Guillermo Restrepo

Topological study of the periodic system.

We carried out a topological study of the Space of Chemical Elements, SCE, based on a clustering analysis of 72 elements, each one defined by a vector of 31 properties. We looked for neighborhoods, boundaries, and other topological properties of the SCE. Among the results one sees the well-known patterns of the Periodic Table and relationships such as the Singularity Principle and the Diagonal Relationship, but there appears also a robustness property of some of the better-known families of elements. Alkaline metals and Noble Gases are sets whose neighborhoods have no other elements besides themselves, whereas the topological boundary of the set of metals is formed by semimetallic elements.

Three dissimilarity measures to contrast dendrograms

We discussed three dissimilarity measures between dendrograms defined over the same set, they are triples, partition, and cluster indices. All of them decompose the dendrograms into subsets. In the case of triples and partition indices, these subsets correspond to binary partitions containing some clusters, while in the cluster index, a novel dissimilarity method introduced in this paper, the subsets are exclusively clusters. In chemical applications, the dendrograms gather clusters that contain similarity information of the data set under study. Thereby, the cluster index is the most suitable dissimilarity measure between dendrograms resulting from chemical investigation. An application example of the three measures is shown to remark upon the advantages of the cluster index over the other two methods in similarity studies. Finally, the cluster index is used to measure the differences between five dendrograms obtained when applying five common hierarchical clustering algorithms on a database of 1000 molecules.

Chemotopological study of positron emitter radionuclides used in PET diagnostic imaging: physical, physico-chemical, dosimetric, quantum and nuclear properties

We carried out a chemotopological study of two groups of positron emitter radionuclides of current and potential use in positron emission tomography (PET) diagnostic imaging. The aim was to look for potential β+ radionuclides not yet in use for PET imaging, taking into account the similarities of these radionuclides with radionuclides already used in PET imaging. The similarity was studied through physical, physico-chemical, dosimetric, quantum and nuclear properties of the radionuclides. We applied cluster analysis (two similarity functions and three grouping methodologies) and generated six dendrograms. One dendrogram was selected in each group, which was used to build up a basis for a topology. From the calculation of topological properties of several subsets of interest, we propose β+ radionuclides 47V, 48V, 63Zn, 70As, 90Nb, 106Ag, 115Sb, 116Sb, 120Sb, 130Cs, 134La and 140Pr, as potential new radionuclides of use in PET imaging.

Comparison of QSARs and Characterization of Structural Basis of Bioactivity Using Partial Order Theory and Formal Concept Analysis: A Case Study with Mutagenicity

Fifteen quantitative structure-activity relationship (QSAR) models developed by various authors for the prediction of mutagenicity of aromatic and heteroaromatic amines were analyzed and thirteen of them, based on 95 amines, were compared using their respective statistics and order theory (Hasse Diagram Technique, HDT) to obtain an ordering of QSAR models. The technique of Formal Concept Analysis (FCA) was applied to the set of 95 amines to extract concepts and, in general, knowledge about the relationship between structural attributes and mutagenicity. HDT may be useful as a general tool for the comparison of different classes of QSAR models. FCA turns out to be a novel mathematical technique for seeking for relationships between molecular structure and activity.

Relating β + radionuclides’ properties by order theory

We studied 27 β+ radionuclides taking into account some of their variants encoding information of their production, such as integral yield, threshold energy and energy of projectiles used to generate them; these radionuclides are of current use in clinical diagnostic imaging by positron emission tomography (PET). The study was conducted based on physical, physico-chemical, nuclear, dosimetric and quantum properties, which characterise the β+ radionuclides selected, with the aim of finding meaningful relationships among them. In order to accomplish this objective the mathematical methodology known as formal concept analysis was employed. We obtained a set of logical assertions (rules) classified as implications and associations, for the set of β+ radionuclides considered. Some of them show that low mass defect is related to high and medium values of maximum β+ energy, and with even parity and low mean lives; all these parameters are associated to the dose received by a patient subjected to a PET analysis.

