A gel as an array of channels

We consider the theory of charged point molecules ('probes') being pulled by an electric field through a two-dimensional net of channels that represents a piece of gel. Associated with the position in the net is a free energy of interaction between the probe and the net; this free energy fluctuates randomly with the position of the probe in the net. The free energy is intended to represent weak interactions between the probe and the gel, such as entropy associated with the restriction of the freedom of motion of the probe by the gel, or electrostatic interactions between the probe and charges fixed to the gel. The free energy can be thought of as a surface with the appearance of a rough, hilly landscape spread over the net; the roughness is measured by the standard deviation of the free-energy distribution. Two variations of the model are examined: (1) the net is assumed to have all channels open, or (2) only channels parallel to the electric field are open and all the cross-connecting channels are closed. Model (1) is more realistic but presents a two-dimensional mathematical problem which can only be solved by slow iteration methods, while model (2) is less realistic but presents a one-dimensional problem that can be reduced to simple quadratures and is easy to solve by numerical integration. In both models the mobility of the probe decreases as the roughness parameter is increased, but the effect is larger in the less realistic model (2) if the same free-energy surface is used in both. The mobility in model (2) is reduced both by high points in the rough surface ('bumps') and by low points ('traps'), while in model (1) only the traps are effective, since the probes can flow around the bumps through the cross channels. The mobility in model (2) can be made to agree with model (1) simply by cutting off the bumps of the surface. Thus the simple model (2) can be used in place of the more realistic model (1) that is more difficult to compute.