A probabilistic model for fitting MWC polynomials in protein-ligand binding

Given a binding polynomial in Adair form, A(x) = 1 + beta 1 x + ... + beta n x n, beta i greater than or equal to 0, a basic problem is to determine a method of fitting a model polynomial to A(x) and a quantitative measure of the goodness of fit. This paper presents such a method for fitting Monod-Wyman-Changeux (MWC) model polynomials when A(x) is of degree three or four. The method of fitting is based on the property that the zeros of an MWC polynomial of any degree lie on a circle in the complex plane. The parameters in the MWC model are determined so that if possible this circle coincides with the circle on which lie the zeros of A(x). The measure of goodness of fit is provided by a probabilistic model which gives the probability that a binding polynomial has its zeros on a circle on which lie the zeros of an MWC polynomial and if so, the probability that the juxtaposition of the two sets of zeros can occur by chance alone.     

Application of a generalized MWC model for the mathematical simulation of metabolic pathways regulated by allosteric enzymes

In our effort to elucidate the systems biology of the model organism, Escherichia coli, we have developed a mathematical model that simulates the allosteric regulation for threonine biosynthesis pathway starting from aspartate. To achieve this goal, we used kMech, a Cellerator language extension that describes enzyme mechanisms for the mathematical modeling of metabolic pathways. These mechanisms are converted by Cellerator into ordinary differential equations (ODEs) solvable by Mathematica. In this paper, we describe a more flexible model in Cellerator, which generalizes the Monod, Wyman, Changeux (MWC) model for enzyme allosteric regulation to allow for multiple substrate, activator and inhibitor binding sites. Furthermore, we have developed a model that describes the behavior of the bifunctional allosteric enzyme aspartate kinase I-homoserine dehydrogenase I (AKI-HDHI). This model predicts the partition of enzyme activities in the steady state which paves the way for a more generalized prediction of the behavior of bifunctional enzymes.     

Nested MWC model describes hydrolysis of GroEL without assuming negative cooperativity in binding

Folding assistance and ATPase activity of GroEL are based on the existence of different conformations. In order to characterise these conformations, published data on steady state ATPase activity in the absence of GroES were reanalysed simultaneously in terms of the Nested MWC model. This model is a hierarchical extension of the symmetry-model of Monod et al. [J. Mol. Biol. 12 (1965) 88]. An unique set of GroEL specific parameters was obtained. This set was supported by comparison of predictions arising from this set of values with experimental data for hydrolysis of ATP in the presence of ADP and ATPgammaS, binding of ATPgammaS and ADP to GroEL in the absence of ATP, and binding of ATP as monitored by fluorescence labelling. Thus, for the first time, multiple data sets for the interaction of nucleotides with GroEL are described quantitatively by an allosteric model. A noteworthy feature of our model is that no negative cooperativity in ATP binding occurs in accordance to experimental observations. Furthermore, the model also includes the existence of a conformation with very high ATPase activity. Such a conformation might be of importance at a certain stage in the folding cycle.     

Oxygen binding to sickle cell hemoglobin.

The extent of oxygen binding and light scattering of concentrated solutions of hemoglobin S have been determined as a function of oxygen partial pressure using a thin film optical cell. Nearly reversible oxygen binding is observed as witnessed by the small hysteresis found between slow deoxygenation and reoxygenation runs. High co-operativity is noted from unusually large concentration-dependent Hill coefficients when aggregated hemoglobin S is present. The application of linkage theory with the inclusion of non-ideal solution properties permits a test of various simple models for oxygen binding to both the monomer (α₂β₂^s) and polymer (aggregated) phase. It is concluded that oxygen binding to the polymer is either negligible or small under present experimental conditions. Phase diagrams of the solution concentration in equilibrium with polymer phase as a function of oxygen partial pressure are derived using best fit values of polymer parameters.     

