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Computational Biology and Chemistry 35 (2011) 19–23

Contents lists available at ScienceDirect

Computational Biology and Chemistry

journa l homepage: www.e lsev ier .com/ locate /compbio lchem

esearch Article

lobal energy minimization of alanine dipeptide via barrier function methods

ien Ming Ng ∗, Muthu Solayappan, Kim Leng Pohepartment of Industrial and Systems Engineering, National University of Singapore, Singapore

r t i c l e i n f o

rticle history:eceived 11 February 2010eceived in revised form

a b s t r a c t

This paper presents an interior point method to determine the minimum energy conformation of ala-nine dipeptide. The CHARMM energy function is minimized over the internal coordinates of the atomsinvolved. A barrier function algorithm to determine the minimum energy conformation of peptides is

6 December 2010ccepted 29 December 2010

eywords:arrier function methodnergy minimization

proposed. Lennard–Jones 6-12 potential which is used to model the van der Waals interactions in theCHARMM energy equation is used as the barrier function for this algorithm. The results of applying thealgorithm for the alanine dipeptide structure as a function of varying number of dihedral angles arereported, and they are compared with that obtained from genetic algorithm approach. In addition, theresults for polyalanine structures are also reported.

© 2011 Elsevier Ltd. All rights reserved.

rotein structure predictionlobal optimization

. Introduction

The problem of energy minimization refers to determining thelobal minimum potential energy conformation of proteins andeptides. The thermodynamical hypothesis proposed by Anfinsen,tates that the native structure of protein would be at its globalree energy minimum (Anfinsen, 1973). Thus the importance ofnding the global minimum of an energy function is associatedith determining the three-dimensional structure of protein in itsative state. Currently, protein structure is determined throughime-consuming and expensive experimental techniques, such as-ray crystallography and nuclear magnetic resonance (NMR) spec-

roscopy. Hence, the development of computational techniques toddress the problem of protein structure prediction is of paramountmportance.

A major challenge in the computational methods lies in solvingarge-scale global optimization problems arising from minimiz-ng energy functions. No efficient mathematical basis exists foretermining the global minimizer and the method often requiresnormous computational time for large problems. Pardalos et al.1994) and Das et al. (2003) explore the optimization methods thatre relevant in the area of energy function minimization.

Thorough reviews of recent advances in the area of protein

tructure prediction are provided in Bonneau and Baker (2001),loudas et al. (2006) and Floudas (2007). In particular, we focus onhe ab initio method, also referred to as first principle method, as its one of the important methods of protein structure prediction. The

∗ Corresponding author. Tel.: +65 6516 5541; fax: +65 6777 1434.E-mail address: isenkm@nus.edu.sg (K.M. Ng).

476-9271/$ – see front matter © 2011 Elsevier Ltd. All rights reserved.oi:10.1016/j.compbiolchem.2010.12.003

importance is underscored by the fact that these methods do notresort to any database or use any other information to determinethe three-dimensional structure of proteins from its amino acidsequence. It is particularly important to have a set of low energyconformations if a number of populated states are present (Wilsonand Cui, 1990). First pass optimization methods play a vital role inidentifying a set of low energy conformations. These low energyconformations can be used to approximate the entropic contribu-tions associated with the stability of the molecule. Once a sufficientensemble of low energy minima has been identified, a statisticalanalysis can be used to estimate the relative entropic contributions(Klepeis and Floudas, 1999). Methods such as the one proposed inthis paper help to identify both the stable three-dimensional struc-ture (global minimum), as well as a set of low energy conformations(local minimum). The advantages of ab initio methods as proposedby McAllister and Floudas (2010) lies in its ability to (1) predictstructures when a related structural homologue is not available,(2) extend the predictions to different environments, and (3) pro-vide insight into the mechanism, thermodynamics, and kinetics ofprotein folding. Moreover, new structures continue to be discov-ered, which would not be possible by methods that rely only oncomparison to known structures (Floudas et al., 2006).

Interior point methods, frequently used to solve nonlinear andnonconvex problems, is uncommon in the area of protein structureprediction via ab initio methods. However, interior point methodsare used in the area of protein threading to solve the linear pro-

gramming formulation (Wagner et al., 2004; Meller et al., 2002).Other than these two works, to the best of our knowledge, we arenot aware of any other reported work in the application of inte-rior point methods to the problem of protein structure prediction,especially with respect to ab initio methods.

