The Poisson distribution played a key role in experiments that had a historic role in the development of molecular biology. In particular, the interpretation and design of experiments elucidating the actions of bacteriophages and their host bacteria during the infection process were based on the parameters of the Poisson distribution. I briefly review three of the most important of these experiments and offer suggestions on the use of the Poisson distribution in statistical calculations in biology laboratory exercises reflecting the use of the Poisson distribution in current biological investigations.

## Introduction

Molecular biology as a discipline evolved out of the interweavings of several older paths of research, including biochemistry, genetics, biophysics, and cell biology. No research system was more important in its evolution, however, than the bacteriophage. The confirmation of DNA as the genetic material (Hershey & Chase, 1952), the confirmation of the triplet genetic code (Crick et al., 1961), the codetermination of the roles of operons in genetic control (Jacob & Monod, 1961), and the demonstration of the existence of messenger RNA (Brenner et al., 1961) are just a few examples of the bacteriophage's central role in key experiments of fundamental importance.

Students know that biology is a broad discipline filled with many aesthetic experiences, whether the flitting of butterflies around flowers or the massive migrations of flocks of cranes. To truly understand the mechanisms of structures, movement, and behavior, however, one must be able to describe the phenomenon mathematically. This allows the observer to determine the contributing factors, to share the results with other investigators, and to predict the effects of future perturbations of the system. For these reasons and more, many voices have called for the strengthening of the use of mathematics in introductory biology courses (National Research Council, 2003; Labov et al., 2010; Woodin et al., 2010; AAAS, 2011; Feser et al., 2013). Core Ideas in the Next Generation Science Standards (NGSS Lead States, 2013) that are inherent in the type of exercises and problems presented here are “Inheritance and Variation of Traits” and “Analyzing and Interpreting Data.”

A review of the very earliest experiments that opened the door to the use of phages as such a rich experimental system reveals that from the beginning, molecular biology was strongly dependent on mathematical treatment of experimental data. Three key experiments are discussed here, with emphasis on the concepts that mathematics formalized. These three are chosen because they all derive their power from the use of a single probability expression, the Poisson distribution, which has broad application to the study of biology.

Following these descriptions of how a statistical distribution can have such an impact on our understanding of a single model system, I present the results of a student laboratory exercise employing the Poisson, two sample problems, and a reference to Poisson's use in an epidemiological case study.

The distribution was derived by Siméon-Denis Poisson (1837) and describes the probability of occurrences (successes or failures) within intervals in which the probability of successes in one interval is independent of the probability of success in other intervals, the probability of success in any given interval is low, and the mean number of successes is known.

The equation is given as

$P(r)=nr×e−n/r!$

where P(r) is the probability of occurrence of r successes in one trial and n is the mean number of successes per trial

Each of the phage experiments described below represents the application of the Poisson distribution, either conceptually or formally, to a better understanding of the behavior of bacteriophages.

## D'Herelle & the Particulate Nature of Bacteriophages

In 1917 Félix d'Herelle reported the existence of plaques (areas of lysis) on plated bacteria collected from patients with dysentery (d'Herelle, 1917). D'Herelle rapidly interpreted the existence of plaques and of serial propagation as evidence that he had observed a discrete microbe. At the time, d'Herelle had two methods for observing the lysis of bacteria: the clearing of bacterial suspensions after inoculation and the formation of plaques on bacterial lawns on plates. D'Herelle reasoned that he had found some particulate microbe.

In further analysis, he developed the concept of plaque-forming units per volume, by counting the number of plaques found per volume of added lytic agent. D'Herelle did serial dilutions of the agent; and at dilutions at which he expected an average of 0.5 plaque-forming units per added volume, he inoculated a series of 10 broth cultures of bacteria. Some of the cultures lysed, but, importantly, some did not. D'Herelle proposed that this could not happen if the lytic agent were a fluid substance, such as an enzyme. Thus, he reasoned, bacterial lysis was due to an “ultramicrobe,” which he named “bacteriophage.”

The logic of this broth lysis experiment expressed by d'Herelle represents what Summers (1999) refers to as an “intuitive” understanding of the probability, which is expressed formally as the Poisson distribution.

