Numerical Solution of a Nonlinear Diffusion Model with Memory

Finite difference approximation of a nonlinear integro-differential equation associated with the penetration of a magnetic field into a substance is studied. Here we discuss the model described by a nonlinear integro-differential equation. The system of time dependent ordinary differential equations is solved using Runge-Kutta method with adaptive step size. The time integral make this a non-trivial application of the Matlab code ODE45. Eight examples are given with mostly homogeneous boundary conditions. The results show that when the analytic solution is not growing in time, then the solution decays at a rate proven theoretically in the literature.Approved for public release; distribution is unlimited

Clusterin, a gene enriched in intestinal stem cells, is required for L1-mediated colon cancer metastasis

Hyperactive Wnt signaling is a common feature in human colorectal cancer (CRC) cells. A central question is the identification and role of Wnt/β-catenin target genes in CRC and their relationship to genes enriched in colonic stem cells, since Lgr5+ intestinal stem cells were suggested to be the cell of CRC origin. Previously, we identified the neural immunoglobulin-like adhesion receptor L1 as a Wnt/β-catenin target gene localized in cells at the invasive front of CRC tissue and showed that L1 expression in CRC cells confers enhanced motility and liver metastasis. Here, we identified the clusterin (CLU) gene that is also enriched in Lgr5+ intestinal stem cells, as a gene induced during L1-mediated CRC metastasis. The increase in CLU levels by L1 in CRC cells resulted from transactivation of CLU by STAT-1. CLU overexpression in CRC cells enhanced their motility and the reduction in CLU levels in L1 overexpressing cells suppressed the ability of L1 to confer increased tumorigenesis and liver metastasis. Genes induced during L1-mediated CRC cell metastasis and enriched in intestinal stem cells might be important for both CRC progression and colonic epithelium homeostasis