Optimal allocation of replicates for measurement evaluation studies

Optimal experimental design is important for the efficient use of modern highthroughput technologies such as microarrays and proteomics. Multiple factors including the reliability of measurement system, which itself must be estimated from prior experimental work, could influence design decisions. In this study, we describe how the optimal number of replicate measures （technical replicates） for each biological sample （biological replicate） can be determined. Different allocations of biological and technical replicates were evaluated by minimizing the variance of the ratio of technical variance （measurement error） to the total variance （sum of sampling error and measurement error）. We demonstrate that if the number of biological replicates and the number of technical replicates per biological sample are variable, while the total number of available measures is fixed, then the optimal allocation of replicates for measurement evaluation experiments requires two technical replicates for each biological replicate. Therefore, it is recommended to use two technical replicates for each biological replicate if the goal is to evaluate the reproducibility of measurements.     

Error structure as a function of substrate and inhibitor concentration in enzyme kinetic experiments.

Optimal design of experiments as well as proper analysis of data are dependent on knowledge of the experimental error. A detailed analysis of the error structure of kinetic data obtained with acetylcholinesterase showed conclusively that the classical assumptions of constant absolute or constant relative error are inadequate for the dependent variable (velocity). The best mathematical models for the experimental error involved the substrate and inhibitor concentrations and reflected the rate law for the initial velocity. Data obtained with other enzymes displayed similar relationships between experimental error and the independent variables. The new empirical error functions were shown superior to previously used models when utilized in weighted non-linear-regression analysis of kinetic data. The results suggest that, in the spectrophotometric assays used in the present study, the observed experimental variance is primarily due to errors in determination of the concentrations of substrate and inhibitor and not to error in measuring the velocity.     

On the analysis of fluorescence decay kinetics by the method of least-squares

Analysis of fluorescence decay kinetics aims at the determination of the analytic expression and the numerical values of the pertinent parameters which describe the decay process. In the well-known method of least-squares, one assumes a plausible functional form for the decay data and adjusts the values of the parameters until the statistically best fit is obtained between the data and the calculated decay function, i.e., until the sum of the weighted squares of the residuals is at a minimum. It is shown that proper weighting of the squares of the residuals may markedly improve the quality of the analysis. Such weighting requires information about the character of the experimental noise, which is often available, e.g., when the noise is due to counting error in photon-counting techniques. Furthermore, dramatic improvements in the accuracy of the analysis may often be achieved by use of auxiliary information available about the system studied. For example, the preexponents in a multiexponential fluorescence decay of a mixture of chromophores (such as tryptophan residues in a protein molecule) may sometimes be estimated independently; much higher accuracy can then be attained for the decay lifetimes by analysis of the decay kinetics. It is proposed that the shape of the autocorrelation function of the weighted residuals may serve as a convenient criterion for the quality of fit between the experimental data and the decay function obtained by analysis. The above conclusions were reached by analysis of computer-simulated experiments, and the usefulness of this approach is illustrated. The importance of stating the uncertainties in the estimated parameters inherent in the analysis of decay kinetics is stressed.     

Evidence for the two-step binding of ATP to myosin subfragment 1 by the rapid-flow-quench method.

1. The initial steps on the myosin ATPase (EC 3.6.1.3) pathway are taken to be: (formula; see text) A two-step binding for ATP is assumed, but the evidence for it is unconvincing; because of the rapidity of the process unambiguous values for K1 and K2 are not available. 2. We investigated the myosin mechanism by the chemical flow-quench technique. Reaction mixtures containing [gamma-32P]ATP plus myosin subfragment 1 were quenched in unlabelled ATP (ATP chase) or acid (Pi burst). 3. We show that the ATP-chase method can lead directly to unambiguous values for K1 and k+2. 4. The binding process was slowed down by 40% ethylene glycol. It was studied as a function of the ATP concentration. A limiting plateau resulted, showing a two-step binding for ATP, and values for K1 and k+2 were obtained. 5. K1 and k+2 are rather sensitive to the experimental conditions. Ethylene glycol and lowering of the pH decrease both constants, but an increase in KCl concentration increases them. This suggests that the binding of ATP to myosin is of an electrostatic nature. 6. The Pi-burst method can lead directly to k+3 + k-3, but under certain conditions the kinetics are governed by K1 and k+2. This uncertainty of the interpretation of Pi-burst experiments is discussed.     

