## B1 Closed Form Theoretical Models

We can generalize from the preceding discussion that a nonlinear regression model must contain a mathematical description of the signal resulting from the sample in addition to a description of the background, unless the background can be accurately subtracted from the raw data. Theoretical equations describing the instrumental response are excellent starting points for such regression models. When used with the appropriate background terms, computations are usually fast, and physically...

## B1 Reversible Electron Transfer Single Potential Step

In principle, it should be possible to pulse the potential to values for which conversions of O to R, or R to O, are rapid and diffusion controlled. The diffusion coefficient can be determined from a single potential step Table 12.1 Model for Reversible Electron Transfer in Single Potential Step Chronocoulometry 2, 3 Assumptions Reversible electron transfer, linear and edge diffusion Regression equation 2(0 b0 + b 112 + b2t bo Qai, b, 2FADmCn-y2 b2 aFADC(itr)-m Qii double layer charge F...

## Fundamentals of NMR Relaxation

Relaxation is a general term used to describe the movement of a system toward equilibrium after an injection of energy. When atomic nuclei in a molecule are excited in a pulsed nuclear magnetic resonance (NMR) experiment, the excited state may exist for a rather long time, often seconds to minutes 1 , Characteristic relaxation times estimated in an NMR experiment can be used to provide information about molecular motion. In the following paragraphs, we provide a simplified summary of pulsed NMR...

## C 1 Reversible and Diffusion Kinetic Control

Cyclic voltammetry is the combination of two linear sweep experiments. The potential is initially swept in a forward direction, and the second scan is the reverse of the first. CV provides signals from the electrolysis of reactants on the forward scan and gives signals from the electrolysis products on the reverse scan 1, 2, 19 . Figure 11.14 shows a typical cyclic voltammogram for a Co(II) complex controlled by electrode kinetics and diffusion. On the forward scan, a peak for the Co(II) Co(I)...

## B3a Rotating Disk and Ultramicroelectrode Voltammetry

If the speed of the electrochemical experiment is so fast that the electron transfer step does not remain at equilibrium, the shapes of steady state voltammograms are controlled by kinetics as well as diffusion. A simple electrode reaction with slow kinetics with respect to the measurement time, for example, Figure 11.7 Steady state voltammogram at 7 mV s 1 of 1 mM ferrocene at a carbon microdisk electrode of 6 jam radius in acetonitrile with no added supporting electrolyte. The solid line...

## Steady State Voltammetry

The term steady state voltammetry is used in this book to denote any voltammetric method in which a steady state is achieved in some range of applied potential between the rate of electrolysis and some other feature of the experiment, such as the rate of diffusion or the rate of a limiting chemical reaction. This condition is signaled by the appearance of a plateau or limiting current in the voltammogram (cf. Figures 11.2 and 11.4). Steady state techniques include rotating disk voltammetry, dc...

## A1 Sedimentation Velocity

At the start of an ultracentrifugation experiment, the macromolecules are distributed homogeneously in the test solution. When the rotor begins to spin, the molecules begin to be pushed toward the bottom of the sector-shaped cell (Figure 10.1). This creates a moving boundary that travels Figure 10.1 Diagram of a sector shaped cell (Adapted from Van Holde, Physical Biochemistry, 1971, 99. Reprinted by permission of Prentice-Hall, Inc., Englewood Cliffs, N.J.) Figure 10.1 Diagram of a sector...

## B3b Models for Deuterium Relaxation in Protein Solutions

The simplest model of protein hydration is a two-state one, where water (i.e., D20) exchanges chemically between a bound and a free state. Using the general approach to equilibrium problems described in Chapter 6 and making the reasonable assumption that for bound water is much less than that of free water, the observed longitudinal relaxation time is 7 where the symbols are defined as follows ru relaxation time of 2D for bound water, T ,o relaxation time of 2D for free water, tm lifetime of a...

