🔬 Mathematics for Microbiologists

Bacteriophage Ecology Group News (BEG News) was published mostly quarterly as an online newsletter for a total of 24 issues, July 1999 through April 2005. As follows is a reprint of the editorial from Volume 11. The newsletter’s successors are the ongoing Phage.org website, phage-therapy.org, and the Bacteriophage Ecology Group Facebook page.

Microbiologists are not a terribly mathematically inclined bunch. If we were, we probably wouldn’t be microbiologists. Indeed, microbiology is difficult enough without having to worry about mathematics. Nevertheless, some math is unavoidable, particularly if one is interested in the population-level or ecological behavior of microorganisms. This short piece is a brief introduction to three mathematical concepts that arise regularly in phage ecology: exponential growth, Poisson statistics, and dimensional analysis.

Exponential Growth

Exponential growth occurs when the rate of growth of a quantity is proportional to the current size of that quantity. For bacteria and phages growing in favorable conditions, this is the normal mode of population growth. The key parameter is the doubling time — the time it takes a population to double. Phage populations can have remarkably short doubling times, sometimes on the order of minutes.

The mathematics of exponential growth: N(t) = N₀ × e^(rt), where N(t) is the population size at time t, N₀ is the initial size, r is the intrinsic rate of increase, and e ≈ 2.718. Understanding exponential growth is essential for predicting how quickly a phage population can expand within a bacterial culture and for interpreting one-step growth experiments.

Poisson Statistics

The Poisson distribution describes the probability of discrete events occurring independently within a fixed interval. In phage biology it underlies the calculation of multiplicity of infection (MOI) and the interpretation of plaque assays. When phages are mixed with bacteria, the number of phages that adsorb to any given bacterium follows a Poisson distribution. At an average MOI of one, approximately 37% of bacteria receive no phages, 37% receive exactly one, and 26% receive two or more.

Dimensional Analysis

Dimensional analysis is the practice of tracking units of measurement through a calculation to ensure the result makes physical sense. For phage ecologists working with adsorption rates, phage densities, and bacterial growth rates, keeping track of units — per milliliter, per minute, per cell — is essential. A phage adsorption rate constant has units of mL/min; multiplying by a phage titer (phages/mL) and bacterial density (bacteria/mL) yields encounters per minute per bacterium.

These three tools — exponential growth, Poisson statistics, and dimensional analysis — form much of the quantitative foundation of phage ecology.

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