Non-linear curve-fitting

The best-fitting set of parameters can be found by minimization of the objective function (Section 13.2.8.2). This can be performed only by iterative procedures. For this purpose several minimization algorithms can be applied, for example, Simplex, Gauss-Newton, and the Marquardt methods. It is not the aim of this chapter to deal with non-linear curve-fitting extensively. For further reference, excellent papers and books are available [18]. 

By providing the computer with starting estimates of the values and the data for [M]totai, [L]totai, and pH at each titration point, a non-linear curve fitting procedure—such as the method of Newton-Raphson—results in the minimization of the least squares function R, which is the sum of the squares of the difference between the experimental amount of alkali titrated and the calculated amount for each experimental point. 

Kinetic Analysis. The kinetic parameters were obtained by iterative non-linear curve fitting of raw data (current generated versus the substrate concentration).The data fitted a modified Michaelis-Menten equation ... 

IC50 values are determined from the non-linear curve fitting of concentration-effect relationships. IC50 is defined as the concentration of test drug for half maximal inhibition of aggregation. 

Before the widespread availability of non-linear curve-fitting software, the individual parameters and fccat were separated by means of a Lineweaver-Burk plot, in which 1/v was plotted against 1/[S]. The plot should be a straight line with a y intercept of x intercept of—l/K, ... 

Perform non-linear curve fitting to each spectrum using n peaks. 

Fig. 8.1. Typical results obtained in a microcalorimetric titration experiment (a) heat effect observed upon each injection of titrant solution and (b) non-linear curve fitting of the experimental results using a simple 1 1 model.

Figure 8. pH dependence of the Ey (Fc /Fc) value for FcCOOH in an MeCN/H20 solution (4 1), 0.1 M in BU4NQ04, at 25 C. The pH of each solution was adjusted with a buffer. The solid line has been obtained through a non-linear curve fitting procedure on equation (2). 

First, however, we note that, although Tyworth and Ruiz-Torres (2000) utilize a non-linear curve-fitting process to build the rate fimction, we will consider a computationally simpler linear regression approach. Since we wish to build an expression of the form... 

Total dissolved inorganic carbon can also be evaluated from the potentiometric titration method described in Section 8.4.1, if it is performed in a closed cell. However, this technique does not give the highest precision possible and thus the coulometiic method first described by Johnson et al. (1985) and outlined here is nowadays the method of choice. Nevertheless, Ct can be evaluated from the potentiomettic titration curve, either Gran function linearization (Section 8.4.2.3 Gran evaliiation ) or by a curve fitting procedure (Section 8.4.2.3 Non-linear curve-fitting ). 

The interrelationship of octanol/water partition coefficients (Kg ). bioconcentration factors (Kg) and toxicity was used, in combination with first-order, one-compartment model assumptions, to estimate toxicant kinetic parameters. This was accomplished via non-linear curve fitting to time-toxicity data from typical LC50 toxicity tests. Preliminary results for a limited data set are presented. It appears that ... 

Estimated threshold LC50 value calculated by non-linear curve fitting as described in the text from data of Brooke et al. (1984) and Walbridge et al. (1983). 

Most widely used methods are based on (i) the competition of protons with the bound metal ions for thiolate ligands, i.e., the pH stability of the metal-thiolate clusters or (ii) the competition with a metal chelator. In the former case the pH stability of the Cd-thiolate clusters is followed by absorption spectroscopy of the CysS-Cd(II) LMCT band at 250 nm. From the apparent p/Ca values the apparent stability constants of the clusters can be derived [103]. The apparent p/sTa values are determined either by taking the pH values of half-maximum absorbance or using a non-linear curve fit of the pH plot [104]. The other method is based on the competition for a single metal ion between the chelator 5F-BAPTA (1,2-bis-(2-amino-5-fluorophenoxy)ethane-A,W,A ,A -tetraacetic acid) and the protein followed by NMR spectroscopy [71]. Although this method was established for the zinc metalloforms of MTs, its applicability to cadmium metalloforms has also been demonstrated [105]. In this case, by analogy with zinc finger proteins a lower affinity for mixed Cys/His coordination of Cd(II) in MTs compared to sole Cys coordination has been shown. 

Through numerical integration of the right hand side of Equ.(4), a relation between foam, core, and unfoamed polymer densities is obtained once the parametas m and n are known fi om experimental data on density profile and non-linear curve fitting. 

Parameter estimates for y, and S were obtained by nonlinear regression analysis of the stress and strain curves using the non-linear curve fitting tool in the software program OriginPro . These values were then used to calculate G, G" and / from Equations (4)-(6). 

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