Mathematics of Ions in Channels and Solutions: Stochastic Derivations, Direct, Variational and Inverse Solutions that fit Data
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Abstract
Ion channels have a role in biology like the channels of field effect transistors in computers: both are valves for electricity controlling nearly everything. Ion channels are proteins with a hole down their middle that catalyze the movement of sodium, potassium, calcium and chloride ions across the otherwise impermeable membranes that define cells. Once a channel opens, it has a single structure on the biological time scale slower than say 2 microseconds. The ions present around every cell and molecule in biology are hard spheres and so the calculation of how hard spheres go through a channel of one structure is a central problem in a wide range of biology. Literally thousands of biologists study the properties of channels in experiments every day. My collaborators and I have shown how the relevant equations can be derived (almost) from stochastic differential equations, and how they can be solved in inverse, variational, and direct problems using models that describe a wide range of biological situations with only a handful of parameters that do not change even when concentrations change by a factor of 10^7. Variational methods hold particular promise as a way to solve problems outstanding for more than a century because they describe interactions of 'everything with everything' else that characterise ions crowded into channels.
Bio
Bob Eisenberg is interested in studying ion channels as physical objects, trying to use the tools of physics, chemistry, engineering, and applied mathematics to understand how they work. Ion channels are proteins with a hole down their middle that are the gatekeepers for cells. Ion channels control an enormous range of biological function in health and disease. But ion channels have simple enough structure that they can be analyzed with the usual tools of physical science. With that analysis in hand, Bob and John Tang, with gifted collaborators, are trying to design practical machines that use ion channels.
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Department of Mathematics Computational and Applied Mathematics Seminar
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REC 316, Purdue University, West Lafayette, IN