A study of statistical error in isothermal titration calorimetry

In isothermal titration calorimetry (ITC), the two main sources of random (statistical) error are associated with the extraction of the heat q from the measured temperature changes and with the delivery of metered volumes of titrant. The former leads to uncertainty that is approximately constant and the latter to uncertainty that is proportional to q. The role of these errors in the analysis of ITC data by nonlinear least squares is examined for the case of 1:1 binding, M+X right arrow over left arrow MX. The standard errors in the key parameters-the equilibrium constant Ko and the enthalpy DeltaHo-are assessed from the variance-covariance matrix computed for exactly fitting data. Monte Carlo calculations confirm that these "exact" estimates will normally suffice and show further that neglect of weights in the nonlinear fitting can result in significant loss of efficiency. The effects of the titrant volume error are strongly dependent on assumptions about the nature of this error: If it is random in the integral volume instead of the differential volume, correlated least-squares is required for proper analysis, and the parameter standard errors decrease with increasing number of titration steps rather than increase.     

Analysis of multitemperature isothermal titration calorimetry data at very low c: Global beats van't Hoff

Isothermal titration calorimetry data for very low c (≡K[M]₀) must normally be analyzed with the stoichiometry parameter n fixed — at its known value or at any reasonable value if the system is not well characterized. In the latter case, ΔH° (and hence n) can be estimated from the T-dependence of the binding constant K, using the van't Hoff (vH) relation. An alternative is global or simultaneous fitting of data at multiple temperatures. In this Note, global analysis of low-c data at two temperatures is shown to estimate ΔH° and n with double the precision of the vH method.     

Using nonlinear least squares to assess relative expression and its uncertainty in real-time qPCR studies

Relative expression ratios are commonly estimated in real-time qPCR studies by comparing the quantification cycle for the target gene with that for a reference gene in the treatment samples, normalized to the same quantities determined for a control sample. For the “standard curve” design, where data are obtained for all four of these at several dilutions, nonlinear least squares can be used to assess the amplification efficiencies (AE) and the adjusted ΔΔC_q and its uncertainty, with automatic inclusion of the effect of uncertainty in the AEs. An algorithm is illustrated for the KaleidaGraph program.     

Calibration in isothermal titration calorimetry: heat and cell volume from heat of dilution of NaCl(aq)

An isothermal titration calorimeter of the perfusion type (MicroCal model VP-ITC) is calibrated using the heat of dilution of NaCl in water. The relative apparent molar enthalpy function (L(phi)) for NaCl(aq) varies strongly and nonlinearly with concentration in the low-concentration region (<0.2M) that is sampled easily and extensively in a single program of injections of NaCl solution into water. This nonlinearity makes it possible to calibrate with respect to two quantities: the measured heat and the active cell volume. The heat factor is determined with typical standard error 0.003; its value in the current case is 0.987. The cell volume factor is 0.93 but is quite sensitive to possible systematic errors in the temperature and in the literature values for L(phi). Both correction factors are closely tied to the delivered volume from the injection syringe, which required a correction factor of 0.973, attributed to an instrumental gear ratio error. Temperature calibration of the instrument showed a small offset of 0.12K at the temperature 25 degrees C of the experiments, but the error increased to more than 1K at 46 degrees C. The experiments were not able to distinguish clearly between mixing algorithms that assume instantaneous mixing on injection and those that assume instantaneous injection followed by mixing; however, examination of these algorithms has revealed an error in a program widely used to analyze isothermal titration calorimetry data.     

Open-ITC: an alternate computational approach to analyze the isothermal titration calorimetry data of complex binding mechanisms

Isothermal titration calorimetry (ITC) is an important technique used in quantitatively analyzing the global mechanism of protein-protein or protein-ligand interactions through thermodynamic measurements. Among different binding mechanisms, the parallel and ligand induced protein oligomerization mechanisms are technically difficult to analyze compared with a sequential binding mechanism. Here, we present a methodology implemented as a program "Open-ITC" that eliminates the need for exact analytical expressions for free ligand concentrations [L] and mole fractions of bound ligand θ that are required for the thermogram analysis. Adopting a genetic algorithm-based optimization, the thermodynamic parameters are determined, and its standard error is evaluated at the global minimum by calculating the Jacobian matrix. This approach yielded a statistically consistent result for a single-site and a two-site binding protein-ligand system. Further, a comparative simulation of a two-step sequential, a parallel, and a ligand induced oligomerization model revealed that their mechanistic differences are discernable in ITC thermograms, only if the first binding step is weaker compared with the second binding step (K(1) 相似文献   

A comparison of simultaneous autoregressive and generalized least squares models for dealing with spatial autocorrelation

Aim  In their recent paper, Kissling & Carl (2008 ) recommended the spatial error simultaneous autoregressive model (SAR_err) over ordinary least squares (OLS) for modelling species distribution. We compared these models with the generalized least squares model (GLS) and a variant of SAR (SAR_vario). GLS and SAR_vario are superior to standard implementations of SAR because the spatial covariance structure is described by a semivariogram model.
Innovation  We used the complete datasets employed by Kissling & Carl (2008 ), with strong spatial autocorrelation, and two datasets in which the spatial structure was degraded by sample reduction and grid coarsening. GLS performed consistently better than OLS, SAR_err and SAR_vario in all datasets, especially in terms of goodness of fit. SAR_vario was marginally better than SAR_err in the degraded datasets.
Main conclusions  GLS was more reliable than SAR-based models, so its use is recommended when dealing with spatially autocorrelated data.     

