Protein-ligand binding with a missing species

Considerable experimental evidence has been produced recently that shows that in the binding of oxygen or carbon monoxide to certain tetrameric hemoglobins, the triply-ligated species is virtually non-existent. The binding polynomial representing this phenomenon for the general case is P(x) = 1 + beta 1x + ... + beta n-1xn-1 + beta nxn, where beta n-1 is nearly zero. The zeros, factorization and associated Hill plots of such binding polynomials with beta n-1 = 0 are investigated for the general case, and are analyzed in detail for n = 3 and n = 4. These results are then compared with the results obtained from experimental data on a number of tetrameric hemoglobins for which beta 3 is small. One concludes that, apart from the slope of the high-saturation asymptote of the Hill plot, a small perturbation of beta 3 from zero produces small changes in other properties associated with the binding process, such as fractional saturation, maximum Hill slope, and zeros and factorization of the binding polynomial.     

The relationship between zeros and factors of binding polynomials and cooperativity in protein-ligand binding

Cooperativity in the protein-ligand binding process is discussed in terms of the zeros of the binding polynomial and the corresponding possible factorizations of the binding polynomial into polynomials having non-negative coefficients. Particular attention is paid to the case in which the real parts of all zeros are negative (Hurwitz polynomial) and the case in which the binding polynomial admits no positive factorization (positive irreducible polynomial). Such factorizations are then interpreted as site linkage patterns and related to cooperativity. The possible combinations of zeros of the binding polynomials for the MWC and KNF tetrahedral, square and linear models are determined and the corresponding factorization and linkage patterns analyzed. An application and interpretation are then made for data obtained from Trout I hemoglobin.     

Analysis of oxygen equilibria of the giant hemoglobin from the earthworm Eisenia foetida using the Adair model

Oxygen equilibrium curves of the giant hemoglobin from the earthworm Eisenia foetida were determined at various concentrations of cations. Using the Adair model of 12 oxygenation steps, we succeeded in fitting the data better than the simple concerted model (MWC model). Analysis of the Adair constants (K1 to K12) indicated that the increase in oxygen affinity occurs in the last six steps (K7 to K12) of the oxygen binding and that it is enhanced by increase in Ca2+ concentration. The Hill coefficient (nmax) at pH 7.5 attained a maximum value of 9.76 at 20 mM CaCl2. In the presence of physiological levels of Ca2+ (5 mM), the Bohr effect was similar to that seen in vertebrates. The data were consistent with the release of two Bohr protons being accompanied by the oxygen-linked binding of one Ca2+. Mg2+ and Na+ exerted a similar effect on the hemoglobin, though to a lesser extent. The stoichiometry of Ca2+ binding of the hemoglobin revealed the presence of two classes of binding sites, of which the affinities are high (Ka = 8.8 x 10(3) +/- 103 M-1) and low. The number of high affinity sites per heme was found to be 0.3, comparable to the number of oxygen-linked Ca2+ binding sites.     

Moment arms and musculotendon lengths estimation for a three-dimensional lower-limb model

This paper presents a set of polynomial expressions that can be used as regression equations to estimate length and three-dimensional moment arms of 43 lower-limb musculotendon actuators. These equations allow one to find, at a low computational cost, the musculotendon geometric parameters required for numerical simulation of large musculoskeletal models. Nominal values for these biomechanical parameters were established using a public-domain musculoskeletal model of the lower limb (IEEE Trans. Biomed. Eng. 37 (1990) 757). To fit these nominal values, regression equations with different levels of complexity were generated, based on the number of generalized coordinates of the joints spanned by each musculotendon actuator. Least squares fitting was used to identify regression equation coefficients. The goodness of the fit and confidence intervals were assessed, and the best fitting equations selected.     

The model of Monod, Wyman, and Changeux generates two sets of parameters when applied to oxygen binding in hemoglobin

The model of Monod, Wyman and Changeux is applied to binding phenomena where the Mass Law and its expansion are employed. In this communication the model of Monod, Wyman and Changeux (MWC) is applied to analyze the oxygen binding reaction in hemoglobin. The symmetrical structure of the MWC model with its three parameters is such that two sets of these parameters, rather than one, fit experimental data for the binding of oxygen to hemoglobin.     