## B2 Peak Shaped Data

Many experiments in chemistry and biochemistry give peak-shaped signals. For methods such as chromatography, overlapped peaks result from poor separation of components of the sample during the analysis. Overlapped peaks in UV-VIS or infrared absorbance, fluorescence, X-ray photo-electron (XPS), and nuclear magnetic resonance (NMR) spectroscopy represent overlapped spectral features. Hence, there is a real need to separate overlapped peaks in a reliable way to extract the information relevant to...

## Square Wave Voltammetry

Square wave voltammetry (SWV) is much more sensitive and has better resolution than cyclic or normal pulse voltammetry. For these reasons, SWV may be the method of choice for systems with low concentrations of electroactive material or with overlapped signals. In this technique (Table II.1), a square potential pulse of frequency consisting of a forward and a reverse component is superimposed on top of an increasing potential staircase and applied to the electrochemical cell (Figure 11.17) 13 ....

## B1 NMR Relaxation of aD and 1H in Aqueous Solutions of 3Lactoglobulin

Proton, deuterium, and oxygen-17 NMR relaxation can be used to investigate the amount and mobility of water bound to proteins. In this example, we illustrate the analysis of pulsed NMR relaxation data for 2D and 'H nuclei of water in a protein solution for such a system. Figure 8.2 shows typical proton relaxation results for an aqueous solution of the protein 3-lactoglobulin A. The transverse relaxation data gave a Figure 8.2 Proton NMR peak intensities vs. delay time r for 0.061 g...

## C2 Electrochemical Catalysis

As implied in Section B.4.a, the special conditions necessary for steady state catalytic voltammograms in linear sweep or cyclic voltammetry are Figure 11.15 Cyclic voltammogram (background subtracted) of 1 mM Co(II)tetraphenyl-porphyrin in 0.1 M TBABr at 25 C showing the results of fitting the data using the DIGI-SIM package. Table 11.10 Parameters Obtained by Nonlinear Regression Analysis of Cyclic Voltammogram of Co(II)TPP at 1 and 5 V s1 Table 11.10 Parameters Obtained by Nonlinear...

## General Characteristics of Voltammetry

Voltammetric methods are a collection of electroanalytical techniques in which the potential of an electrolytic cell is varied toward negative or positive values and the current is measured 1,2 . The output of the experiment is the current vs. the potential applied to the cell. Typical cells consist of reference and counter electrodes and a working electrode. The reference electrode is designed to hold a constant potential, and a potentiostatic circuit is used so that a variation in the...

## A2 Orthogonalization of Parameters to Remove Correlation

If all else fails, serious partial correlation of parameters can be removed by orthogonalization of the parameter space. This orthogonalization method is general. In principle, it should be applicable to a wide variety Table 4.3 Results of Model in Eq. (3.6) Fit onto Steady State Voltammetry Data with Background with Different Initial Parameters from Those in Table 3.7 Table 4.3 Results of Model in Eq. (3.6) Fit onto Steady State Voltammetry Data with Background with Different Initial...

## B1 Lysozyme

Determination of Component Bands The protein spectrum is considered to be the sum of the individual absorption bands arising from specific structural components, such as a helices, 3 sheets, and turns. Fitting such a spectrum directly without prior knowledge of the unknown number of Gaussian bands would be a daunting task. To aid in this task, as mentioned previously, we first examine the second derivative of the spectrum (Figure 7.3). The negative peaks in the second derivative spectrum (cf....

## B2 Nonlinear Regression Algorithms

General programs that require the user to supply only the model and the data are available for nonlinear regression. Some of these are available in several commercial graphics software packages, and some have been written by independent authors 1, 6-8 , Many of these programs use numerical differentiation of the model so that analytical derivatives need not be provided by the user. Commercial software packages for mathematics such as Mathcad and Matlab have built in minimization functions that...

## B3b Normal Pulse Voltammetry

This technique has advantages in the estimation of electron transfer rate constants compared to other voltammetric methods. The signal to noise ratio of the pulse techniques is generally better than in methods in which current is measured continuously. NPV is particularly suitable for regression analysis because it has a closed form model for voltammograms under diffusion-kinetic control. In normal pulse voltammetry, a series of periodic pulses of linearly increasing or decreasing potential are...

