Non-Michaelis-Menten kinetics model for conductance of low-conductance potassium ion channels

Explicit Treatment of Non-Michaelis-Menten and Atypical Kinetics in Early Drug Discovery*

Biological systems are highly regulated. They are also highly resistant to sudden perturbations enabling them to maintain the dynamic equilibrium essential to sustain life. This robustness is conferred by regulatory mechanisms that influence the activity of enzymes/proteins within their cellular context to adapt to changing environmental conditions. However, the initial rules governing the study of enzyme kinetics were mostly tested and implemented for cytosolic enzyme systems that were easy to isolate and/or recombinantly express. Moreover, these enzymes lacked complex regulatory modalities. Now, with academic labs and pharmaceutical companies turning their attention to more-complex systems (for instance, multiprotein complexes, oligomeric assemblies, membrane proteins and post-translationally modified proteins), the initial axioms defined by Michaelis-Menten (MM) kinetics are rendered inadequate, and the development of a new kind of kinetic analysis to study these systems is required. This review strives to present an overview of enzyme kinetic mechanisms that are atypical and, oftentimes, do not conform to the classical MM kinetics. Further, it presents initial ideas on the design and analysis of experiments in early drug-discovery for such systems, to enable effective screening and characterisation of small-molecule inhibitors with desirable physiological outcomes.

Contributions of HERG K+ current to repolarization of the human ventricular action potential

Action potential repolarization in the mammalian heart is governed by interactions of a number of time- and voltage-dependent channel-mediated currents, as well as contributions from the Na+/Ca2+ exchanger and the Na+/K+ pump. Recent work has shown that one of the K+ currents (HERG) which contributes to repolarization in mammalian ventricle is a locus at which a number of point mutations can have significant functional consequences. In addition, the remarkable sensitivity of this K+ channel isoform to inhibition by a variety of pharmacological agents and clinical drugs has resulted in HERG being a major focus for Safety Pharmacology requirements. For these reasons we and others have attempted to define the functional role for HERG-mediated K+ currents in repolarization of the action potential in the human ventricle. Here, we describe and evaluate changes in the formulations for two K+ currents, IK1 and HERG (or IK,r), within the framework of ten Tusscher model of the human ventricular action potential. In this computational study, new mathematical formulations for the two nonlinear K+ conductances, IK1 and HERG, have been developed based upon experimental data obtained from electrophysiological studies of excised human ventricular tissue and/or myocytes. The resulting mathematical model provides much improved simulations of the relative sizes and time courses of the K+ currents which modulate repolarization. Our new formulation represents an important first step in defining the mechanism(s) of repolarization of the membrane action potential in the human ventricle. Our overall goal is to understand the genesis of the T-wave of the human electrocardiogram.

Unified modeling of conductance kinetics for low- and high-conductance potassium ion channels

A kinetics model is proposed for the description of ion conductance of low- and high-conductance potassium ion channels. The model describes ion permeation through the selectivity filter, which is assumed to be the only conductance determining part of the open channel. The filter occupancy can vary from zero to three ions, affecting the ion entry and exit rates. Ion motion between the binding sites inside the filter is assumed fast compared to the latter rates allowing averaging the equilibrium entry and exit rate constants over the possible ion configurations in the filter with a particular occupancy. Averaged rate constants related to a pair of adjacent occupancy states characterize a particular ion permeation mechanism. An expression for the channel conductance as a function of the symmetrical external ion concentration is derived. It comprises a sum of concentration independent conductance amplitudes for different ion permeation mechanisms weighted by the equilibrium filter occupancy probabilities. It is shown that each amplitude (i.e., maximum contribution to the channel conductance from each conductance mechanism) is proportional to an averaged exit rate constant and to quantities characterizing the effect of the applied electric field on the rate constants and the equilibrium ion distribution in the filter. The conductance expression derived provides a good description of the experimentally observed conductance-concentration curves for low-conductance (e.g., Kir2.1) and high-conductance (e.g., KcsA) potassium channels. It enables one to obtain equilibrium ion binding constants at different filter occupancies and to calculate the average number of ions in the selectivity filter for a given external ion concentration. For KcsA this number (2.0 at 200 mM) is in a good agreement with the available experimental value (2.1 at 200 mM). For the high-conductance potassium channels the net negative electrical charge around the selectivity filter increases the ion binding constants, thereby causing the larger occupancy probabilities to occur at smaller external ion concentrations compared to the low-conductance channels. This substantially increases the contributions of the two- and three-ion permeation mechanisms, with the larger conductance amplitudes leading to increased channel conductance compared to the low-conductance channels.

Incidence of partial charges on ion selectivity in potassium channels

Potassium channels are membrane proteins known to select potassium over sodium ions at a high diffusion rate. We conducted ab initio calculations on a filter model of KcsA of about 300 atoms at the Hartree-Fock level of theory. Partial charges were derived from the quantum mechanically determined electrostatic potential either with Merz-Kollman or Hinsen-Roux schemes. Large polarization and/or charge transfer occur on potassium ions located in the filter, while the charges on sodium ions remain closer to unity. As a result, a weaker binding is obtained for K(+) ions. Using a simplified version of a permeation model based on the concerted-motion mechanism for ion translocation within the single-file ion channel [P. H. Nelson, J. Chem. Phys. 117, 11396 (2002)], we discuss how differences in polarization effects in the adducts with K(+) and Na(+) can play a role as for ionic selectivity and conductance.