Motivated by the recent development of advanced experimental techniques in molecular biology, this paper focuses on the study of the dynamical properties of a two-gene regulatory network. A mathematical model for a two-gene regulatory network is derived and several of their properties are analyzed. Due to the presence of mixed continuous/discrete dynamics and hysteresis, we employ hybrid systems models to capture the dynamics of the system. The proposed model incorporates binary hysteresis with different thresholds capturing the interaction between the genes. We analyze properties of the solutions and asymptotic stability of equilibria in the system as a function of their parameters. As a difference to previous efforts employing piecewise-linear models, the analysis of our hybrid system model reveals the presence of limit cycles for a certain range of parameters, a behavior that is associated with the presence of hysteresis. The set of points defining the limit cycle is characterized and its asymptotic stability properties are studied. Furthermore, we determine conditions under which the stability properties of the limit cycle are robust to changes of parameters. Numerical simulations are presented to illustrate the findings. © 2014 Elsevier Inc.

A mathematical model for a two-gene regulatory network is derived and several of its properties are analyzed. Due to the presence of continuous dynamics and binary hysteresis, we propose a hybrid system model. Binary hysteresis with different thresholds captures the interaction between the genes. We analyze properties of the solutions and asymptotic stability of equilibria in the system as a function of its parameters. Our analysis reveals the presence of limit cycles for a certain range of parameters, a behavior that is associated with the presence of binary hysteresis. The set of points defining the limit cycle is characterized and its asymptotic stability properties are studied. Numerical simulations are presented to illustrate some of the results. ©2013 IEEE.

Myopia in human and animals is caused by the axial elongation of the eye and is closely linked to the thinning of the sclera which supports the eye tissue. This thinning has been correlated with the overproduction of matrix metalloproteinase (MMP-2), an enzyme that degrades the collagen structure of the sclera. In this short paper, we propose a descriptive model of a regulatory network with hysteresis, which seems necessary for creating oscillatory behavior in the hybrid model between MMP-2, MT1-MMP and TIMP-2. Numerical results provide insight on the type of equilibria present in the system.

We developed a tumor vaccine consisting of fixed hepatocellular carcinoma (HCC) cells/tissue fragments, biodegradable microparticles encapsulating granulocyte‐macrophage‐colony stimulating factor and interleukin‐2, and an adjuvant. The vaccine protected 33% of syngeneic mice from HCC cell challenge. The vaccine containing human autologous HCC fragments showed essentially no adverse effect in a phase I/IIa clinical trial and 8/12 patients developed a delayed‐type hyper‐sensitivity (DTH) response against the fragments. Although 2 of 4 DTH‐response‐negative patients had recurrence after curative resection, the DTH‐response‐positive patients had no recurrence. The time before the first recurrence in the vaccinated patients was significantly longer than that in 24 historical control patients operated in the same department (P<0.05). This formulation is a promising candidate to prevent recurrence of human HCC.