Daniel Romero

Mentor: Dr. Scott McKinley
College of Liberal Arts and Sciences
 
"I have a strong curiosity for mathematics. After spending so many years studying it, I've always wondered how did all these theorems and proofs come about. So one reason for getting involved with research so that I have the opportunity to be apart of that process and witness through first hand experience how mathematical research is done. The other reason being that I want to broaden my knowledge of mathematics as much as possible. The more exposure to each of its different branches I have, the more ready and competent I feel myself to be to tackle other unsolved problems. "

Major

Mathematics

Minor

N/A

Research Interests

  • General Mathematics

Organizations

  • University Mathematics Society

Hobbies and Interests

  • Learning about mathematics
  • Drawing

Research Description

A Mathematical Analysis of HIV : A study in the optimal number and distribution of envelope spikes
One can think of a virus as genetic information contained in a viral envelope. On the surface of this envelope are numerous binding sites used to bind to the membrane of a target cell. These envelope binding sites are commonly referred to as envelope “spikes.” With enough spikes bound to receptors on the surface of a target cell, a virion can fuse with the target membrane and inject its genetic information, ultimately leading to the production of more virions. As a defense mechanism though, the immune system releases antibodies that can neutralize the virus through different methods depending on the type of antibody. Recent studies have shown that one type of antibody, IgA, can bind multiple virions together. They collectively become a nonfunctional mass of cells, and can no longer infect the target cells. The recent work of Dr. McKinley and colleagues has shown that another type of antibody, IgG, can bind simultaneously to virions and to mucin fibers in the local environment, leaving the antibody-virion complex immobile and thus unable to reach target cells.
 
The “goal” of the virus is to make contact with enough receptors on the cell membrane so that infection initiates. This can depend on both the number of free envelope spikes and their spatial distribution. It should be noted that there are costs and benefits associated with having a large number of binding sites on the viral envelope. The presence of more spikes allows for a higher probability of infection, but also leads to a higher probability that a virion will be affected by bound antibodies. On the other hand, having fewer spikes reduces the chance of antibody neutralization, but at the same time it reduces infectivity rate. Furthermore, it stands to reason that the spike spatial distribution may play a role. Clustering of spikes may allow for a more rapid succession of binding, leading to either faster rates of infection or neutralization. Meanwhile, scattered, dispersed spikes prevent multiple sites from being neutralized by a single antibody, but also decrease the probability of binding to a receptor of a cell upon contact.
 
So far we have spoken in generalities, but in many ways HIV is a unique case. Studies have shown that there is a strong correlation between virion size and the number of envelope spikes. However, HIV breaks significantly from that pattern in that HIV virions have very few envelope spikes. The goal of this project is therefore to analyze the optimal number and spatial distribution of viral spikes in the general case; and then describe where HIV stands in comparison. We will approach this problem by the use of mathematical models that will describe the random distribution of the envelope spikes using spatial point processes, the dynamic binding and unbinding of antibodies using Ordinary Differential Equations and then combine these into a comprehensive probabilistic model for infection.