Distance matrix error

We tested our new potential by applying a local optimization procedure to the potential of some proteins, starting with the native structure as given in the Brookhaven Protein Data Bank, and observing how far the coordinates moved through local optimization. For a good potential, one expects the optimizer to be close to the native structure. As in Ulrich et al. [34], we measure the distance between optimizer B and native structure A by the distance matrix error... 

Table 1. Distance matrix errors DME (in A) between optimizers and native structures...

The so-called distance matrix error is a shape similarity measure based on rms deviations.220-222 jn this case, the distance between two configurations a and b is defined as ... 

Nonlinear mapping (NLM) as described by Sammon (1969) and others (Sharaf et al. 1986) has been popular in chemometrics. Aim of NLM is a two-(eventually a one- or three-) dimensional scatter plot with a point for each of the n objects preserving optimally the relative distances in the high-dimensional variable space. Starting point is a distance matrix for the m-dimensional space applying the Euclidean distance or any other monotonic distance measure this matrix contains the distances of all pairs of objects, due. A two-dimensional representation requires two map coordinates for each object in total 2n numbers have to be determined. The starting map coordinates can be chosen randomly or can be, for instance, PC A scores. The distances in the map are denoted by d t. A mapping error ( stress, loss function) NLm can be defined as... 

A corresponding inverse triangle inequality can be applied to each triplet to raise values in the lower bound matrix L. Now, a distance matrix D, usually referred to as the trial distance matrix, can be constructed by simply choosing elements dt/ randomly between w/ and lif and used to construct a metric matrix G. A matrix so constructed might be some approximation to the distances in the real molecule, but probably not a very good one. Clearly, every time an element d is selected, it puts limits on subsequent selected distances. This problem of correlated distances is discussed further in the section Systematic Errors and Bias. 

Distances for each atom pair are randomly chosen between their lower and upper bounds. These distances are then converted into three-dimensional coordinates and refined against a simple error function made up of contributions from upper and lower bound violations and chiral constraint violations to ensure that the structure meets all distance and chiral constraints. The details of converting the distance matrix to three-dimensional coordinates are beyond the scope of this chapter but are provided in Crippen s text (126) and in an upcoming review article (133). 

Here, 8 and 5, are variable parameters, which are determined during the regression so that the standard error of estimate for a studied property is as small as possible. The augmented distance matrix has been used for generation of a number of variable distance indices (Randid and Pompe, 2001 Lucic et al., 2003) the variable Wiener index, the variable hyper-Wiener index, the variable Balaban index, and variable complements of these indices based on the augmented distance-complement matrix. 

The aim of non-linear Sammon s mapping (NLM) is to represent a p-dimensional space, containing n objects, by a two-dimensional map that preserves the distances between the objects optimally. This optimization problem is not easy and requires extensive computing time because of the large number (2n) of parameters. The starting point is a distance matrix for the p-dimensional space with the Euclidean distance most widely applied. The initial 2n map co-ordinates can be chosen randomly but the scores of the first and second principal component may be used favourably. A mapping error E that reflects the differences of the distances between two objects i and J in p-space (c/,/) and in the 2-dimensional map (dij) has to be minimized (equation 35) ... 

To consider the effect of the thickness, thickness factor is used in calculating the distance matrix, and the factor was estabhshed through the numerical tests for various models. Figure 1 shows shortest paths of A-A and B-B for typical mesh. A-A and B-B have same lengths, but A-A is calculated as about 1.15 times of B-B with Dijkstra s algorithm, so that new scheme was developed to adjust the errors. 

Cross-relaxation rates and interproton distances in cyclo(Pro-Gly) from the full matrix analysis of NOESY spectrum recorded at Tm = 80 ms and T = 233 K. Cross-relaxation rates are obtained from the volumes shown in table 2 according to eq. (11) by Matlab (Mathworks Inc). Error limits were obtained from eq. (27) with Aa = 0.015 (table 2). 

Because experimental error is always present in a measured data matrix, the corresponding row-mode eigenvectors (or eigenspectra) form an orthonormal set of basis vectors that approximately span the row space of the original data set. Figure 4.14 illustrates this concept. The distance between the endpoints of a and a is equal to the variance in a not explained by x and y, that is, the residual variance. 

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