## Ellis-Delbrück & the One-Step Growth Curve

Other researchers began to explore the use of phages for the investigation of genetic mechanisms. Emory Ellis examined the reproductive cycle of the virus. In an earlier publication, d'Herelle (1926) had already shown the lytic cycle of his bacteriophage throughout three “bursts” of lysis. Ellis was joined by Max Delbrück, and they began a very careful analysis of the so-called one-step growth curve of the bacteriophage they had isolated. One of the first steps was the determination of the number of phages that infected bacteria during a lytic cycle. As Ellis (2007) described, however, it was necessary to know how many phages were bound to viable bacteria, as opposed to being lost to ineffectual binding to cellular debris. So, taking a cue from d'Herelle, they did multiple infections of aliquots of the same bacterial culture and counted the number of cultures infected (lysed) and not lysed. Delbrück found the Poisson distribution as an expression that described their experimental circumstances, so their data were analyzed by that expression, using the number of nonlysed cultures to calculate the actual mean number of phages per infected cell. From that, they determined that only one phage was necessary to lyse a bacterium (Ellis & Delbrück, 1939).

## Luria-Delbrück & the Fluctuation Test

A phenomenon that was observed from the very early days of d'Herelle's work was the occurrence, on bacterial plates that were infected with high amounts of phages so as to completely lyse the bacteria, of bacterial colonies that were resistant to phage infection. Ultimately, the question became “Are these bacterial mutants that exist prior to phage infection, or is the resistance somehow caused by the infection process itself, in a few of the bacteria?”

Salvador Luria (1984) conceived of an experiment that would distinguish these possibilities. Luria reasoned that if he divided a bacterial culture into multiple aliquots and grew them individually, then, if mutation happened to bacteria without the presence of phages, the process would occur randomly; some of these cultures would have no mutations, but in others mutation might occur soon after growth began, and those mutants would reproduce throughout the incubation of the culture, producing a clone of phage-resistant bacteria. If he then plated these separate cultures along with high numbers of phages, cultures that had no or late-occurring mutations would show, on average, only a few resistant colonies. However, any cultures in which mutation occurred early in the incubation period would show a large number of phage-resistant colonies because the early-forming mutants had time to produce many descendants during the incubation.

If, on the other hand, resistance to phage lysis occurred only by interaction with the phage, those events would be rare and independent, and their distribution among the bacterial plates would follow the Poisson distribution – with no plates showing large numbers of phage-resistant colonies. Luria performed the experiment multiple times, and the distribution showed occasional plates with very large numbers of phage-resistant colonies, the expected result for the hypothesis that bacteria mutate to phage resistance independent of the presence of phages. Delbrück, with whom Luria had been collaborating, formalized the mathematical treatment and published the work, now known as the “fluctuation test” (Luria & Delbrück, 1943). This paper was a watershed in the development of molecular biology because it showed the existence of independent mutation in bacteria (Luria, 2007).

## Applications

I have made use of the Poisson distribution as employed by Ellis-Delbrück by having students plate bacteriophages, using a kit available from Carolina Biological Supply (catalog no. HB-124315). The stock phage solution provided coliphage T2 in a titer of approximately 1 × 109 PFU/mL. To get a more accurate estimate of the phage titer, each student group did a serial dilution of the phage in 0.9% NaCl and plated 1 mL each of dilutions of 10−2, 10−4, 10−6, 10−8, and 10−9 dilutions with one drop of E. coli and 3 mL of 0.9% agar. Out of nine groups, due to student error two groups obtained results that did not follow the serial dilution and were discarded. Of the remaining seven groups, the numbers of plaque-forming units on the 10−8 and 10−9 plates were as shown in Table 1.

Table 1.
T2 bacteriophage counts from serial dilution plating.
Group10−810−9
16
Group10−810−9
16

Thus, 4 of 7 groups had no plaques on the 10−9 plate. Therefore, P(0) = 4/7 = 0.571. The Poisson equation for P(0) is

$P(0)=n0×e−n/0!=e−n$

Thus, ln P(0) = −n.

In this lab exercise, ln P(0) = ln (0.571) = −0.560, and n = 0.560.

Multiplying this value for n times its dilution factor gives an estimated phage titer of 5.6 × 108, a more accurate measurement than the information provided with the sample of “approximately” 1 × 109 PFU/mL.

