A new approximate function for estimation of conformational potential from the excluded volume inside the sphere centered at the geometrical centre of molecule (model of overlapping spheres) is proposed. The value of the function was successfully correlated with the conformational energies of six simple alkanes (from butane to isoheptane). The iterative procedure based on the minimisation of excluded volume is discussed. The method was checked on n-decane and five of its branched derivatives (up to C18H38). The method appears to be very efficient in finding the low-energy conformations of normal alkanes but, for branched molecules, it yielded conformations of higher energy in some cases. The method can be regarded as a simple and fast procedure for finding lowenergy conformations

Nenad Raos

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ISSN-0011-1643CCA-2609 Original Scientific PaperIterative Method for Finding Low-EnergyConformations Based on the Model of OverlappingSpheres: Application to AlkanesNenad RaosInstitute for Medical Research and Occupational Health,P.O. Box 291, HR-10001 Zagreb, CroatiaReceived November 13, 1998; revised February 3, 1999; accepted March 5, 1999A new approximate function for estimation of conformational po-tential from the excluded volume inside the sphere centered at thegeometrical centre of molecule (model of overlapping spheres) isproposed. The value of the function was successfully correlatedwith the conformational energies of six simple alkanes (from bu-tane to isoheptane). The iterative procedure based on the minimi-sation of excluded volume is discussed. The method was checkedon n-decane and five of its branched derivatives (up to C18H38).The method appears to be very efficient in finding the low-energyconformations of normal alkanes but, for branched molecules, ityielded conformations of higher energy in some cases. The methodcan be regarded as a simple and fast procedure for finding low-energy conformations.Key words: iterative method, molecular mechanics, excluded volume,low-energy conformations, model of overlapping spheres, alkanesINTRODUCTIONThe problem of finding the global minimum in conformational analysis,i.e. the search for the most stable conformer(s), is the central problem ofconformational analysis of our times.1,2 There are many approaches to theproblem; they can be generally classified into three groups: (1) finding thecharacteristic conformational patterns from experimental data (e.g. Rama-chandran plot3), (2) building up the final molecular conformation from mo-lecular segments (subconformations),4,5 and (3) finding the global minimumCROATICA CHEMICA ACTA CCACAA 72 (4) 727¿736 (1999)by analysing the potential energy surface (molecular dynamics,6 Monte Car-lo method,7–9 simulated annealing,10 low-mode search method11). Recently,approaches based on genetic algorithm have become operational.12Model of overlapping spheres (OS) recently used for estimation of con-formational energy13 was based on the method proposed by A. J. Hopfin-ger14 and H. A. Scheraga15 for the estimation of hydration (solvation) energyand QSAR analysis.16 While Hopfinger’s model assumes that the solvationenergy of a molecule is dependent on excluded volume in a sphere aroundthe solvated atoms in the molecule, my model13 assumes that the conforma-tional energy of the molecule is roughly proportional to the excluded volumeinside the sphere centered at a specific place in the molecule. This assump-tion simply rationalises the chemical intuition (less strained conformers areless »crowded«) and therefore