tork: conformational analysis method for molecules and complexes

Tork: Conformational Analysis Method forMolecules and Complexes

CHIA-EN CHANG,1,2 MICHAEL K. GILSON1

1Center for Advanced Research in Biotechnology, University of Maryland BiotechnologyInstitute, 9600 Gudelsky Drive, Rockville, Maryland 20850

2Department of Chemistry and Biochemistry, University of Maryland at College Park,College Park, Maryland 20742

Received 31 January 2003; Accepted 23 May 2003

Abstract: A conformational search method for organic molecules and bimolecular complexes is presented. Themethod, termed Tork, uses normal-mode analysis in bond–angle–torsion coordinates and focuses on a key subset oftorsional coordinates to identify natural molecular motions that lead the initial conformation to new energy minima. Newconformations are generated via distortion along these modes and their pairwise combinations, followed by energyminimization. For complexes, special treatment is accorded to the six coordinates that specify the position andorientation of one molecule relative to the other. Tests described here show that Tork is highly efficient for cyclic,acyclic, and mixed single molecules, as well as for host–guest complexes.

© 2003 Wiley Periodicals, Inc. J Comput Chem 24: 1987–1998, 2003

Key words: host-guest; conformational search; normal mode; internal coordinates; cartesian

Introduction

Conformational search methods are useful in a range of chemicaldesign applications, including drug discovery and the design oftargeted chemical hosts. Considerable progress has been made insearch algorithms for single molecules, as recently summarized.1

Some methods, such as Metropolis Monte Carlo in torsional co-ordinates and the Mining Minima algorithm,2 are appropriate foracyclic molecules, while others are preferred for molecules withflexible rings and macrocycles.3–7 However, many drug-like com-pounds possess both cyclic and acyclic moieties, and chemicalhosts often include large, flexible macrocycles. There is thereforea need for methods that are effective for compounds with flexibleacyclic parts, semirigid rings, and/or flexible macrocyclic structures.

In addition, surprisingly little work has been done on theapplication of conformational search techniques to host–guestcomplexes. Most computational methods for bimolecular com-plexes aim to solve the “docking” problem of fitting a proteintogether with another protein or with a drug-like ligand. In suchapplications, the complexity of the proteins mandates severe sim-plifications, such as keeping much or all of the protein rigid. Muchless attention has been paid to methods of docking chemical guestsinto hosts, a problem of considerable interest as an aid to thedesign of such systems.

Low-mode search (LMOD)1 is an advanced conformationalsearch algorithm in which natural motions of the molecule are

identified by normal-mode analysis; distortion of the moleculealong the soft modes guides the search toward new low-energystructures. Tests on single molecules suggest that LMOD is sig-nificantly more efficient than the SUMM method,8 a torsionalMonte Carlo procedure known to be highly effective. LMOD alsohas been applied to bimolecular complexes,9,10 albeit withoutdetailed testing. However, analysis of LMOD indicates that thereare several avenues for improvement. First, LMOD calculatesnormal modes based on Cartesian coordinates, but a normal modein Cartesian coordinates is a linear combination of Cartesiandisplacements so distortion along a Cartesian mode can onlyproduce linear motions of atoms. Such a distortion can nevergenerate a full bond rotation and will always produce bondstretches (see Fig. 1). As a consequence, distortions along Carte-sian modes do not lead smoothly to new energy minima. Second,we observed that soft modes in Cartesian coordinates often corre-spond to trivial motions that are unable to carry the molecule intoa new energy minimum, as discussed in the Results section.

This article describes Tork, a conformational search methodthat also uses normal-mode analysis but overcomes the limitationsof LMOD discussed in the previous paragraph and is thereforemore efficient. Tork describes the conformation of the molecule interms of bond lengths, bond angles, and torsion angles. Normalmodes constructed as linear combinations of these bond–angle–

Correspondence to: M. K. Gilson; e-mail: [email protected]

© 2003 Wiley Periodicals, Inc.

torsion (BAT) coordinates allow bond rotations to be expressedaccurately as circular motions rather than linear displacements. Inaddition, Tork does not diagonalize the entire second-derivativematrix of the potential energy but instead focuses on a small set oftorsion angles that determine the essential conformation. As aconsequence, a much larger fraction of normal modes correspondsto displacements that are able to carry the molecule into newenergy minima. Tests detailed here indicate that Tork is efficientfor cyclic and acyclic molecules, mixed cyclic/acyclics, macro-cycles, and host–guest complexes.

Computational Methods

Overview of Tork

The Tork algorithm may be summarized as follows. To begin, themolecule in any reasonable starting conformation is energy mini-mized to a local energy minimum, termed the initial conformation.An internal coordinate system built of bond lengths, bond angles,and torsion angles is established and a set of torsional degrees offreedom that are critical determinants of the conformation is iden-tified. The second derivative matrix of the potential energy withrespect to these key torsions is computed and diagonalized and theresulting eigenvectors adjusted to account for the correlated mo-tions of atoms whose torsions share a single bond rotation, asdetailed below. The adjusted eigenvectors, termed “drivers,” rep-resent natural motions of the molecule. The initial conformation isthen distorted along the driver vectors, and/or linear combinationsthereof, and energy minimized to generate new low-energy con-formations.

