Mendelian dynamics and Sturtevant’s paradigm

Abstract

This a brief introduction to the formal genetics centered around two mathematical ideas going back to Gregor Mendel and Alfred Sturtevant.Preambule. The road from biology (or from any branch of science on the fundamental level) to mathematics goes in several ( often Brownian rather than straight) paths in parallel.• Identifying a class of phenomena-"particular trees in a forest"-that appear with a regularity suggesting an underlying (mathematical) structure. (Are there non-mathematical structures?)•• Designing and performing experiments/observations purifying and amplifying what is seen by the naked eye (e.g. by planting our "trees" to an "artificial soil").••• Making (often implicitly) ad hoc hypotheses, (e.g. continuity, symmetry, functoriality)-that provide a logical framework for the experimental data.For example, the "theory of coin tossing" derives its mathematical beauty and the (probabilistic) predictive power not from such "definitions" as "the probability is a measure of uncertainty" but from the assumption that the the probability distribution on the space Z n 2 of the imaginary outcomes (binary nsequences) equals the (normalized) Haar measure that is, moreover, invariant under the permutation group S n . Such hypotheses (assumptions), fragments of the grammer of the language in which Nature delievers her messages, are what a mathematician is primerly interested in, while a scientist is concerned with the "meaning" of a message-the structure that is harder to formilize.Desiphering the grammer of Nature, or, biologically speaking, guessing the design of a seed by looking at (sample branches of) the grown tree, is rarely (if ever) done by mathematicians (Newtons do not count), even when the experimental data are abundant (as in the present day molecular biology). The past mathematical experience channals your imagination toward the old rather than new mathematical concepts.Even rigorously reformulating such hypotheses is not a straighforward task. For example, the Dirac δ-function needed the theory of distributions to be accepted by mathematicians and the functoriality of the (derivation of the) Boltzmann equation was recognized (albeit not much exploited till now) only with the advent of the "functoriality paradigm" within pure mathematics.When a "seed" is cultivated in a "mathematical soil", what grows out of it-mathematician's tree-might look not quite as the real one. But a math-1

Crystals, proteins, stability and isoperimetry

“…(More recent and more relevant examples are seen in the slow start in the mathematical development of Mendelian genetics and in the failure of identifying general mathematical principles underlying Sturtevant's reconstruction of the linear structure on the set of genes on a chromosome of Drosophila melanogaster from samples of a probability measure on the space of gene linkages [13].) The brain is hardly able to generate mathematical concepts by itself; it needs an input of "raw structures", and Nature has much to offer.…”