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-
Systems with inheritance: dynamics of distributions with conservation of support, natural selection and finite-dimensional asymptotics
If we find a representation of an infinite-dimensional dynamical system as a nonlinear kinetic system with {\it conservation of supports} of distributions, then (after some additional technical steps) we can state that the asymptotics is finite-dimensional. This conservation of support has a {\it quasi-biological interpretation, inheritance} (if a gene was not presented initially in a isolated population without mutations, then it cannot appear at later time). These quasi-biological models can describe various physical, chemical, and, of course, biological
systems. The finite-dimensional asymptotic demonstrates effects of {\it ``natural" selection}. The estimations of asymptotic dimension are presented. The support of an individual limit distribution is almost always small. But the union of such supports can be the whole space even for one solution. Possible are such situations: a solution is a finite set of narrow peaks getting in time more and more narrow, moving slower and slower. It is possible that these peaks do not tend to fixed positions, rather they continue moving, and the path covered tends to infinity at $t \rightarrow \infty$. The {\it drift equations} for peaks motion are obtained. Various types of stability are studied.
In example, models of cell division self-synchronization are
studied. The appropriate construction of notion of typicalness in infinite-dimensional spaces is discussed, and the ``completely thin" sets are introduced.
Phenomenology and homeopathy.
There is a great overlap between the way of seeing the world in clinical homeopathy and in the technical philosophical system known as phenomenology. A knowledge of phenomenologic principles reveals Hahnemann to have been an unwitting phenomenologist. The ideas of phenomenology as applied to medicine show that homeopathy is the ideal medical system to fulfill the goals of coming ever closer to true patient concerns and experience of illness.