This algorithm is similar to LMOD because it also uses normalmodes to guide the distortions of the molecule. However, it differs

from LMOD in important respects. First, BAT coordinates areused as the basis set instead of Cartesian coordinates. As a con-sequence, the molecular distortions can generate bond rotationswithout the severe bond stretching associated with Cartesian dis-tortions (see Fig. 1), and the search is therefore more effectivelyguided toward new low-energy conformations. Second, becausethe modes are based upon only a few torsion angles, the matrix ofsecond derivatives is small and only a few eigenvectors are gen-erated. In contrast, LMOD uses all atomic coordinates and there-fore generates 3Natom � 6 modes from which a much smallernumber of “low” modes must be selected as a basis for theconformational search. As discussed below, these low modes canbe similar to each other. Finally, in Tork multiple distortions of asingle initial conformation usually suffice to generate an extensiveset of low-energy conformations. LMOD, in contrast, alwaysneeds to carry out the search in successive steps, using the newconformations in each step as starting points for the next genera-tion. This requirement presumably results from the fact that dis-tortions based upon Cartesian coordinates, and hence restricted tolinear motions of atoms, cannot produce extensive conformationalchange.

Tork Algorithm

Tork operates in BAT coordinates11,12 in which the position ofeach atom i � 3 is specified by its bond length (bi), bond angle(�i), and dihedral angle (�i) with respect to three other atoms thatare bonded in sequence and whose coordinates are already defined.(For atoms i � 3, one or more coordinates are identified asexternal.) In the example shown in Figure 2, atom 1 is defined asthe origin; atom 2 is defined by a bond length (b2); atom 3 isdefined by a bond length and a bond angle (b3, �3); and atoms 4,5, and 6 are defined by (b4, �4, �4), (b5, �5, �5), and (b6, �6,�6), respectively, where the � are dihedral angles and �5 and �6

are both defined with respect to atoms 2, 3, and 4. The internalcoordinates of this molecule can be written as a column vectorwith 3Natom � 6 elements, where Natom is the number of atoms:QT � [b2, b3, �3, b4, �4, �4, b5, �5, �5, b6, �6, �6]. For abimolecular complex, six additional coordinates are introduced to

Figure 1. Consequences of describing a bond torsion as a linearcombination of Cartesian components. Solid lines and filled circlesshow the equilibrium conformation of a molecule, where the stationaryatoms of the torsion are in the x-z plane. Rotation of the indicatedtorsion turns the right atom along the circular path. A Cartesianprojection of this rotation (open circles) proves to be a linear displace-ment of the atom along the y-axis, which cannot generate a full bondrotation but instead stretches the bond and thus sharply raises theenergy.

Figure 2. Internal BAT coordinates for a small molecule.

1988 Chang and Gilson • Vol. 24, No. 16 • Journal of Computational Chemistry

define the position (X, Y, Z) and orientation (�, �, �) of onemolecule relative to the other.

The matrix of the second derivatives of the potential energy Ewith respect to BAT coordinates,

HBAT � ��2E

�b2�b2

�2E

�b2�b3

· · ·�2E

�b2��n

�2E

�b3�b2

· · · · · · · · ·

· · · · · · · · ·�2E

��n��n

�2E

��n�b2

· · · · · ·�2E

��n��n

�, (1)

is computed as

HBAT � C�1 � HAC � �C�1�T, (2)

where HAC is the Hessian in terms of anchored Cartesian coordi-nates11,12 and C is a matrix with elements Cij � � xBAT, j/� xAC,i,where xBAT, j and xAC,i are coordinates j and i in the BAT andanchored Cartesian representations, respectively. Both the Hessianmatrix in Cartesian coordinates HAC and the transformation matrixC are computed analytically.

Next, one dihedral angle �i is selected for each bond in themolecule whose rotation is expected to be an important conforma-tional determinant. For example, the bonds joining atoms 2 with 3and 3 with 4 in Figure 2 can be represented by dihedral angles �4

and �5, respectively. Note that �6 is excluded at this stage. Thesubmatrix H� of HBAT that includes only derivatives in thesetorsional coordinates is then extracted. In the present example,only the key torsions �4 and �5 are selected, yielding

H� � ��2E

��4��4

�2E

��4��5

�2E

��5��4

�2E

��5��5

�. (3)

More generally, n torsions may be included in H�. H� is thendiagonalized to yield n eigenvectors e1, e2, . . . , en and theircorresponding eigenvalues �1, �2, . . . , �n. In the present exam-ple, the eigenvalues �1, �2 correspond to two (normalized) eigen-vectors e1, e2, e.g., e1 � [0.88, �0.48], e2 � [0.48, 0.88]. Theseshort eigenvectors are then embedded into full-length coordinatevectors e� with the other coordinates held fixed and their compo-nents hence set to 0. Thus, e1 and e2 are expanded to e�1 � [0.0, 0.0,0.0, 0.0, 0.0, 0.88, 0.0, 0.0, �0.48, 0.0, 0.0, 0.0] and e�2 � [0.0, 0.0,0.0, 0.0, 0.0, 0.48, 0.0, 0.0, 0.88, 0.0, 0.0, 0.0].

To generate smooth bond rotations, the elements of each e�i thatcorrespond to the same rotatable bond as the key torsion angles areassigned the same values as the key torsions, to form vectorstermed “drivers,” di. Thus, because �5 and �6 share the samerotatable bond vectors e�1 and e�2 are modified to yield

d1 � 0.0, 0.0, 0.0, 0.0, 0.0, 0.88, 0.0, 0.0, �0.48, 0.0, 0.0, �0.48

d2 � 0.0, 0.0, 0.0, 0.0, 0.0, 0.48, 0.0, 0.0, 0.88, 0.0, 0.0, 0.88.