the method of overlapping spheres can be re-garded as a heuristic approach to the problem of the global minimum. Themodel proved successful for estimation of conformational energy of cop-per(II) chelates with N-alkylated amino acids,13 selecting the conformerswith energy difference > 3–5 kJ mol–1.The OS approach for finding the low-energy conformers has two generaladvantages: (1) it is computationally simple; calculation of the excluded vol-ume is thousands of times faster than the full energy minimisation (molecu-lar mechanics calculations), and (2) the excluded volume is not very depend-ent on the exact geometry of the molecule. But the model, as it was origi-nally proposed,13 was lacking generality. Namely, it was developed for thevery narrow class of molecules (square-planar coordination compounds) andcould hardly be applied to other compounds without substantial modifica-tions. The aim of this paper is to propose a more general approach, whichshould be applicable at least to the acyclic alkanes.METHODSThe OS approach is based on the evaluation of the function:13,14V* = jv jV S s( ) (1)where V* is the volume of overlapping spheres, Sv having the radius Rv andthe van der Waalas sphere sj of j-th atom, respectively. In the new approach,presented for the first time in this paper, the centre of the sphere is situatedat the geometrical centre of the molecule:xi = 1/N jN1xji i = 1, 2, 3 (2)728 N. RAOSwhere the xji is the i-th coordinate of the j-th atom. Then, a new »confor-mational potential« is proposed:VOS = k V* + Vb + V (3)where k is an energy parameter and Vb and V are bonding and valence anglebending potential, respectively. The value of energy parameter k is not verycritical, but it has to be properly chosen to enable rapid convergentionwithout big distortion of bond lengths and angles. Throughout this paper,an arbitrary value 21 MJ mol–1 nm–3 was used. (Introduction of non-bondedpotential in Eq. (3) did not alter the results, but it considerably prolonged thecalculations.) Function (3) was subjected to minimisation by the steepest--descent method.Calculations were performed with a program developed by Kj. Rasmussenand co-workers,17–19 which was modified to deal with function (3). For mo-lecular mechanics calculations, the simple valence-bond force field for alka-nes20 was used.RESULTS AND DISCUSSIONAnalysis of V* FunctionExcluded volume (i.e. volume of overlapping spheres), as defined in Eqs.(1) and (2), was calculated for all conformers of six normal and branched al-kanes (Table I) at three values of the sphere radius (Rv). In five cases, theITERATIVE METHOD FOR FINDING LOW-ENERGY CONFORMATIONS 729TABLE IConformational analysis of alkanesaMolecule Numberof non-equivalentconformersVT max–1kJ molV VT min–1*kJ mol( )Rv/ nm 0.25 0.30 0.35n-butanen-pentanen-hexane2-methylbutane2,3-dimethylbutane2,3-dimethylpentane2472292.54.66.02.40.412.60.0 0.0 0.00.0 0.0 0.00.0b0.0b0.0b0.0 0.0 0.00.0 0.0 0.02.5b2.5b0.1baVT, conformational energy; V*, excluded volume; Rv, sphere radius.bDecreasing series of conformational energies does not entirely correspond to the decreasingseries of OS volumes.conformer with the lowest energy (global minimum) had the lowest overlap-ping volume. In only one case (2,3-dimethylpentane), the minimum of V*did not correspond to the global minimum of energy, but the conformer withthe lowest V* had an energy < 2.5 kJ mol–1 above the global minimum. Inmost cases, decreasing values of conformational potential follow decreasingvalues of excluded (OS) volume.The correlation of OS volume with the conformational potential