Curve fitting with nonlinear regression

Curve Fitting with Nonlinear Regression Analysis... 

Fig. 6.13 Measurement of FAC data for a range of pyrimethamine concentrations applied to sol-gel-entrapped dehydrofolate reductase (A) overlay of breakthrough curves and (B) nonlinear regression analysis of the fit to the measured breakthrough volumes from A [28]. Adapted with permission from the American Chemical Society.

FIGURE 16.1 Hyperbolic saturation curve of CYP2C19-catalyzed (5)-mephenytoin 4 -hydroxylation in human liver microsomes. Am and Vmax values were calculated with Equation 16.1 (Am = 23.7 2.15 jlM and Fmax = 0.36 0.01 nmol/min/mg). Data were fitted by nonlinear regression (Eq. 16.1, SigmaPlot 9.0). 

Equation [8.89] is linear with respect to parameters 4> and 0, but nonlinear with respect to q, and therefore the data must be fitted to this calibration function using nonlinear least-squares regression (Section 8.3.8) it is emphasized that it is very important to ensure that the initial estimates for the unkown parameters should be reasonably close to the final best estimates (see the text box dealing with nonlinear regression). In the present example (Equation [8.89]) excellent initial estimates can be obtained experimentally (see below) but if this is not possible tricks can be employed to obtain reasonable first estimates. One way is to plot the experimental data for Ra VRsis s Qa VQsis" nd draw an approximate curve though the points by hand. Experimental data expected to be well represented by Equation [8.89] should extrapolate to a value of (Ra /Rsis ) = 0 as (Qa"/Qsis ) zero, and to (Ra VRsis") = as (Qa"/Qsis") becomes... 

The time course of HPOD formation was examined with nonlinear regression analysis over a 6 h period at 15°C using IMM-LOX that contained 3.0 mg protein. Curve fit estimates were obtained using Equation 1,... 

When estimates of k°, k, k", Ky, and K2 have been obtained, a calculated pH-rate curve is developed with Eq. (6-80). If the experimental points follow closely the calculated curve, it may be concluded that the data are consistent with the assumed rate equation. The constants may be considered adjustable parameters that are modified to achieve the best possible fit, and one approach is to use these initial parameter estimates in an iterative nonlinear regression program. The dissociation constants K and K2 derived from kinetic data should be in reasonable agreement with the dissociation constants obtained (under the same experimental conditions) by other means. 

Toward these ends, the kinetics of a wider set of reaction schemes is presented in the text, to make the solutions available for convenient reference. The steady-state approach is covered more extensively, and the mathematics of other approximations ( improved steady-state and prior-equilibrium) is given and compared. Coverage of data analysis and curve fitting has been greatly expanded, with an emphasis on nonlinear least-squares regression. 

Solutions are presented in the form of equations, tables, and graphs—most often the last. Serious numerical results generally have to be obtained with computers or powerful calculators. The introductory chapter describes the numerical procedures that are required. Inexpensive software has been used here for integration, differentiation, nonlinear equations, simultaneous equations, systems of differential equations, data regression, curve fitting, and graphing. 

Hyperbolic curve fits to control enzymatic data and to data obtained in the presence of a competitive inhibitor. Curve fitting to the Michaelis-Menten equation results in two different values for Km- However, Km does not, in actuality, change, and the value in the presence of inhibitor (15 uiM) is an apparent value. Fitting with the correct equation, that for turnover in the presence of a competitive inhibitor ( Eq. 5), results in plots identical in appearance to those obtained with the Michaelis-Menten equation. However, nonlinear regression now reveals that Km remains constant at 5 ulM and that [l]/Ki = 2.5 with knowledge of [/], calculation of K is straightforward... 

It is apparent that O Eq. 5 is a variation of the MichaeUs-Menten equation. The inhibitor data shown in O Figure 4-7 can instead be fitted to O Eq. 5, holding Km (and Vjnax) constant to their control values (5 pM and 20 nmol/min/mg, respectively). The curve obtained is identical to that fitted with the Michaelis-Menten equation (O Figure 4-7), but nonlinear regression now yields the information that K, of the inhibitor equals 40% of the concentration at which it was included in the assay to obtain the best-fit curve. In other words, if the concentration of inhibitor present in the experiment shown in O Figure 4-7 was 25 pM, the Ki for the inhibitor is 10 pM. 

Representative data for [ H]acetylcholine binding to the membrane-bound Torpedo nAChR. Bindng was measured either by equilibrium dialysis (closed circles) as described in Protocol 4.1 or by centrifugation (open squares, see Protocol 4.2). Estimated Kd values from nonlinear regression curve fitting were 12 nM and 10 nM, respectively with corresponding Rq values of 0.14 ulM and 0.135 ulM... 

Calibration curves were fitted, and EC50 values were derived using the nonlinear regression package pro Fit 5.5 (QuantumSoft, Zurich, Switzerland). The results of the calibration curve measurements were fitted to a sigmoidal dose-response function of the following form with a slope faetor of 1 ... 

Figure 8 shows a plot of the concentration of methanol produced by the hydrolysis of SiQAC at pH 4.07 in water and the nonlinear regression curve of equation (18) assuming three consecutive, irreversible first-order reactions. A summary of the observed rate constants at each pH studied is shown in Table 4. Regression fits produced R2 values of better than 0.99 for all the pH values investigated. Plots of the observed values of k, k2, and k3 vs. pH are linear in all cases, with R2 values greater than 0.99, and with slopes of -0.997, -0.992 and -0.999, respectively. The ratio of kt k2 k3 is approximately 20 3 1. 

This process is partially overlapped with the next process, the j3 relaxation. To analyze the loss permittivity in the subglass zone in a more detailed way, the fitting of the loss factor permittivity by means of usual equations is a good way to get confidence about this process [69], Following procedures described above Fig. 2.42 represent the lost factor data and deconvolution in two Fuoss Kirwood [69] as function of temperature at 10.3 Hz for P4THPMA. In Fig. 2.43 show the y and relaxations that result from the application of the multiple nonlinear regression analysis to the loss factor against temperature. The sum of the two calculated relaxations is very close to that in the experimental curve. 

The final nonlinear equation, containing K, Cl, and Ires parameters to be determined, requires data input from measured FA fluorescence intensities (I and Iq), and total metal ion additions at each titration interval (Cm)- Equation 6 may be solved by nonlinear regression. FA fluorescence quenching curves for titrations with Cu at pH 5, 6, and 7 are shown in figure 2. The lines through the data points represent the best fit curve of equation 6 to each set of data. 

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