A theoretical approach to the bell-shaped dependency of cell proliferation on the hormone concentration

Binding of human growth hormones (hGH) to the receptors was studied with a theoretical, molecular-level model. An hGH has two sites bindable to different receptors with different binding energies. In the model the hGHs diffusively moved in a box (i.e. the volume of solution), and receptors on the bottom face the box (i.e. a membrane). The system consisted of a number of hGHs and receptors, which could form hGH-[receptor]2 or hGH-receptor complexus. In a complex, small inter-molecular positional fluctuations were allowed with keeping the inter-molecular binding. Partition function of the system was calculated. In a low hGH-concentration range, free receptors were dominant on the membrane; in a medium concentration range, hGH-[receptor]2 complexus, which induce cell-proliferation, were dominant, and in a high concentration range, hGH-receptor complexus, which inhibit the proliferation, were dominant. This dependency (bell-shaped dependency) of formation of hGH-[receptor]2 complex on the hGH concentration agreed well with experimental observation. The values of EC50 (hGH concentration at that the cell-proliferation rate rose to 50% of the maximum rate by the formation of hGH-[receptor]2 complexus) and IC50 (hGH concentration at that the proliferation rate decreased to 50% of the maximum by the formation of hGH-receptor complexus) from my method were 18 pM and 2.2 microM, respectively. By calculating thermodynamic quantities (i.e. entropy and enthalpy), factors that determine the bell-shaped dependency were obtained. At the medium concentration, the entropy of free hGHs played an important role in stabilizing the hGh-[receptor]2 complex. Small changes in binding energies or in inter-molecular positional fluctuations largely changed the dependency of the complex formation on the hGH concentration. This method is useful in explaining the experimental results that small molecular modification largely changes the formation of the complex.