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Signal Transduction Pathways

Describes the sequential steps by which a cell converts an external stimulus into a functional response, involving receptor activation and downstream signaling cascades.
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The statement of the theorem

Let C(t)=([M1],[M2],…,[MN])\mathbf{C}(t) = \left( [M_1], [M_2], \dots, [M_N] \right) be the vector of concentrations of NN molecular species at time tt. The dynamics of the pathway are governed by the system of Ordinary Differential Equations (ODEs): \n\nd[Mi]dt=∑j(kon,ij[Mj][L]−koff,ij[Mi])+∑k(vk−kdeg,k[Mi])\frac{d[M_i]}{dt} = \sum_{j} \left( k_{on, ij} [M_j] [L] - k_{off, ij} [M_i] \right) + \sum_{k} \left( v_{k} - k_{deg, k} [M_i] \right) \n\nwhere kon,ijk_{on, ij} and koff,ijk_{off, ij} are the association and dissociation rate constants for the interaction between MjM_j and LL (ligand/precursor), and vkv_k represents the production rate from a preceding step kk. The overall pathway response RR is a function of the final activated species: R=f(C(t))R = f(\mathbf{C}(t)).