Q2. What is the purpose of this survey?
The purpose of this survey is to show how research on the Molecular Distance Geometry Problem, which establishes embeddings of molecules in Euclidean space using NMR data, evolved from purely continuous search methods to mixed-discrete (via concepts in graph rigidity) to almost exclusively combinatorial — which can be applied to proteins.
Q3. What is the way to test the pruning of a molecule?
Since equations cannot be verified exactly in floating point arithmetic, it is important to set up a tolerance ε accurately, because excessively small tolerances could force the pruning of all the atomic positions, whereas excessively large ones could allow infeasible positions to be accepted.
Q4. What are the main realms of application of DGPs?
The three main realms of application of DGPs are:• the Molecular Distance Geometry Problem (MDGP) and its variants;• the (wireless) Sensor Network Localization Problem (SNLP);• Graph Drawing (GD).
Q5. How do you compute partial embeddings for each cluster?
The partial embeddings for each cluster are computed by first solving an SDP relaxation of the quadratic system (3) restricted to edges in the cluster, and then applying a local NLP optimization algorithm that uses the optimal SDP solution as a starting point.
Q6. What is the way to find a sBB?
Since searches in nonlinear manifolds of a continuous space can rarely find exact optima, the best one can do in such cases is ε-approximation; this is way the sBB is usually referred to as an exact GO method.•
Q7. What is the natural pruning test?
While the natural pruning test just checks that distances are contained in certain intervals in O(K), essentially O(1) if K = 3, shortest path computations are of order O(n2) in general.
Q8. How can the authors find a shortest path between vertex i and vertex k?
One way for computing a reasonably tight upper bound D(i, k) to ||xi− xk|| is by finding the shortest path between vertex i and vertex k of the graph G.When both pruning tests are used together [28], the BP algorithm is able to find valid embeddings in a smaller number of steps.
Q9. What is the definition of the DMDGP?
Whereas the polynomial method in [13] requires conditions that are too strict to be applied in practice, it is very likely that most protein backbones satisfy the DMDGP definition.
Q10. What is the basic idea of the Stochasting Proximity Embedding (?
If the constraint is violated, the positions of the two atoms are changed according to explicit formulae in order to improve the current embedding.
Q11. What is the simplest way to see the intersection of two spheres in R3?
In order to see this in the case K = 3, recall that the intersection of two spheres (spherical surfaces) in R3 is either empty, or a single point, or a circle in space; intersecting these sets with a third sphere might yield the empty set, a singleton, two distinct points, or the whole circle.
Q12. What is the inverse of the rk(A)?
If rk(A) < 2, either rk(A) = 0 (which means that x1 = x2 = x3, which in particular implies that x1, x2, x3 are collinear, violating the strict triangular inequality) or rk(A) = 1, which implies x1−x3 and x2−x3 are linearly dependent, i.e. x1, x2, x3 are collinear, again violating strict triangular inequality.
Q13. Why is the sBB faster than the exact algorithm?
due to the usual trade-off between effort and results, heuristics are expected to perform better, CPU time-wise, than the exact algorithm.