Diffusion-limited binding explains binary dose response for local arterial and tumour drug delivery

Background:  Local drug delivery has transformed medicine, yet it remains unclear how drug efficacy depends on physicochemical properties and delivery kinetics. Most therapies seek to prolong release, yet recent studies demonstrate sustained clinical benefit following local bolus endovascular delivery.
Objectives:  The purpose of the current study was to examine interplay between drug dose, diffusion and binding in determining tissue penetration and effect.
Methods:  We introduce a quantitative framework that balances dose, saturable binding and diffusion, and measured the specific binding parameters of drugs to target tissues.
Results:  Model reduction techniques augmented by numerical simulations revealed that impact of saturable binding on drug transport and retention is determined by the magnitude of a binding potential, B_p , ratio of binding capacity to product of equilibrium dissociation constant and accessible tissue volume fraction. At low B_p (< 1), drugs are predominantly free and transport scales linearly with concentration. At high B_p (> 40), drug transport exhibits threshold dependence on applied surface concentration.
Conclusions:  In this paradigm, drugs and antibodies with large B_p penetrate faster and deeper into tissues when presented at high concentrations. Threshold dependence of tissue transport on applied surface concentration of paclitaxel and rapamycin may explain threshold dose dependence of in vivo biological efficacy of these drugs.     

The determination of positive and negative co-operativity with allosteric enzymes and the interpretation of sigmoid curves and non-linear double reciprocal plots for the MWC and KNF models

(i) It is proved that only four independent constants can ever be obtained by extrapolation procedures applied to non-hyperbolic steady-state or binding data, (ii) Analysis of the algebraic graphs $\text{y}\text{x}$, $\text{(1/y)}\text{(1/x)}$, $\text{y}\text{(}\text{y}\text{x}\text{)}$ and ($\text{x}\text{y}\text{)/x}$ is shown to require a knowledge of the sign of six curve shape determinants. In each case, the sign is a necessary and sufficient condition for a specific curve shape feature, (iii) The precise graphical effect of positive and negative co-operativity then requires the definition of two reference curves, the osculating hyperbola at zero substrate concentration, OH(0), and the osculating hyperbola at infinite substrate concentration OH(∞). These are better first order approximations than the Hill equation, (iv) Rules for determining unambiguously the sign of initial, final and overall co-operativity coefficients by inspection of non-hyperbolic binding curves are then possible, (v) These rules require that saturation data for:

$\text{y=}\text{∑}\text{i=1}\text{n}\text{a}{}_{\mathrm{i}}\text{x}{}^{\mathrm{i}}\text{∑}\text{i=0}\text{n}\text{β}{}_{\mathrm{i}}\text{x}{}^{\mathrm{i}}$

be fitted by computer for low concentrations to the hyperbola:

$\text{OH}\text{(o)=}\text{(}\text{-a}{}_1{}^2\text{ψ}{}_{11}{}^{20}\text{)x}\text{[(}\text{-a}{}_1\text{β}{}_0\text{ψ}{}_{11}{}^{20}\text{)+x]}$

while regression of high substrate concentration data is to:

$\text{OH}\text{(∞)=}\text{(}\text{a}{}_{\mathrm{n}}\text{β}{}_{\mathrm{n}}\text{)x}\text{[(}\text{φ}{}_{\mathrm{n,n-1}}\text{a}{}_{\mathrm{n}}\text{β}{}_{\mathrm{n}}\text{)+x]}$

. Comparisons of the best fit pseudo-kinetic constants then gives the type of co-operativity present in an unambiguous way with no assumptions as to molecular mechanism, (vi) These rules are then applied to the MWC and KNF allosteric models of ligand binding and the constraints necessary for specific curve shape effects are given, (vii) The graphical expression of positive or negative final co-operativity depends only on events at high substrate concentration but overall and initial co-operativities produce specific geometric effects depending upon the difference between behaviour of saturation data at both extremes of concentration, (viii) This apparent anomaly is explained by a discussion of the relationships between the osculating hyperbolae, the theoretical parent hyperbola and the Hill plot asymptotes.     