is alsovisible in the plot of V* vs. central torsion angle in 2-methybutane and 2,3-dimethylbutane (Figure 1). For both molecules, the global (trans) minimumof conformational potential was obtained after minimising the energy of theconformation with the minimal overlapping volume. The minimum of the V*function for 2,3-dimethylbutane corresponded to the trans-conformation, butfor 2-methylbutane the conformation with minimal OS volume was a tran-sition state which, upon minimisation, yielded two equivalent trans-confor-mations.Iterative ProcedureThe first step of the iterative procedure is the steepest-descent minimi-sation of function (3) starting from an arbitrary conformation (due to sim-plicity of function (3), this step needs less CPU time than the usual molecu-lar mechanics minimisation). After the first step, a minimisation of totalconformational energy is performed (molecular mechanics step). It is not al-730 N. RAOSFigure 1. Plot of overlapping volume (V*) vs. central torsion angle at Rv = 0.25 nmfor 2-methylbutane (a) and 2,3-dimethylbutane (b).a bways necessary to perform MM calculations after the minimisation of func-tion VOS (Eq. (3)); if there was no essential change of conformation, or theobtained conformation was close to an already known conformation or totransition state (very hard to minimise), it is better to change Rv and repeatthe procedure. The »global« minimum obtained in the described procedurehas to be tested in a full range of Rv values (e.g. 0.20–0.60 nm).The iterative method was checked on n-decane and its four derivatives(Table II). In all (four) runs, the global minimum for n-decane (all-trans con-formation) was obtained in at least five steps, only once changing the cen-tral sphere radius (Rv = 0.50 nm). Similarly, n-hexane yielded the globalminimum starting from any conformation after performing one to three cy-cles of 1000 steepest-descent iterations (Rv = 0.30 nm).The methods did not prove so successful for branched alkanes becauseconformers of higher energies were also obtained. Moreover, for the highlybranched molecule of 3-ethyl-5,7-dipropyldecane it was impossible to obtainmore than one minimum of lower energy from any initial conformation.The method seems to be more suitable for more symmetric molecules (com-pare 4-ethyldecane with 5-butyldecane and 5-butyl-7-propyldecane with3-ethyl-5,7-dipropyldecane). This finding could be attributed to some kindof conformational heterogeneity of branched molecules, namely to the factthat they are composed of more or less idependent, differently »crowded«segments.Success of minimisation is not dependent on the energy of initial confor-mation for either kind of alkanes. The global minimum of n-decane was ob-tained in one step starting from a conformer with VT = 15.34 kJ mol–1, butin two steps from a conformer with VT = 3.98 kJ mol–1. The most importantfactor for the success of the iterative OS method seems to be some kind ofconformational flexibility, i.e. ability of a conformer to easily change its con-formation (by altering up to four torsion angles, in our examples).To check the influence of sphere radius (Rv) on the course of minimisa-tion, Rv was systematically varied in the range from 0.2 to 0.6 nm (TableIII). It is obvious that some conformations of 3-methyldecane (e.g. 2, 5) aremore suitable for the iterative approach than others (e.g. 1, 7). Higher val-ues of Rv are generally more efficient than the lower ones. In the 0.55–0.60nm range, the lowest conformers (14 and 15) are reachable in one step fromseven initial conformations (3, 4, 6, 10, 11, 