Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4YF, UK

Background

All cellular activations are regulated by various signal transduction pathways, which are the network of interacting proteins used to carry over signals in the cell's environment for producing associate responses. The MAPK (mitogen-activated protein kinase) or its synonymous ERK (extracellular signal regulated kinase) pathway is one of the major signal transduction systems which regulates the cellular growth control of all eukaryotes like cell proliferation or apoptosis. The complex structure of this regulatory mechanism whose main components are Ras, Raf, and MEK proteins (see Figure

Simple representation of the structure of MAPK/ERK pathway

Simple representation of the structure of MAPK/ERK pathway.

Due to the importance in the cellular lifecycle, the MAPK/ERK pathway has been intensively studied, thereby a number of qualitative descriptions of this regulatory mechanism are available in the literature. However none of the sources describe the system by an explicit set of reactions. Here we combine these qualitative sources for a representation of the pathway as a list of (quasi) reactions which is used to produce a basis for stochastic simulation. For defining our reaction set we denote all components by simple notations and use multiple parametrizations to indicate different localization of the molecules in the cell and to describe the protein using different binding sites as well as various phosphorylations.

Modelling by diffusion approximation

Gene regulation is commonly modelled via ordinary differential equations (ODEs). Although ODEs are successful to represent some reactions like linear production and degradation, they cannot describe the small system variability of the actual reactions. For biochemical systems, stochastic processes are a natural choice as these kinds of dynamic formalization take into account the probabilistic manner of the different biological activations. In this study under the assumption that the probability distribution of the number of the molecules of each species at ^{1/2 }(_{
t
}, Θ) and _{
t
}, Θ)}_{1},...,Θ_{
r
})' explicitly. Θ_{
j
}(_{
t
}, Θ) describes the hazard of each reaction at time

Diffusion approximation for inference

For estimating the model parameters, i.e. the stochastic rate constants, we apply the discretized version of diffusion approximation, which is known as Euler approximation, Δ_{
t
}= _{
t
}, Θ)Δ^{1/2}(_{
t
}, Θ)Δ_{
t
}, where Δ_{
t
}shows a

where _{
t
}= _{
t + Δt
}- _{
t
}. As can be seen from equation 1, the conditional posterior density of reaction rates Θ does not have a known distribution. We compute the posterior distribution of Θ using the MCMC method. Moreover to decrease the bias causing by discretization we augment our observations by putting extra time states between given measurements. Then conditional on accepted Θ, we simulate and update the missing states by implementing the Metropolis-Hastings algorithm as one block of