The Poisson distribution is not, of course, limited to the study of phages. In my biostatistics laboratory course and in my undergraduate research, I have crafted problems and exercises that employ the use of Poisson in one- and two-dimensional space. For example, I have had students process data on the distribution of yeast cells on a hemacytometer grid (indeed, I later found that Gossett – of Student's t-test fame – wrote his first paper on the use of the Poisson distribution in counting blood cells in a hemacytometer; Student, 1907).

Many problems have been and can be written that demonstrate the utility of the Poisson in one- and two-dimensional space as well as in time-dependent situations (Triola, 2007; Doane & Seward, 2010). Given a set of such data, there are three considerations or parameters that affect the outcome of a Poisson, and each can be used as an investigatory tool: (1) the requirement for independence of trial success, (2) the requirement for the number of successes per unit to be rare, and (3) the requirement that the mean number of successes per unit be known. Conformity of a set of data to a Poison distribution is taken to mean that considerations 1 and 2 hold true for these data and that consideration 3 can be determined with confidence.

The class's phage-plaque-count data above show how (the Poisson nature of phage infection having been shown before) the mean can be more accurately determined – the same logic as was used in the one-step growth curve (Ellis & Delbrück, 1939). The fluctuation test shows how a set of data can reveal that independence and rarity do not hold and the biological consequences of that fact. A problem set that demonstrates the same idea follows.

## Problem Set A

A student goes into a small field and sets up a series of transects such that she creates 100 quadrats, each 1 m2 in area, creating a 10 × 10 plot. In random order, she then removes 1 kg of soil from the center of each quadrat and stores and labels the samples in separate containers. When back at the lab, the student counts the number of pill bugs in each sample, and the data set shown in Table 2 is obtained.

1. Calculate and plot the Poisson distribution as a line graph – to plot actual numbers of pill bugs expected, multiply each P(r) value (from the equation, or there are several online Poisson calculators that can be found) times the total (213) for these data. Now plot the actual data as a bar graph. Does the distribution of data closely follow the Poisson equation?

2. What does the answer to that question say about the distribution of the pill bugs in the field? Is it random? Clustered? Uniform?

3. What does that say about the interactions of the pill bugs?

Table 2.
Problem Set A (total number of pill bugs = 213; total number of trials = 100; n = 213/100 = 2.13).
43
10
11
43
10
11

## Problem Set B

A student quickly sets up a linear array of 1 m2 quadrats along a transect on a beach front. He has 100 such quadrats in the array. He spends 2.0 minutes at each quadrat collecting shells and placing them into separate labeled containers. After breaking down the quadrats, he counts the shells in each container and obtains the data set shown in Table 3.

1. Calculate the Poisson distribution for this data set as a line graph, as in Problem Set A. Now plot the actual data as a bar graph.

2. Does the distribution of the data follow the Poisson equation? What do you expect?

3. If the distribution of the data does not follow the Poisson, what might be mitigating factors?

Table 3.
Problem Set B (total number of shells = 213; total number of quadrats = 100; n = 213/100 = 2.13).
11 10
29
26
16
11 10
29
26
16

These problems only assume an assessment of fit by observation of graphical data distribution vs. theoretical curve (a more statistical answer could be obtained by a chi-square comparison of distributions).

These problems suggest exercises by which students can generate data for comparison with the Poisson distribution. Others include adding a small number of yeast to a hemacytometer and counting the number of yeast cells per square, counting absences per classroom throughout a month, or counting dandelions in a field per unit area.

A real-life example of how the Poisson distribution was used to actually calculate a probability, given that all relevant conditions were satisfied, is in the statistical treatment of the Woburn, Massachusetts, leukemia case (Cutler et al., 1986; De Veaux et al., 2006). What appeared to parents to be an unusually high frequency of leukemia cases resulted in an extended trial against a chemical company, a book titled A Civil Action, and a movie of the same name. Online Poisson calculations using Excel are available (Letkowski, 2012).

I believe that these examples of the incorporation of mathematics from the history of biology, as well as from approaches in medicine and wildlife biology today, provide a nonthreatening way to increase mathematical exposure and illuminate some of the key experiments from the early evolution of molecular biology.

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