Sensitivity of the non-linear dynamics of lysozyme “Liesegang rings” to small asymmetries

This paper deals with the analysis of the sensitivity of the non-linear dynamics of the crystallization process of lysozyme and related “Liesegang rings” phenomena to small asymmetries that may characterize the geometry and/or the boundary conditions of the system under investigation. Mathematical models and appropriate numerical methods are introduced to handle the complex phenomena related to protein nucleation and further precipitation (or resolution) according to the concentration distribution. The configuration under investigation consists of a protein chamber and a salt chamber separated by an “interface”. The interface is strictly related to the presence of agarose gel in the protein chamber. Different models of the interface are considered. For the first group of simulations the deformation of the interface due to surface tension effects is neglected. For the second group of simulations this deformation is taken into account. The distribution of salt at the initial time is supposed to follow the shape of the gel meniscus whose interface cannot be horizontal due to surface tension effects. The shape is modeled using a sin function in order to have a minimum protruding in the protein chamber at the mean point along the horizontal length of the chamber. For the last group of numerical computations the gel meniscus is supposed to be not symmetrical with respect to this point in order to simulate small experimental imperfections. The numerical simulations show that neglecting the interface deformation leads to 1D results. The phenomenon is characterized by a certain degree of periodicity in time and along the vertical dimension (Liesegang patterns). The bands of Liesegang patterns are not spatially uniform. New solid particles are created on the lower boundary of depleted bands if the local concentration of salt reaches a value to let the local protein concentration overcome the “supersaturation limit”. The numerical simulations show that the space distance between two consecutive layers tends to decrease towards the bottom of the protein chamber. If the shape of the meniscus is taken into account, lysozyme reacts to produce crystals distributed along both the horizontal and vertical directions. The effect of a non-planar boundary seems to be very important even if the deviation from the planar condition is very small. The third group of simulations finally shows that even if the asymmetry introduced in the shape of the meniscus is very small, the effect on the crystal pattern may be appreciable.