Numerical computation for free boundary of ligand and signal transduction associated with the invadopodia formation

Abstract

Cancer cell invasion in the metastasis process contributes to the high death cases among cancer patients. The spread of tumors from one part to another in the body is a result of the existence of finger-like protrusions or known as invadopodia. The formation of invadopodia involves several molecular processes that include the activity of matrix metalloproteinases (MMPs) in degrading the extracellular matrix (ECM), the creation of ligand, stimulation of signal transduction from the binding of ligand with epidermal growth factor receptor, up-regulation of MMPs, and actin polymerization. The purpose of this study is to investigate the emergence of invadopodia on the plasma membrane through the mathematical model of quasistatic and unsteady cases involving ligand-protein and signal transduction processes. The degradation of the ECM by the MMPs is the starting point for the occurrence of invadopodia formation where the density of MMPs is taken as a trigonometric function. The creation of invadopodia is a result of actin polymerization activity that moves the plasma membrane. Hence, the movement is assumed as the membrane velocity and is accounted for as without and with jump velocity approaches where the jump from ligand to signal occurs. The method of level set is emphasized to detect the movement of the free boundary plasma membrane and is considered as a zero level set function. In addition, the location of the plasma membrane leads to the occurrence of regular points (a point that is far from the interface) and neighboring points (a point that is near to the interface). These points are solved using the second-order centered finite difference method and ghost fluid with the linear extrapolation method. The results showed that the mentioned integrated methods effectively describe the movement of the free boundary plasma membrane and this directly points out the formation of protrusions (invadopodia) on the plasma membrane. Furthermore, the size of the protrusions is observed to become longer as time increases. However, the aggressive (longer) protrusion is detected in the quasi-static model, whereas only small protrusions are spotted in the unsteady model. It is also observed that the disconnection of the plasma membrane happened in the quasi-static model and without jump velocity approach compared to the other problems. Nevertheless, in all problems conducted, the density of ligand and signal is the highest on the interface due to the stimulation of signal through the binding between ligand and membrane-associated receptor that is happening here. Besides, the numerical errors are compared for the three sizes of meshes. The simulation results demonstrated that for all profiles of level set, ligand, and signal, the higher size of meshes provides a smaller value of error compared to the lower size of meshes.