The Mathematical Model

GRF (R, concentration r) binds reversibly to receptors (F, density f) on the pituitary somatotroph:

The rate of the forward reaction is kx. The rate of backward reaction depends on the concentration of SRIF, s, being kb= k2 + k3 ^(s), where ^(s) = 1 / (1 + exp[-(s - s0)/50]) is a sigmoidal function of s, rising from zero to unity over a range of concentrations of SRIF of s0 ± 360. The authors thus get the equations f = -ki (r + r0) f + \_k2 + ks^(s)]u (1)

dt where u reflects the number of postactivated, desensitized GRF receptors, r is the concentration of GRF in the extracellular space, and r0 reflects constitutive activation of the GH release mechanism. The step in the sigmoidal function, ^(s), can be made more or less steep by reducing or increasing 60. If k3 is positive, the rate of the backward reaction is greater in the presence of somatostatin. The rate of release of growth hormone is proportional to the rate of binding (the constant of proportionality being a function of s) and given by

Thus, in the presence of very high levels of somatostatin, the rate of release is k4(r + r0)f, and in the absence of SRIF it is (k4 + k5)(r + r0)f. To simulate the model we need a differential equation for s,

dt where Is is the rate of release (of infusion) of SRIF, and k7 its decay rate. The dynamics of SRIF concentration are separate from the other dynamical variables: SRIF affects the dynamics of the interaction between GRF and its receptors, and growth hormone release, but there are no effects of these on SRIF release or decay. A further differential equation in r could be added dr = Ir- kl (r + r0) f - ker dt where k6 is the relative rate of decay of GRF in the absence of binding, and Ir is the input of GRF. However, other factors frequently have a substantial effect on r and it is simpler for the purposes of initial modeling to assume that external factors maintain the level of r either as a constant infusion or a pulsatile pattern of stimulation. Understanding the model may be easier if SRIF is thought of as acting as a switch that changes the behavior of the somatotroph when present at suprathreshold concentration (substantially above s0). In the absence of SRIF successive pulses of GRF desensitize the somatotroph as the receptor-effector mechanism is progressively inactivated. Somatostatin resensitizes the somatotroph by speeding up the recovery of the receptor-effector mechanism to the free, available state. The variable u, being proportional to the number of postactivated, desensitized GRF receptors, can be thought of as an index of the extent of desensitization of the release system. If the further assumption is made that the total number of receptors is constant (i.e., in the short term, receptors are neither created nor destroyed), and the total concentration of sensitized receptors is fT, then u = fT - f can be written, and, therefore, f can equally well be regarded as a complementary index of sensitivity: the ability of the somatotrophs to respond to stimulation by GRF.

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