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In this paper, we present an interior point algorithm based onhe barrier function method to determine the minimum energyonformation of alanine dipeptide by minimizing the CHARMMMackerell et al., 1998) free energy equation. An important featuref our algorithm is the choice of barrier function to be have utilized.t is normal to use a reciprocal function or a logarithmic functionf the constraints as a barrier term for the problem (Solayappant al., 2008). However, a minor reformulation of the original prob-em helped us to identify the barrier function that is inherentlyresent in the objective function (free energy equation). Though it

s not mandatory, identifying barrier terms in such a way worksell with the nature of the problem structure and no additional

onstraints are enforced on the problem.The rest of the paper is organized as follows: Section 2 presents

he problem formulation and energy function used, while Section 3roposes a solution method to solve the energy minimization prob-

em. The computational details and results are provided in Sectionsand 5, respectively. Section 6 provides some concluding remarks.

. Problem formulation

Ab initio method determines the native structure of a proteinirectly from its amino acid sequence by minimizing the relevantnergy function. It is exactly due to this that the ab initio method ishe most difficult and yet highly preferred method for determininghe structure of proteins and other macromolecules (Bonneau andaker, 2001). In our work, we have used the CHARMM (Chemistryt HARvard Molecular Mechanics) energy function, which is givenelow:

=∑bonds

Kb(b− b0)2 +∑UB

KUB(S − S0)2 +∑

angles

K�(� − �0)2

+∑

dihedrals

K�(1+ cos(n� − ı))

+∑

improper dihedrals

Kimp(ϕ − ϕ0)2

+∑

nonbonded paris

{εij

[(Rminij

rij

)12

−(Rminij

rij

)6]+ qiqjε1rij

}, (1)

here rij is the Euclidean distance between atoms i and j, and qi, ishe partial charge of atom i, and Kb, KUB, K� , K� , Kimp,Rminij εij ε1 andıre constants. The function described in Eq. (1) computes the poten-ial energy V as a function of cartesian coordinates of atoms. In thease of problems pertaining to protein structure, the energy func-ion is generally used as a function of internal coordinates, i.e., bondengths b, bond angles �, dihedral angles � and improper dihedralngles ˚ The notations with subscript zero represent the equilib-ium values of the corresponding terms. The general assumption inhe bio-chemistry community is that the energy required to perturbhe bond length and the bond angles from their equilibrium valuess relatively large and can be assumed to be of constant value (Byrdt al., 1996). We also adopt the same assumption, and hence the

rotein conformation problem is addressed as a function of dihedralngles in our work.

Based on the CHARMM potential energy function and assump-ions thereof, the problem of energy minimization by ab initio

ethods could be formulated as a nonconvex nonlinear optimiza-

and Chemistry 35 (2011) 19–23

tion problem as shown in (2):

MinimizeV(˚)Subject to−� ≤ �ij ≤ �, i = 2, . . . , N,

j = 3, . . . , N,j = i+ 1,

˚∈�N−1,

(2)

where V is the expression for the total potential energy of the pro-tein as a function of its dihedral angles as given in Eq. (1),˚={�ij :i = 2, ..., N, j = 3, ..., N, j = i + 1}∈� N−1 is a vector of dihedral angles foratoms i and j, while N is the total number of atoms in the proteinconsidered. Here, we adopt a single variable representation for thedihedral angles irrespective of the atom type involved.

The objective function, V(˚), accounts for both the bondedand the non-bonded interactions. However, in the case of bondedatoms, since we are assuming equilibrium values for bond lengthsand bond angles, only the interaction energy of atoms that form adihedral plane is included. In the case of non-bonded interactions,the van der Waals and electrostatic potential of atoms that are sep-arated by at least three covalent bonds are considered. We considerlong-range interactions for all atoms and do not use any cut-off dis-tance to simplify the problem structure. Due to the nonconvexityof the objective function, it is possible to have a multitude of localminimum points even for small peptide systems.

3. Proposed solution method

Though a plethora of methods are available to solve noncon-vex optimization problems that are similar to the one that weencounter in protein structure prediction, interior point methodsare uncommon in the area of ab initio methods. We propose asolution technique based on barrier function methods to solve theformulation shown in (2). This involves using the steepest descentmethod for minimizing the transformed objective function.