Automated affinity chromatography measurements of compound mixtures using a lab-on-valve apparatus coupled to electrospray ionization mass spectrometry

We report a fully automated affinity chromatography system using a lab-on-valve (LOV) apparatus coupled to an electrospray ionization ion-trap mass spectrometer (ESI-MS). The system allows simultaneous measurements of multiple ligand affinities to proteins immobilized on beads. Bead regeneration, column repacking, and repetitive measurements are achieved on the time scale of several minutes. In this study, the system was used to screen the binding of a peptide mixture to human and Trypanosoma brucei (T. brucei) truncated Pex5 (tPex5) proteins. Equilibrium dissociation constants (K(d)) were measured for T. brucei tPex5 and compared to the values obtained by a fluorescence-based competition assay. The three peptides that showed affinity toward tPex5 had K(d) values that were comparable in magnitude (within a factor of 5) and showed the same ranking order as those from manual fluorescence measurements. With 12 min of sample infusion, the entire sample-to-sample cycle takes about 15 min and can be repeated without any preparation between runs. For T. brucei tPex5 affinity measurements, 1 mg of protein was sufficient for 35 repetitive analyses in the automated LOV-ESI-MS apparatus. The system allows rapid determination of K(d) in the range of 10(-5)-10(-7) M for sample mixtures and is suitable for screening a large number of compounds against multiple proteins.     

A new data analysis method to determine binding constants of small molecules to proteins using equilibrium analytical ultracentrifugation with absorption optics.

In principle, equilibrium analytical ultracentrifugation (AU) can be used to quantify the binding stoichiometry and affinity between small-molecule ligands and proteins in aqueous solution. We show here that heteromeric binding constants can be determined using a data-fitting procedure which utilizes a postfitting computation of the total amount of each component in the centrifuge cell. The method avoids overconstraining the fitting of the radial concentration profiles, but still permits unique binding constants to be determined using measurements at a single wavelength. The computational program is demonstrated by applying it to data obtained with mixtures of a 500-Da molecule and interleukin-2, a 16-kDa protein. The 1:1 binding stoichiometry and heteromeric dissociation constants (K(ab)) determined from centrifuge data at two different wavelengths are within the 4-9 microM range independently determined from a functional assay. Values for K(ab) have been obtained for ligands with affinities as weak as 500 microM. This AU method is applicable to compounds with significant UV absorbance (approximately 0.2) at concentrations within approximately 5- to 10-fold of their K(ab). The method, which has been incorporated into a user procedure for IgorPro (Wavemetrics, Oswego, OR), is included as supplementary material.     

Finger Muscle Attachments for an OpenSim Upper-Extremity Model

We determined muscle attachment points for the index, middle, ring and little fingers in an OpenSim upper-extremity model. Attachment points were selected to match both experimentally measured locations and mechanical function (moment arms). Although experimental measurements of finger muscle attachments have been made, models differ from specimens in many respects such as bone segment ratio, joint kinematics and coordinate system. Likewise, moment arms are not available for all intrinsic finger muscles. Therefore, it was necessary to scale and translate muscle attachments from one experimental or model environment to another while preserving mechanical function. We used a two-step process. First, we estimated muscle function by calculating moment arms for all intrinsic and extrinsic muscles using the partial velocity method. Second, optimization using Simulated Annealing and Hooke-Jeeves algorithms found muscle-tendon paths that minimized root mean square (RMS) differences between experimental and modeled moment arms. The partial velocity method resulted in variance accounted for (VAF) between measured and calculated moment arms of 75.5% on average (range from 48.5% to 99.5%) for intrinsic and extrinsic index finger muscles where measured data were available. RMS error between experimental and optimized values was within one standard deviation (S.D) of measured moment arm (mean RMS error = 1.5 mm < measured S.D = 2.5 mm). Validation of both steps of the technique allowed for estimation of muscle attachment points for muscles whose moment arms have not been measured. Differences between modeled and experimentally measured muscle attachments, averaged over all finger joints, were less than 4.9 mm (within 7.1% of the average length of the muscle-tendon paths). The resulting non-proprietary musculoskeletal model of the human fingers could be useful for many applications, including better understanding of complex multi-touch and gestural movements.     

A sol-gel-integrated protein array system for affinity analysis of aptamer-target protein interaction

A sol-gel microarray system was developed for a protein interaction assay with high activity. Comparing to 2-dimensional microarray surfaces, sol-gel can offer a more dynamic and broad range for proteins. In the present study, this sol-gel-integrated protein array was used in binding affinity analysis for aptamers. Six RNA aptamers and their target protein, yeast TBP (TATA-binding protein), were used to evaluate this method. A TBP-containing sol-gel mixture was spotted using a dispensing workstation under high-humidity conditions and each Cy-3-labeled aptamer was incubated. The dissociation constants (K(d)) were calculated by plotting the fluorescent intensity of the bound aptamers as a function of the TBP concentrations. The K(d) value of the control aptamer was found to be 8?nM, which agrees well with the values obtained using the conventional method, electric mobility shift assay. The sol-gel-based binding affinity measurements fit well with conventional binding affinity measurements, suggesting their possible use as an alternative to the conventional method. In addition, aptamer affinity measurements by the sol-gel-integrated protein chip make it possible to develop a simple high-throughput affinity method for screening high-affinity aptamers.     

