On the number of realizations of certain Henneberg graphs arising in protein conformation

Abstract

Several application fields require finding Euclidean coordinates consistent with a set of distances. More precisely, given a simple undirected edge-weighted graph, we wish to find a realization in a Euclidean space so that adjacent vertices are placed at a distance which is equal to the corresponding edge weight. Realizations of a graph can be either flexible or rigid. In certain cases, rigidity can be seen as a property of the graph rather than the realization. In the last decade, several advances have been made in graph rigidity, but most of these, for applicative reasons, focus on graphs having a unique realization. In this paper we consider a particular type of weighted Henneberg graphs that model protein backbones and show that almost all of them give rise to sets of incongruent realizations whose cardinality is a power of two.

Minimal NMR distance information for rigidity of protein graphs

“…Another advantage of the DMDGP order is that it is enough to apply the BP (or other algorithm) to find only one solution, since all the others can be easily obtained using symmetry properties defined in the BP tree [50,53]. These properties are also related to the cardinality of the DMDGP solution set, which can be computed based on the DMDGP graph [46], prior to actually finding realizations.…”

“…These properties are also related to the cardinality of the DMDGP solution set, which can be computed based on the DMDGP graph [46], prior to actually finding realizations. In [50], possible extensions of this result when distances are not precise are also discussed.…”

“…For example, if the graph is rigid, the solution set is finite (up to translations and rotations). In this case, a combinatorial search is better suited than a continuous one, because in addition to the accuracy and efficiency of combinatorial methods, graph rigidity allows us to obtain more information about the cardinality and the structure of the solution set X [41,50] (in Section 3, we will see that these results change when distance values are not precise).…”

A New Approach to the Discretization of Multidimensional Scaling

“…The ε value is set to 0.001 in all the experiments, even if e = 0, in order to deal with some errors introduced by imprecise computer arithmetics. In all generated instances, the total number of solutions is always 2 (when G is embeddable in dimension k, then the total number of solutions is 2, because all distances are available [18]). We point out however that, in general when an approximated embedding is searched, the introduced errors in the distances may lead to the identification of multiple solutions.…”

The Number of Realizations of a Laman Graph

“…Let us just highlight three application areas that are covered there: In materials science the rigidity of crystals, noncrystalline solids, glasses, silicates, etc., is studied; among the numerous publications in this area we can mention [3,21]. In biotechnology one is interested in possible conformations of proteins and cyclic molecules [22], in particular to the enumeration of such conformations [14,25]. In robotics, one aims at computing the configurations of mechanisms, such as 6R chains or Stewart--Gough platforms.…”