A new measure of cooperativity in protein-ligand binding

An allosteric binding system consisting of a single ligand and a nondissociating macromolecule having multiple binding sites can be represented by a binding polynomial. Various properties of the binding process can be obtained by analyzing the coefficients of the binding polynomial and such functions as the binding curve and the Hill plot. The Hill plot has an asymptote of unit slope at each end and the departure of the slope from unity at any point can be used to measure the effective interaction free energy at that point. Of particular interest in detecting and measuring cooperativity are extrema of the Hill slope and its value at the half-saturation point. If the binding polynomial is symmetric, then there is an extremum of the Hill slope at the half-saturation point. This value, the Hill coefficient, is a convenient measure of cooperativity. The purpose of this paper is to express the Hill coefficient for symmetric binding polynomials in terms of the roots of the polynomial and to give an interpretation of cooperativity in terms of the geometric pattern of the roots in the complex plane. This interpretation is then applied to the binding polynomials for the MWC (Monod-Wyman-Changeux) and KNF (Koshland-Nemethy-Filmer) models.     

The binding of W7, an inhibitor of striated muscle contraction, to cardiac troponin C

W7 is a well-characterized calmodulin antagonist. It decreases the maximal tension and rate of ATP hydrolysis in cardiac muscle fibers. Cardiac troponin C (cTnC) has been previously implicated as the mechanistically significant target for W7 in the myofilament. Two-dimensional NMR spectra ({1H,15N}- and {1H,13C}-HSQCs) were used to monitor the Ca2+-dependent binding of W7 to cTnC. Titration of cTnC x 3Ca2+ with W7 indicated binding to both domains of the protein. We examined the binding of W7 to the separated domains of cTnC to simplify the spectral analysis. In the titration of the C-terminal domain (cCTnC x 2Ca2+), the spectral peaks originating from a subset of residues changed nonuniformly, and could not be well-described as single-site binding. A global fit of the cCTnC x 2Ca2+ titration data to a two-site, sequential binding model (47 residues simultaneously fit) yielded a dissociation constant (Kd1) of 0.85-0.91 mM for the singly bound state, with the second dissociation constant fit to 3.40-3.65 mM (> or = 4 x Kd1). The titration data for the N-terminal domain (cNTnC x Ca2+) was globally fit to single-site binding model with a Kd of 0.15-0.30 mM (41 residues fit). The data are consistent with W7 binding to each domain's major hydrophobic pocket, coordinating side chains responsible for liganding cTnI. When in muscle fibers, W7 may compete with cTnI for target sites on cTnC.     

Acetylcholine and epibatidine binding to muscle acetylcholine receptors distinguish between concerted and uncoupled models.

The muscle acetylcholine receptor (AChR) has served as a prototype for understanding allosteric mechanisms of neurotransmitter-gated ion channels. The phenomenon of cooperative agonist binding is described by the model of Monod et al. (Monod, J., Wyman, J., and Changeux, J. P. (1965) J. Mol. Biol. 12, 88-118; MWC model), which requires concerted switching of the two binding sites between low and high affinity states. The present study examines binding of acetylcholine (ACh) and epibatidine, agonists with opposite selectivity for the two binding sites of mouse muscle AChRs. We expressed either fetal or adult AChRs in 293 HEK cells and measured agonist binding by competition against the initial rate of 125I-alpha-bungarotoxin binding. We fit predictions of the MWC model to epibatidine and ACh binding data simultaneously, taking as constants previously determined parameters for agonist binding and channel gating steps, and varying the agonist-independent parameters. We find that the MWC model describes the apparent dissociation constants for both agonists but predicts Hill coefficients that are far too steep. An Uncoupled model, which relaxes the requirement of concerted state transitions, accurately describes binding of both ACh and epibatidine and provides parameters for agonist-independent steps consistent with known aspects of AChR function.     

Noninvasive determination of upper airway resistance and flow limitation.