Normalization and analysis of residual variation in two-dimensional gel electrophoresis for quantitative differential proteomics

Although two-dimensional gel electrophoresis (2-DE) has long been a favorite experimental method to screen proteomes, its reproducibility is seldom analyzed with the assistance of quantitative error models. The lack of models of residual distributions that can be used to assign likelihood to differential expression reflects the difficulty in tackling the combined effect of variability in spot intensity and uncertain recognition of the same spot in different gels. In this report we have analyzed a series of four triplicate two-dimensional gels of chicken embryo heart samples at two distinct development stages to produce such a model of residual distribution. In order to achieve this reference error model, a nonparametric procedure for consistent spot intensity normalization had to be established, and is also reported here. In addition to variability in normalized intensity due to various sources, the residual variation between replicates was observed to be compounded by failure to identify the spot itself (gel alignment). The mixed effect is reflected by variably skewed bimodal density distributions of residuals. The extraction of a global error model that accommodated such distribution was achieved empirically by machine learning, specifically by bootstrapped artificial neural networks. The model described is being used to assign confidence values to observed variations in arbitrary 2-DE gels in order to quantify the degree of over-expression and under-expression of protein spots.     

Frontal affinity chromatography coupled to mass spectrometry for screening mixtures of enzyme inhibitors.

Frontal affinity chromatography coupled online to mass spectrometry (FAC/MS) has previously been used to estimate binding constants for individual protein ligands present in mixtures of compounds. In this study FAC/MS is used to determine enzyme substrate kinetic parameters and binding constants for enzyme inhibitors. Recombinant human N-acetylglucosaminyltransferase V was biotinylated and adsorbed onto immobilized streptavidin in a microcolumn (20 microL). The enzyme was shown to be catalytically competent transferring GlcNAc from the donor UDP-GlcNAc to beta-d-GlcpNAc-(1-->2)-alpha-d-Manp-(1-->6)-beta-d-Glcp-OR acceptor giving beta-d-GlcpNAc-(1-->2)-[beta-d-GlcpNAc-(1-->6)]-alpha-d-Manp-(1-->6)-beta-d-Glcp-OR as the reaction product. The kinetic parameters K(m) and V(max) for the immobilized enzyme could be determined by FAC/MS and were comparable to those measured in solution. Analysis of a mixture of eight trisaccharide analogs in a single run yielded K(d) values for each of the eight compounds ranging from 0.3 to 36 microM. These K(d) values were 2 to 10 times lower than the inhibition constants, K(I)'s, determined in solution using a standard radiochemical assay. However, the ranking order of K(d)'s was the same as the ranking of K(I) values. FAC/MS assays can therefore be employed for the rapid estimation of inhibitor K(d) values making it a valuable tool for enzyme inhibitor evaluations.     

Statistical error in isothermal titration calorimetry: variance function estimation from generalized least squares

The method of generalized least squares (GLS) is used to assess the variance function for isothermal titration calorimetry (ITC) data collected for the 1:1 complexation of Ba(2+) with 18-crown-6 ether. In the GLS method, the least squares (LS) residuals from the data fit are themselves fitted to a variance function, with iterative adjustment of the weighting function in the data analysis to produce consistency. The data are treated in a pooled fashion, providing 321 fitted residuals from 35 data sets in the final analysis. Heteroscedasticity (nonconstant variance) is clearly indicated. Data error terms proportional to q(i) and q(i)/v are well defined statistically, where q(i) is the heat from the ith injection of titrant and v is the injected volume. The statistical significance of the variance function parameters is confirmed through Monte Carlo calculations that mimic the actual data set. For the data in question, which fall mostly in the range of q(i)=100-2000 microcal, the contributions to the data variance from the terms in q(i)(2) typically exceed the background constant term for q(i)>300 microcal and v<10 microl. Conversely, this means that in reactions with q(i) much less than this, heteroscedasticity is not a significant problem. Accordingly, in such cases the standard unweighted fitting procedures provide reliable results for the key parameters, K and DeltaH(degrees) and their statistical errors. These results also support an important earlier finding: in most ITC work on 1:1 binding processes, the optimal number of injections is 7-10, which is a factor of 3 smaller than the current norm. For high-q reactions, where weighting is needed for optimal LS analysis, tips are given for using the weighting option in the commercial software commonly employed to process ITC data.     

