Mendelian Dynamics and Sturtevant ’ s Paradigm
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 Z2 of the imaginary outcomes (binary nsequences) equals the (normalized) Haar measure that is, moreover, invariant under the permutation group Sn. 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-
Understanding bistability in complex enzyme-driven reaction networks.
Much attention has been paid recently to bistability and switch-like behavior that might be resident in important biochemical reaction networks. There is, in fact, a great deal of subtlety in the relationship between the structure of a reaction network and its capacity to engender bistability. In common physicochemical settings, large classes of extremely complex networks, taken with mass action kinetics, cannot give rise to bistability no matter what values the rate constants take. On the other hand, bistable behavior can be induced in those same settings by certain very simple and classical mass action mechanisms for enzyme catalysis of a single overall reaction. We present a theorem that distinguishes between those mass action networks that might support bistable behavior and those that cannot. Moreover, we indicate how switch-like behavior results from a well-studied mechanism for the action of human dihydrofolate reductase, an important anti-cancer target.
Algebras in genetics.
In this section we provide a short introduction to “algebras in genetics”. We explain how algebras arise in population genetics and we construct some examples of gametic algebras. These examples will be reconsidered and discussed in a wider frame-work in section 7. We introduce zygotic and copular algebras and we discuss the construction of these algebras from gametic algebras. The exposition will reveal various elementary though fundamental properties of these algebras.