Discriminating between Concerted and Sequential Allosteric Mechanisms by Comparing Equilibrium and Kinetic Hill Coefficients

Relationship between Decimal Hill Coefficient, Intermediate Processes, and Mesoscopic Fluctuations in Gene Expression

The Hill function is relevant for describing enzyme binding and other processes in gene regulatory networks. Despite its theoretical foundation, based on the mechanism of ligand-receptor binding, it is often used as a proper fitting function with a noninteger Hill coefficient in the description of gene expression. In this study, we explicitly considered intermediate processes in the transcription factor binding sites and mesoscopic concentration fluctuations, which, in contrast to the case of a single binding site without conformal states or all-or-none binding, lead to a noninteger Hill coefficient for the transcription rate. The relationships between the intermediate processes and the decimal Hill coefficient were established through a direct relationship between the dissociation constants, both with and without fluctuations. This outcome contributes to a deeper understanding of the underlying processes associated with the decimal Hill coefficient of gene expression rates while also enabling the prediction of an effective value of the Hill coefficient from the underlying mechanism. This procedure provides a simplified and effective description of the complex mechanisms that underlie gene expression.

Distinguishing between concerted, sequential and barrierless conformational changes: Folding versus allostery

Characterization of transition and intermediate states of reactions provides insights into their mechanisms and is often achieved through analysis of linear free energy relationships. Such an approach has been used extensively in protein folding studies but less so for analyzing allosteric transitions. Here, we point out analogies in ways to characterize pathways and intermediates in folding and allosteric transitions. Achieving an understanding of the mechanisms by which proteins undergo allosteric switching is important in many cases for obtaining insights into how they function.

Mathematical models describing oxygen binding by hemoglobin

Despite the fact that the investigation of the structural and functional properties of hemoglobin dates back more than 150 years, the topic has not lost its relevance today. The most important component of these studies is the development of mathematical models that formalize and generalize the mechanisms determining the cooperative binding of ligands based on data on the structural and functional state of the protein. In this work, we review the mathematical relationships describing oxygen binding by hemoglobin, ranging from the classical Hüfner, Hill, and Adair equations to the Szabo-Karplus and tertiary two-state mathematical models based on the Monod-Wyman-Changeux and Koshland-Némethy-Filmer concepts. The generality of the considered equations as mathematical functions, bearing in their basis a power dependence, is demonstrated. The problems and possible solutions related to approximation of experimental data by the oxygenation equations with correlated fitting parameters are noted. Attention is paid to empirical equations, extended versions of the Hill equation, where the coefficient of cooperation is modulated by Gauss and Lorentz distributions as functions of partial oxygen pressure.

Cooperative & competitive binding of anti-myosin tail antibodies revealed by super-resolution microscopy

The MF30 monoclonal antibody, which binds to the myosin subfragment-2 (S2), was found to increase the extent of myofibril shortening. Yet, previous observations found no effect of this antibody on actin sliding over myosin during in vitro motility assays with purified proteins in which myosin binding protein C (MyBPC) was absent. MF30 is hypothesized to enhance the availability of myosin heads (subfragment-1 or S1) to bind actin by destabilizing the myosin S2 coiled-coil and sterically blocking S2 from binding S1. The mechanism of action likely includes MF30's substantial size, thereby inhibiting S1 heads and MyBPC from binding S2. Hypothetically, MF30 should enhance the ON state of myosin, thereby increasing muscle contraction. Our findings indicate that MF30 binds preferentially to the unfolded heavy chains of S2, displaying positive cooperativity. However, the dose-response curve of MF30's enhancement of myofibril shortening did not suggest complex interactions with S2. Single, double, and triple-stained myofibrils with increasing amounts of antibodies against myosin rods indicate a possible competition with MyBPC. Additional assays revealed decreased fluorescence intensity at the C-zone (central zone in the sarcomere, where MyBPC is located), where MyBPC may inhibit MF30 binding. Another monoclonal antibody named MF20, which binds to the light meromyosin (LMM) without affecting myofibril contraction, showed less reduction in fluorescence intensity at the C-zone in expansion microscopy than MF30. Expansion microscopy images of myofibrils labeled with MF20 revealed labeling of the A-band (anisotropic band) and a slight reduction in the labeling at the C-zone. The staining pattern obtained from the expansion microscopy image was consistent with images from photolocalization microscopy which required the synthesis of unique photoactivatable quantum dots, and Zeiss Airyscan imaging as well as alternative expansion microscopy digestion methods. Consistent with the hypothesis that MF30 competes with MyBPC binding to S2, cardiac tissue from MyBPC knockout mice was stained more intensely, especially in the C-zone, by MF30 compared to the wild type.

