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Derivation and application of a dimensionless quantity for understanding viral evolution

Studies of viral evolution have traditionally been both pathogen and host specific. Titles of published manuscripts clearly illustrate this tendency in focus. Examples include: 'Evolution of the hemagglutinin of equine H3 influenza virus' and 'Mechanisms of GII.4 Norovirus Persistence in Human Populations'. Determining how a specific virus x evolves in a specific host population y is of course an important avenue of research. However, we are now at a point in time when we can synthesize across these research findings to identify general patterns of viral evolution and what processes shape them. An effective approach towards the identification of these general patterns and processes requires more than listing specific examples and qualitatively categorizing them. It requires a quantitatively rigorous framework. Here, we propose the development of this framework, which shares fundamental similarities with an approach that has been extremely effective in the physical field of fluid mechanics.

The field of fluid mechanics focuses on understanding the dynamics of fluid flows. Flows are categorized into two general types: laminar (smooth) flow and turbulent flow. When does a fluid flowing through or over an object exhibit laminar flow and when does it exhibit turbulent flow? To predict this, physicists use a quantity called the Reynolds number. The Reynolds number is a dimensionless quantity, computed as the ratio of inertial forces to viscous forces. Inertial forces depend both of the velocity of the fluid and the characteristic length of the object, while viscous forces depend only on the properties of the fluid. As a ratio between these two quantities, the Reynolds number therefore depends both on the fluid and the object through which it flows. To compute the Reynolds number for a specific system, physicists simply measure these necessary parameters for any specific fluid and any specific object. The value of the Reynolds number allows for an immediate prediction of the type of flow dynamic that will result: low Reynolds numbers produce laminar flows, while high Reynolds numbers produce turbulent flows.

The Reynolds number is a powerful tool in fluid mechanics, resulting in three distinct advantages. First, the Reynolds number allows for a greater understanding of which parameters are important for impacting flow conditions. Second, the Reynolds number enables reliable, quantitative predictions of flow dynamics, even for cases of fluid flow that have not been experimentally determined. Third, the Reynolds number facilitates the development of general engineering strategies for controlling flow. These three advantages of the Reynolds number in the field of fluid mechanics are common to dimensionless quantities more generally: they allow for greater understanding, prediction, and control. Dimensionless quantities have these advantages because their structure allows for a mathematically elegant way to move from the specifics of systems to comparative analyses across systems. In essence, they are similar to a common language which allows for communication across different cultures. They have been extensively used in physics, and their use in biology has started to rise. This increase has occurred as biologists have become more quantitative and as the benefit of these dimensionless numbers in other fields has become increasingly evident.

Here, we propose the use of a dimensionless number to understand patterns of viral evolution and the processes driving them. We propose this approach because we see two striking parallels between fluid flow and viral evolution. First, as mentioned, fluid flows can be categorized into two distinct dynamical regimes: laminar flow and turbulent flow. Similarly, viral evolution in host populations can be categorized into two distinct dynamical regimes, based on their phylogenies: cactus-like evolution and acacia-like evolution. Viruses exhibiting cactus-like evolution have limited genetic diversity over time. Imagine a cactus and its branches: the number of branches at any level of the cactus is approximately the same. Similarly, the amount of genetic diversity at any point in time is approximately the same for viruses exhibiting cactus-like evolution. Examples include influenza A virus in human hosts, HIV evolution within a single infected human, and norovirus evolution in humans. In contrast, viruses exhibiting acacia-like evolution exhibit continued growth in their genetic diversity over time. Again, imagine an acacia tree and its branches: the number of branches increases towards the top of the tree. Similarly, the amount of genetic diversity increases with time for viruses exhibiting acacia-like phylogenies. Examples include measles in humans and influenza A virus evolution in pigs. Second, whether a fluid flow is laminar or turbulent depends both on the fluid and on the object through which it passes. Similarly, whether a virus exhibits cactus-like evolution or acacia-like evolution depends not only on the virus itself, but also the host population in which it is evolving. In lieu of viscous and inertial forces, viral evolution depends on parameters such as the virus's mutation rate, the degree of cross-immunity between different strains, host lifespan, and the virus's reproductive rate in the host population.

The project proposed here consists of two parts. Part I focuses on the mathematical derivation of an appropriate dimensionless number for viral evolution. Part II focuses on three applications of this derived dimensionless quantity. The first application is the testing of this dimensional quantity with observed virological data. Does computing this dimensionless parameter for several host-virus systems enable us to correctly predict their evolutionary patterns? The second application is the comparison between viral evolution at different scales. Can a virus chronically infecting a single host, like HIV, evolve similarly to a virus that is evolving at the level of the whole population, like influenza? The third application is to determine whether the value of the dimensionless quantity is changing over time for specific host-virus systems. If so, do we see a transition from one type of evolution (e.g., a cactus-like one) to a different type (e.g., an acacia-like one)?

In sum, a dimensionless quantity for viral evolution, akin to the Reynolds number in fluid mechanics, would result in improved understanding, prediction, and control. We would have a better understanding of why certain viruses in certain host populations evolve similarly to or differently from other viruses in other host populations. We would have better predictive capabilities of viral evolution when a new pandemic strain entered our population. And, finally, we would employ better viral control strategies by sharing strategies across different systems and by understanding the leverage points of viral evolution.

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