Ligand binding and the gelation of sickle cell hemoglobin.

The solubility equilibrium between monomer and polymer which has been shown to exist in deoxyhemoglobin S solutions is examined in solutions partially saturated with carbon monoxide. The total solubility is found to increase monotonically with increasing fractional saturation. At low fractional saturations the increase is nearly linear, amounting roughly to an increase of 0.01 g cm^?3 in solubility for each 10% increase in fractional saturation. Linear dichroism measurements on the spontaneously aligned polymer phase are used to examine the composition of the polymer as a function of the fractional saturation of the corresponding solution phase. The dichroism experiments show that the polymer phase contains less than 5% of CO-liganded hemes even at supernatant fractional saturations in excess of 70%. The polymer selects against totally liganded hemoglobin molecules by a minimum factor of 65 and against singly liganded molecules by a factor of at least 2.5. Consequently, polymerized hemoglobin S has a ligand affinity which is significantly lower than that of monomeric hemoglobin S in the deoxy quaternary structure.The kinetics of the polymerization reaction in the presence of CO are similar to those observed in pure deoxyhemoglobin S solutions. The polymerization is preceded by a pronounced delay, the duration of which, t_d, is proportional roughly to the 30th power of the solubility. At low fractional saturations, this amounts to a tenfold increase in t_d for each 10% increase in the fractional saturation.These results show that the polymerization reaction is nearly specific for deoxyhemoglobin. Models for the dependence of the solubility and the polymer saturation on ligand partial pressure demonstrate the importance of solution phase non-ideality in determining the solubility of mixtures. The results require selection against partially liganded species which is significantly greater than is predicted by the two-state allosteric model. The data are compatible with either sequential or allosteric models in which the major polymerized component is the unliganded hemoglobin molecule.     

Catch-bond model derived from allostery explains force-activated bacterial adhesion

High shear enhances the adhesion of Escherichia coli bacteria binding to mannose coated surfaces via the adhesin FimH, raising the question as to whether FimH forms catch bonds that are stronger under tensile mechanical force. Here, we study the length of time that E. coli pause on mannosylated surfaces and report a double exponential decay in the duration of the pauses. This double exponential decay is unlike previous single molecule or whole cell data for other catch bonds, and indicates the existence of two distinct conformational states. We present a mathematical model, derived from the common notion of chemical allostery, which describes the lifetime of a catch bond in which mechanical force regulates the transitions between two conformational states that have different unbinding rates. The model explains these characteristics of the data: a double exponential decay, an increase in both the likelihood and lifetime of the high-binding state with shear stress, and a biphasic effect of force on detachment rates. The model parameters estimated from the data are consistent with the force-induced structural changes shown earlier in FimH. This strongly suggests that FimH forms allosteric catch bonds. The model advances our understanding of both catch bonds and the role of allostery in regulating protein activity.     

Linked functions in allosteric proteins: Extension of the concerted (MWC) model for ligand-linked subunit assembly and its application to human hemoglobins