We have shown that a polynomial equation, FP = AP3 + BP2 + CP + D, where F is flow and P is pressure, can accurately determine the presence of inspiratory flow limitation (IFL). This equation requires the invasive measurement of supraglottic pressure. We hypothesized that a modification of the equation that substitutes time for pressure would be accurate for the detection of IFL and allow for the noninvasive measurement of upper airway resistance. The modified equation is Ft = At3 + Bt2 + Ct + D, where F is flow and t is time from the onset of inspiration. To test our hypotheses, data analysis was performed as follows on 440 randomly chosen breaths from 18 subjects. First, we performed linear regression and determined that there is a linear relationship between pressure and time in the upper airway (R2 0.96 +/- 0.05, slope 0.96 +/- 0.06), indicating that time can be a surrogate for pressure. Second, we performed curve fitting and found that polynomial equation accurately predicts the relationship between flow and time in the upper airway (R2 0.93 +/- 0.12, error fit 0.02 +/- 0.08). Third, we performed a sensitivity-specificity analysis comparing the mathematical determination of IFL to manual determination using a pressure-flow loop. Mathematical determination had both high sensitivity (96%) and specificity (99%). Fourth, we calculated the upper airway resistance using the polynomial equation and compared the measurement to the manually determined upper airway resistance (also from a pressure-flow loop) using Bland-Altman analysis. Mean difference between calculated and measured upper airway resistance was 0.0 cmH2O x l(-1) x s(-1) (95% confidence interval -0.2, 0.2) with upper and lower limits of agreement of 2.8 cmH2O x l(-1) x s(-1) and -2.8 cmH2O x l(-1) x s(-1). We conclude that a polynomial equation can be used to model the flow-time relationship, allowing for the objective and accurate determination of upper airway resistance and the presence of IFL.     

Sigmoidicity in Allosteric Models

We show results complementary to papers by Bardsley and Waight [2, 3], Gibson and Levin [13], Goldbeter [14], and Karlin [19] on sigmoidicity as an essential feature of allosteric models, possibly leading to a criterion of choice between these. In particular, we give the explicit form for the second derivative of the saturation function in the MWC case, and also the calculation of the binding polynomial in the circular KNF case. First, we give analytical conditions of sigmoidicity for each characteristic function in the MWC and KNF allosteric models. In the MWC model, the regions of sigmoidicity are different for the state and saturation functions, and when the catalytic activities of the two conformational states are different, the area of sigmoidicity is significantly larger for the steady-state rate function than for the saturation function. Furthermore, we rigorously prove the existence of mixed kinetic cooperativity in certain conditions. In the KNF model, the state and saturation functions are the same and their sigmoidicity depends only on the degree of coupling between subunits and on the relative stability of the asymmetrically induced subunit interactions. Finally, we suggest a theoretical criterion for discrimination between allosteric models.     

An analytic solution to the Monod-Wyman-Changeux model and all parameters in this model.

Starting from the Monod-Wyman-Changeux (MWC) model (Monod, J., J. Wyman, and J. P. Changeux. 1965. J. Mol. Biol. 12:88-118), we obtain an analytical expression for the slope of the Hill plot at any ligand concentration. Furthermore, we derive an equation satisfied by the ligand concentration at the position of maximum slope. From these results, we derive a set of formulas which allow determination of the parameters of the MWC model (kR, C, and L) from the value of the Hill coefficient, nH, the ligand concentration at the position of maximum slope [( A]0), and the value of nu/(n-nu) at this point. We then outline procedures for utilizing these equations to provide a "best fit" of the MWC model to the experimental data, and to obtain a refined set of the parameters. Finally, we demonstrate the applicability of the technique by analysis of oxygen binding data for Octopus hemocyanin.     

An adjusted coefficient of determination (R2) for generalized linear mixed models in one go

The coefficient of determination (R²) is a common measure of goodness of fit for linear models. Various proposals have been made for extension of this measure to generalized linear and mixed models. When the model has random effects or correlated residual effects, the observed responses are correlated. This paper proposes a new coefficient of determination for this setting that accounts for any such correlation. A key advantage of the proposed method is that it only requires the fit of the model under consideration, with no need to also fit a null model. Also, the approach entails a bias correction in the estimator assessing the variance explained by fixed effects. Three examples are used to illustrate new measure. A simulation shows that the proposed estimator of the new coefficient of determination has only minimal bias.     

Multi-well chip for forming a uniform embryoid body in a tiny droplet with mouse embryonic stem cells

A multi-well chip (MWC) is described by which mouse embryonic carcinoma (EC) stem cells form a comparatively more rapid and uniform embryoid body (EB) over the conventional hanging drop (HD) method. The newly developed MWC consists of an array of extruded through-holes, each of which holds a droplet of the cell suspension. The study found that the small curvature radius of the droplet in the MWC improved the EB formation rate of a hanging drop from 70% to 98%. Furthermore, the EBs formed by the MWC were uniformly round in shape regardless of the number of suspended cells ranging from 0.5 x 10(3) to 20 x 10(3). The ratio of beating colonies from the MWC was over 2-fold larger than that from HD. The experiments demonstrate that the MWC will be a valuable experimental tool for robust and reproducible EB-based differentiation of a defined number of ES cells.     

