Ionic Interactions in Biological and Physical Systems: a Variational Treatment

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All biology occurs in ionic mixtures loosely called Ringer solutions. Pure water is lethal to cells and most proteins. Biology depends on interactions of ions. Interactions of Na+, K+, and Ca2+ with channel proteins produce electrical signals of nerves and coordinate muscle contraction including the heart. Proteins, channels, and nucleic acids concentrate small ions to number densities greater than 10 M because their active sites have large densities of acids and bases with permanent charge. Interactions dominate such concentrated solutions but biochemical and biophysical theories rarely include ion-ion interactions. Classical theories use rate constants independent of interactions even in highly concentrated solutions. Enzymes and transporters are analyzed with the theory of ideal uncharged gases, without physical interactions between reactants. In classical theories, the concentration of one reactant does not change the free energy of another. In experiments, classical and modern, the concentration of one reactant does change the free energy of another. In classical theories, interactions between ions must appear as interactions between ion and protein because ion-ion interactions do not exist. Classical theories invoke conformation changes of proteins or complex schemes of chemical reactions when models (without ion-ion interactions) fail to fit experiment.

Ion-ion interactions have been ignored (in my view) because no one knew how to deal with them. Variational methods that allow interactions to be analyzed in conservative systems have not been available for dissipative systems like ionic solutions. These mathematical problems are now resolved in the theory of complex fluids, electro-rheology. An Energetic Variational Approach to dissipative systems has been developed by Chun Liu, more than anyone else. Existence and uniqueness have been proven and Navier Stokes equations have been derived. If a component is added to a variational model, the resulting Euler Lagrange differential equations automatically describe new interactions with minimal new parameters. Thus, variational methods are quite specific when confronted with new ions in solution, or additional forms of transport, like convection or heat flow, along with the usual diffusion and electrical migration.

A variational ‘primitive’ model of finite size ions in ion channel proteins has been successfully constructed and more atomic detail can be added as needed. Numerical inefficiencies are being removed in a variety of ways, but variational methods have not yet been applied to bulk solutions to predict the consequences of finite ion size in a range of experiments and conditions.


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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Researchers should cite this work as follows:

  • Bob Eisenberg (2014), "Ionic Interactions in Biological and Physical Systems: a Variational Treatment,"

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WTHR 201, Purdue University, West Lafayette, IN