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1. Introduction. The purpose of the present paper is to study blowup mechanism for a system of parabolic equations. It arises in mathematical biology to describe the chemotactic feature of slime molds.We take the form proposed by Nanjundiah [20], simplifying the one previously given by Keller and Segel [14]. It is stated as follows, where u = u (x, t) and v = v(x, t) stand the density of slime molds and the concentration chemical substances secreted by them, respectively:

Lipid rafts have attracted much attention because of their significant functional roles in membrane-associated processes. It is thought that sphingomyelin and cholesterol are essential for forming lipid rafts; however, their motion characteristics are not fully understood despite numerous studies. Here we show accurate local motions encompassing an entire sphingomyelin molecule, which were captured by measuring quadrupole splittings for 19 kinds of site-specifically deuterated sphingomyelins (that is, molecular motion capture of sphingomyelin). The quadrupole splitting profiles, which are distinct from those reported from perdeuterated sphingomyelins or simulation studies, reveal that cholesterol enhances the order in the middle parts of the alkyl chains more efficaciously than at the shallow positions. Comparison with dimyristoylphosphocholine bilayers suggests that cholesterol is deeper in sphingomyelin bilayers, which likely explains the so-called umbrella effect. The experiments also demonstrate that (i) the C2'-C3' bond predominantly takes the gauche conformation, (ii) the net ordering effect of cholesterol in sphingomyelin bilayers is not larger than that in phosphatidylcholine bilayers, (iii) cholesterol has no specific preference for the acyl or sphingosine chain, (iv) the acyl and sphingosine chains seem mismatched by about two methylene lengths, and (v) the motion of the upper regions of sphingomyelin chains is less temperature dependent than that of lower regions probably due to intermolecular hydrogen bond formation among SM molecules. These insights into the atomic-level dynamics of sphingomyelin provide critical clues to understanding the mechanism of raft formation.

MT1-MMP is a potent invasion-promoting membrane protease employed by aggressive cancer cells. MT1-MMP localizes preferentially at membrane protrusions called invadopodia where it plays a central role in degradation of the surrounding extracellular matrix (ECM). Previous reports suggested a role for a continuous supply of MT1-MMP in ECM degradation. However, the turnover rate of MT1-MMP and the extent to which the turnover contributes to the ECM degradation at invadopodia have not been clarified. To approach this problem, we first performed FRAP (Fluorescence Recovery after Photobleaching) experiments with fluorescence-tagged MT1-MMP focusing on a single invadopodium and found very rapid recovery in FRAP signals, approximated by double-exponential plots with time constants of 26 s and 259 s. The recovery depended primarily on vesicle transport, but negligibly on lateral diffusion. Next we constructed a computational model employing the observed kinetics of the FRAP experiments. The simulations successfully reproduced our FRAP experiments. Next we inhibited the vesicle transport both experimentally, and in simulation. Addition of drugs inhibiting vesicle transport blocked ECM degradation experimentally, and the simulation showed no appreciable ECM degradation under conditions inhibiting vesicle transport. In addition, the degree of the reduction in ECM degradation depended on the degree of the reduction in the MT1-MMP turnover. Thus, our experiments and simulations have established the role of the rapid turnover of MT1-MMP in ECM degradation at invadopodia. Furthermore, our simulations suggested synergetic contributions of proteolytic activity and the MT1-MMP turnover to ECM degradation because there was a nonlinear and marked reduction in ECM degradation if both factors were reduced simultaneously. Thus our computational model provides a new in silico tool to design and evaluate intervention strategies in cancer cell invasion.