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Mathematics
[PAST EVENT] Mathematics Colloquium and EXTREEMS-QED Lecture: Greg Smith (William & Mary)
October 23, 2015
2pm - 3pm
Abstract: Terrell Leslie Hill is well-known among physical chemists and biophysicists for his work on the thermodynamics of small systems, free energy transduction, and biochemical cycle kinetics. Textbooks emphasize Hill's elegant description of how the hydrolysis of ATP produces mechanical work (via biased transitions through states of differing free energy).
While developing the "sliding filament model" into a quantitative description of muscle action, Hill conceived of a graphical formalism for representing fluxes in unimolecular systems. The formalism is an application of the Markov Chain Tree Theorem, a classical mathematical result that expresses the steady-state probability distribution of an irreducible Markov chain in terms of directed spanning trees of its associated state-transition graph. Hill's diagrammatic method is readily applied to models of ion channel gating and ligand-receptor binding.
To efficiently apply Hill's diagrammatic method to clusters of interacting ligand-gated ion channels, it has been necessary to characterize the structural properties of "reduced graph powers" - denoted G^(N) - that are the transition graphs for the master Markov chain for N identical (but not independent) M-state automata with transition graph G of size |G|=M.
Richard Hammack (Virginia Commonwealth University) and I have obtained results related to the structural properties of reduced graph powers. Most significantly, we provide a construction of minimum cycle bases of G^(N). The minimal cycle basis construction would appear to be prerequisite to the application of Hill's diagrammatic method to the analysis of non-equilibrium steady-states of clusters of interacting ligand-gated ion channels. For example, the construction provides conditions that ensure against violations of microscopic reversibility. It is also an interesting combinatorial problem in its own right.
While developing the "sliding filament model" into a quantitative description of muscle action, Hill conceived of a graphical formalism for representing fluxes in unimolecular systems. The formalism is an application of the Markov Chain Tree Theorem, a classical mathematical result that expresses the steady-state probability distribution of an irreducible Markov chain in terms of directed spanning trees of its associated state-transition graph. Hill's diagrammatic method is readily applied to models of ion channel gating and ligand-receptor binding.
To efficiently apply Hill's diagrammatic method to clusters of interacting ligand-gated ion channels, it has been necessary to characterize the structural properties of "reduced graph powers" - denoted G^(N) - that are the transition graphs for the master Markov chain for N identical (but not independent) M-state automata with transition graph G of size |G|=M.
Richard Hammack (Virginia Commonwealth University) and I have obtained results related to the structural properties of reduced graph powers. Most significantly, we provide a construction of minimum cycle bases of G^(N). The minimal cycle basis construction would appear to be prerequisite to the application of Hill's diagrammatic method to the analysis of non-equilibrium steady-states of clusters of interacting ligand-gated ion channels. For example, the construction provides conditions that ensure against violations of microscopic reversibility. It is also an interesting combinatorial problem in its own right.