Least-Squares Unfolding

A different approximation for the source spectrum, used with neutrons, assumes that S E) can be represented as a sum of NS discrete components. Therefore, one can write 

Because of the difficulties of matrix inversion, a least-squares solution has been attempted with NR NS. If NR NS, no unique solution exists, but an acceptable one has been obtained. 

The least-squares unfolding starts with Eq. 11.53 and minimizes the quantity 

The weighting factors w, are usually taken to be the inverse of the variance of Mj. The minimization is achieved by setting 

Computer round-off errors in completing the matrix inversion shown by Eq. 11.58 lead to large oscillations in the solution X. The oscillations can be reduced if the least-squares solution is constrained. Details of least-squares unfolding with constraints are given in Refs. 6 and 7. 

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The MCR-ALS decomposition method applied to three-way data can also deal with nontrilinear systems [81]. Whereas the spectrum of each compound of the columnwise augmented matrix is considered to be invariant for all of the matrices, the unfolded C matrix allows the profile of each compound in the concentration direction to be different for each appended data matrix. This freedom in the shape of the C profiles is appropriate for many problems with a nontrilinear structure. The least-squares problems solved by MCR-ALS, when applied to a three-way data set, are the same as those in Equation 11.11 and Equation 11.12 the only difference is that D and C are now augmented matrices. The operating procedure of the MCR-ALS method has already been described in Section 11.5.4, but some particulars regarding the treatment of three-way data sets deserve further comment. 

The energy values have been obtained by unfolding the spectra by a least-squares fit using Voigt profiles, the main criterium being the minimization of the number of profiles. 

The resulting matrix is then unfolded into a one-dimensional vector, which can be merged with the shape description, and is suitable for multivariate statistics analysis such as principal component analysis (PGA) and partial least squares (PLS). 

Figure 4. Gdn-HCl-induced unfolding of Y50F measured by fluorometry. The dotted line was obtained by nonlinear least-square fit of the data as described in Materials and Methods.

Figure 2. Dependence of unfolding rate constants on pH. Wild type tailspike protein was prepared in 50 mM Tris, 1.7 mM 2-mercaptoethanol and 2% SDS and adjust to different pH values by 1 N HCl. Thermal unfolding was done at 65°C and followed by SDS-PAGE at about 20°C. Sample pH values shown here have been corrected to 65°C. kj (a) and k2 ( ) shown in log are the thermal unfolding rate constants for the conversions from N to I and from I to M, respectively. The linear lines through the data points are the results of least-square fit to each individual pH phase for both kj and k2 data. The calculated slopes of the fitting lines for kj are -0.46 and 0.35 for the low and high pH phases, respectively and for k2 are -1.9 and 1.1 for the low and high pH phases, respectively.

The calculation of (.E) based on Eq. 14.41 is another case of unfolding. Usually the flux is expressed in terms of a number of energy groups G. If G < n, unfolding of Eq. 14.41 is a simple case of least-squares fit. Unfortunately, in most cases of practical interest, G > n, and the only way to obtain E) is to assume a certain a priori form for it and then try to improve upon this initial guess. The result depends on the choice of the input spectrum, the set of threshold reactions chosen, the errors of the measured activities, and the uncertainties of the cross sections involved. The several unfolding codes that are used differ mainly in the choice of the input spectrum. A brief description of four such codes, SAND-II, SPECTRA, relative deviation minimization method (RDMM)," and LSL-M2 is given next. 

Fig. 7. Conservation of the unfolding/folding mechanism of cold-shock proteins (Csp) from B. subtilis (Bs), B. caldolyticus (fid), and Thermotoga maritima (Tm). (a) Equilibrium unfolding transitions of Csp from Bs (A), Be ( ), and Tm ( ) induced by GdmCI at 25° and monitored by intrinsic fluorescence. Least-squares fit analyses based on the two-state model yield stabilization energies AGstab = 11.3,20.1, and 26.2 kJ/mol for Csp from Bs, Be, and Tm, respectively, (b) Kinetics of unfolding (open symbols) and refolding (closed symbols) of Bs (A, A), Be ( , ) and T Csp (O, ), respectively. The apparent rate constants, X, are plotted against the GdmCI concentration. The fits are on the basis of the linear two-state model. ...

Figure 4 Data for the Gdn-HCl induced unfolding of Staphylococcal nuclease A, monitored by changes in CD signal at 222 nm and by the fluorescence intensity at 340 nm (excitation at 295 nm). The solid cun/es are a global nonlinear least-squares fit of a two-state model to the data with fitting parameters = 5.32 kcal/mol and m = 5.83 kcal/mol-M.

To calculate the values of AH and that best describe the folding curve, initial values of AH, ACp, Of and On are estimated, and Equation [6] is fitted to the experimentally observed values of the change in ellipticity as a function of temperature, by a nonlinear least-squares curve-fitting routine such as the Levenberg-Marquardt algorithm. Similar equations can be used to estimate the thermodynamics of folding of proteins and peptides that undergo folded multimer to unfolded monomer transitions. 

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