I. Introduction to Conformation Searching

2(a)(b)Figure 2. Potential energy surface of n-pentane as a function of its two flexible torsionalangles in (a) a contour format and (b) a surface format (from Molecular Modelling: Principlesand Applications, A. R. Leach, Addison Wesley Longman Limited, Essex, England, 1996.)A systematic conformation search involves varying the flexible geometrical parameters of a molecule (in this case,the two torsional angles C1-C2-C3-C4 and C2-C3-C4-C5) in a regular way and calculating the single point energyfor each configuration. To generate the potential energy surfaces for n-butane or n-pentane, for example, themolecular mechanics energy was evaluated from 0 to 360 degrees in increments of 20 degrees. For n-butane, thiscorresponds to 18 points, while for n-pentane, 18 2 = 324 points are required. Then, energy minimizations are carriedout for each set of initial torsional angles to determine the unique stable conformations of the molecule.For larger molecules with many flexible torsional angles (such as proteins), systematic determination of the potentialenergy surface quickly becomes computationally prohibitive. For a molecule with just 6 torsional angles, 18 6 (or3.4×10 7 ) points are required to generate the torsional potential energy surface using 20-degree increments! Even ifthe angular increment is increased from 20 degrees to 60 degrees, 6 6 or 46,656 points must still be determined.Conformation Searching of PolypeptidesConsider a fragment of a polypeptide backbone that might be found in a protein shown in Figure 3.OHCφCψNCNCCHOFigure 3. A backbone of a polypeptide.The O-C-N-H torsional angles of the peptide bond are rigid due to resonance. The torsional angles of thepolypeptide backbone, denoted Φ and Ψ, are flexible, however. These torsional angles can be treated much like thetwo torsional angles of n-pentane, and a potential energy surface can be mapped. For example, for alaninedipeptide, a plot of the potential energy surface contours is shown in Figure 4.

3Figure 4. Contours of alanine dipeptide potential energy surface (from Molecular Modelling:Principles and Applications, A. R. Leach, Addison Wesley Longman Limited, Essex,England, 1996.)In Figure 4, the angle Φ is plotted on the x-axis and the angle Ψ is shown on the y-axis. This plot shows thedifferent energy minima of alanine dipeptide. For a polypeptide with a geometry described by many more torsionalangles it is computationally difficult to map the full potential energy surface, and a systematic conformer search isnot feasible. For larger systems, other conformational searching techniques will be necessary.For a wide variety of known systems, it has been shown that the different Φ and Ψ angles in each protein fall inspecific ranges. When the different Φ and Ψ angle pairs found in many proteins are plotted, the graph produced iscalled a Ramachandran plot (Figure 5). Note that there are two major basins in which most of the Φ and Ψ anglepairs fall: one centered around Φ = –120º, Ψ = 135º and a second basin centered at about Φ = –90º, Ψ = –30º.There is also a much smaller minor basin centered at about Φ = 70º, Ψ = 45º.Figure 5. Ramachandran plot of Φ and Ψ angles from crystal structures of a variety ofproteins (from Molecular Modeling and Simulation, T. Schlick, Springer, New York, 2002.).