3.1. Barrier function algorithm

From Eq. (1),we can hypothetically treat the energy function as acombination of just the dihedral and electrostatic interactions andformulate the problem as given in (3). Note that rij is a function ofthe dihedral angle �:

Hypothetical Primal Problem

Minimize f (˚) =∑

dihedrals

k�(1+ cos(n� − ı))+∑

nonbonded pairs

qiqjε1rij

Subject torij(˚) ≥ 0,− � ≤ ˚ ≤ �,

(3)

To handle the constraints in (3), a barrier function method is used.Barrier function methods transform a constrained problem intosolving a sequence of unconstrained problems. When added to theobjective function, barrier functions prevent the generated pointsfrom leaving the feasible region. They generate a sequence of fea-sible points whose limit is a solution to the original problem. Therequirement of a barrier function is that it should be continuousin the interior of the feasible region and it takes a value of ∞on its boundary. This would make sure that successive feasiblepoints that are generated stay within the feasible region (Bazaraa

et al., 1993). In our problem, the term for van der Waals interactionturns out to be a good candidate for such a function. The van derWaals interaction term is continuous over the region, {˚ : r(˚) > 0},and approaches ∞ as the boundary of the region {˚ : r(˚)≥0} isreached. If � is the barrier parameter and the van der Waals inter-

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a

K.M. Ng et al. / Computational Bi

ction term is used as the barrier function, B(˚), then the barrierroblem can be formulated as follows:

Hypothetical Barrier Problemmin

��(˚,�) = inf{f (˚)+�B(˚) : rij(˚) ≥ 0,−� ≤ ˚ ≤ �}

where B(˚) =∑

nonbonded pairs

εij

[(Rminij

rij(˚)

)12

−(Rminij

rij(˚)

)6].

(4)

ote that the constraints present in the original formulation (2)ave been included in the objective function using the barrier func-ion and only an unconstrained minimization problem, as shownn (4) needs to be solved. Thus a series of unconstrained problemsre solved by decreasing the value of barrier parameter � from aarge initial value at every iteration and the optimal solution of theth iteration is used as an initial solution for the (i + 1)th iteration.lgorithm 1 shows the barrier function algorithm (BFA) that weave proposed.

lgorithm 1 (Barrier function algorithm (BFA)).nitialization step:et ε> 0 be a termination scalar. Let �1 > 1, ˇ∈ (0, 1) and k = 1. Lethe randomly generated torsion angle˚1 be the starting solution.

Step 1:Starting with ˚k, �k, solve the following problem using the

ethod of steepest descent:

min˚

�(˚,�)

Let˚k+1 be a solution to the barrier problem; Go to Step 2.Step 2:If �k < 1, solve the barrier problem using ˚k+1 and �k+1 = 1 as

he initial points and stop. Otherwise let �k+1 =ˇ�k, k← k + 1 ando to Step 1.

. Computational details

There are several factors that need to be considered beforeolving the problem of minimum energy conformation. The typef peptide to be modeled, its corresponding data set for thearameters involved and the means to implement the coordinateonversions should be taken care of. In the following section, wexplain the various factors and implementation details requiredor setting up the problem.

Alanine dipeptide is one of the smallest peptides and is fre-uently used to test the efficiency of optimization methods thatre proposed. Due to blocking of amino and carboxyl end groups,ifferent forms of alanine dipeptide are available and we considerhe alanine dipeptide formed when two alanine amino acids areoined together by a peptide bond.

Alanine dipeptide on the whole has 23 atoms which are con-ected by 22 bonds. It has 39 triples (bond angles) and 49 dihedrals.he equation for the energy function involves many constants thatre specific to the type of atoms that are involved in a particu-ar interaction. Moreover, bond lengths and bond angles of atomsre also required to model and solve the problem. Values for theseonstants and other parameters are determined via experimentalechniques or ab initio methods and is a complex process by itself.uch parametrization is available for different energy functions ande have used the one that is consistent with the CHARMM forceeld. In order to generate the required values, Tinker v4.2, a pub-

icly available software suite developed by Ponder (Ponder, 2004) issed. We use the CHARMM27 parametrization data that is providedy the software for our calculations.

The term rij, which represents the Euclidean distance betweentoms i and j in the objective function, is a function of the internal

and Chemistry 35 (2011) 19–23 21

coordinates (bond lengths, angles and dihedrals). However, com-puting distances using the internal coordinates is difficult and notadvocated for optimization problems where it has to be executedrepeatedly. Hence, conversion to a Cartesian system of coordinatesis imperative. One of the efficient algorithms for this has been pro-posed in Thompson (1967), and we have used it for performing theconversions.

5. Computational results

The proposed algorithms have been tested with the alaninedipeptide structure discussed above. There are a total of 49 dihe-dral angles present in alanine dipeptide, including the backbonedihedral angles. We consider different numbers of dihedral anglesas variables to test the computational efficiency of the algorithmdeveloped. Such an experiment also helps to identify several min-imal energy conformations of the peptide that is considered.