The allosteric model of Monod et al. (1965) (MWC) has been extended to take into account the effects of subunit dissociation. The problem is formulated theoretically in terms of a general model for two allosteric species (dimers and tetramers) linked by a polymerization reaction. Relationships are presented for interpreting the dimer-tetramer association constants in terms of allosteric model parameters.Sub-cases of the general model were tested against recent experimental data on the oxygenation-linked dimer-tetramer equilibria in normal human hemoglobin and in the variant hemoglobin Kansas (β₁₀₂, Asp → Thr). The objectives of these analyses were: (1) to find the simplest models capable of describing the linked dimer-tetramer equilibria in the two hemoglobin systems, and (2) to evaluate the corresponding model parameters so that allosteric properties of the two hemoglobins may be compared.In the simplest version of the model, the dimer is half of an R-state tetramer. This model was found to be excluded unequivocally by the data for both normal hemoglobin and hemoglobin Kansas when the α and β chains have equal binding affinities. When this two-state model was modified to permit non-equivalent affinities for the chains, the model could be fitted to hemoglobin Kansas, but not to hemoglobin A. A model, in which the dimers are allowed to exist in a state different from the tetramer R state, was found to be consistent with the data for hemoglobin A, with equivalent binding by the α and β chains. For hemoglobin A, the unliganded R-state tetramers have a different subunit dissociation energy from that of fully liganded R-state tetramers. The simplest model capable of describing both hemoglobin A and hemoglobin Kansas was obtained by extending this three-state model to permit (but not require) functional non-equivalence of the α and β chains. For these MWC models, unique estimates were obtained for the model parameters.The allosteric constants for tetrameric hemoglobins A and Kansas are approximately equal. The value obtained from hemoglobin A is similar to previous estimates, whereas the value for hemoglobin Kansas is lower than previously estimated (Edelstein, 1971) by approximately two orders of magnitude. The low affinity of hemoglobin Kansas tetramer does not arise from an unusually high allosteric constant favoring the T-state species. It is largely the consequence of a greatly reduced oxygen affinity of β chains in the T state, and reduced values for the ratio between affinities in the R and T states.     

A monomer-dimer model explains the results of radiation inactivation: binding characteristics of insulin receptor purified from human placenta

The technique of radiation inactivation has been used on highly purified human placental insulin receptor in order to determine the functional molecular size responsible for the insulin binding and to evaluate the "affinity regulator" hypothesis, which has been proposed to explain the increase in specific insulin binding to rat liver membranes observed at low radiation doses [Harmon, J. T., Hedo, J. A., & Kahn, C. R. (1983) J. Biol. Chem. 258, 6875-6881]. Three different types of inactivation curves were observed: (1) biphasic with an enhanced binding activity after exposure to low radiation doses, (2) nonlinear with no change in binding activity after exposure to low radiation doses, and (3) linear with a loss in the binding activity with increasing radiation exposures. A monomer-dimer model was the simplest model that best described the three types of radiation inactivation curves observed. The model predicts that an increase in insulin binding activity would result after exposure to low radiation doses when the initial dimer/monomer ratio is equal to or greater than 1 and a monomer is more active than a dimer. The monomer size of the binding activity was estimated to be 227,000 daltons by this model. This value most likely reflects the size of the monomeric alpha beta form. To substantiate this model, the purified receptor was fractionated by Sepharose CL-6B chromatography. The insulin binding profile of this column indicated two peaks.(ABSTRACT TRUNCATED AT 250 WORDS)     

A mechanistic motor-clutch model that explains cell shape dynamics to cyclic stretch

Many cells in the body experience cyclic mechanical loading, which can impact cellular processes and morphology. In vitro studies often report that cells reorient in response to cyclic stretch of their substrate. To explore cellular mechanisms involved in this reorientation, a computational model was developed by adapting previous computational models of the actin–myosin–integrin motor-clutch system developed by others. The computational model predicts that under most conditions, actin bundles align perpendicular to the direction of applied cyclic stretch, but under specific conditions, such as low substrate stiffness, actin bundles align parallel to the direction of stretch. The model also predicts that stretch frequency impacts the rate of reorientation and that proper myosin function is critical in the reorientation response. These computational predictions are consistent with reports from the literature and new experimental results presented here. The model suggests that the impact of different stretching conditions (stretch type, amplitude, frequency, substrate stiffness, etc.) on the direction of cell alignment can largely be understood by considering their impact on cell–substrate detachment events, specifically whether detachments preferentially occur during stretching or relaxing of the substrate.     