12, and 13). In this range of Rv,no conformers of higher energy were obtained (with the exception of 15),and only in six cases was it not possible to change conformation. However,in the range Rv = 0.20–0.25 nm, iterative procedure yielded conformers oflower energy only in seven cases.ITERATIVE METHOD FOR FINDING LOW-ENERGY CONFORMATIONS 731732 N. RAOSTABLE IIGlobal optimisation of n-decane and its derivativesaStep Rv/ nm VT/ kJ mol–1Conformationn-decane0 – 8.37 g–g–g–g–g–g–g–1 0.50 4.27 g–tg–tg–tg–2 0.50 2.65 g–tg–tttg–3 0.50 0.72 g–tttttg–4 0.50 –1.73 ttttttg–5 0.60 –4.18 ttttttt0 – 3.98 tg–g–g–tg–g–1 0.50 3.14 ttg–tg–tg2 0.50 –4.18 ttttttt0 – 15.34 tg–g–g–tgg–1 0.50 –4.18 ttttttt0 – 3.41 tg–g–g–ttg–1 0.50 0.54 ttg–tttg–2 0.50 –4.18 ttttttt(S)4-ethyldecane0 – 21.22 g–g–(t)g–g–g–tg–1 0.55 6.61 tg–(t)g–g–g–tg–2 0.35 6.33 tg–(t)g–tg–tg–3 0.35 5.25 tg–(t)g–tttg–4 0.55 4.97 tg–(g)g–tttg–5 0.60 2.61 tg–(g)g–tttt0 – 14.17 tt(g)g–tttt1 0.6 11.88 tt(g)ttttt0 – 8.02 tt(t)g–tg–tg–1 0.65 5.75 tt(t)g–g–g–tt5-butyldecane0 – 25.33 g–g–g–(tg–g–)g–g–gt1 0.60 19.03 tg–g–(tg–g–)g–g–g–t2 0.60 5.08 ttg–(tg–g–)g–g–tt3 0.60 3.19 ttg–(tg–t)g–ttt(continued)ITERATIVE METHOD FOR FINDING LOW-ENERGY CONFORMATIONS 733Step Rv/ nm VT/ kJ mol–1Conformation4 0.60 0.95 ttg–(ttt)g–ttt0 – 32.66 ttg–(g–g–g–)g–g–g–t1 0.60 31.13 ttg–(g–g–g–)g–g–tt2 0.60 1.72 ttg–(gtt)g–g–t0 – 59.03 g–g–g–(g–g–g–)g–g–g–t1 0.60 17.98 ttt(g–g–g–)g–g–tt2 0.60 10.73 ttt(gtt)g–tg–t3 0.60 8.53 ttt(gtt)g–ttt5-butyl-7-propyldecane0 – 33.91 ttg–(g–g–g–)g–g–(g–g–)g–t1 0.60 33.32 ttg–(g–g–g–)g–g–(g–g–)tt2 0.60* 26.50 tg–t(gtt)g–t(gt)tt3 0.60 25.11 ttt(gtt)g–t(gt)tt4 0.60* 23.63 ttt(gtt)g–g–(tt)tt5 0.20 17.18 tg–t(ttt)g–g–(tt)tt6 0.60 16.68 ttt(ttt)g–g–(tt)tt0 – 20.83 ttg–(gg–g–)g–g–(g–g–)tt1 0.60 8.40 ttt(ttt)g–g–(g–g–)tt2 0.60 7.61 ttt(ttt)g–g–(g–t)tt0 – 8.02 ttg–(tg–g–)g–g–(g–g–)g–t1 0.60 6.88 ttg–(ttt)g–g–(g–g–)tt3-ethyl-5,7-dipropyldecane0 – 39.93 t(g–)tg–(gt)g–g–(g–g–)tt1 0.55 33.32 t(g–)tg–(gt)g–g–(g–t)tt0 – 43.34 t(g–)gg(gg–)g–g–(gg–)tt1 0.60 30.10 t(g–)gg(gt)g–g–(g–t)tt0 – 43.04 t(g–)gg(gg–)tg–(g–g–)tg–1 0.60* 23.79 g–(g–)g–t(tt)g–g–(g–t)g–t0 – 34.38 t(g)gg(gg–)g–g–(g–g–)tg–1 0.60 23.20 t(g)gg(gt)g–g–(g–g–)tg–0 – 23.64 t(g)gg(gt)tg–(g–g–)tt1 0.60 23.10 t(g)gg(gt)g–g–(g–g–)ttaFor meaning of symbols see Table I.* 2000 steepest-descent iterations were performed, otherwise 1000.TABLE II (cont.)CONCLUSIONTo perform iterative search for the conformers of lower energy, the fol-lowing procedure is recommended:(1) Construct an arbitrary conformation and obtain the local minimumby minimising its strain energy.(2) Perform 1000 steepest-descent iterations on the conformation, set-ting up Rv at its maximum value.(3) If torsion angles were altered, find the minimum of conformationalenergy starting from this conformation.(4) If a conformation with lower energy was obtained in step (3), performagain procedures (2) and (3).734 N. RAOSTABLE IIIConformations of (S)3-methyldecane obtained after minimising the VOS function*No. VT/ kJ mol–1 Rv/ nm0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.601 17.34 x x 16 12 11 11 11 11 x2 9.26 3 4 x x 17 6 x 5 213 9.06 x x 18 7 19 20 20 15 144 6.33 13 10 10 13 13 13 13 15 135 6.10 x x 12 x 8 8 8 8 x6 5.79 22 10 13 13 13 13 13 13 157 5.64 x x x x x x 23 24 118 5.61 x x x 25 25 25 25 9 269 4.43 x x x 25 12 12 12 12 x10 3.18 x x x x 27 13 13 15 1311 3.05 x 28 x x x x x 14 1412 2.79 x x x x x x x 15 1413 1.07 x 10 x x x x x x 1514 0.66 x x x x x x x x x15 0.41 x 29 x x x x x x 14* VT, strain energy; x, no change of conformation. Conformers with higher energy than