It is common to consider only 2–5 variables for determining theminimum energy conformation of alanine dipeptide. This is doneto reduce the computational load and the accurate empirical valueof energy function is derived by interfacing the solution methoddeveloped with other force field programs available. We vary thenumber of dihedrals (variables) considered for each experimentand do not interface with any of the force field programs avail-able. The energy value reported is completely calculated using thesolution method developed. The dihedrals, van der Waals and elec-trostatic interaction energy are calculated only for the number ofparticipating dihedral angles and it is due to this that the energyvalues are different in all the four cases. Moreover, we allow thetorsional angles to take on any value between −� and � to deter-mine the minimum energy configuration. As explained in Section3, we use the BFA algorithm to determine the optimal solution andthe results obtained are tabulated in Table 1. The computationswere carried out on a PC with Intel Core 2 Duo processor running at1.83 GHz and 1 GB of memory. The algorithms were implementedin MATLAB Version 7.2.

The Var column in Table 1 refers to the number of dihedral anglesconsidered for that experiment, while Vstart and Vend refer to theenergy values in kcal/mol of the starting and ending conformation,respectively. The number of atomic interactions that were consid-ered for each experiment is listed under the column heading Intns.The values of dihedral angles� and are also reported for the mini-mum energy conformation found. The last column refers to the totalnumber of iterations required to determine the reported minimumenergy value. The number of atomic interactions reported here isimportant because it forms a core component of the total energyfunction. Moreover, for each interaction considered, the distancebetween the end atoms (rij) has to be calculated, thereby increasingthe computational cost.

For the 2-variable problem, we consider only the backboneatoms, excluding the side chain atoms, and fix the torsion aroundthe peptide bond, ω, to 180◦. In the case of 5 variables, we includethe two side chain carbon atoms and also allow ω to vary between−� and �. For the 15-variable problem, we include the end grouphydrogen atoms and oxygen atoms along with the hydrogen andoxygen atoms that form the peptide plane. The complete structureof alanine dipeptide is considered for the 49-variable case. Gener-ally the hydrogen bond interactions are not included and a cut-offdistance is also used to reduce the computational load. However,we do not consider such assumptions so that we could study thestructure in its entirety.

The difference in the energy between the starting conformationand the optimal conformation, as presented in Table 1, shows theefficiency of the BFA algorithm. The barrier parameter, �, in theBFA algorithm is reduced from 100 to 1 by 5% at every iteration. Ina general barrier function method, the barrier parameter is usually

22 K.M. Ng et al. / Computational Biology and Chemistry 35 (2011) 19–23

Table 1Minimum energy values of alanine dipeptide computed via BFA.

Var Vstart (kcal/mol) Vend (kcal/mol) Time (s) Intns � (◦) (◦) Its

8 6 174.73 176.90 9012 13 −179.52 −176.98 9054 73 112 68 9067 192 −85.33 −53.40 90

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osairaatapfitcsiv

tnfa

Table 2Comparison of BFA and GA.

Variables Energy (kcal/mol) Time (s)

BFA GA BFA GA

2 27.86 58.52 8 144

AcNH–(Ala)n–CONHCH3, where n is the number of alanine residuesconsidered in the study. For each alanine residue, two dihedralangles (�/ ) are varied and the minimum energy conformation isfound. Similar structures were solved using the BFA algorithm and

2 27.88 27.865 27.05 25.11

25 −32.75 −149.54 349 −56.05 −229.89 36

educed to close to zero, at which point, the augmented objectiveunction becomes close to the original objective function and theolution obtained at that instance is considered to be an approxi-ate solution for the original problem. In our case, since we use the

an der Waals function which is inherently present in the objectiveunction as the barrier function, allowing the barrier parame-er to converge to zero would not solve the original problem.ence, the algorithm is terminated when � reaches 1, i.e., when

he augmented objective function resembles the original objectiveunction. In order to confirm this, we performed some experimentsn which we allowed � to approach 0, and the solution obtained

as used as an initial solution to solve the original problem. Thesexperiments show that the points were almost always trapped atocal minima and this supports terminating the algorithm whenhe barrier parameter reaches 1. Such an early termination alsoas the advantage of avoiding ill-conditioning issues encoun-ered in barrier function methods when the barrier parameterpproaches 0.

