Analytical Solution of the Model

One particular advantage of the very simple model formulation outlined above is that for periods when r and s are constant, the differential equations can be solved to give explicit expressions for the extent of desensitization and for the rate of GH release. Making the substitution u = fT - f (where fT is the total concentration of free receptors in the completely sensitized state, i.e., the maximal concentration of free receptors) and kb = k2 + k3^(s) in Eq. (1), the result is f = - kl (r + ro) f + *b( fT - f)

= kb fT - Kf where K = kl(r + r0) + kb. The equilibrium values of f, f*, is given by solving df/dt = 0, obtaining f * = kbT = kbfT = Ik + k3^(4 fT

Thus, f * is an irreducible minimum sensitivity below which the system will not fall. This is a dynamic equilibrium or minimum, though, since it depends on both r and s and will therefore change as they do. Rewriting the differential equation in terms off *, the result is f = -K f- f *) (6)

assuming that the initial concentration of free receptors at time t = o is fo. Thus, after a change in the r,s environment, the excess sensitivity of the system over the new equilibrium declines exponentially with time, and the relative rate of decline is K = ki(r + ro) + kb, which equals kl(r + ro)+ k2 + k3^(s). Therefore, the release rate is given by p(r,s,f) = [k4 + k5(1 -^[s])] (r + r0)f (8)

kr = k4 + k5(1 -^[s]) is defined to be the rate constant for release, and therefore p(r,s,f) = kr (r + ro)f. We can therefore calculate the cumulative amount released as the integral of this, obtaining

P(r,s,t) = kr (r + ro) [f*t + (fo - f*)(1 -e-Kt)/K] (9)

if the initial free receptor concentration is fo, These expressions enable theoretical dose/ response curves for any concentrations of r, s to be obtained. These can then be fitted to dose/response curve data, first of all as a test of the model, and second to estimate the model parameters.

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