initialstructure are underlined. Conformations (VT /kJ mol–1):1 ttgg–g–tg–, 2 g–g–g–g–g–g–g–, 3 g–g–tgtg–g–, 4 g–g–ttgtg–, 5 tg–g–g–g–tg–, 6 g–g–tg–tgt, 7 g–ttttg–g–,8 tg–tg–tg–g–, 9 tg–tg–ttg–, 10 g–g–ttgtt, 11 ttttttg–, 12 tg–ttttg–, 13 g–g–ttttt, 14 ttttttt, 15 tg–ttttt,16 ttttg–tt (2.95), 17 g–gtg–tg–g–(18.47), 18 g–g–ttttg (3.45), 19 g–tttttg–(6.08), 20 g–g–ttttg–(3.46),21 tg–g–g–g–g–t (6.43), 22 g–g–ttttt (1.07), 23 g–tttttg–(3.88), 24 ttttttg–(3.12), 25 tg–tttg–g–(4.52),26 tg–tg–tg–g–(5.61), 27 g–g–ttttg (3.45), 28 ttttg–tt (2.95), 29 tg–ttgtt (2.77).(5) If step (3) yielded the conformer of higher energy, or step (2) failed togive a substantial change of torsion angles, repeat procedures (2) – (4), de-creasing the Rv value in each step, until the conformer with lower energy isobtained.Obviously, the iterative OS method may be combined with a randomsearch for initial conformations. Moreover, for systematical application ofthe method, it is necessary to find the optimal strategy for the studied classof compounds. The drawbacks of the method, viz. that the global minimumis not reachable from any conformation and that the minima of higher ener-gies could also be obtained, call for its development. This will be the aim ofmy further research.REFERENCES1. A. E. Howard and P. A. Kollman, J. Med. Chem. 31 (1988) 1669–1675.2. Y. C. Martin and C. T. Lin, Three-Dimensional Quantitative Structure – ActivityRelationship: D2 Dopamine Agonists as an Example, in: The Practice of MedicalChemistry (Ed. C. G. Wermuth), Academic Press, London, 1996, pp. 459–483.3. G. N. Ramachandran, Struct. Chem. Mol. Biol. (1968) 77–87.4. A. R. Leach and K. Prout, J. Comput. Chem. 11 (1990) 1193–1205.5. N. Raos and Vl. Simeon, Croat. Chem. Acta 57 (1984) 1217–1224.6. A. Warshel and M. Levitt, J. Mol. Biol. 106 (1976) 421–437.7. G. Chang, W. C. Guida, and W. C. Still, J. Am. Chem. 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Rasmussen, »The ConsistentForce Field: Development of Potential Energy Functions in Conformational Analy-sis«, Recent Experimental and Computational Advances in Molecular Spectros-copy, Proceedings of the NATO Advanced Study Institute on MOLECULARSPECTROSCOPY, Ponta Delgada, Sao Miguel, Acores, Portugal, 1992, R. Fausto(Ed.), NATO ASI Series C: Mathematical and Physical Sciences, Vol. 406, KluwerAcademic Publishers, Dodrecht, 1993, pp. 381–419.20. N. Raos, S. R. Niketi}, and Vl. Simeon, J. Inorg. Biochem. 16 (1982) 1–19.ITERATIVE METHOD FOR FINDING LOW–ENERGY CONFORMATIONS 735SA@ETAKIterativna metoda za pronala`enje konformacija niske energijetemeljena na modelu preklapanja kugliNenad RaosPredlo`ena je aproksimativna funkcija za procjenu konformacijske energije izprora~una isklju~enog volumena unutar kugle smje{tene u geometrijskom sredi{tumolekule (model preklapanja kugli). Vrijednost te funkcije uspje{no je uspore|ena skonformacijskim energijama konformera {est jednostavnih alkana (od butana do izo-heptana). Prodiskutirana su svojstva iterativne metode utemeljene na minimalizacijiisklju~enog volumena. Ta je metoda isku{ana na n–dekanu i njegovim petorim raz-granatim derivatima (sve do C18H38). Metoda je vrlo uspje{na za pronala`enje kon-formera niske energije normalnih alkana, ali je za razgranane molekule pokatkaddala konformacije vi{e energije. Predlo`ena iterativna metoda nudi jednostavan i brzpostupak za pronala`enje konformacija niske energije.736 N. RAOS

Iterative Method for Finding Low-Energy Conformations Based on the Model of Overlapping Spheres: Application to Alkanes

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