In this paper, we consider a three-dimensional model of cell signal transduction. In this model, the deactivation of signalling proteins occur throughout the cytosol and activation is localized to specific sites in the cell. We use matched asymptotic expansions to construct the dynamic solutions of signalling protein concentrations. The result of the asymptotic analysis is a system of ordinary differential equations. This reduced system is compared to numerical simulations of the full three-dimensional system. As well, we consider the stability of equilibrium solutions. We find that the systems under consideration may undergo sustained oscillations, hysteresis and other complex behaviors. The simulations of the full three-dimensional system agree with simulations of the reduced ordinary differential equations.

Intracellular signalling molecules form pathways inside the cell. These pathways carry a signal to target proteins which results in cellular responses. We consider a spherical cell with two internal compartments containing localized activating enzymes where as deactivating enzymes are spread uniformly through out the cytosol. Two diffusible signalling molecules are activated at the compartments and later deactivated in the cytosol due to deactivating enzymes. The two signalling molecules are a single link in a cascade reaction and form a self regulated dynamical system involving positive and negative feedback. Using matched asymptotic expansions we obtain approximate solutions of the steady state diffusion equation with a linear decay rate. We obtain three-dimensional concentration profiles for the signalling molecules. We also investigate an extension of the above system which has multiple cascade reactions occurring between multiple signalling molecules. Numerically, we show that the speed of the signal is an increasing function of the number of links in the cascade.

In this paper, we consider a three-dimensional model of cell signal transduction with delay. The deactivation of signalling proteins occurs throughout the cytosol and activation is localized to specific sites in the cell. The enzyme kinetic functions employ a constant delay to model the time lapse during reactions and also the recovery times associated with conformational changes. We use matched asymptotic expansions to construct the dynamic solutions of signalling protein concentrations. The result of the asymptotic analysis is a system of delayed differential algebraic equations. This reduced system is compared to numerical simulations of the full three-dimensional system. As well, we consider the stability of equilibrium solutions. We find that the systems under consideration may undergo Hopf bifurcations for certain delay values. In these cases sustained oscillations are observed. The Poincaré-Lindstedt3 method is used to improve upon the asymptotic approximations. The simulations of the full three-dimensional system correspond well with simulations of the reduced delayed differential algebraic equations.

Plasmacytoid urothelial carcinoma (PUC) is a rare variant of bladder cancer characterized by distinct histopathology and advanced stage at diagnosis. Multimodal treatment is usually indicated. We present a case of PUC causing bilateral ureteral obstruction with subsequent renal failure followed shortly by malignant small bowel obstruction, demonstrating the need for a high degree of clinical suspicion in diagnosis of this aggressive subtype. Moreover, the local invasiveness of the disease cannot be understated, given that it can rapidly spread with little radiologic evidence of progression until it is at an advanced stage.

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