A Mathematical Model for Mycolactone Toxin Reaction and Diffusion in Cell Cytoplasm

Abstract

The study of mechanisms and transport of proteins inside cells provide understanding of the dynamics of biological systems and give clues to biological effect of drugs on our system. Reaction-diffusion mechanisms inside cells control various cell functions from adhesion, haptotaxis, chemotaxis to cytoskeletal rearrangement. The theoretical model presented in this work aims to model activation of proteins in a biological cell. In our previous paper on mathematical model for mycobacterium invasion of tissues, (Nyarko et. al. 2017), we reported that mycolactone activation of theWASP in the cytoplasm of cells leads to tissue necrosis in the Buruli ulcer disease. We present two approaches to model reaction-diffusion in the cytoplasm: first the system of reaction-diffusion arising from mycolactone and WASP interactions are solved in a 3D domain with mycolactone toxin allowed to diffuse into the cytoplasm through the cell membrane to bind the WASP whiles defining a flux boundary condition on the cell membrane for the mycolactone, and restricting the WASP to the cytoplasm compartment. In the second model, we use the idea of periodicity to develop a topological setup for a layer of tissue that constitutes aggregation of similar cells. The model is solved using the coefficient form of PDE in Comsol Multiphysics. In the numerical simulation we compare the performance of three sparse direct solvers (UMFPACK, SPOOLES and PARDISO) on a 64-bit windows HP-Z1 workstation machine with 16GB RAM. The numerical results show the concentration distribution of mycolactone to be maximum at the center of the cytoplasm. The study shows that, the PARDISO solver performs better than the UMFPACK and SPOOLES solvers in terms of memory requirements and computational time on a fine mesh. The model is a contribution to the understanding of drug diffusion in the cell and most importantly to the understanding of the etiology of Buruli ulcer disease.