The 48 kDa Ca2+-binding protein of bovine brain.

The integrated rate equation for reactions with stoichiometry A----P + Q is: e0t = -Cf . ln(1-delta P/A0) + C1 delta P + 1/2C2(delta P)2 where the coefficients C are linear or quadratic functions of the kinetic constants and the initial substrate and product concentrations. I have used the 21 progress curves described in the accompanying paper [Cox & Boeker (1987) Biochem. J. 245, 59-65] to develop computer-based analytical and statistical techniques for extracting kinetic constants by fitting this equation. The coefficients C were calculated by an unweighted non-linear regression: first approximations were obtained from a multiple regression of t on delta P and were refined by the Gauss-Newton method. The procedure converged in six iterations or less. The bias in the coefficients C was estimated by four methods and did not appear to be significant. The residuals in the progress curves appear to be normally distributed and do not correlate with the amount of product produced. Variances for Cf, C1 and C2 were estimated by four resampling procedures, which gave essentially identical results, and by matrix inversion, which came close to the others. The reliability of C2 can also be estimated by using an analysis-of-variance method that does not require resampling. The final kinetic constants were calculated by standard multiple regression, weighting each coefficient according to its variance. The weighted residuals from this procedure were normally distributed.     

Comparison of Amersham and Agilent microarray technologies through quantitative noise analysis

We carried out a series of replicate experiments on DNA microarrays using two cell lines and two technologies--the Agilent Human 1A Microarray and the GE Amersham Codelink Uniset Human 20K I Bioarray. We demonstrated that quantifying the noise level as a function of signal strength allows identification of the absolute and differential mRNA expression levels at which biological variability can be resolved above measurement noise. This represents a new formulation of a sensitivity threshold that can be used to compare platforms. It was found that the correlation in expression level between platforms is considerably worse than the correlation between replicate measurements taken using the same platform. In addition, we carried out replicate measurements at different stages of sample processing. This novel approach enables us to quantify the noise introduced into the measurements at each step of the experimental protocol. We demonstrated how this information can be used to determine the most efficient means of using replicates to reduce experimental uncertainty.     

Genome-wide linkage analysis of systolic blood pressure: a comparison of two approaches to phenotype definition

Problem 1 of the Genetic Analysis Workshop 13(GAW13) contains longitudinal data of cardiovascular measurements from 330 pedigrees. The longitudinal data complicates the phenotype definition because multiple measurements are taken on each individual. To address this complication, we propose an approach that uses generalized estimating equations to obtain residuals for each time point for each person. The mean residual is then taken as the new phenotype with which to use in a variance components linkage analysis. We compare our phenotype definition approach to an approach that first reduces the multiple measurements to a single measurement and then models these summary statistics as regression terms in a variance components analysis. For each approach, multipoint linkage analysis was performed using the residuals and the SOLAR computer program. Our results show little difference between the methods based on the LOD scores.     

Affinity analysis of non-steady-state data obtained under mass transport limited conditions using BIAcore technology

Binding data obtained with Biacore instrumentation is often evaluated using a kinetic transport model where reaction rate constants and a mass transport coefficient are used to describe the interaction. Here the use of a simplified model, an affinity transport model, for determination of the affinity (K(D)) but not the kinetics (k(a), k(d)) has been investigated. When binding rates were highly governed by mass transport effects the two models returned the same affinity and gave similar residuals, but k(a) and k(d) values found with the kinetic transport model were unreliable. On the other hand the affinity transport model failed to describe the data when binding curves were less influenced by mass transport effects. Under such circumstances the kinetic transport model returned correct k(a) and k(d) values. Depending on the outcome of the analysis the affinity transport model can therefore be used to reduce uncertainties of the kinetic parameters or as an easy way to determine K(D) values from non-steady-state data. The use of the affinity transport model is illustrated with simulated data and with binding data obtained for the interaction between a 439 Da thrombin inhibitor and immobilized thrombin. Since it is more difficult to resolve high k(a) values for low molecular weight analytes, the affinity transport model may be particularly useful for affinity analysis involving fast reactions between such analytes and immobilized protein targets.     

