Cellular responses are usually induced by detection of extracellular signalling molecules, called ligands,  through specialized cell-surface receptors. Blood vessel development and vascular repair are regulated by ligands called growth factors (GF). There are many types of GFs and receptors associated with them. Our focus is on vascular endothelial growth factors (VEGFs) and VEGF receptors (VEGFR).

VEGF is a large family of bivalent molecules. It occurs in different isoforms of different length. The mounting evidence suggests that various isoforms are involved in different cellular responses such as migration and proliferation. VEGF is also an indicator of all solid tumour growth and wound healing, therefore research is conducted to develop therapies that target VEGF pathways.

VEGFR belongs to the receptor tyrosine kinase family and, like other members of that family, requires dimerisation to be active. The receptor population is involved in coupling with other receptors or membrane associated molecules, internalization, recycling, degradation and synthesis - the pathway termed trafficking. Both VEGFR monomers and VEGFR dimers undergo internalization by the same mechanism. The molecules are absorbed and transferred to the early endosome, in a process called endocytosis.

Our aim is to build a comprehensive mathematical model for the purpose of simulating cell fate. Signalling molecules can be effective at very low concentration. To handle that behaviour, probabilistic methods have been used.

We developed stochastic models of the trafficking and binding dynamics of VEGFR to the bivalent VEGF in terms of Markov processes. We use the moment generating function with moment closure techniques to study average number of each complex involved in the process and fluctuations around it. On the other hand, the Van Kampen expansion is used to analyse stochastic and deterministic behaviour of our model.

We also study the descriptors of interest in the model such as time until having a definite number of an observed molecule or the stationary distribution of the system. The descriptors allow us to analyse the state of the system in the long term and give some intuition of how much time the system spends until reaching the steady state.  Moreover, the dynamics of the system until reaching steady state is studied by applying moment closure techniques over Kolmogorov equations, that permit us to shed some light on the behaviour of the process until reaching the steady state.

The model allows us to track the number of each complex, including phosphorylated molecules on the cell surface and in the endosome. The different locations of phosporylated units may cause different cellular responses. The spatial distribution of receptors on the cell without any stimulation has an impact on cell decision, after it is stimulated. It can be also studied in our model. The model can be used to study the response for different isoforms of VEGF and for VEGFR1,VEGFR2 competition.

VEGF; VEGFR; stochastic modelling