Comparative investigation of eigenvalue-based molecular descriptors

Molekulski deskriptori su brojevi ili nizovi brojeva koji se koriste za kvantifikovanje molekulske strukture. Posebna klasa molekulskih deskriptora su grafovske invarijante. Poznate su i kao topoloski molekulski deskriptori. Izvo ˇ denje ovih deskriptora omoguceno je zamenom ´ molekula molekulskim grafom. Mnoge korisne matematicke veli ˇ cine mogu se izra ˇ cunati iz ˇ molekulskog grafa, npr. sopstvene vrednosti. Stoga je postalo moguce konstruisati molekulske ´ deskriptore koji se zasnivaju na sopstvenim vrednostima. Oni se nazivaju topoloski molekulski ˇ deskriptori zasnovani na sopstvenim vrednostima. Danas ih ima mnostvo. Samo nekoliko ˇ njih koristi sopstvene vrednosti dobijene iz klasicne matrice susedstva. Me ˇ du njima se isticuˇ energija grafa, Estradin indeks i rezolventna energija. U okviru ove doktorske disertacije izvrseno je uporedno ispitivanje ovih deskriptora. ˇ Prvi deo poglavlja Rezultati i diskusija izvestava o rezultatima u vezi sa istra ˇ zivanjem rela- ˇ cija izmedu energije grafa, Estradinog indeksa i rezolventne energije. Tri topoloska molekulska ˇ deskriptora zasnovana na sopstvenim vrednostima uporedena su pomocu nekoliko skupova ´ alkana i benzenoidnih ugljovodonika. Otkrivene su i diskutovane relacije medu njima. Identifikovani su strukturni parametri koji upravljaju ovim odnosima i dobijene su odgovarajuce´ formule zasnovane na visestrukoj linearnoj regresiji. Pokazalo se da sva tri istra ˇ zena indeksa ˇ kodiraju gotovo iste strukturne informacije o molekulu. Oni se razlikuju samo po stepenu osetljivosti na grananje molekula i po broju nevezivnih molekulskih orbitala. Dalja analiza energije grafa, Estradinog indeksa i rezolventne energije vezana je za degenerativnost ovih deskriptora. Da bi se testirao diskriminativni potencijal ovih deskriptora, korisˇceno je nekoliko klasa izomera hemijskih stabala. U ovim skupovima broj atoma ugljeni- ´ ka se kretao od 9 do 20. Kvantifikovanje degenerativnosti je uradeno pomocu dobro utvr ´ dene velicine. Rezultati pokazuju da energija grafa i Estradin indeks imaju sli ˇ can nivo degenera- ˇ tivnosti. Nagla promena degenerativnosti rezolventne energije u slucaju hemijskih stabala ˇ zahtevala je dodatno ispitivanje. Dobijeni rezultati su pokazali da postoji mnogo hemijskih stabala sa istom rezolventnom energijom. Ona se zovu 푟–ekvienergetska hemijska stabla. Zatim su predstavljeni podaci vezani za pretrazivanje rezolventnih ekvienergetskih hemijskih ˇ stabala. Treci deo poglavlja Rezultati i diskusija donosi rezultate o strukturnoj osetljivosti ener- ´ gije grafa, Estradinog indeksa i rezolventne energije na nekoliko serija katakondenzovanih i perikondenzovanih izomernih benzenoidnih ugljovodonika. Strukturna osetljivost je jedno od najvaznijih i najmanje istra ˇ zenih svojstava grafovskih invarijanti. Nedavno je predstavlje- ˇ na nova metoda za procenu strukturne osetljivosti topoloskih molekulskih deskriptora. Ovaj ˇ algoritam se sastoji od nekoliko razlicitih koraka. Zasnovan je na Tanimoto indeksu i Morga- ˇ novim kruznim fingerprintovima. Utvr ˇ deno je da energija grafa, Estradin indeks i rezolventna energija imaju slicnu strukturnu osetljivost na katakondenzovane izomere. Energija grafa je ˇ najosetljivija na male promene u perikondenzovanim benzenoidnim ugljovodonicima. Pored toga, osetljivost ovih deskriptora testirana je na katakondenzovanim izomerima sa razlicitim ˇ brojem zaliva, uvala i fjordova. Otkriveno je da se vrednost ovih deskriptora postepeno menja sa postepenim povecanjem broja ovih strukturnih detalja. Estradin indeks i rezolventna ´ energija se slicno pona ˇ saju, i u nekim slu ˇ cajevima pokazuju istu strukturnu osetljivost. To se ˇ moze pripisati visokoj korelaciji izme ˇ du njih. U cetvrtom delu poglavlja Rezultati i diskusija predstavljeni su rezultati ispitivanja uticaja ˇ cikla na vrednost energije grafa, Estradinog indeksa i rezolventne energije. Naime, pokazano je da indeksi dobro opisuju fine strukturne detalje, te se moze pretpostaviti da ukoliko znamo ˇ kako je deskriptor koreliran sa strukturom onda mozemo da saznamo i kako osobine zavise ˇ od strukture. U cilju ispitivanja uticaja cikla na vrednost molekulskih deskriptora zasnovanih na sopstvenim vrednostima dizajnirana su tri in silicio eksperimenta . Poslednji deo ovog poglavlja predstavlja rezultate potencijalne hemijske primenljivosti nasih deskriptora. Ta ˇ cnije, ispitan je potencijal predvi ˇ danja fizicko–hemijskih osobina. Ener- ˇ gija grafa, Estradin indeks i rezolventna energija testirani su kao orude za predvidanje tacke ˇ kljucanja, toplote obrazovanja i koeficijenta raspodele oktanol/voda alkana. Pokazano je da ˇ se molekulski deskriptor zasnovan na sopstvenim vrednostima ne moze pojedina ˇ cno koristiti ˇ za uspesno predvi ˇ danje ovih fizicko–hemijskih osobina. Prvi zagreba ˇ cki indeks, broj nula u ˇ spektru i broj metil grupa, takode, moraju biti ukljuceni u modele. Dobijene statisti ˇ cke ve- ˇ licine pokazuju da su modeli konstruisani pomo ˇ cu Estradinog indeksa i rezolventne energije ´ znatno bolji od modela sa energijom grafa. Takav trend je jos izra ˇ zeniji u slu ˇ caju koeficijenta ˇ raspodele