions. The quasispecies model thus provides a more refined view of evolution, where the goal is not simply to maximize fitness but to sustain a dynamic equilibrium between mutation and selection.
In conclusion, this thesis contributes to the field of evolutionary biology by applying rigorous mathematical models to the study of viral evolution. It provides valuable insights into how RNA viruses evolve, highlighting the role of mutation in shaping their genetic diversity. The quasispecies model offers a powerful framework for understanding the interplay between mutation and selection, challenging traditional views of evolution as a process of optimization.
The study begins by introducing the Eigen model, a foundational model in evolutionary biology that describes the behavior of populations over time when treating self replicating molecules. The Eigen model plays a central role in understanding how populations evolve when errors in replication (mutations) occur. The thesis presents two versions of this model: one in continuous time and one in discrete time. Both variations are used to explain how mutation and natural selection jointly influence the distribution of different species or genomes within a population, providing insights into the underlying evolutionary dynamics.
An important part of this analysis is the application of the Perron-Frobenius theorem, a mathematical tool that ensures the existence and uniqueness of equilibrium solutions under certain conditions. This theorem is crucial in determining the steady-state or equilibrium distribution of species within a population subjected to both mutation and selection processes. In this steady state, which is called the “quasispecies”, a specific distribution of the organisms tends to dominate the population. This concept contrasts with the traditional view where evolution leads to the survival of the single “fittest” organism; instead, the quasispecies framework suggests that evolution results in a population made up of a spectrum of organisms.
The thesis also explores an alternative model of viral evolution, the Crow-Kimura model. This model takes a different approach by decoupling the processes of mutation and replication, treating them as independent events. Unlike the Eigen model, where mutations occur during the replication process, the Crow-Kimura model allows mutations to happen at any point in the life cycle of an organism. This alternative perspective provides a broader understanding of how these processes might influence the evolution of populations in a different context.
The thesis then explores the two major results derived from the quasispecies theory. These results are key to understand the complexity of evolutionary dynamics in systems where mutation plays a significant role:
- Absence of an Evolutionary Optimization Principle: In classical evolutionary theory, particularly in Fisher’s fundamental theorem of natural selection, it is said that a population will tend to increase its average fitness over time. This idea suggests that natural selection alone drives a population toward optimal fitness. However, the quasispecies model challenges this principle by showing that the presence of mutations can avoid that a population ever achieves an optimal fitness state. The continuous introduction of mutations into the population prevents the dominance of the fittest genome, leading instead to a more diverse distribution of genomes.
- Error Threshold Phenomenon: This is a crucial aspect of the quasispecies model. The error threshold refers to the maximum mutation rate that a population can sustain before losing genetic information. If the mutation rate goes above this limit, the replication process makes too many mistakes to reliably pass on working genetic material. This leads to a breakdown in the population’s ability to maintain its evolutionary advantage. This phenomenon is particularly relevant to RNA viruses, which are known for their high mutation rates.
The mathematical analysis of these models is complemented by various simulations that help illustrate the behavior of populations under different initial states. Some of these simulations demonstrate how, under certain conditions, small variations in the initial distribution of genomes can lead to drastically different outcomes in the population’s long-term equilibrium state. Meanwhile, other simulations show how, under the complementary conditions, the steady state only depends on the underlying system and all the possible initial states lead to the same stationary distribution.
Furthermore, the thesis highlights the importance of the balance between mutation and selection. While natural selection tends to favor organisms with higher fitness, mutations introduce variability that can prevent a population from being solely composed of the most fit organisms. This balance is essential for maintaining genetic diversity within the population, which in turn allows the population to adapt to changing environmental conditions. The quasispecies model thus provides a more refined view of evolution, where the goal is not simply to maximize fitness but to sustain a dynamic equilibrium between mutation and selection.
In conclusion, this thesis contributes to the field of evolutionary biology by applying rigorous mathematical models to the study of viral evolution. It provides valuable insights into how RNA viruses evolve, highlighting the role of mutation in shaping their genetic diversity. The quasispecies model offers a powerful framework for understanding the interplay between mutation and selection, challenging traditional views of evolution as a process of optimization.
Tipo de Trabajo Académico: Trabajo Fin de Grado
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