A new rapid and simple method to determine the kinetics of electrode reactions of biologically relevant compounds from the half-peak width of the square-wave voltammograms

A new method is introduced to determine the kinetic parameters of electron transfer reactions of biologically important compounds, based on the measurements of the half-peak width (DeltaE(p/2)) of the square-wave voltammograms. A simple surface (diffusionless) redox reaction, and a simple electrode reaction occurring from dissolved state are considered as model systems. In the region of quasireversible electron transfer, the half-peak widths of theoretical square-wave voltammograms are linear functions of the logarithm of the dimensionless kinetic parameter ln(K) that characterizes the rate of the electron transfer reaction. The dimensionless kinetic parameter K is defined as K=k(s)(fD)(-0.5) for the redox reaction taking place from dissolved state, whereas for the surface redox reaction K is defined as K=k(s)/f (k(s) is the standard rate constant of electron transfer, f is the SW frequency, and D is the diffusion coefficient). A set of linear regression equations for the dependences DeltaE(p/2)vs. ln(K) are derived, which can be used for rapid and precise determination of the charge-transfer kinetic parameters. The estimated values for the standard rate constants of various biologically relevant redox systems using this approach are in very good agreement with the experimental values determined by other square-wave voltammetric methods. The square-wave voltammetric half-peak width method can be used as a simple and reliable alternative to other voltammetric methods developed for the kinetic characterization of electron transfer rates.     

Photophysics of the fluorescent K+ indicator PBFI.

The fluorescent indicator PBFI is widely used for the determination of intracellular concentrations of K+. To investigate the binding reaction of K+ to PBFI in the ground and excited states, steady-state and time-resolved measurements were performed. The fluorescence decay surface was analyzed with global compartmental analysis yielding the following values for the rate constants at room temperature in aqueous solution at pH 7.2: k01 = 1.1 x 10(9) s-1, k21 = 2.7 x 10(8) M-1s-1, k02 = 1.8 x 10(9) s-1, and k12 = 1.4 x 10(9) s-1. k01 and k02 denote the respective deactivation rate constants of the K+ free and bound forms of PBFI in the excited state. k21 represents the second-order rate constant of binding of K+ to the indicator in the excited state whereas k12 is the first-order rate constant of dissociation of the excited K(+)-PBFI complex. From the estimated values of k12 and k21, the dissociation constant Kd* in the excited state was calculated. It was found that pKd* (-0.7) is smaller than pKd (2.2). The effect of the excited-state reaction can be neglected in the determination of Kd and/or the K+ concentration. Therefore, intracellular K+ concentrations can be accurately determined from fluorimetric measurements by using PBFI as K+ indicator.     

Determination of the parameters of self-association by direct fitting of the omega function

Nonlinear regression is used to fit the omega function vs. protein concentration curves (first described by B.K. Milthorpe, P.D. Jeffrey and L.W. Nichol, Biophys. Chem. 3 (1975) 169) obtained from sedimentation equilibrium experiments on self-associating macromolecules. Nonlinear regression allows the direct fit of these curves with discrete or indefinite self-association reaction models in order to obtain values for the equilibrium constants and second virial coefficient. The method is independent of the choice of reference concentration and avoids the original method of extrapolating an omega function curve to zero concentration and then using the extrapolated value to construct a monomer activity curve used for analysis. This extrapolation can become very difficult for mild to strong self-associations where incorrectly extrapolated values lead to systematic error in the monomer activity curves. The method is applied to results from a mild, indefinite self-association, exemplified by the self-association of human spectrin, and to computer-simulated data of weak, mild and strong, indefinite self-associations.     

A strategy for efficient characterization of macromolecular heteroassociations via measurement of sedimentation equilibrium.

A method is proposed for the selection of experimental conditions for sedimentation equilibrium experiments that will provide maximal information about the values of equilibrium association constants within a given scheme for heteroassociation of two solute components. A discriminator function is proposed that indicates the sensitivity of the experimentally observed gradient or gradients to alterations in the underlying association constants. The value of this function is plotted or tabulated as a function of the concentrations of the two components, over a broad range of solution compositions. It is suggested that experiments performed with loading compositions corresponding to large absolute values of the discriminator function will yield the most information with respect to determination of the underlying association constants. This method was tested by predicting optimal conditions for three different types of sedimentation equilibrium experiments: (i) measurement of total (natural) solute absorbance; (ii) measurement of individual component gradients via measurement of tracer absorbance; and (iii) global analysis of multiple experiments. Experimental data resulting from sedimentation equilibrium experiments carried out under the specified conditions were simulated by addition of realistic levels of random error to calculated equilibrium gradients. The simulated data were then analyzed exactly as real experimental data, i.e., without prior knowledge of the underlying association constants. It was found that the highest accuracy and precision in determination of heteroassociation constants are obtained by global analysis of multiple experiments performed using significantly different loading compositions, each of which is selected from 'sensitive' regions of the discriminator map.