oktanol/voda alkanaMolecular descriptors are numbers or series of numbers used for quantification of molecular structure. A special class of molecular descriptors are graph invariants. They are also known as topological molecular descriptors. The derivation of these descriptors has been enabled by the substitution of molecule by a molecular graph. Many useful mathematical quantities may be calculated from a molecular graph, e.g., eigenvalues. Therefore, it became possible to construct molecular descriptors that are using eigenvalues. These are called eigenvalue–based topological molecular descriptors. Today, there are plethora of them. Only few of them are using eigenvalues obtained from the ordinary adjacency matrix. The graph energy, Estrada index, and resolvent energy are the most prominent among them. Within this doctoral dissertation comparative investigation of these descriptors have been performed. The first part of Results and discussion chapter reports results regarding investigation of relationships among graph energy, Estrada index, and resolvent energy. Three eigenvalue– based topological molecular descriptors are compared using several datasets of alkanes and benzenoid hydrocarbons. The relations among them are found and discussed. Structural parameters that govern these relations are identified and the corresponding formulae based on multiple linear regression have been obtained. It has been shown that all three investigated indices are encoding almost the same structural information of a molecule. They differ only by the extent of the sensitivity on a structural branching of a molecule and on the number of non–bonding molecular orbitals. Further analysis of the graph energy, the Estrada index, and the resolvent energy is concerned with the degeneracy of these descriptors. To test discriminative potential of these descriptors, several classes of chemical-tree-isomers have been employed. In these sets number of carbon atoms ranged from 9 up to 20. The quantification of degeneracy has been done using well–established measure. The results show that graph energy and Estrada index exert similar degeneracy level. The specious degeneracy of the resolvent energy in the case of chemical trees is discussed. Obtained results indicated that there are many chemical trees with the same resolvent energy. These are called 푟–equienergetic chemical trees. Then, the results on searching for resolvent equienergetic chemical trees are given. The third part of Results and discussion chapter brings results on structural sensitivity of the graph energy, the Estrada index, and the resolvent energy on several series of catacondensed and pericondensed isomeric benzenoid hydrocarbons. Structural sensitivity is one of the most important and the least investigated property of graph invariants. Recently, a novel method for assessing the structural sensitivity of topological molecular descriptors was put forward. This algorithm consists of several different steps. It is based on Tanimoto index and Morgan circular fingerprints. It was found that graph energy, Estrada index, and resolvent energy exert similar structural sensitivity on catacondensed isomers. The graph energy showed best performance on pericondensed molecules. Additionally, the sensitivities of these descriptors were tested on the catacondensed isomers with the increasing number of bays, coves, and fjords. It was revealed that these descriptors gradually change with the increasing number of these structural features. The Estrada index and resolvent energy perform similarly and in some cases with the same structural sensitivity. This may be attributed to the high correlation between them. The fourth part of the Results and discussion chapter presents the results of the examination of the influence of the cycle on the value of the graph energy, the Estrada index, and the v resolvent energy. Namely, it has been shown that indices describe fine structural details well, so it can be assumed that if we know how the descriptor correlates with the structure then we can also find out how the properties depend on the structure. In order to examine the influence of a cycle on the value of molecular descriptors based on the eigenvalues, three 푖푛 푠푖푙푖푐표 experiments were designed. The last part of this chapter presents results of potential chemical applicability of our descriptors. More precisely, predictive potential of eigenvalue–based topological molecular descriptors was examined. The graph energy, the Estrada index, and the resolvent energy were tested as parameters for the prediction of boiling points, heats of formation, and octanol/water partition coefficients of alkanes. It was shown that an eigenvalue–based molecular descriptor cannot be individually used for successful prediction of these physico–chemical properties. The first Zagreb index, the number of zeros in the spectrum and the number of methyl groups must be also involved in the models. Performed statistics showed that the models constructed using the Estrada index and the resolvent energy are significantly better than ones with the graph energy. Such trend is even more noticeable in the case of octanol/water partition coefficients of alkanes