Biophysics applications of Seiberg-Witten theory
Ion channels of cell membranes is a perfect example of a 1D Coulomb gas. Statistical mechanics of the latter may be mapped onto certain non-Hermitian quantum mechanics. We use this example to develop instanton calculus on Riemann surfaces of constant energy. Borrowing from the Seiberg-Witten analysis, we show that the actions along principal cycles of such surfaces obey ODE in the moduli space, known as Picard-Fuchs equation. We derive and solve Picard-Fuchs equations for Coulomb gases of various charge content. Analysis of monodromies of these solutions yields semiclassical spectra as well as instanton effects. Both are shown to be in an excellent agreement with numerical simulations.