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-
Crystals, proteins, stability and isoperimetry
We attempt to formulate several mathematical problems suggested by structural patterns present in biomolecular assemblies. Our description of these patterns, by necessity brief and over-concentrated in some places, is self-contained, albeit on a superficial level. An attentive reader is likely to stumble upon a cryptic line here and there; however, things will become more transparent at a second reading and/or at a later point in the article.
The incidence of dead and dying fish in the Moray Firth, September 1963
Dead and dying fish were reported from the Moray Firth during a period of 15 days in September 1963. The initial mortalities were in the inner Firth, but they later spread outwards, especially eastwards along the southern shores of the Firth. Mass mortalities of the lugworm (Arenicola marina) also occurred in Scottish waters in 1962 and 1963, and at Balintorein Easter Ross fish and lugworm mortalities were contemporary.