4II. Conformation Searching MethodsSystematic MethodsSystematic conformation searching is a method used to find stable conformers of relatively small molecules. In thismethod, all the flexible torsional angles in the molecule are varied in a systematic fashion in order to generate a setof initial structures. Each initial structure is then energy minimized and the stable conformations are enumerated.The advantage of a systematic search is that all the global and local minima will be found as long as the step-size inthe torsional angle is not too large. However, systematic methods are very difficult to apply to molecules with morethan 7 or 8 flexible torsional angles because the number of initial structures generated becomes enormous and theamount of CPU time to energy minimize each becomes prohibitive.The “difficulty” of a systematic search can be related to the number of structures to be energy minimized. For linearacyclic systems, the difficulty is 6 Nt , where N t is the number of flexible torsions present. A list of calculateddifficulties is presented in Table 2.Table 2. Difficulties of systematic searches involving acyclic systems.Length of Chain Number of Flexible Difficulty(Atoms)Torsions, N t5 2 366 3 2167 4 12968 5 77769 6 46,65610 7 280,00011 8 1,680,00012 9 1.0×10 7: : :17 14 7.8×10 10For cyclic systems, the difficulty is 6 Nt –5Nr , where N t is the number of flexible torsions present and N r is thenumber of rings. A list of calculated difficulties for cyclic systems is presented in Table 3 for N r =1.Table 3. Difficulties of systematic searches involving cyclic systems with one ring.Size of Ring Number of Flexible Difficulty(Atoms)Torsions, N t5 5 16 6 67 7 368 8 2169 9 129610 10 777611 11 46,65612 12 280,000: : :17 17 2.2×10 9

5Monte Carlo MethodsMonte Carlo conformation searching methods are used to find stable conformers of large molecules for whichsystematic searches are not feasible. A Monte Carlo search can be performed by randomly varying the Cartesiancoordinates of a molecule or by randomly varying selected dihedral angles. In these methods, a new structure isgenerated by either shifting the Cartesian coordinates of the atoms by a random amount or by varying the dihedralangles by a random amount. A flow chart for these procedures is shown in Figure 6.Figure 6. Flow chart for Monte Carlo conformation search (from Molecular Modelling:Principles and Applications, 2 nd edition. A. R. Leach, Prentice Hall, Harlow, England, 2001.)Monte Carlo methods that involve randomly varying the cartesian coordinates of a molecule are sometimes referredto as cartesian stochastic searches. An example of one step in such a procedure is shown in Figure 7.Figure 7. The process for a cartesian stochastic search.

6Example: CycloheptadecaneThe objective of the cycloheptadecane paper [M. Saunders, K. N. Houk, Y.-D. Wu, W. C. Still, M. Lipton, G.Chang, and W. C. Guida, J. Am. Chem. Soc. 1990, 112, 1419-1427] is to benchmark various conformation searchingmethods. The authors wanted to find out which methods could find most of the low energy conformers ofcycloheptadecane (within 3 kcal/mol of the global minimum). For each method, the authors compiled how manyconformers were located and how much computer (CPU) time the calculation took.A typical cycloheptadecane conformer is shown in Figure 8. Table 4 summarizes the numbers of low-energyconformers of cycloheptadecane found using the MM2 force field.Figure 8. A typical cycloheptadecane conformer.Table 4. Numbers of conformations of cycloheptadecane found relative to the global minimum.within 1 kcal/mol within 2 kcal/mol within 3 kcal/mol11 69 262Five different types of methods were employed for the conformation search: stochastic (random) cartesiancoordinate searches, systematic torsional tree searches, Monte Carlo torsional searches, distance geometry methods,and molecular dynamics. A summary of the results is given in Table 5.Table 5. Summary of conformation searching methods for cycloheptadecane.Method# Low EnergyConformers Found(% of total)CPU time(days)Stochastic Cartesian222Coordinate(85%)Stochastic Cartesian178Coordinate - improved(68%)Torsional Tree Search I 138(53%)Torsional Tree Search II 211(81%)Monte Carlo I 237(90%)Monte Carlo II 249(95%)Distance Geometry 176(67%)Molecular Dynamics 169(65%)CPU time/conformer90 0.4115 0.109 0.06530 0.1430 0.1330 0.1230 0.1730 0.18Note that not only is it important for a conformation search to locate low-energy conformers quickly, but it is alsoimportant that the search be able to find a significant proportion of the total. Therefore, while the distance geometryand molecular dynamics methods are not too much slower than some of the other methods, that they find less than70% of the total conformers is problematic. For this system, Monte Carlo methods provide a good combination ofefficiency and completeness, locating over 90% of the conformers with a time per conformer of 0.12-0.13.