Reaction-Diffusion Modeling ERK- and STAT-Interaction Dynamics

<p/> <p>The modeling of the dynamics of interaction between ERK and STAT signaling pathways in the cell needs to establish the biochemical diagram of the corresponding proteins interactions as well as the corresponding reaction-diffusion scheme. Starting from the verbal description available in the literature of the cross talk between the two pathways, a simple diagram of interaction between ERK and STAT5a proteins is chosen to write corresponding kinetic equations. The dynamics of interaction is modeled in a form of two-dimensional nonlinear dynamical system for ERK&#8212;and STAT5a &#8212;protein concentrations. Then the spatial modeling of the interaction is accomplished by introducing an appropriate diffusion-reaction scheme. The obtained system of partial differential equations is analyzed and it is argued that the possibility of Turing bifurcation is presented by loss of stability of the homogeneous steady state and forms dissipative structures in the ERK and STAT interaction process. In these terms, a possible scaffolding effect in the protein interaction is related to the process of stabilization and destabilization of the dissipative structures (pattern formation) inherent to the model of ERK and STAT cross talk.</p

Qualitative Modelling of Quasi-homogeneous Effects in ERK and STAT Interaction Dynamics

On the basis of qualitative analysis of author's model published in previous paper, the stability and temporal behaviour of quasi-homogeneous distributions of ERK-protein concentrations are analyzed in terms of corresponding reaction-diffusion problem. The stable quasi-homogeneous distributions are treated as a dynamical basis of pathway compartmentalization. It is also shown, that a crowding effect exists in the form of loss of pathway stability. An experimentally verifiable issue for possible existence of protein scaffolding mechanism is derived on the basis of its qualitative correspondence with the pattern formation and molecular crowding effects inherent to the considered model. Moreover, it is demonstrated, that the predicted ERK and STAT pathway instability can be interpreted as traveling wave propagation of molecular concentration drop and jump from the nucleus membrane to the cell one and vice versa

Quantum Two-State Dynamics Driven by Stationary Non-Markovian Discrete Noise: Exact Results

We consider the problem of stochastic averaging of a quantum two-state dynamics driven by non-Markovian, discrete noises of the continuous time random walk type (multistate renewal processes). The emphasis is put on the proper averaging over the stationary noise realizations corresponding, e.g., to a stationary environment. A two state non-Markovian process with an arbitrary non-exponential distribution of residence times (RTDs) in its states with a finite mean residence time provides a paradigm. For the case of a two-state quantum relaxation caused by such a classical stochastic field we obtain the explicit exact, analytical expression for the averaged Laplace-transformed relaxation dynamics. In the limit of Markovian noise (implying an exponential RTD), all previously known results are recovered. We exemplify new more general results for the case of non-Markovian noise with a biexponential RTD. The averaged, real-time relaxation dynamics is obtained in this case by numerically exact solving of a resulting algebraic polynomial problem. Moreover, the case of manifest non-Markovian noise with an infinite range of temporal autocorrelation (which in principle is not accessible to any kind of perturbative treatment) is studied, both analytically (asymptotic long-time dynamics) and numerically (by a precise numerical inversion of the Laplace-transformed averaged quantum relaxation).Comment: Chemical Physics, in pres