Optimization of the stability constants of the ternary system of diclofenac/famotidine/β-cyclodextrin by nonlinear least-squares method using theoretical equations

This study aimed to establish a new method for determining the stability constants of drug/β-cyclodextrin (β-CD) complexes when multiple drugs interacting with each other coexist in the solution of complexation. The basic drug famotidine (FAM) and the acidic drug diclofenac (DIC) were used as model drugs, their solubility decreasing owing to their mutual interaction. The dissolution of both FAM and DIC was characterized by A_L-type phase solubility diagrams in the presence of the other's 1:1 complex with β-CD. When the stability constant was calculated from the slope of the phase solubility diagram using the conventional phase solubility diagram method, it was modified in the presence of the other drug. However, by performing optimization calculations that considered the interactions between the drug/β-CD complex and the drug, drug/β-CD complexes, and drugs, we were able to accurately calculate the stability constant of DIC/β-CD and FAM/β-CD complexes even in the presence of FAM and DIC, respectively. The results of the solubility profile indicated that various molecular species, which are attributed to drug-drug and drug/β-CD interactions, interfere with the values of the dissolution rate constants and saturated concentration in the solubility profiles.

A New Model of Hemoglobin Oxygenation

The study of hemoglobin oxygenation, starting from the classical works of Hill, has laid the foundation for molecular biophysics. The cooperative nature of oxygen binding to hemoglobin has been variously described in different models. In the Adair model, which better fits the experimental data, the constants of oxygen binding at various stages differ. However, the physical meaning of the parameters in this model remains unclear. In this work, we applied Hill's approach, extending its interpretation; we obtained a good agreement between the theory and the experiment. The equation in which the Hill coefficient is modulated by the Lorentz distribution for oxygen partial pressure approximates the experimental data better than not only the classical Hill equation, but also the Adair equation.

An SPR-based method for Hill coefficient measurements: the case of insulin-degrading enzyme

Insulin-degrading enzyme (IDE) is a highly conserved zinc metallopeptidase and is capable to catalytically cleave several substrates besides insulin, playing a pivotal role in several different biochemical pathways. Although its mechanism of action has been widely investigated, many conundrums still remain, hindering the possibility to rationally design specific modulators which could have important therapeutical applications in several diseases such as diabetes and Alzheimer's disease. In this scenario, we have developed a novel surface plasmon resonance (SPR) method which allows for directly measuring the enzyme cooperativity for the binding of insulin in the presence of different IDE activity modulators: carnosine, ATP, and EDTA. Results indicate that both positive and negative modulations of the IDE activity can be correlated to an increase and a decrease of the measured Hill coefficient, respectively, giving a new insight into the IDE activity mechanism. The use of the IDE R767A mutant for which oligomerization is hindered confirmed that the positive allosteric modulation of IDE by carnosine is due to a change in the enzyme oligomeric state occurring also for the enzyme immobilized on the gold SPR chip.

Partitioning the Hill coefficient into contributions from ligand-promoted conformational changes and subunit heterogeneity

Heterooligomers that undergo ligand-promoted conformational changes are ubiquitous in nature and involved in many essential processes. Conformational switching often leads to positive cooperativity in ligand binding that is reflected in a Hill coefficient with a value greater than one. The subunits comprising heterooligomers can differ, however, in their affinity for the ligand. Such so-called site heterogeneity results in apparent negative cooperativity that is reflected by a Hill coefficient with a value less than one. Consequently, positive cooperativity due to the ligand-promoted allosteric switch can be masked, in cases of such heterooligomers, by apparent negative cooperativity owing to site heterogeneity. Here, we derived expressions for the Hill coefficient, in the case of a heterodimer, in which the contributions from the ligand-promoted allosteric switch and site heterogeneity are separated. Using these equations and simulations for higher order oligomers, we show under which conditions site heterogeneity can significantly mask the extent of observed positive cooperativity.

On the Emergence of Orientational Order in Folded Proteins with Implications for Allostery

The beautiful structures of single- and multi-domain proteins are clearly ordered in some fashion but cannot be readily classified using group theory methods that are successfully used to describe periodic crystals. For this reason, protein structures are considered to be aperiodic, and may have evolved this way for functional purposes, especially in instances that require a combination of softness and rigidity within the same molecule. By analyzing the solved protein structures, we show that orientational symmetry is broken in the aperiodic arrangement of the secondary structure elements (SSEs), which we deduce by calculating the nematic order parameter, P2. We find that the folded structures are nematic droplets with a broad distribution of P2. We argue that a non-zero value of P2, leads to an arrangement of the SSEs that can resist external forces, which is a requirement for allosteric proteins. Such proteins, which resist mechanical forces in some regions while being flexible in others, transmit signals from one region of the protein to another (action at a distance) in response to binding of ligands (oxygen, ATP, or other small molecules).