A Bayesian extension of phylogenetic generalized least squares: Incorporating uncertainty in the comparative study of trait relationships and evolutionary rates

Phylogenetic comparative methods use tree topology, branch lengths, and models of phenotypic change to take into account nonindependence in statistical analysis. However, these methods normally assume that trees and models are known without error. Approaches relying on evolutionary regimes also assume specific distributions of character states across a tree, which often result from ancestral state reconstructions that are subject to uncertainty. Several methods have been proposed to deal with some of these sources of uncertainty, but approaches accounting for all of them are less common. Here, we show how Bayesian statistics facilitates this task while relaxing the homogeneous rate assumption of the well-known phylogenetic generalized least squares (PGLS) framework. This Bayesian formulation allows uncertainty about phylogeny, evolutionary regimes, or other statistical parameters to be taken into account for studies as simple as testing for coevolution in two traits or as complex as testing whether bursts of phenotypic change are associated with evolutionary shifts in intertrait correlations. A mixture of validation approaches indicates that the approach has good inferential properties and predictive performance. We provide suggestions for implementation and show its usefulness by exploring the coevolution of ankle posture and forefoot proportions in Carnivora.     

Weighted averaging partial least squares regression (WA-PLS): an improved method for reconstructing environmental variables from species assemblages

Weighted averaging regression and calibration form a simple, yet powerful method for reconstructing environmental variables from species assemblages. Based on the concepts of niche-space partitioning and ecological optima of species (indicator values), it performs well with noisy, species-rich data that cover a long ecological gradient (>3 SD units). Partial least squares regression is a linear method for multivariate calibration that is popular in chemometrics as a robust alternative to principal component regression. It successively selects linear components so as to maximize predictive power. In this paper the ideas of the two methods are combined. It is shown that the weighted averaging method is a form of partial least squares regression applied to transformed data that uses the first PLS-component only. The new combined method, ast squares, consists of using further components, namely as many as are useful in terms of predictive power. The further components utilize the residual structure in the species data to improve the species parameters (optima) in the final weighted averaging predictor. Simulations show that the new method can give 70% reduction in prediction error in data sets with low noise, but only a small reduction in noisy data sets. In three real data sets of diatom assemblages collected for the reconstruction of acidity and salinity, the reduction in prediction error was zero, 19% and 32%.     

Least median of squares and iteratively re‐weighted least squares as robust linear regression methods for fluorimetric determination of α‐lipoic acid in capsules in ideal and non‐ideal cases of linearity

This study outlines two robust regression approaches, namely least median of squares (LMS) and iteratively re‐weighted least squares (IRLS) to investigate their application in instrument analysis of nutraceuticals (that is, fluorescence quenching of merbromin reagent upon lipoic acid addition). These robust regression methods were used to calculate calibration data from the fluorescence quenching reaction (?F and F‐ratio) under ideal or non‐ideal linearity conditions. For each condition, data were treated using three regression fittings: Ordinary Least Squares (OLS), LMS and IRLS. Assessment of linearity, limits of detection (LOD) and quantitation (LOQ), accuracy and precision were carefully studied for each condition. LMS and IRLS regression line fittings showed significant improvement in correlation coefficients and all regression parameters for both methods and both conditions. In the ideal linearity condition, the intercept and slope changed insignificantly, but a dramatic change was observed for the non‐ideal condition and linearity intercept. Under both linearity conditions, LOD and LOQ values after the robust regression line fitting of data were lower than those obtained before data treatment. The results obtained after statistical treatment indicated that the linearity ranges for drug determination could be expanded to lower limits of quantitation by enhancing the regression equation parameters after data treatment. Analysis results for lipoic acid in capsules, using both fluorimetric methods, treated by parametric OLS and after treatment by robust LMS and IRLS were compared for both linearity conditions.     

Can you trust the parametric standard errors in nonlinear least squares? Yes,with provisos

Background

Questions about the reliability of parametric standard errors (SEs) from nonlinear least squares (LS) algorithms have led to a general mistrust of these precision estimators that is often unwarranted.