## A3 What Happens to Errors in y When Nonlinear Models Are Linearized

Linear least squares methods can be used for all linear models. Recall that the equations for parameters obtained from unweighted linear least squares are derived by assuming that there are no errors in the Xj and that the errors in the y (meas) have equal variances. This assumes explicitly that ey has about the same value for a series of measured yI, independent of the actual value of yr In the Beer's law example, we estimated a standard error in the absorbance of 0.003 units. This means that,...

## Sb Bovine Isocitrate Dehydrogenase Citrate

The velocity of the isocitrate dehydrogenase catalysis vs. concentration of the inhibitor Mn2+-citrate shows a complex variation with C . Initial curvature upward is followed by gradual decline with a shoulder for CI between 2 to 4 mM. Preliminary analysis showed that the model was more complex than that in Table 6.13. Testing with a variety of models suggested that the data contained three distinct regions of response to Mn2+-citrate. The set of chemical equations in Box 6.4 was found to lead...

## A 1 Titrations of Weak Acids and Bases

Models to analyze acid-base and other titration data employ the usual theory of equilibrium 1 , In general, an equation describing the measured response in terms of the initial concentration of analyte (c), concentration of titrant (cb), initial volume of analyte (V ), and volume of titrant added (Vb) is used as the model. Weak Acids Consider a potentiometric titrations of a weak monobasic acid with a strong base that is, The exact equation 3 for the concentration of hydrogen ion during the...

## B4 Example Regression Analysis for an Exponential Decay

The fluorescence intensity of a reactant is measured to obtain its concentration in millimoles vs. time for a suspected first-order decomposition reaction. The data are read from a file in the form of y reactant concentration in mM and time in seconds. Assuming that errors in y are independent of y i.e., errors are absolute , use nonlinear regression to see if first-order decay holds and to estimate the rate constant and the pre-exponential factor. This example demonstrates how...

## A3 Gram Schmidt OrthogonalizationAn Example

We now discuss the details of orthogonalization by using the example of the ECE model for chronocoulometry in eq. 4.5 . Direct use of this equation led to slow convergence using steepest descent or Marquardt- Levenberg algorithms, and indications of partial parameter correlation were obtained from correlation matrices. A model can be considered a linear combination of functions that may contain nonlinear parameters. Gram-Schmidt orthogonalization 3 leads to an equivalent model from the explicit...

## A 1 Models for Peak Shape

This section discusses analysis of data from single channel chromatography. In these experiments, the effluent from the column passes through a single detector and the output of the analysis is the response of the detector vs. time. The time of introduction of the sample onto the column is t 0. In Section 3.B.2 we discussed various models for peak-shaped data. Because of the wide use of chromatography in modern science, we now focus specifically on separation of overlapped peaks in this...

## B 2 a Analysis of Scattering Data

The caseins occur in cow's milk as large colloidal complexes of protein and salts, commonly called casein micelles. Removal of calcium results in the dissociation of the micelle into smaller protein complexes called submicelles. The submicelles consist of four proteins, called al s-, a2s-, fi-, and K-casein, in the ratios of 4 1 4 1. These proteins have average monomer molecular weights of 23,300 and are considered to have few specific secondary structural features. The isolated fractions...

## A 1 Main Characteristics of Infrared Spectra of Proteins

The backbone of a polypeptide chain absorbs infrared radiation, which excites vibrational modes of the peptide bonds. Infrared spectroscopy measures the amount of light absorbed by these vibrations over a range of frequencies of the incident light. The positions and intensities of the infrared absorbance bands of these vibrational modes are sensitive to the protein's secondary structure. Band frequencies are characteristic of specific structural units, and their areas are proportional to the...

## C1 Goodness of Fit Criteria

In many situations, the exact model for a set of data has not been established. The goodness of fit criteria for a series of nonlinear regression analyses to all of the possible models can be used to find the model that best fits the data. The usual goodness of fit criteria, deviation plots, standard deviations SD , and error sums S , are compared for this task. Section 2.B.4 illustrated the use of SD, standard errors of parameters, and deviation plots for assessing goodness of fit. A SD...