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-
Crystalls , Proteins and Isoperimetry
We attempt to formulate several mathematical problems suggested by structural patterns present in bio-molecular assemblies. Our description of these assemblies, by necessity brief, is self-contained, albeit on a superficial level. If a reader stumbles upon something confusing, this, hopefully, will become clearer at a later point in the article. 1 Is there Mathematics in Biology? Day-Night, Day-Night, Day-Night. Summer-Winter, Summer-Winter... . What is the structure behind this. Where should we look to uncover it? Is it in the singing of birds, in the flows of rivers, in the changes of temperature? We know, with hindsight, what we should do. Look up into the night sky, find a few specks of light – planets – that slowly crawl amongst the unmoving stars, imagine how their routes appear if seen from the sun, invent calculus, guess differential equations for the planetary routes, unravel symmetries of these.... and the world of mathematical wonders opens to you: Lie groups, algebraic varieties, symplectic manifolds... . Now try: Alive-Dead, Alive-Dead, Alive-Dead,... . Where do we go from here? ”... living matter, while not eluding the ”laws of physics” as established up to date, is likely to involve ”other laws of physics” hitherto unknown, which
Minimally disturbed, multicycle, and reproducible synchrony using a eukaryotic "baby machine".
A eukaryotic "baby machine" has been developed that produces synchronized cultures that display up to four synchronous cell cycles. That such cells can be produced implies that methods unable to produce successive synchronized cell cycles may not actually synchronize cells. But most important, the baby machine method now opens the way for the study of the cell cycle of minimally disturbed, artifact-free, well-synchronized, mammalian cells.