Some mathematical models of intermolecular autophosphorylation

Derivation of stationary distributions of biochemical reaction networks via structure transformation

Long-term behaviors of biochemical reaction networks (BRNs) are described by steady states in deterministic models and stationary distributions in stochastic models. Unlike deterministic steady states, stationary distributions capturing inherent fluctuations of reactions are extremely difficult to derive analytically due to the curse of dimensionality. Here, we develop a method to derive analytic stationary distributions from deterministic steady states by transforming BRNs to have a special dynamic property, called complex balancing. Specifically, we merge nodes and edges of BRNs to match in- and out-flows of each node. This allows us to derive the stationary distributions of a large class of BRNs, including autophosphorylation networks of EGFR, PAK1, and Aurora B kinase and a genetic toggle switch. This reveals the unique properties of their stochastic dynamics such as robustness, sensitivity, and multi-modality. Importantly, we provide a user-friendly computational package, CASTANET, that automatically derives symbolic expressions of the stationary distributions of BRNs to understand their long-term stochasticity.

Autophosphorylation and the Dynamics of the Activation of Lck

Lck (lymphocyte-specific protein tyrosine kinase) is an enzyme which plays a number of important roles in the function of immune cells. It belongs to the Src family of kinases which are known to undergo autophosphorylation. It turns out that this leads to a remarkable variety of dynamical behaviour which can occur during their activation. We prove that in the presence of autophosphorylation one phenomenon, bistability, already occurs in a mathematical model for a protein with a single phosphorylation site. We further show that a certain model of Lck exhibits oscillations. Finally, we discuss the relations of these results to models in the literature which involve Lck and describe specific biological processes, such as the early stages of T cell activation and the stimulation of T cell responses resulting from the suppression of PD-1 signalling which is important in immune checkpoint therapy for cancer.

A mathematical model of Aurora B activity in prophase and metaphase

Aurora B kinase is a protein that controls several processes in mitosis when it is found associated with INCENP, Survivin and Borealin in a complex known as the Chromosomal Passenger Complex. Aurora B in complex with INCENP is phosphorylated on three sites, resulting in the full activation of Aurora B. In prophase and metaphase, Aurora B is activated at centromeres, the region of chromatin linking sister chromatids, due to an autophosphorylation mechanism, and it has been hypothesised that Aurora B is activated throughout the cytoplasm due to its concentration at centromeres. In this article, we first develop a time-dependent model of Aurora B activation that does not incorporate spatial variation. This model is used to demonstrate the various qualitative behaviours that the activation of Aurora B is capable of displaying for different model parameters. Next, we develop a spatio-temporal model of Aurora B activation that includes diffusion of soluble Aurora B and binding of Aurora B to immobile centromeric binding sites. This model describes the activation of Aurora B throughout the cytoplasm due to its concentration-dependent activation at centromeres. The models demonstrate the effects that a soluble phosphatase concentration, multisite phosphorylation and diffusion have on the activation of Aurora B.