(4)

Finally, each driver di is assigned a force constant ki obtained bynumerical evaluation of the derivative of the energy with respect todistortion along the driver.

The drivers are used to construct distortion directions D fordistorting the initial conformation Q° toward new low-energyconformations,

Qdist � Q� � �D.

The distortion directions are either pure drivers D � �di or linearcombinations of two different drivers di and dj, where the weight-ing is based upon the ratio of force constants:

D � �di ki

kjdj. (5)

This weighting causes greater displacement to occur along thesofter of the two drivers. Thus, each driver generates two puredistortion directions, depending upon the sign, and each combina-tion of drivers generates four distortion directions, depending uponthe signs selected in eq. (5). As a consequence, the total number ofdistortion directions for a system with n drivers is 2n 2n(n �1). If this number is not too large, then all are explored. Otherwise,the pure drivers are explored and then a tractable number ofcombined distortion directions is constructed from random pairs ofdrivers according to eq. (5). This approach to constructing distor-tion directions differs significantly from that of LMOD, whichgenerates 3Natom � 6 vibrational modes and distorts along linearcombinations of the modes whose force constants are below anarbitrary cutoff. Tork uses a smaller number of more highlyfocused search directions.

Empirically, the distortion directions just described do notadequately mobilize bimolecular complexes. For complexes,therefore, the combined distortion directions [eq. (5)] are modifiedby adding a separate random offset to each of the six coordinates(X, Y, Z, �, �, �) that describe the position and orientation of onemolecule relative to the other. The random offsets are uniformlydistributed in the ranges �2 Å for the translational coordinates and�90° for the rotational coordinates.

Once distortion directions have been selected for a given initialconformation, the search proceeds by varying � [eq. (5)] in astepwise fashion until either � � �2 or the rise in potential energyexceeds a user-defined threshold. Except as otherwise noted, astepsize of 0.3 and an energy threshold of 1000 kcal/mol are usedfor cyclic compounds; the corresponding parameters for acycliccompounds are 0.5 and 300 kcal/mol energy, respectively. Wheneither stopping criterion is reached, the distorted conformation isenergy minimized, first with the conjugate gradient algorithm andthen with the Newton–Raphson algorithm. Multiple successiveenergy thresholds can also be employed (e.g., 500, 1000, and 1500kcal/mol). After the 500-kcal/mol threshold, for example, has beenreached and the corresponding structure has been energy mini-mized, distortion resumes from the unminimized conformation andcontinues until the 1000-kcal/mol stopping point is reached (or �

Tork Analysis of Molecules and Complexes 1989

is incremented by �2). The new conformation is then energyminimized and distortion resumes until all energy thresholds (orother stopping criteria) are reached. The use of multiple energythresholds drives torsion angles through large rotational ranges,avoiding the need to distort from multiple initial conformations.

As each new conformation is generated, it is compared with itspredecessors and eliminated if it is a repeat. Dihedral anglesassociated with freely rotating bonds are considered distinct ifdifferent by greater than 60°; dihedrals within rings are considereddistinct if different by greater than 15°. For symmetrical cyclic andacyclic molecules, cyclic permutations and other symmetries aretested.13

LMOD and Quenched Dynamics

This article compares Tork with a local implementation ofLMOD.1 The parameters for LMOD are based upon previouswork1: step size 2.5, energy threshold 2500 kcal/mol, and searchdirections constructed from modes with frequencies �250 cm�1,based upon a Hessian that is not mass weighted. Because bothLMOD and Tork find new minima by distorting molecules alongnormal modes, it was of concern that they might share an inabilityto identify certain energy minima. Quenched molecular dynamics(QMD)14 was therefore used as an unbiased (but inefficient) ref-erence method that provides some assurance that no importantminima are missed by Tork and LMOD. In each case, two molec-ular dynamics calculations were carried out at temperatures of 600and then 800 K, with a timestep of 1 fs and a dielectric constant of1.0. Coordinates were saved every 0.5 ps from each run and laterenergy minimized by the same strategy used to minimize distortedconformations generated in Tork. To match the Tork calculations,a generalized Born15 electrostatics model was used during mini-mization of menthol, cyclophane, and their complex, while vac-uum conditions were used for the nonpolar test compounds.

Computational Details

Initial coordinates for all molecules were generated with the pro-gram QUANTA16; for the host–guest complex, the guest wasplaced inside the host with a docking program.17,18 Potentialenergy was computed with the CHARMM 22 parameter set. Eachconformation was energy minimized by the conjugate gradient andthen the Newton–Raphson method until the energy gradient was�10�2 kcal/mol/Å. The generalized Born solvent model was usedduring energy minimization for menthol, cyclophane, and theircomplex, while vacuum conditions were used for the nonpolarcompounds. All calculations were done on a PC with a Pentium III733-MHz processor.

Results and Discussion

This section evaluates Tork’s efficiency for a series of isolatedmolecules and for one bimolecular complex. The first two mole-cules—n-octane and cyclononane—are of particular interest be-cause they have been used to test other conformational searchmethods.1,13,19 They test the performance of the algorithm for therather different cases of an acylic compound and a moderately

flexible ring. The third test case, (�)-menthol, has both cyclic andacyclic parts. The fourth test case is a macrocycle, a cyclophanethat binds menthol from aqueous solution.20 The fifth test case isthe menthol–cyclophane complex, a system of especial interestbecause it is necessary to identify the low-energy conformations ofcomplexes to compute binding affinities. For each test system,comparison is made with LMOD, which is known to comparefavorably with other methods.1 Quenched molecular dynamicscalculations are used to verify that the Tork and LMOD calcula-tions are not missing conformations.

