Although systems thinking and quantitative methods are distinct themes with their own separate histories in biology, the latter is often essential for the former. Some biologists are fond of emphasizing the descriptive, non-quantitative nature of their subject and the blissful absence, and, indeed, uselessness, of "theory". The great microscopist, Santiago Ramon y Cajal, who first clarified the nature of nerve cells and their interconnections, wrote a guide for the young scientist1 with a chapter entitled "Diseases of The Will: Contemplators; Bibliophiles and Polyglots; Megalomaniacs; Instrument addicts; Misfits; Theorists". One can only wonder what he would have said of Hodgkin & Huxley. Many students seem drawn to biology as much to get away from mathematics as for any intrinsic love of taxonomic complexity. (The converse is also true!) These attitudes are suprising because the history of biology reveals some of the most striking examples of how quantitative reasoning helps us to understand the world around us.
Gregor Mendel is the first case in point. Many biologists of his time undertook experiments in crossing plants and other organisms—Charles Darwin, for one—but Mendel did something nobody else did: he counted and sought an explanation for the patterns that emerged among the numbers2. This allowed him to predict the existence of entities, subsequently called genes, whose chemical identity would not be worked out for another hundred years.
If Mendel's calculations were trivial in comparison to, say, Paul Dirac's prediction of the positron, the scope of his achievement was at least as remarkable. In terms of bang-for-the-buck, Mendel is in a class of his own. Biologists, even those like Thomas Hunt Morgan who were initially suspicious of Mendel's rediscovered "theory", quickly came to appreciate its implications. Morgan associated genes with the aniline-dye staining structures called chromosomes in the nuclei of cells but it was his brilliant student, Alfred Sturtevant, who really grasped the power of Mendelian calculations. Sturtevant produced the first genetic map of a chromosome3 while still a graduate student.
What Mendel did for genetics, Luria & Delbrück did for bacterial genetics4. They founded the subject by showing that bacterial resistance to infection by phages arises spontaneously in bacteria, implying the existence of genes that can be mutated, rather than being induced by contact between virus and bacterium.
Imagine several vats of bacterial culture, growing under identical conditions. An aliquot is taken from each vat and plated onto a viral lawn. Resistant bacteria give rise to colonies on the plate. If resistance arises by induction after contact, then the number of colonies should follow a Poisson distribution. Hence, the mean number of colonies should equal the variance. If, however, resistance arises from spontaneous mutation of bacteria in the vats prior to coming into contact with virus, then the number of initial occurrences in a vat would follow a Poisson distribution but each occurrence would give rise to a clone of resistant bacteria within the corresponding vat. An aliquot from the vat would then have resistant bacteria whose number would depend on how long the clonal population(s) had been growing. This distribution is not easy to calculate; Delbrück (presumably) spent much of the paper4 doing it. The essential point is that the distribution is no longer Poisson but something like a large multiple of a Poisson distribution. (The distribution itself is needed if you want to estimate the mutation rate5, not just infer the existence of mutations.) Accordingly, the variance of the distribution should now be much larger than its mean. A straightforward calculation of the mean and variance of the number of resistant colonies will therefore distinguish induction from mutation. A simple idea but one with profound consequences for the development of molecular biology.
As the examples of Mendel and Luria & Delbrück suggest, genetics has always been particularly quantitative. It was the mathematical analyses of Haldane, Fisher and Wright that first showed how genetic mutations could become fixed in a population and thereby provide a basis for the heritable variation on which Darwinian natural selection could act. Their unification of Darwin and Mendel laid one of the foundation stones of modern biology. A more recent example of quantitative reasoning having a deep imact on biological thinking is Alfred Knudson's demonstration of the "two-hit hypothesis" in cancer genetics6.
Mendel studied mathematics and physics at the University of Vienna and subsequently taught physics at the Realschule in Brunn. Delbrück was a theoretical physicist and one of the founders of the "phage school". Luria, in contrast, is reported to have said7 "If you have to use statistics to prove something, it isn't worth knowing", which must have made the drafting of his most famous paper something of a trial.
1Santiago Ramon y Cajal, Advice for a Young Investigator, MIT Press, 2004.
2Gregor Mendel, "Versuche über Pflanzen-Hybriden (Experiments in plant hybridization)", Verhandlungen des Naturforschenden Vereins (Proceedings Natural History Society), Brünn, 1866. HTML
3 Eric Kandel, "Thomas Hunt Morgan at Columbia University: genes, chromosomes, and the origins of modern biology", Columbia University Alumni Magazine. HTML. Alfred H Sturtevant, "The linear arrangement of six sex-linked factors in Drosophila, as shown by their mode of association", J Exp Zoo 14:43-59 1913. PDF
4 Salvador Luria and Max Delbrück, "Mutations of bacteria from virus sensitivity to virus resistance", Genetics 28:491-511 1943.
5W S Kendall and P Frost, "Pitfalls and perils of Luria-Delbrück fluctuation analysis", Cancer Research 48:1060-5 1988. PubMed. A Dewanji et al, "A generalized Luria-Delbruck model", Mathematical Biosciences 197:140-52 2005. PubMed
6Alfred G Knudson, "Mutation and cancer: statistical study of retinoblastoma", PNAS 68:820-3 1971. PubMed
7David Nanney, "Metaphor and mechanism:'Epigenetic Control Systems' reconsidered", Symposium on The Epigenetics of Cell Transformation and Tumor Development American Association for Cancer Research, 8th Annual Meeting San Francisco, California, May 26, 1989. HTML