# References for the Molecular Distance Geometry Problem

Note: This question has also been posted here.

The molecular distance geometry problem (MDGP) consists of two sub-problems:

1. Given observations of noisy distances between atoms in a molecule, estimate the values of the true distances.
2. Given these estimated distances, compute the locations of the atoms.

More formally, the first sub-problem can be stated as:

Given the datasets $$\mathcal{D}_1,\mathcal{D}_2,\dots,\mathcal{D}_n$$ of noisy distances for the atoms defined by the points $$\mathcal{S} = \{x_1,x_2,\dots,x_n\}$$, estimate the $$n \times n$$ symmetric distance matrix $$\mathbf{A} = (d_{ij})$$, where $$d_{ij} = \lvert\lvert x_i - x_j\rvert\rvert$$ and $$x_i \in \mathbb{R}^K$$ for $$i,j \in \{1,2,...,n\}$$.

The second sub-problem can then be formulated as:

Given $$\hat{\mathbf{A}} = (\hat{d}_{ij})$$, which is an estimate of $$\mathbf{A} = (d_{ij})$$, find the points $$x_1,x_2,...,x_n$$ such that $$\lvert\lvert x_i - x_j\rvert\rvert = \hat{d}_{ij} \ \forall \ i,j$$.

The second sub-problem is well-studied in the literature. If all distances $$d_{ij}$$ are given, and if $$K=3$$, then this problem can be solved using a linear order of operations [1]. Alternatively, if only a small subset of these distances are given, then it is possible to infer the rest of the unknown distances using specific geometrical constraints, such as the triangle inequality [2].

I am currently interested in the first sub-problem. More precisely, are there references that explore different noise models for the distances between atoms and references that attempt to estimate these distances?

• You posted the same question on Matter Modelling SE : mattermodeling.stackexchange.com/questions/5088/… . Note that cross-posting the same question on different sites is not recommended. Jun 3, 2021 at 17:20
• Sorry, I wasn't sure which site would be more appropriate. Which one do you think is more appropriate for this question? Jun 3, 2021 at 18:04
• Personally, I would leave both questions as they are, but edit them to add the link to the question on the other site so that people know there is a cross post. I am not a moderator though, so it's just my personal opinion. Jun 3, 2021 at 21:19