For each test system, the main table of results starts withinformation regarding the total number of local energy minima thatwere found in extensive searches by multiple methods and/or thenumber within cutoff energy ranges of the global energy mini-mum. Subsequent lines list data for specific search methods. Ineach case, the total number of conformations listed includes theinitial conformation in the count. For menthol, cyclophane, and thementhol–cyclophane complex, multiple search generations werecarried out to perform the complete search, that is, the initialconformations were provided by an external program (QUANTAor the docking method cited above) and subsequent generations ofthe search were initiated from the new conformations generated bythe previous search. The method used to check whether a confor-mation has been previously generated is described in the Compu-tational Methods section.

n-Octane

Extensive conformational searches with both Tork and QMDyielded 129 energy minima for n-octane. As an initial comparisonbetween Tork and LMOD, the starting conformation fromQUANTA was subjected to 50 distortions and energy minimiza-tions with each method. In applying Tork to n-octane, methylrotations were not considered to generate new conformations. Fivedihedrals were therefore selected as key dihedrals, generating fivedrivers. The distortion directions were constructed of all puredrivers and all combinations of pairs of drivers [eq. (5)] for a totalof 50 distortion directions. As shown in Table 1, this calculationled to a total of 36 distinct conformations whereas the correspond-ing LMOD search yielded only 10.

More thorough searches were then done. The Tork searcheswere carried out with 20 different energy thresholds for eachdistortion direction, for a total of 20 � 50 � 1000 distortedstructures that were subjected to energy minimization and overlapchecking. Searches starting from three different energy minimayielded all but one or two of the remaining conformations, asshown in Table 1; using 25 energy thresholds enabled Tork todiscover all 129 conformations via distortions from the singleinitial conformation. For comparison, an LMOD search that alsoinvolved 1000 minimizations was done by generating 50 differentdistortions for each of 20 different starting structures; this searchyielded 95 new conformations, significantly fewer than generatedby Tork calculations of similar length. When LMOD is restartedfrom the 95 new conformations it generates all but 2 of theremaining conformations of the molecule, but the computationalcost is considerable.

Interestingly, a Tork calculation with 20 different initial con-formations and only one energy threshold per distortion direction


generated fewer new minima (115) than the runs in which a singleconformation was distorted to many energy thresholds, eventhough 1000 distorted structures still were generated and mini-mized.

Cyclononane

The conformations of cyclononane have been studied previously,although with different force fields.4,13 Here, we focus on identi-fying all local energy minima whose energies are within 50 kcal/mol of the global minimum. The results, based upon theCHARMM force field, are compared with those of Saunders, whoreported eight local minima for cyclononane with the MM2 andMM3 force fields.13 Extensive searching with Tork and our im-plementation of LMOD yields eight conformers similar to thosefound by Saunders. Thus, the global energy minimum listed inTable 2 and depicted in Figure 3 matches that of Saunders if theangles �127.4 and 56.5 are substituted globally for �125.4 and

56.1, respectively. The other conformations also are closely relatedto those listed in the previous study. However, we find a ninthconformation that is also within 50 kcal/mol of the global energyminimum, as listed in Table 2. Long quenched dynamics calcula-tions yield only six distinct conformations, all of which are amongthose identified by Tork and LMOD.

An initial comparison between Tork and LMOD was done bygenerating an equal number of distortions from the same singlestarting conformation. In Tork, eight key torsional degrees offreedom within the ring (Fig. 3) were used to define the reducedHessian matrix (see Computational Methods). The 8 pure drivers(hence 16 distortion directions) were used along with 24 distortiondirections from randomly selected combinations of drivers, for atotal of 40 distortion directions for each energy threshold. InLMOD, the 20 lowest modes were used and the initial conforma-tion was distorted in either the positive or negative direction, so 40distorted samples were generated. A conformation with potential

Table 1. Performance of TORK, LMOD, and QMD for Conformational Search of n-Octane.

MethodNumber of initial

conformationsEnergy

thresholdsEnergy of initialconformations

Energyminimizations

Totalconformations CPUa (s)

Extensive search 129TORK 1 1 2.516 50 36 24

1 20 �0.679 1000 127b 4891 20 2.516 1000 127c 4711 20 6.792 1000 128d 4761 25 �0.679 1250 129 611

20 1 1000 115 484

LMOD 1 1 2.516 50 10 2220 1 1000 95 43595 1 4750 127 2041

QMD 3 120,000 125

aCPU times for LMOD do not include overlap checking.bMissed minima at 6.613 and 6.616 kcal/mol.cMissed minima at 1.047 and 8.283 kcal/mol.dMissed minimum at 2.126 kcal/mol.

Table 2. Ring Dihedral Angles (Degrees) of the Nine Conformations of Cyclononane Found with theCHARMM Force Field, Ranked by Potential Energy (kcal/mol).