Studying multisite binary and ternary protein interactions by global analysis of isothermal titration calorimetry data in SEDPHAT: application to adaptor protein complexes in cell signaling

Multisite interactions and the formation of ternary or higher-order protein complexes are ubiquitous features of protein interactions. Cooperativity between different ligands is a hallmark for information transfer, and is frequently critical for the biological function. We describe a new computational platform for the global analysis of isothermal titration calorimetry (ITC) data for the study of binary and ternary multisite interactions, implemented as part of the public domain multimethod analysis software SEDPHAT. The global analysis of titrations performed in different orientations was explored, and the potential for unraveling cooperativity parameters in multisite interactions was assessed in theory and experiment. To demonstrate the practical potential and limitations of global analyses of ITC titrations for the study of cooperative multiprotein interactions, we have examined the interactions of three proteins that are critical for signal transduction after T-cell activation, LAT, Grb2, and Sos1. We have shown previously that multivalent interactions between these three molecules promote the assembly of large multiprotein complexes important for T-cell receptor activation. By global analysis of the heats of binding observed in sets of ITC injections in different orientations, which allowed us to follow the formation of binary and ternary complexes, we observed negative and positive cooperativity that may be important to control the pathway of assembly and disassembly of adaptor protein particles.     

Statistical uncertainty and its propagation in the analysis of quantitative polymerase chain reaction data: Comparison of methods

Most methods for analyzing real-time quantitative polymerase chain reaction (qPCR) data for single experiments estimate the hypothetical cycle 0 signal y₀ by first estimating the quantification cycle (C_q) and amplification efficiency (E) from least-squares fits of fluorescence intensity data for cycles near the onset of the growth phase. The resulting y₀ values are statistically equivalent to the corresponding C_q if and only if E is taken to be error free. But uncertainty in E usually dominates the total uncertainty in y₀, making the latter much degraded in precision compared with C_q. Bias in E can be an even greater source of error in y₀. So-called mechanistic models achieve higher precision in estimating y₀ by tacitly assuming E = 2 in the baseline region and so are subject to this bias error. When used in calibration, the mechanistic y₀ is statistically comparable to C_q from the other methods. When a signal threshold y_q is used to define C_q, best estimation precision is obtained by setting y_q near the maximum signal in the range of fitted cycles, in conflict with common practice in the y₀ estimation algorithms.     

Influences of error distributions of net ecosystem exchange on parameter estimation of a process-based terrestrial model: A case of broad-leaved Korean pine mixed forest in Changbaishan, China

Model predictions can be improved by parameter estimation from measurements. It was assumed that measurement errors of net ecosystem exchange (NEE) of CO₂ follow a normal distribution. However, recent studies have shown that errors in eddy covariance measurements closely follow a double exponential distribution. In this paper, we compared effects of different distributions of measurement errors of NEE data on parameter estimation. NEE measurements in the Changbaishan forest were assimilated into a process-based terrestrial ecosystem model. We used the Markov chain Monte Carlo method to derive probability density functions of estimated parameters. Our results showed that modeled annual total gross primary production (GPP) and ecosystem respiration (Re) using the normal error distribution were higher than those using the double exponential distribution by 61–86 gC m^?2 a^?1 and 107–116 gC m^?2 a^?1, respectively. As a result, modeled annual sum of NEE using the normal error distribution was lower by 29–47 gC m^?2 a^?1 than that using the double exponential error distribution. Especially, modeled daily NEE based on the normal distribution underestimated the strong carbon sink in the Changbaishan forest in the growing season. We concluded that types of measurement error distributions and corresponding cost functions can substantially influence the estimation of parameters and carbon fluxes.     

Hierarchy of simulation models in predicting molecular recognition mechanisms from the binding energy landscapes: structural analysis of the peptide complexes with SH2 domains.

Computer simulations using the simplified energy function and simulated tempering dynamics have accurately determined the native structure of the pYVPML, SVLpYTAVQPNE, and SPGEpYVNIEF peptides in the complexes with SH2 domains. Structural and equilibrium aspects of the peptide binding with SH2 domains have been studied by generating temperature-dependent binding free energy landscapes. Once some native peptide-SH2 domain contacts are constrained, the underlying binding free energy profile has the funnel-like shape that leads to a rapid and consistent acquisition of the native structure. The dominant native topology of the peptide-SH2 domain complexes represents an extended peptide conformation with strong specific interactions in the phosphotyrosine pocket and hydrophobic interactions of the peptide residues C-terminal to the pTyr group. The topological features of the peptide-protein interface are primarily determined by the thermodynamically stable phosphotyrosyl group. A diversity of structurally different binding orientations has been observed for the amino-terminal residues to the phosphotyrosine. The dominant native topology for the peptide residues carboxy-terminal to the phosphotyrosine is tolerant to flexibility in this region of the peptide-SH2 domain interface observed in equilibrium simulations. The energy landscape analysis has revealed a broad, entropically favorable topology of the native binding mode for the bound peptides, which is robust to structural perturbations. This could provide an additional positive mechanism underlying tolerance of the SH2 domains to hydrophobic conservative substitutions in the peptide specificity region.