A ternary complex model explains the agonist-specific binding properties of the adenylate cyclase-coupled beta-adrenergic receptor

The unique properties of agonist binding to the frog erythrocyte beta-adrenergic receptor include the existence of two affinity forms of the receptor. The proportion and relative affinity of these two states of the receptor for ligands varies with the intrinsic activity of the agonist and the presence of guanine nucleotides. The simplest model for hormone-receptor interactions which can explain and reproduce the experimental data involves the interaction of the receptor R with an additional membrane component X, leading to the agonist-promoted formation of a high affinity ternary complex HRX. Computer modeling of agonist binding data with a ternary complex model indicates that the model can fit the data with high accuracy under conditions where the ligand used is either a full or a partial agonist and where the system is altered by the addition of guanine nucleotide or after treatment with group-specific reagents, e.g. p-hydroxymercuribenzoate. The parameter estimates obtained indicate that the intrinsic activity of the agonist is correlated significantly with the affinity constant L of the component X for the binary complex HR. The major effect of adding guanine nucleotides is to destabilize the ternary complex HRX from which both the hormone H and the component X can dissociate. The modulatory role of nucleotides on the affinity of agonists for the receptor is consistent with the assumption that the component X is the guanine nucleotide binding site. The ternary complex model was also applied successfully to the turkey erythrocyte receptor system. The model provides a general scheme for the activation by agonists of adenylate cyclase-coupled receptor systems and also of other systems where the effector might be different.     

Evidence for carbon monoxide binding to sickle cell polymers during melting.

The melting of sickle cell hemoglobin (HbS) polymers was induced by rapid dilution using a stopped-flow apparatus. The kinetics of polymer melting were monitored using light scattering. Polymer melting in the absence of any hemoglobin ligand was compared to melting when the diluting buffer was saturated with carbon monoxide (CO). In this way the role of CO in polymer melting could be assessed. The data were analyzed using models that assumed that melting occurs only at the ends of polymers. It was further assumed that CO could only bind to HbS in the solution phase. However, our data could not be fitted to this model, where CO cannot bind directly to the polymer. Thus, CO probably binds directly to the polymers during our melting experiments. This result is discussed in terms of oxygen induced polymer melting and polymerization processes in sickle cell disease     

A multistranded polymer model explains MinDE dynamics in E. coli cell division

In Escherichia coli, the location of the site for cell division is regulated by the action of the Min proteins. These proteins undergo a periodic pole-to-pole oscillation that involves polymerization and ATPase activity of MinD under the controlling influence of MinE. This oscillation suppresses division near the poles while permitting division at midcell. Here, we propose a multistranded polymer model for MinD and MinE dynamics that quantitatively agrees with the experimentally observed dynamics in wild-type cells and in several well-studied mutant phenotypes. The model also provides new explanations for several phenotypes that have never been addressed by previous modeling attempts. In doing so, the model bridges a theoretical gap between protein structure, biochemistry, and mutant phenotypes. Finally, the model emphasizes the importance of nonequilibrium polymer dynamics in cell function by demonstrating how behavior analogous to the dynamic instability of microtubules is used by E. coli to achieve a sufficiently rapid timescale in controlling division site selection.     

Oxygen binding to partially oxidized hemoglobin. Analysis in terms of an allosteric model

We report on oxygen binding to partially oxidized (aquomet) hemoglobin. The fractional saturation with oxygen is evaluated by deconvoluting the optical absorption spectra, in the 500-700 nm wavelength region, in terms of oxyhemoglobin, deoxyhemoglobin and methemoglobin spectral components. Experiments have been performed with auto-oxidized samples and with samples obtained by mixing ferrous hemoglobin with fully oxidized hemoglobin (mixed samples). An increase in oxygen affinity and a decrease in cooperativity are observed on increasing the amount of ferric hemoglobin in the sample. A high cooperativity (nH approximately 2) is maintained even in the presence of 50-60% ferric hemes. Moreover, for equal amounts of methemoglobin the oxygen affinity is lower and the cooperativity higher for mixed samples than for those auto-oxidized. The results are analyzed within the framework of a modified Monod-Wyman-Changeux allosteric model taking into account the effects brought about by the presence of oxidized hemes and of alpha betta dimers. The distribution of ferric subunits within the tetramers in fully deoxygenated and fully oxygenated samples, as derived from the model, provides details on the cooperative behavior of partially oxidized hemoglobin.     