Conformation Energy Ring dihedral angles

1 16.069 �127.4 56.5 56.5 �127.4 56.5 56.5 �127.4 56.5 56.52 16.437 �122.9 87.0 �75.0 119.6 �65.1 �65.1 119.6 �75.0 �87.03 17.193 �104.7 90.4 �104.7 48.9 73.0 �68.3 �68.3 73.0 48.94 17.937 �105.8 98.6 �47.4 �59.3 148.8 �91.8 56.8 �89.3 119.85 19.521 �39.3 107.0 �129.9 51.3 59.4 �70.6 �51.4 135.6 �50.76 21.390 �85.7 130.4 �123.8 40.9 40.9 �123.8 130.4 �85.7 70.67 29.147 61.4 �45.3 �53.0 �1.5 91.4 �1.5 �53.0 �45.3 61.48 43.311 �75.1 33.8 170.6 170.6 33.8 �75.1 103.3 �107.9 103.39 54.619 �69.4 39.4 179.4 �172.0 �49.4 31.1 52.2 �118.7 104.6


energy 16.437 kcal/mol was used to initiate the searches with bothTork and LMOD. Tork generated seven new conformations thatincluded the global energy minimum, while LMOD generated fournew conformations that did not include the global energy mini-mum (Table 3). In Tork, increasing the number of energy thresh-olds from one to three leads to generation of eight or nine of thenine conformations, depending upon the choice of initial confor-mation, as shown in Table 3; a similar result can be obtained byusing three different initial conformations. However, LMOD doesnot generate more than seven of the nine conformations until morethan three initial conformations are used.

Note that starting Tork from one initial conformation andsampling through multiple energy thresholds works as well for thiscyclic compound as it does for linear octane. Indeed, as themolecule is distorted along Tork’s linear combinations of torsionalcoordinates the energy rises and then falls (data not shown),indicating that the distortions successfully move the conformationacross energy barriers and into new low-energy conformations.This helps account for the fact that Tork can generate a full set ofconformations via distortions from a single initial conformation.

The cyclononane case also illustrates the advantages of limitingthe Hessian to the key torsional degrees of freedom. If all 22

dihedral degrees of freedom are included in the Hessian (but nobond stretches or angle bends), then 22 internal modes are obtainedby diagonalization. When distortions are generated based upon theeight modes with the smallest force constants, generating 40 dis-torted samples, only 5 distinct conformations are found, and thesedo not include the global energy minimum. Examination of themodes suggests that the degradation in performance relative toTork results from the fact that many of the modes are similar toeach other. Clearly, there is no value to including similar modes.As an example, compare the distortions resulting from the seventhand eighth modes of this 22-mode system [Fig. 4(a)]. Distortingalong both modes generates similar conformations and is thusunproductive. For comparison, Figure 5 shows that the most sim-ilar pair of modes used in Tork (modes 6 and 7), with its smallerbasis set, are markedly different. In addition, some modes generatedistortions that are ineffective in shifting the conformation of thering because they move only hydrogen atoms. Thus, Figure 4(b)shows that the sixth of the 22 modes only causes 2 hydrogen atomsto pivot around a ring carbon, a movement that is not useful ingenerating a new minimum. LMOD, which operates in all internaldegrees of freedom, generates similar unproductive modes, such asthose shown in Figure 6. This excessive attention to hydrogendisplacements might be remedied by the use of a mass-weightedHessian. On the other hand, it is not clear in general that favoringmotions of massive moieties over light ones is always beneficial.

Menthol

Menthol is of interest because it contains freely rotatable bondsalong with single bonds constrained in a ring. Extensive confor-mational searches of (�)-menthol reveal 60 distinct energy min-ima ranging from 0.726–13.11 kcal/mol. The global minimum hasa chair conformation with all substituents equatorial [Fig. 7(A)].Seven and 11 minima lie within 3 and 5 kcal/mol, respectively, of

Figure 3. Global energy minimum of cyclononane, showing the eightkey dihedral angles used in the conformational search.

Table 3. Performance of TORK, LMOD, and QMD for Conformational Search of Cyclononane.

Method

Number ofinitial

conformationsEnergy

thresholds

Energy ofinitial

conformationsEnergy

minimizationsTotal

conformationsConformations

founda CPUb (s)

Extensive search 9TORK 1 1 16.4 40 7 1–6, 7 16

1 3 16.4 120 9 1–9 641 3 17.1 120 8 1–8 681 3 17.9 120 9 1–9 643 1 120 8 1–8 61

LMOD 1 1 16.4 40 4 2, 4, 6, 7 153 1 120 7 1–7 527 1 280 9 1–9 120

QMD 3 12,000 6 1–6

aSee Table 2 for details of potential energies.bCPU times for LMOD do not include overlap checking.


the global minimum. (These numbers include the global mini-mum.)

An initial comparison between Tork and LMOD was againdone via an equal number of distortions from a single initialconformation, that provided by QUANTA. In this conformation,the cyclohexyl ring has a chair conformation with all substituentsin high-energy axial (ax) positions [Fig. 7(B)]; the potential energyis 5.691 kcal/mol. Tork calculations starting from this structureused 40 distortion directions based upon 7 torsion angles: 5 in thering and 1 each in the hydroxyl and isopropyl groups. The distor-tion directions were based upon the 7 pure drivers along with 26random combinations. Calculations with 1 energy threshold perdirection, and hence 40 minimizations, yielded 11 new conforma-tions, of which 4 were within 5 kcal/mol of the global energyminimum. LMOD searches were done with the 10 lowest Carte-sian modes in all degrees of freedom, for a total of 20 minimiza-tions. This search yielded 10 new conformations, of which 2 werewithin 5 kcal/mol of the global energy minimum. Repeating thesearch with the 20 lowest mode and hence 40 minimizations did

not generate additional conformations. The Tork results are thussomewhat, but not dramatically, superior to LMOD.