Rules relating hepatotoxicity with structural attributes of drugs

The relationship between molecular structures of drugs and their hepatotoxicity was studied by characterizing their structure in a new way and using formal concept analysis, a mathematical technique to condense knowledge into particular rules, which does not imply linearly assumptions as many conventional statistical techniques. The structural characterization was based on molecular descriptors and molecular frameworks, further decomposed into structural elements, rings, and bridges. The methodology was applied to drugs in the liver toxicity knowledge base database with the potential to cause drug-induced liver injury. Numbers of atoms and bonds along with the aromatic ratio were suitable descriptors for such drugs. The higher the number of rings and asymmetric structural elements in their terminal ring systems, the higher is the probability of hepatotoxicity. Rules were found which may help to design drugs which are unlikely to be hepatotoxic.

The Journal of Mathematical Chemistry: a bibliometric profile

This paper describes the bibliometric characteristics of 2398 articles published in the Journal of Mathematical Chemistry in the period 1987–2015. These articles have been analysed using data from the Web of Science Core Collection and demonstrate the contribution of the journal not only to mathematical chemistry but also to science more generally.

How frequently do clusters occur in hierarchical clustering analysis? A graph theoretical approach to studying ties in proximity

BackgroundHierarchical cluster analysis (HCA) is a widely used classificatory technique in many areas of scientific knowledge. Applications usually yield a dendrogram from an HCA run over a given data set, using a grouping algorithm and a similarity measure. However, even when such parameters are fixed, ties in proximity (i.e. two equidistant clusters from a third one) may produce several different dendrograms, having different possible clustering patterns (different classifications). This situation is usually disregarded and conclusions are based on a single result, leading to questions concerning the permanence of clusters in all the resulting dendrograms; this happens, for example, when using HCA for grouping molecular descriptors to select that less similar ones in QSAR studies.ResultsRepresenting dendrograms in graph theoretical terms allowed us to introduce four measures of cluster frequency in a canonical way, and use them to calculate cluster frequencies over the set of all possible dendrograms, taking all ties in proximity into account. A toy example of well separated clusters was used, as well as a set of 1666 molecular descriptors calculated for a group of molecules having hepatotoxic activity to show how our functions may be used for studying the effect of ties in HCA analysis. Such functions were not restricted to the tie case; the possibility of using them to derive cluster stability measurements on arbitrary sets of dendrograms having the same leaves is discussed, e.g. dendrograms from variations of HCA parameters. It was found that ties occurred frequently, some yielding tens of thousands of dendrograms, even for small data sets.ConclusionsOur approach was able to detect trends in clustering patterns by offering a simple way of measuring their frequency, which is often very low. This would imply, that inferences and models based on descriptor classifications (e.g. QSAR) are likely to be biased, thereby requiring an assessment of their reliability. Moreover, any classification of molecular descriptors is likely to be far from unique. Our results highlight the need for evaluating the effect of ties on clustering patterns before classification results can be used accurately.Graphical abstractFour cluster contrast functions identifying statistically sound clusters within dendrograms considering ties in proximity

Quantifying Complexity of Partially Ordered Sets

We discuss two complexity indicators reported in the literature for partially ordered sets (posets), the first one based on linear extensions and the second one on incomparabilities. Later, we introduce a novel indicator that combines comparabilities and incomparabilities with a Shannon’s entropy approach. The possible values the novel complexity indicator can take are related to the partitions of the number of order relationships through Young diagrams. Upper and lower bounds of the novel indicator are determined and analysed to yield a normalised complexity indicator. As an example of application, the complexity is calculated for the ordering of countries based on their performance in chemical research. Finally, another complexity indicator is outlined, which is based on comparabilities, incomparabilities, and equivalences.

Advances in Mathematical Chemistry and Applications

MOLGEN 5.x combines the efficiency of the molecular generator MOLGEN 3.5 and the flexibility of MOLGEN 4.x. To achieve this, the software was reimplemented based on a totally new concept. The most visible new features are fuzzy molecular formula input and explicit use of atom state patterns. We here describe the first version MOLGEN 5.0 of this new series.