Triangle inequality limits

The three-dimensional distance limits are the minimum and maximum value that each distance can assume in any three-dimensional structure all the distances of which lie between their given lower and upper bounds. The computation of these distance limits is a difficult and unsolved problem, but loose approximations are available. By loose , we mean that each approximate upper limit is at least as large as the true upper limit, while each approximate lower limit is no larger than the true lower limit. The most important of these approximate limits are called the triangle inequality limits. The reason they are the most important is that they can be computed rapidly and reliably, even for very large... 

Figure 1 The upper (top) and lower (bottom) triangle inequality limits. The heavy solid line denotes the distance at its lower bound, while a light solid line denotes the distance at its upper bound the dashed line denotes the associated limit...

Figure 2 The digraph whose shortest paths determine the triangle inequality limits. The two-headed arrows in the left and right halves have length equal to the upper bound between the corresponding pair of atoms, while the one-headed arcs going from left to right have length equal to the negative of the lower bound. Note that not all possible arcs are present...

The triangle inequality limits are the minimum and maximum values that the distances can assume in any metric space consistent with the triangle inequality limits themselves. 

Note that although the bounds on the distances satisfy the triangle inequalities, particular choices of distances between these bounds will in general violate them. Therefore, if all distances are chosen within their bounds independently of each other (the method that is used in most applications of distance geometry for NMR strucmre determination), the final distance matrix will contain many violations of the triangle inequalities. The main consequence is a very limited sampling of the conformational space of the embedded structures for very sparse data sets [48,50,51] despite the intrinsic randomness of the tech-... 

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. 

The set of all metric spaces satisfying the distance limits, being the intersection of the box defined by the limits and the convex cone defined by the triangle inequalities, is itself a convex set. Thus every value of every distance between its lower and upper limits is attained in some metric space consistent with all the limits. 

The regions in which the separation can be achieved are limited by the constraints m2 < fl2 and m3 > fli. If either the former or the latter inequality is not fulfilled, the extract or the raffinate, respectively, is flooded with solvent and no separation is obtained. The region of the (m2, m3) plane where the separation can take place includes three other regions, besides the triangle-shaped complete-separation region. In the part where m3 > fl2 and U < mz < az the constraint 17.59c is not fulfilled. Therefore, the extract is insufficiently retained, it is carried forward, pollutes the raffinate stream, and drives its purity below 100%, whereas constraint 17.59d being satisfied, the purity of the extract is still 100%. For similar reasons, the region where mz < fli and < uz corresponds to operation... 

When one of the inequalities (8.7) becomes an equality, that is, when the largest of the three tensions equals the sum of the two smaller, the Neumann triangle degenerates to a line the vertex that was previously opposite the longest side comes to lie on that side, as the altitude of the triangle, measured from that vertex to that side, vamshes. Suppose is the largest tension. Then in the limit we are now contemplating... 

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