Density-Profile Processes Describing Biological Signaling Networks: Almost Sure Convergence to Deterministic Trajectories

We introduce jump processes in R(k), called density-profile processes, to model biological signaling networks. Our modeling setup describes the macroscopic evolution of a finite-size spin-flip model with k types of spins with arbitrary number of internal states interacting through a non-reversible stochastic dynamics. We are mostly interested on the multi-dimensional empirical-magnetization vector in the thermodynamic limit, and prove that, within arbitrary finite time-intervals, its path converges almost surely to a deterministic trajectory determined by a first-order (non-linear) differential equation with explicit bounds on the distance between the stochastic and deterministic trajectories. As parameters of the spin-flip dynamics change, the associated dynamical system may go through bifurcations, associated to phase transitions in the statistical mechanical setting. We present a simple example of spin-flip stochastic model, associated to a synthetic biology model known as repressilator, which leads to a dynamical system with Hopf and pitchfork bifurcations. Depending on the parameter values, the magnetization random path can either converge to a unique stable fixed point, converge to one of a pair of stable fixed points, or asymptotically evolve close to a deterministic orbit in Rk. We also discuss a simple signaling pathway related to cancer research, called p53 module.

Immobilization of Catalysts of Biological Interest on Porous Oxidized Silicon Surfaces

The present paper deals with the immobilization of redox mediators and proteins onto protected porous silicon surfaces to obtain their direct electrochemical reactions and to retain their bioactivities. This paper shows that MP-11 and viologens are able to establish chemical bonds with 3-aminopropyltriethoxylsilane-modified porous silicon surface. The functionalization of the surfaces have been fully characterized by energy dispersive X-ray analysis (EDX) and X-ray photoelectron spectroscopy (XPS) to examine the immobilization of these mediators onto the solid surface. Amperometric and open circuit potential measurements have shown the direct electron transfer between glucose oxidase and the electrode in the presence of the viologen mediator covalently linked to the 3-aminopropyltriethoxylsilane (APTES)-modified porous silicon surfaces.