Radiation-related posterior lenticular opacities in Hiroshima and Nagasaki atomic bomb survivors based on the DS86 dosimetry system

This paper investigates the quantitative relationship of ionizing radiation to the occurrence of posterior lenticular opacities among the survivors of the atomic bombings of Hiroshima and Nagasaki suggested by the DS86 dosimetry system. DS86 doses are available for 1983 (93.4%) of the 2124 atomic bomb survivors analyzed in 1982. The DS86 kerma neutron component for Hiroshima survivors is much smaller than its comparable T65DR component, but still 4.2-fold higher (0.38 Gy at 6 Gy) than that in Nagasaki (0.09 Gy at 6 Gy). Thus, if the eye is especially sensitive to neutrons, there may yet be some useful information on their effects, particularly in Hiroshima. The dose-response relationship has been evaluated as a function of the separately estimated gamma-ray and neutron doses. Among several different dose-response models without and with two thresholds, we have selected as the best model the one with the smallest x2 or the largest log likelihood value associated with the goodness of fit. The best fit is a linear gamma-linear neutron relationship which assumes different thresholds for the two types of radiation. Both gamma and neutron regression coefficients for the best fitting model are positive and highly significant for the estimated DS86 eye organ dose.     

Osmometric analysis of quasi-indefinite mixed associations of the first kind

Mixed associations of the type A + B----AB, A + AB----A2B, ..., A + Ai-1 B----AiB, ... are readily analyzed by osmometric methods. The equilibrium molar concentration of A, mA, is obtained very simply from mA = meq-m0B; here meq = c/Meqn is the equilibrium molar concentration of all associating species and m0B denotes the stoichiometric or original molar concentration of B. The quantity mB can then be obtained from methods developed by Steiner. The value of the binding polynomial lambda is given by lambda = m0B/mB; lambda is a function of mA only. In principle, one can evaluate the equilibrium constants (kA,B,etc.) by fitting lambda to the appropriate polynomial in mA of degree n (n = 2, 3, ...). The binding polynomial lambda is analogous to polynomials encountered in the analysis of self-associations. By making some simple assumptions one can develop four analogs of two sequential, equal equilibrium constant (SEK) or two attenuated equilibrium constant (AK) models. With the aid of r (the number average degree of binding), g (the osmotic coefficient), lambda, as well as mA and mB, one can evaluate the equilibrium constant or constants. The methods developed here can be extended to the nonideal case.     

Thermodynamics of nucleotide binding to the chaperonin GroEL studied by isothermal titration calorimetry: evidence for noncooperative nucleotide binding.

We characterized the thermodynamics of binding reactions of nucleotides ADP and ATPgammaS (a nonhydrolyzable analog of ATP) to GroEL in a temperature range of 5 degrees C to 35 degrees C by isothermal titration calorimetry. Analysis with a noncooperative binding model has shown that the bindings of nucleotides are driven enthalpically with binding constants of 7x103 M-1 and 4x104 M-1 for ADP and ATPgammaS, respectively, at 26 degrees C and that the heat capacity change DeltaCp is about 100 cal/mol.K for both the nucleotides. The stoichiometries of binding were about 8 and 9 molecules for ADP and ATPgammaS, respectively, per GroEL tetradecamer at 5 degrees C, and both increased with temperature to reach about 14 (ADP) and 12 (ATPgammaS) for both nucleotides at 35 degrees C. The absence of initial increase of binding heat as well as Hill coefficient less than 1.2, which were obtained from the fitting to the model curve by assuming positive cooperativity, showed that there was virtually no positive cooperativity in the nucleotide bindings. Incorporating a difference in affinity for the nucleotide (ADP and ATPgammaS) between the two rings of GroEL into the noncooperative binding model improved the goodness of fitting and the difference in the affinity increased with decreasing temperature.     

