“Wave” asymptotic solution of isoelectric focusing problem

The article is concerned with asymptotic analysis of Isoelectric Focusing (IEF) mathematical model in the so-called “anomalous” regimen. The integro-differential problem is a mathematical description of the IEF problem. It consists of N differential equations, algebraic equations, and N integral equations. Solutions of the problem are analytical concentrations of ampholytes. Its graphics (so-called profiles of ampholytes) in trivial regimen have the form of the Gaussian curves. In “anomalous” regimen, under high current densities of J, a breakdown of the Gaussian distribution takes place. The “plates’’ on the tops of the graphics appear and widen with the increase of J. As long as Gaussian distribution is one of the most important laws of mathematical physics, its breakdown constitutes an essential scientific interest. Is the Gaussian distribution a partial case of a more common distribution? What formula is expressed in its distribution? In the present article the asymptotic solution of the IEF problem is obtained. It establishes the form of the non-Gaussian distribution by concentrations in the ''anomalous’’ regimen. The application of the saddle-point method to the integro-differential problem allows provision of its solution in the form of exponent with the power (“wave”) series in the exponent. By means of asymptotic methods it is established that for each value of J (current density) only a finite number has essential contribution to the sum of the series. Remaining terms are negligible as infinitesimal terms. For the uniform distribution of ampholytes the solution is an exponential function with finite number by even degrees (x-xk)2l in exponential. The magnitude l is increased under magnification of the current density J. It is known, that “the plate” on the top of the graphics by function exp(-x2k) widens with high values of l in exponential. The developed analysis explains the behavior of profiles in “anomalous” regimen. It indicates the formula, which is a generalization of Gaussian distribution for integro-differential IEF problem.