More extensive Tork searches done with 15 and 20 differentenergy thresholds for each distortion direction yielded nearly allthe conformations, as shown in Table 4. The search with 15thresholds took 1700 s to generate all conformations within 3kcal/mol of the global energy minimum and all but 1 within 5kcal/mol. It also proved possible to identify all 60 minima effi-ciently via a two-stage process similar to that used in LMOD. First,Tork was run from one initial conformation with drivers basedupon only ring dihedrals, generating nine distinct conformations ofthe ring. Second, these 9 minima were used as initial conforma-tions for a second search in which only the two nonring dihedralswere used to construct drivers, yielding all 60 minima in only479 873 � 1352 s (see Table 4). A less sophisticated two-stepapproach was also tried. Here, the 11 new conformations that weregenerated by the first search listed in Table 4 were used as startingpoints for searches with 2 energy thresholds. As shown in Table 4,this procedure yielded 47 new conformations for a total of 58conformation in 132 2765 s.

A comprehensive LMOD calculation also was carried out,starting with the 10 new conformations generated from the

Figure 4. Results of distorting cyclononane along low modes based upon normal modeanalysis in all 22 torsional degrees of freedom. (A) Black, energy minimum; red and blue,distortions along the similar seventh and eighth modes; the chief differences are the positionsof hydrogen atoms. (B) Black, energy minimum; red, Distortion along the sixth mode, whichonly moves hydrogen atoms.

Figure 5. Results of distorting cyclononane along the two most sim-ilar distortion directions generated by Tork, based upon only eight keytorsion angles. The directions are ranked 6 and 7 according to theirforce constants. Black, energy minimum; red and blue, distortionsalong sixth (red) and seventh (blue) directions. The distorted structuresdiffer substantially from each other in contrast with results in Figure 4.

Figure 6. Results of distorting cyclononane along an unproductivelow mode based upon Cartesian coordinates. Black, energy minimum;red, distorted conformation; the distortion produces little change otherthan shifts in hydrogen positions.


QUANTA starting conformation (see above). As summarized inTable 4, 6 additional rounds of LMOD with 20 minimizations perconformation were carried out, ultimately yielding 59 of the 60

known conformations. The missing minimum is the eighth lowest,lying within 5 kcal/mol of the global minimum. Overall, LMODappears to be somewhat less efficient than Tork for menthol,especially when the ring search is done separately in Tork.

The procedure in which the ring is mobilized separately fromthe rest of the compound is more efficient because it reduces therate at which repeat conformations are generated. For example, ifthe ring has two main conformations, boat and chair, and the sidegroups have four main conformations, then manipulating the sidegroups will quickly generate four conformations for boat and fourfor chair. However, if the ring conformation is allowed to vary aswell, it will sample both conformations while the side-chains varysimultaneously and there will be a greater chance of generatingrepeats. Thus, separating cyclic from acyclic parts effectivelyfactorizes the combinatorics.

Cyclophane

A cyclophane that binds menthol was studied as an example of asynthetic macrocyclic host. Thorough searches with Tork, LMOD,

Figure 7. Menthol. (A) Global minimum with all substituents equa-torial. (B) Initial conformation from the program QUANTA. Substitu-ents are in high-energy axial positions.

Table 4. Performance of TORK, LMOD, and QMD for Conformational Search of Menthol.


conformationsEnergy

thresholdsEnergy

minimizations

Total conformations

CPUc (s)�3a �5a Allb

Extensive search 7 11 60TORK 1d 1 40 1 3 12 132

1d 15 600 7 10 55 17001d 20 800 7 11 57 2217

1e 3 120 1 2 9 4799f 4 288 7 11 60 873

408 7 11 60 1352

1d 1 40 1 3 12 13211d 2 880 7 11 58 2765

920 7 11 58 2897

LMOD 1 1 40 0 2 11 130

1 1 20 0 2 11 7210 1 200 1 3 25 72414 1 280 6 9 46 86024 1 480 7 9 51 14847 1 140 4 5 35 4402 1 40 2 4 14 1251 1 20 0 0 1 68

1180 7 10 59 3773

QMD 3 12,000 7 11 60

Total conformations includes the initial conformation(s).aNumber of minima within 3 or 5 kcal/mol of the global minimum.bAll conformations found.cCPU times for LMOD do not include overlap checking.dTork search with seven key dihedrals.eTork search with drivers based on only ring torsions.fTork search with drivers based on only the two noncyclic torsions.


and quenched dynamics yielded a global energy minimum at87.372 kcal/mol and 4, 10, and 23 minima within 1, 2, and 3kcal/mol, respectively, of the global minimum (Table 5). (Thesecounts include the global minimum itself.) Figure 8 shows theconformation of energy 88.929 kcal/mol that was generated byQUANTA; this was used as the initial search conformation exceptas otherwise noted.

An initial comparison of Tork with LMOD was again basedupon searches from a single conformation. In Tork, the key dihe-drals were taken to be the seven single bonds in the macrocyclethat are not involved in smaller rings such as the benzenes (Fig. 8).

The initial search used distortions along only the pure drivers, fora total of 14 distortion directions, and only 1 energy threshold.This calculation generated seven distinct conformations within 3kcal/mol of the global minimum; these include the second lowestconformation known. An LMOD calculation that also used 14search directions generated only 2 distinct conformations within 3kcal/mol of the global minimum (see Table 5).

