An Invitation to Pharmacostatics.

Pharmacology, the study of interactions between biological processes and therapeutic agents, is traditionally presented as consisting of two subdisciplines: pharmacokinetics, which is about the distribution and metabolism of drugs in organisms, and pharmacodynamics, which is about the organisms' response to drugs. In discovery-stage pharmacology however, one primary concern is what we call pharmacostatics, the characterization of equilibrium parameters and states of core interactions of physiologic and therapeutic interest. This usually takes the form of studying dose-response curves, without consideration for the relevant qualitative properties of the underlying reaction networks, e.g., the existence, multiplicity and asymptotic stability of steady states. Furthermore, steady-state calculations customarily employ manually derived closed-form expressions based on approximating assumptions. While these formulas may seem adequate most of the time, the assumptions need not apply, and there are genuine though seemingly uncommon cases where this approach is not feasible and/or fails to explain non-monotone dose-response curves. It is this paper's aim to stimulate interest in mathematical problems arising in pharmacostatics. We specifically pose two problems about a particular relevant class of networks of reversible binding reactions. The first problem is to exploit a certain fixed-point formulation of the equilibrium equation to devise an algorithmic method that would be compellingly preferable to current practice in the pharmacostatics context. The second problem is to explicitly anticipate the possibility of non-monotone dose-response curves from network topology. Addressing these problems would positively impact biopharmaceutical research, and they have inherent mathematical interest.

A method to calculate binding equilibrium concentrations in the allosteric ternary complex model that supports ligand depletion.

The allosteric ternary complex model is frequently used in pharmacology to represent the interaction of a receptor R with two ligands A and B. Certain well-known formulas are routinely used to calculate the fractions of the receptor bound at equilibrium with A only, B only, and both A and B. However, it is often omitted that these classical formulas presume that there is no ligand depletion, i.e. that the equilibrium concentrations [A] and [B] of the ligands are well approximated by their total concentrations [A](T) and [B](T). We present a calculation method which is applicable without this or any restrictions. The equilibrium concentration [R] of the receptor is implicitly characterized by an equation which is solved with a very simple convergent numerical algorithm. The concentrations [A] and [B] are given by explicit formulas in terms of [R]. The required parameters are the equilibrium dissociation constants K(A) and K(B), the cooperativity factor α, and the total concentrations [R](T), [A](T) and [B](T).

Effect of the calcimimetic R-568 [3-(2-chlorophenyl)-N-((1R)-1-(3-methoxyphenyl)ethyl)-1-propanamine] on correcting inactivating mutations in the human calcium-sensing receptor.

Over 257 mutations in the human calcium-sensing receptor (hCaSR) gene have been reported. Heterozygous inactivating mutations can result in familial hypocalciuric hypercalcemia (FHH), whereas homozygous inactivating mutations can cause life-threatening neonatal severe hyperparathyroidism (NSHPT). Activating mutations in the hCaSR can result in hypercalciuria and hypocalcemia. A recent publication on the successful treatment of a patient suffering from FHH with the hCaSR positive allosteric modulator cinacalcet prompted our interest in exploring the molecular pharmacology of calcimimetics to correct signaling defects associated with inactivating hCaSR mutations. We prepared 11 mutant hCaSRs, previously identified in patients suffering from NSHPT or FHH, and tested their ability to couple to inositol phosphate accumulation and intracellular calcium mobilization in transiently transfected human embryonic kidney 293 and Chines hamster ovary cells using the calcimimetic R-568 [3-(2-chlorophenyl)-N-((1R)-1-(3-methoxyphenyl)ethyl)-1-propanamine]. We found that extracellular Ca(2+) was significantly less potent on the mutated receptors compared with wild-type hCaSR. However, R-568 was able to enhance the potency of extracellular Ca(2+) toward the mutant hCaSRs. Furthermore, R-568 increased the maximal agonist response on several of the mutant CaSRs. We applied a novel operational model of allosteric modulation/agonism that provided a common mechanism to account for the behavior of the wild-type and mutant hCaSRs. The data provide evidence for the potential use of calcimimetics to treat diseases such as FHH and NSHPT where severe hypercalcemia can be life-threatening.

Monotonicity of interleukin-1 receptor-ligand binding with respect to antagonist in the presence of decoy receptor.

We consider the interaction between interleukin-1 IL-1, its receptor IL-1RI, the receptor antagonist IL-1Ra and a decoy receptor (or trap) that binds both with the ligand and the antagonist. We study how the interaction between IL-1Ra and the decoy receptor influences the effect of either reagent on reducing the equilibrium concentration of the receptor-ligand complex. We obtain that, given a certain relationship among the equilibrium constants and the total concentrations of solutes, IL-1Ra can reverse the effect of the decoy receptor of decreasing the equilibrium concentration of the receptor-ligand complex. This finding derives from a mathematical result applicable to any reversible chemical reaction system comprising four species arranged in a square such that each species binds its two immediate neighbors. The result gives the monotonicity of the equilibrium concentrations of the complex species as functions of the total concentrations of the simple species.