Binding of oxygen and carbon monoxide to arthropod hemocyanin: an allosteric analysis

The binding of oxygen and carbon monoxide to hemocyanin from the mangrove crab Scylla serrata and the lobster Homarus americanus has been studied by thin-layer optical absorption and front face fluorescence techniques. Three types of experiments were performed on subunit and oligomeric preparations of each hemocyanin: oxygen binding, carbon monoxide binding, and oxygen-carbon monoxide competition studies. The results obtained from the subunit preparations of dissociated oligomers from both hemocyanins show that the binding site can be ligated by either one oxygen or one carbon monoxide. The binding results obtained with the oligomeric samples of hemocyanin from both species cannot be described by the two-state MWC model [Monod, J., Wyman, J., & Changeux, J. P. (1965) J. Mol. Biol. 12, 88-118] since the data from the three types of binding experiments cannot be fit with a single set of binding constants. The MWC model has been extended by including a third allosteric form, and an analysis based on the three-state model is able to fit the data from the three types of experiments with the same set of binding constants. The comparison of the oxygen to carbon monoxide affinity ratios (kO2/kCO) indicates that the structure around the binding site of subunits in the T form oligomer is similar to that of the free subunits. The oligomeric forms of both these hemocyanins bind carbon monoxide with a weak but definite positive cooperativity. An analysis of the affinity ratios for the T, S, and R forms suggests that the high affinity of the R form results from a specific interaction between oxygen and binding site.     

The effect of overshooting the target strain on estimating viscoelastic properties from stress relaxation experiments

BACKGROUND: Tendon's mechanical behaviors have frequently been quantified using the quasi-linear viscoelastic (QLV) model. The QLV parameters are typically estimated by fitting the model to a single-step stress relaxation experiment. Unfortunately, overshoot of the target strain occurs to some degree in most experiments. This has never been formally investigated even though failing to measure, minimize, or compensate for overshoot may cause large errors in the estimation of parameters. Therefore, the objective of this study was to investigate the effect of overshoot on the estimation of QLV parameters. METHOD OF APPROACH: A simulated experiment was first performed to quantify the effect of different amounts of overshoot on the estimated QLV parameters. Experimental data from tendon was then used to determine if the errors associated with overshoot could be reduced when a direct fit is used (i.e., the actual strain history was used in the curve fit). RESULTS: We found that both the elastic and viscous QLV parameters were incorrectly estimated if overshoot was not properly accounted for in the fit. Furthermore, the errors associated with overshoot were partially reduced when overshoot was accounted for using a direct fit. CONCLUSIONS: A slow ramp rate is recommended to limit the amount of overshoot and a direct fit is recommended to limit the errors associated with overshoot, although other approaches such as adjusting the control system to limit overshoot could also be utilized.     

Functional properties of human skeletal muscle acetylcholine receptors expressed by the TE671 cell line

Functional properties of acetylcholine receptors from intact TE671 human medulloblastoma cells were examined using tracer ion flux, ligand competition against 125I-labeled alpha-bungarotoxin binding, and single channel recording measurements. 125I-Labeled alpha-bungarotoxin binds to surface receptors with the forward rate constant 1.8 X 10(5) M-1 s-1 and dissociates with the rate constant 4.6 X 10(-5) s-1, at 21 degrees C; the apparent dissociation constant is 2.6 X 10(-10) M. alpha-Bungarotoxin binds to at least two sites/receptor, but blocks agonist-induced 22Na+ uptake when bound to only one site. The reversible antagonists, dimethyl-d-tubocurarine and gallamine, occupy two sites which exhibit nearly equivalent affinities, but block agonist-induced uptake by occupying only one site. Strong agonists activate rapid sodium uptake with relatively low affinity, but desensitize with a much higher affinity; among agonists, the ratio of low to high affinity dissociation constants ranges from 1600 to 4000. By using the estimated dissociation constants, the allosteric model of Monod, Wyman, and Changeux (MWC) can be fitted to the concentration dependencies of both steady-state agonist occupancy and desensitization. The fitting analysis discloses an allosteric constant of 3 X 10(-5), which is the ratio of activatable to desensitized receptors in the absence of agonist. The rate of recovery from desensitization can exceed the rate of onset of desensitization elicited by low concentrations of agonist, further supporting the general MWC framework. Single channel recordings show that the channel opening probability is greater than 0.7 at high agonist concentrations. Favorable channel opening is shown to only slightly oppose strong desensitization.