A broader Tork search, starting from the same initial confor-mation, was carried out using the pure drivers along with 26combined distortion directions constructed from randomly selectedpairs of drivers, for a total of 40 distortion directions. This calcu-lation generated 12 conformations whose energies are within 3kcal/mol of the global minimum (Table 5). For comparison, anLMOD search based upon the 20 lowest modes (and hence also 40search directions) yielded only 5 distinct minima within 3 kcal/molof the global minimum (Table 5). However, neither LMOD norTork located the global minimum: LMOD found the third lowestminimum and Tork found the second lowest. A still broader Torkrun starting from the single QUANTA-generated conformation butusing 8 energy thresholds per search direction generated all theknown low-energy conformations in about 11,400 s (see Table 5).

Tork and LMOD were also used to carry out second rounds ofsearches starting from the minima found in the initial comparisons(above). Starting from the 6 conformations other than the initialconformation from QUANTA, and using 2 energy thresholds perdistortion, Tork found 22 distinct low-energy conformations.When overlaps between the minima from the two rounds ofsearches were eliminated, Tork proved to have generated thedesired 23 conformations in a total of 408 14,376 � 14,784 s.Similarly, LMOD was used to carry out three additional rounds of

Table 5. Performance of TORK, LMOD, and QMD for Conformational Search of Cyclophane.


conformationsEnergy

thresholdsEnergy

minimizations

Total conformations

CPUb (s)�1a �2a �3a

Extensive search 4 10 23TORK 1 1 14 3 5 7 408

1 1 40 3 8 12 12171 8 320 4 10 23 11,430

1 1 14 3 5 7 4086 2 480 4 9 22 14,376

492 4 10 23 14,784

LMOD 1 1 14 0 1 2 4191 1 40 2 5 8 974

1 1 14 0 1 2 4191 1 40 1 3 6 10115 1 200 3 8 16 5245

10 1 400 4 10 22 9880654 4 10 23 16,555

QMD 3 12,000 4 10 23

aNumber of conformations within 1, 2, and 3 kcal/mol of the global energy minimum.bCPU times for LMOD do not include overlap checking.

Figure 8. Initial conformation of cyclophane host from the programQUANTA. The seven key dihedral angles used in Tork are indicated.


searching, starting first from the one new minimum from the firstquick search. This 4-round procedure also generated 23 low-energy conformations in a total of 419 1011 5245 9880 �16,555 seconds.

The reason Tork outperforms LMOD for this molecule is thatalthough the cyclophane is cyclic it is nonetheless flexible, soTork’s ability to generate smooth bond rotations is advantageous.In particular, the rotational orientations of the NH groups andbenzene rings are key conformational determinants of this mole-cule; examples of these rotations are shown in Figure 9. Distor-tions along Cartesian modes generate linear atomic displacementsand thus cannot easily carry out these rotations, in contrast to BATcoordinates, which allow these components to rotate smoothly.Indeed, as in the case of cyclononane, motion along Tork’s dis-tortion directions leads first to high energies but then to lowerenergies, indicating successful traversal of energy barriers.

Cyclophane–Menthol Complex

This section describes the conformational search for a relativelycomplicated system, the complex of a synthetic cyclophane with(�)-menthol. Surprisingly little attention has been paid to thechallenge of finding the low-energy conformations of host–guestcomplexes despite the importance of this task in calculating bind-ing affinities. Here, extensive Tork searches (well beyond thosedetailed below) yielded a global energy minimum at 78.088 kcal/mol (Fig. 10) and 14, 63, and 133 minima within 1, 2, and 3kcal/mol, respectively, of the global minimum. (These countsinclude the global minimum itself.) Although the first 2 countsseem likely to be exhaustive, we are not confident that the 133count is complete.

Initial conformations for the complex were prepared using adocking algorithm17,18 to fit the global minimum of isolatedmenthol to four different conformations of cyclophane that liewithin 1 kcal/mol of the global minimum of free cyclophane(Table 5). (Note that the docking algorithm was used only as aconvenient means to place menthol in a reasonable startingposition; the docking optimization was minimal and other meth-ods of preparing the initial conformation of the complex couldalso be used.) These initial complexes were then subjected toenergy minimization, yielding 4 initial conformations with en-ergies of 80.198, 81.026, 81.088, and 83.447 kcal/mol. As forthe free cyclophane receptor, seven single-bond torsion angleswithin the macrocyclic ring were selected as key degrees offreedom for the Tork search. Because the equatorial chair

conformation of menthol’s cyclohexyl ring is markedly morestable than alternative ring conformations, the ring torsionswere not included in the key degrees of freedom for menthol;only the two nonring freely rotatable bonds were included. Sixadditional degrees of freedom corresponding to the translationand rotation of menthol relative to cyclophane were also in-cluded in the reduced Hessian, H�. The Tork search was basedon 15 pure drivers (30 distortion directions) along with 70random combinations of drivers, for a total of 100 distortiondirections. Only one energy threshold was used for each dis-tortion direction in the initial set of calculations.

An initial Tork search starting from the four initial conforma-tions led to a total of 4 of the 14 minima within 1 kcal/mol of theglobal minimum. An LMOD search based upon 50 low modes andthus 100 distortion directions (as for Tork) also was started fromeach initial conformation. The LMOD search yielded no confor-mations within 1 kcal/mol of the global minimum (Table 6). Toimprove the efficiency of the LMOD, subsequent searches used alarger number of initial conformations and 25 instead of 50 lowmodes.