Genetically encoded photo-cross-linkers map the binding site of an allosteric drug on a G protein-coupled receptor

G protein-coupled receptors (GPCRs) are dynamic membrane proteins that bind extracellular molecules to transduce signals. Although GPCRs represent the largest class of therapeutic targets, only a small percentage of their ligand-binding sites are precisely defined. Here we describe the novel application of targeted photo-cross-linking using unnatural amino acids to obtain structural information about the allosteric binding site of a small molecule drug, the CCR5-targeted HIV-1 co-receptor blocker maraviroc.     

Relationship between ligand binding and conformational change of macromolecules in the two-state allosteric model

The relationship between the binding function $\text{Y}$ and the state function $\text{R}$ of an oligomeric protein has been analysed for the general two-state allosteric model. It is shown that this relation is determined by the numerical values of the inherent parameters of the model. The shape of the function $\text{Y}\text{= f (}\text{R}\text{)}$ can therefore be strictly concave, strictly convex or inverse sigmoidal according to the conditions. In the two-state allosteric model only a dimeric protein can display a linear relationship between $\text{Y}$ and $\text{R}$.In the paper general criteria for the estimation of the state function $\text{R}$ from experimentally obtained conformational parameters are discussed.     

Effects of S-nitrosation on oxygen binding by normal and sickle cell hemoglobin.

S-Nitrosated hemoglobin (SNO-Hb) is of interest because of the allosteric control of NO delivery from SNO-Hb made possible by the conformational differences between the R- and T-states of Hb. To better understand SNO-Hb, the oxygen binding properties of S-nitrosated forms of normal and sickle cell Hb were investigated. Spectral assays and electrospray ionization mass spectrometry were used to quantify the degree of S-nitrosation. Hb A(0) and unpolymerized Hb S exhibit similar shifts toward their R-state conformations in response to S-nitrosation, with increased oxygen affinity and decreased cooperativity. Responses to 2, 3-diphosphoglycerate were unaltered, indicating regional changes in the deoxy structure of SNO-Hb that accommodate NO adduction. A cycle of deoxygenation/reoxygenation does not cause loss of NO or appreciable heme oxidation. There is, however, appreciable loss of NO and heme oxidation when oxygen-binding experiments are carried out in the presence of glutathione. These results indicate that the in vivo stability of SNO-Hb and its associated vasoactivity depend on the abundance of thiols and other factors that influence transnitrosation reactions. The increased oxygen affinity and R-state character that result from S-nitrosation of Hb S would be expected to decrease its polymerization and thereby lessen the associated symptoms of sickle cell disease.     

A simple model explains the cell cycle-dependent assembly of centromeric nucleosomes in holocentric species

Centromeres are essential for chromosome movement. In independent taxa, species with holocentric chromosomes exist. In contrast to monocentric species, where no obvious dispersion of centromeres occurs during interphase, the organization of holocentromeres differs between condensed and decondensed chromosomes. During interphase, centromeres are dispersed into a large number of CENH3-positive nucleosome clusters in a number of holocentric species. With the onset of chromosome condensation, the centromeric nucleosomes join and form line-like holocentromeres. Using polymer simulations, we propose a mechanism relying on the interaction between centromeric nucleosomes and structural maintenance of chromosomes (SMC) proteins. Different sets of molecular dynamic simulations were evaluated by testing four parameters: (i) the concentration of Loop Extruders (LEs) corresponding to SMCs, (ii) the distribution and number of centromeric nucleosomes, (iii) the effect of centromeric nucleosomes on interacting LEs and (iv) the assembly of kinetochores bound to centromeric nucleosomes. We observed the formation of a line-like holocentromere, due to the aggregation of the centromeric nucleosomes when the chromosome was compacted into loops. A groove-like holocentromere structure formed after a kinetochore complex was simulated along the centromeric line. Similar mechanisms may also organize a monocentric chromosome constriction, and its regulation may cause different centromere types during evolution.