While seeking to compare the performance of our method withther methods in the literature, we do not find much work thatolves the problem under the same assumptions or conditionsdopted in our work. As an example, even though the˛BB approachn Maranas et al. (1996) belongs to the ab initio methods, the resultseported are for blocked dipeptide structures by interfacing thelgorithm with other energy programs and holding the dihedralngles at known constant values. Moreover, the ˛BB approach useshe ECEPP energy function. Hence, we have instead used a geneticlgorithm approach to compare with the performance of the pro-osed BFA method. The CHARMM energy function was used as thetness function with the variables taking on values between−180◦

o 180◦. The genetic algorithm was implemented with a scatteredrossover function which generates a random binary vector andelects the genes from parent 1 if the component of a random vectors 1, and the genes from parent 2 if the component of that randomector is 0. This crossover operation is illustrated in Fig. 1.

The mutation operation was achieved using a crossover frac-

ion, which determines the percentage of crossover children in theext generation without including the elite children. The crossover

raction is varied from 0 to 1, by a factor of 0.05 at every run of thelgorithm. Starting from an initial population of 20, the algorithm

Fig. 1. Example of crossover operation.

5 25.11 25.13 12 13125 −149.54 −132.54 354 58249 −229.89 −171.69 3667 1530

is terminated when the population size reaches 500. This geneticalgorithm is also implemented on MATLAB.

The results obtained by the genetic algorithm are presented inTable 2 and compared against the results of BFA. It can be inferredfrom the table that the BFA method locates a minimum conforma-tion which is better than the one found by the genetic algorithmmethod. A comparison of the energy values found and the compu-tation time required by both algorithms is shown in Fig. 2. Thoughthe time taken in the case of BFA for 49 variables is more thanthat of the genetic algorithm, it is compensated by the significantimprovement in the energy values identified.

We have also compared our method with Wilson and Cui (1990),which discusses the application of simulated annealing to pep-tides and reports the energy values found for various polyalanines,

Fig. 2. Comparison of BFA and GA for (a) energy value determined (b) computationtime required.

K.M. Ng et al. / Computational Biology and Chemistry 35 (2011) 19–23 23

Table 3Comparison of results for polyalanines.

n No. of variables (dihedrals) SA approach (Wilson and Cui, 1990) BFA Approach

Min energy (kcal/mol) Avg energy (kcal/mol) Time (min) Energy (kcal/mol) Time (min)

2 4 −24.55 −23.86 2.42 −23.91 0.203 6 −36.15 −33.81 3.93 −35.62 0.374 8 −50.20 −48.96 4.35 −50.42 0.605 10 −64.16 −58.07 6.00 −63.25 0.976 12 −79.05 −75.71 15.41 −78.64 1.207 14 −94.04 −90.06 9.98 −91.37 1.638 16 −109.15 −101.90 12.34 −105.67 2.43

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from internal molecular coordinates. Journal of Chemical Physics 47, 3407–3410.

9 18 −124.22 −110 20 −139.43 −120 40 −291.45 −240 80 −528.58 −4

he results obtained are compared with that obtained from Wilsonnd Cui (1990) in Table 3.

In the SA approach, each problem is solved 10 times and theesults are reported for each run. In Table 3, the columns Min energynd Avg energy correspond to the minimum value and the averagealue of the energy found in 10 runs, respectively. The time takener run in minutes is also reported for the SA approach. The lastwo columns in the table report the energy value found and timeaken for the BFA approach. From the results, we can see that theime taken by the BFA approach is lesser than that required by theA approach. Although both approaches use different energy func-ions, the results indicate that the BFA approach is able to obtainomparable energy values.

. Conclusion

In this paper, we have proposed a barrier function algorithmo solve the problem of energy minimization in peptides and pro-eins by using the van der Waals function as a barrier function. Suchmethod eliminates the need for external functions which mighttherwise complicate an already complex objective function. Com-utational results using the proposed method were presented forhe alanine dipeptide structure. The problem was solved by con-idering different number of dihedral angles for each experimentalun. The number of atomic interactions that were considered alsoaried with respect to the dihedral angles involved. Such an exper-ment helps to identify several minimum energy conformations ofhe peptide considered. To further illustrate the applicability of ourroposed method, we have also provided the results for polyala-ines.

The minimum energy conformations that were identified by ourethod, can be used as initial conformers for other programs andould hence reduce the overall computational cost in other appli-

ations, such as protein structure prediction, peptide docking andrug design. The work in this paper also illustrates the possibil-

ty of exploiting the structure of physical functions encounteredo that suitable computational methods can be used to solve thenderlying optimization problem effectively.

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