Tork and LMOD were both used to carry out second and thirdrounds of searches, starting from conformations generated in pre-ceding rounds. A Tork search starting from 5 low-energy confor-mations found in the first search, and hence involving 500 energyminimizations, led to 5 and 24 conformations with energies within1 and 2 kcal/mol, respectively, of the global minimum. An LMODsearch starting from 10 low-energy conformations from the previ-ous LMOD search, and hence also involving 500 energy minimi-zations, led to only 1 and 5 minima within 1 and 2 kcal/mol,respectively, of the global minimum in the second-round search(see Table 6). Additional rounds of Tork and LMOD searchingwere carried out using the new conformations found in the secondround as initial conformations. Together, the three LMODsearches, involving a total of 1400 energy minimizations, yieldedonly 1 minimum within 1 kcal/mol of the global minimum. At79.037 kcal/mol, this is the 13th lowest conformation found in theextensive conformational searches mentioned above. Together, thethree rounds of Tork, involving a total of 1300 energy minimiza-tions, located 10 minima within 1 kcal/mol of the global minimum,including the second lowest (78.3 kcal/mol). For completeness,one additional round of Tork searching was done starting from allof the five new low conformations found in the third round. Thecombined 4-round search generated all 11, 63, and 133 conforma-

Figure 10. Low-energy conformation of the cyclophane–mentholcomplex, generated by Tork.

Figure 9. Illustrations of rotation of (A) an NH group and (B) abenzene ring in the cyclophane.


tions within 1, 2, and 3 kcal/mol of the global minimum that hadbeen identified by more extensive searches.

For this system, Tork runs based upon multiple energythresholds were somewhat less effective than runs based uponmultiple initial conformations. Thus, in a search starting fromthe same initial docked conformations but using two energythresholds Tork found a total of six conformations within 1kcal/mol of the global minimum (Table 6), including the secondlowest. A second round of searching with 2 thresholds, startingfrom 5 of the new low-energy conformations, found the globalminimum, along with 10 conformations within 1 kcal/mol of it.When overlaps between the minima from the 2 rounds ofsearches were eliminated, Tork had generated 10, 55, and 128conformations within 1, 2, and 3 kcal/mol, respectively, of theglobal minimum (see Table 6). Thus, although the two mul-tithreshold runs also involved a total of 1800 energy minimi-zations they located fewer of the low-energy conformationsthan the series of single-threshold searches described in theprevious paragraph. We conjecture that, although Tork cangenerate smooth bond rotations in the macrocyclic ring, theligand may bump or leave the receptor when the complex isstrongly distorted, as occurs when multiple thresholds are used.As a consequence, it seems preferable to use multiple initialconformations rather than multiple energy thresholds when

applying Tork to ligand–receptor complex systems. Note thatusing multiple initial conformations does not significantly in-crease the CPU demands of the method. Thus, for the presentsystem, computing the Hessian in BAT coordinates and diago-nalizing the reduced Hessian [eq. (3)] require only 0.05 and�0.001 s, respectively.

Conclusions

This article describes Tork, a new conformational searchmethod that is highly efficient for cyclic, acyclic, and mixedmolecules. Tork also is effective for host– guest complexes, animportant class of systems for which previous conformationalsearch methods have not been tested in any detail. Like LMOD,Tork uses normal-mode analysis to guide the search for newlow-energy conformations. However, Tork differs in using BATcoordinates rather than Cartesians and in focusing only on a fewkey dihedral torsions rather than on all 3Natom � 6 internalcoordinates. For complexes, Tork also accords special treatmentto the six coordinates that specify the relative position andorientation of the two molecules.

Based upon tests to date, it appears that a full set of confor-mations can be generated for simple molecules—those with fewer

Table 6. Comparison of the Performance of TORK, LMOD and QMD for Conformational Search of theCyclophane Complex.


conformationsEnergy

thresholdsEnergy

minimizations

Total conformations

CPU (s)�1a �2a �3a

Extensive search 14 63 133TORK 4 1 400 4 12 31 39,064

5 1 500 5 24 54 45,5254 1 400 9 29 56 35,660

1300 10 42 98 120,249

4 1 400 4 12 31 39,0645 1 500 5 24 54 45,5254 1 400 9 29 56 35,6605 1 500 11 40 65 43,192

1800 14 63 133 163,441

4 2 800 6 23 59 79,5005 2 1000 10 48 95 87,066

1800 10 55 128 166,566

LMOD 4 1 400 0 3 12 27,80610 1 500 1 5 21 37,51510 1 500 1 10 23 37,221

1400 1 10 29 102,542

QMD 4 16,000 10 32 87

In LMOD, the CPU time does not include the time for overlap checking.aNumber of conformations within 1, 2, and 3 kcal/mol of the global energy minimum. The global minimum is 78.088kcal/mol in the CHARMM force field.


than about 10 key torsions—by distortions that all start from asingle initial conformation. For more complicated molecules andfor complexes, it is preferable to use multiple initial conforma-tions. As noted above, computing and diagonalizing the Hessianmatrix for additional initial conformations adds a negligible com-putational cost. For a single molecule with acyclic and cyclic parts,the search can be further accelerated by factorizing the search intotwo parts. In the first part, the key torsion angles are limited to thering, so a collection of ring conformers is generated. In the secondsearch, conformations with each ring conformer are used to initiatesearches with key torsions active only in the acyclic part of themolecule.

It is anticipated that the Tork algorithm will have a range ofapplications. Its ability to identify low-energy conformations ofbimolecular complexes should be of particular value in the designof molecular hosts and the computation of binding affinities viapredominant states methods.2,12,21

Acknowledgments

The authors thank Dr. M. Saunders for information regarding thetreatment of symmetry in his publication.13 This work was sup-ported by the National Institutes of Health (GM61300).

References

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tork: conformational analysis method for molecules and complexes

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