## A Numerical Method for Determining the Effective Burst Size and Cycle Time of a Virus

- May 30, 2001
- 3:30 p.m.
- LeConte 412

## Abstract

Life is reproduction, mutation, and selection. (This definition is Francis Crick's, from his book, "What is Life?". If you don't like it, go argue with HIM.) Although mutation and selection are regularly quantified nowadays, accurate practical estimates of reproduction are more elusive. The relevant quantity of interest in this talk is the "age-specific fertility" of an average individual. The age-specific fertility is a histogram of reproductive frequency of an individual against age. The area under this histogram from ages t[1] to t[2] on the X-axis is therefore the average number of offspring produced between t[1] and t[2].

Quantifying reproduction is central to several biological subspecialties. In microbiology, for example, it could help assess antiviral or antibacterial therapies, and dissect the biological correlates of microbial virulence. In zoology, it could help determine reproductive success during accidental pest introductions or deliberate wildlife releases. Present practical methods of quantifying reproduction have severe limitations. They either depend on being able to identify individuals (e.g., population or wildlife censuses) or they are forced to make the mathematical assumption of synchronous reproduction (i.e., all individuals in a population have a deterministic cycle time). For many populations of interest (e.g., microbes), individuals can not be identified by experimenters. Moreover, the reproductive cycle-time often varies from individual to individual, with single individuals typically giving rise to their progeny at different times.

The talk gives a renewal-like equation that relates age-specific fertility to experimental observables. The equation suggests an experimental protocol that measures the age-specific fertility in a viral population. First, synchronize founder members of a viral population at the same age by attaching them to cells at 4 degrees C. Then, measure the time-course of a marker (e.g., a viral protein) that can act as a surrogate for the number of founder members and their descendants. Because the renewal-like equation is numerically unstable, its naive solution as a set of simultaneous linear equations generates wild fluctuations in the presence of experimental error. Computer simulations of the experiment show, however, that Tikhonov's method can extract good, practicable approximations for the age-specific fertility. Although the numerical methods and statistical analysis are at present crude, data analyzed from a pilot experiment using laboratory strains of HIV show reasonable agreement with known parameters of HIV growth.

References: Spouge JL and Layne SP (1999) A practical method for simultaneously determining the effective burst sizes and cycle times of viruses. PNAS 96: 7017-7022