Scaling limits of a model for selection at two scales

The dynamics of a population undergoing selection is a central topic in evolutionary biology. This question is particularly intriguing in the case where selective forces act in opposing directions at two population scales. For example, a fast-replicating virus strain outcompetes slower-replicating strains at the within-host scale. However, if the fast-replicating strain causes host morbidity and is less frequently transmitted, it can be outcompeted by slower-replicating strains at the between-host scale. Here we consider a stochastic ball-and-urn process which models this type of phenomenon. We prove the weak convergence of this process under two natural scalings. The first scaling leads to a deterministic nonlinear integro-partial differential equation on the interval $[0,1]$ with dependence on a single parameter, $\lambda$. We show that the fixed points of this differential equation are Beta distributions and that their stability depends on $\lambda$ and the behavior of the initial data around $1$. The second scaling leads to a measure-valued Fleming-Viot process, an infinite dimensional stochastic process that is frequently associated with a population genetics.

Scaling limits of a model for selection at two scales

The dynamics of a population undergoing selection is a central topic in evolutionary biology. This question is particularly intriguing in the case where selective forces act in opposing directions at two population scales. For example, a fast-replicating virus strain outcompetes slower-replicating strains at the within-host scale. However, if the fast-replicating strain causes host morbidity and is less frequently transmitted, it can be outcompeted by slower-replicating strains at the between-host scale. Here we consider a stochastic ball-and-urn process which models this type of phenomenon. We prove the weak convergence of this process under two natural scalings. The first scaling leads to a deterministic nonlinear integro-partial differential equation on the interval $[0,1]$ with dependence on a single parameter, $\lambda$. We show that the fixed points of this differential equation are Beta distributions and that their stability depends on $\lambda$ and the behavior of the initial data around $1$. The second scaling leads to a measure-valued Fleming-Viot process, an infinite dimensional stochastic process that is frequently associated with a population genetics.

Transsynaptic Tracing from Peripheral Targets with Pseudorabies Virus Followed by Cholera Toxin and Biotinylated Dextran Amines Double Labeling.

Transsynaptic tracing has become a powerful tool used to analyze central efferents that regulate peripheral targets through multi-synaptic circuits. This approach has been most extensively used in the brain by utilizing the swine pathogen pseudorabies virus (PRV)(1). PRV does not infect great apes, including humans, so it is most commonly used in studies on small mammals, especially rodents. The pseudorabies strain PRV152 expresses the enhanced green fluorescent protein (eGFP) reporter gene and only crosses functional synapses retrogradely through the hierarchical sequence of synaptic connections away from the infection site(2,3). Other PRV strains have distinct microbiological properties and may be transported in both directions (PRV-Becker and PRV-Kaplan)(4,5). This protocol will deal exclusively with PRV152. By delivering the virus at a peripheral site, such as muscle, it is possible to limit the entry of the virus into the brain through a specific set of neurons. The resulting pattern of eGFP signal throughout the brain then resolves the neurons that are connected to the initially infected cells. As the distributed nature of transsynaptic tracing with pseudorabies virus makes interpreting specific connections within an identified network difficult, we present a sensitive and reliable method employing biotinylated dextran amines (BDA) and cholera toxin subunit b (CTb) for confirming the connections between cells identified using PRV152. Immunochemical detection of BDA and CTb with peroxidase and DAB (3, 3'-diaminobenzidine) was chosen because they are effective at revealing cellular processes including distal dendrites(6-11).