Simulation-Based Predictions of Binding Affinities of Matrix Metalloproteinase Inhibitors

Simulation-Based Predictions of Binding Affinities of MatrixMetalloproteinase InhibitorsAkash Khandelwala, Viera Lukacovaa, Daniel M. Krollb, Dogan fiˆmezc, Soumyendu Rahad, and Stefan Balaz*a

a Department of Pharmaceutical Sciences and Center for Protease Research,b Department of Physics,c Department of Mathematics,d Department of Computer ScienceNorth Dakota State University, Fargo, ND 58105, USA

Full Paper

Inhibitory potencies of 28 hydroxamate derivatives tomatrix metalloproteinase 9 (MMP-9) have been modeledusing the Linear Response method that describes freeenergy of binding as a linear combination of ensembleaverages of van der Waals, electrostatic, and cavity-relatedterms. Individual terms are differences in the respectivequantities between the bound and free ligands, which weredetermined using two methods: a hybrid Monte Carlomethod within the frame of Surface-generalized BornContinuum Solvation Model and Molecular Dynamicssimulation. The MD simulations of ligands in the hydratedMMP-9 active site, as well as of free ligands in water werecarried out for 200 ps. The time-averaged structures of theligands bound with the protein and of the free ligands were

collected at 5, 10, 25, 50, 100, and 200 ps. The bestcorrelations between experimental and calculated bindingenergies, obtained using the training set containing 21structurally diverse compounds, explained about 90% ofexperimental variance. The predicted binding affinities forthe test set of 7 inhibitors were in good agreement with theexperimental data (RMSE� 0.944). Similar RMSE valueswere observed with 200 random selections of the 7-member test set. The loss of polar and nonpolar solventaccessible surface areas upon binding was identified as themain phenomenon contributing to affinity, accompaniedby the enhancement of van der Waals interactions uponbinding.

1 Introduction

About one third of all proteins needmetals for at least someof their functions. Imbalanced activities of metalloproteinsare implicated in diseases like cancer invasion and meta-stasis, arthritis, Alzheimer×s disease, and Creutzfeldt-Jakobdisease. Ligands, which bind to metals coordinated to theproteins, are candidates for therapeutic interventions in thediseases. Matrix metalloproteinases (MMPs), hydrolysingcomponents of extracellular matrix and other extracellularproteins, belong to the most-studied metalloenzymes.Development of MMP inhibitors is complicated by struc-tural relatedness of the MMP family [1], in which somemembers assume normal physiological roles and others arepathological, depending upon the given concentration oractivity. X-ray and/or NMR structures are publicly avail-able for 9 out of 24 known human MMPs. Reliablestructures of the catalytic domains of the remainingMMPs were obtained by comparative modeling [1, 2]utilizing a significant sequence identity in these areas (56 ±64%). Rich structural information makes MMPs an

754 QSAR Comb. Sci. 2004, 23 DOI: 10.1002/qsar.200430896 ¹ 2004 WILEY-VCH Verlag GmbH&Co. KGaA, Weinheim

* To receive all correspondence: tel.: � 1-701-231-7749, [email protected]

Key words: molecular dynamics, linear response approximation,binding affinity, free energy of binding, matrix metalloproteinases

Abbreviations: �, �, �, �, � ± adjustable parameters, �Gb ± freeenergy of binding, ELR ± extended Linear Response (approach),FEP ± Free Energy Perturbation, GB ± generalized Born (model),HMC ± Hybrid MC (simulation), Ki ± the inhibition constant,LOO ± leave-one-out (cross-validation), LR ± Linear Response(approach), LSO ± leave-several-out (cross-validation), MC ±Monte Carlo (simulation), MD ± molecular dynamics, MM ±molecular mechanics, MMPs ± matrix metalloproteinases, NPSA-SA ± nonpolar SASA, PB ± Poisson-Boltzman (equation), PSASA± polar SASA, RMSE ± root mean squared errors, SA or SASA ±solvent-accessible surface area, SGBCSM ± Surface-GeneralizedBorn Continuum Solvation Model, vdW ± van der Waals(interaction).

A. Khandelwal et al.

� ��

exemplary case for development of selective inhibitorsusing computational tools.The zinc atom in the MMP binding site represents an

additional challenge due to the possibility of multi-dentatecoordination bonding of inhibitors that is most appropri-ately treated quantum mechanically [3 ± 12]. Simplifiedforce field-based approaches differ in the level of sophisti-cation. In the nonbonded approach [13, 14], optimizedelectrostatics and van derWaals terms are used tomodel themetal interactions with ligand and proteins [15, 16]. Thetetrahedral coordination geometry was enforced usingdummy cations placed in the vertices of a tetrahedronencompassing the zinc atom [17, 18]. The geometry enforce-ment is more stringent in the bonded model [19 ± 21] thatdescribes the coordination bonds between the metal atomand the ligand/enzyme by the bond terms including bondstretching, angle bending, and torsional terms. The direc-tional force field YETI incorporates more flexibility inselection of appropriate valence [22, 23]. Further force fieldenhancements include addition of polarizable bonds [24,25], directionality based on orbital hybridization [26], andligand field stabilization energy [27, 28]. Unfortunately, themore sophisticated force fields are not readily available forthe use in MMP-ligand modeling.The approaches to prediction of binding affinities to

receptors with known structures can be classified into threecategories using the partitioning of the binding energy intoindividual contributions as the main criterion, and, for thepartitioning approaches, the utilization of ensembles ofstructures as an additional criterion.Non-partitioning methods are represented by Free En-

ergy Perturbation (FEP) [29], Thermodynamic Integration[30], and similar techniques [31]. In FEP, free energy ofbinding (�Gb) is expressed as the difference��Gb between�Gb×s related to the binding of two ligands. Based on athermodynamic cycle, ��Gb is calculated as the differenceof two other free energies, for an abstract mutation ofcomplex 2 into complex 1 and for the mutation of ligand 2into ligand 1. The paths for mutations are dissected intonumerous points that are characterized by a linear combi-nation of initial and final state potentials. While the methodremains an important technique for free energy calculations,its drawbacks (limitation to close homologs and extremedemand on computational resources due to extensivesampling) currently preclude its routine use in drug design.Ensemble-based partitioning approaches wholly rely on

the (sometimes questioned [32]) assumption that differentcontributions to free energy can be calculated separatelyand that they are additive. They are at least ten times fasterthan the non-partitioning approaches, partially due to thenecessity to consider only two states: complexes and freeligands. The contributions to free binding energy areexpressed as the differences � (between the solvated ligandin the bound and free states) in the ensemble averages(denoted by angle brackets). Two main streams in thiscategorydiffermainly in theuseof fitting toexperimentaldata.

In the first stream, representative structures are gener-ated using molecular mechanics (MM)-based moleculardynamics (MD) simulations with explicit water [33,34] orwith the Generalized Born (GB) continuum model [35].Free energy of binding is given as the sum of the differences� for (1) the force-field energy calculated after removal ofsolvent and counter-ions; (2) the polar solvation energy,calculated as a numerical solution of the Poisson-Boltzmann(PB) equation [33, 34] or with GB model [33, 35]; (3) anempirical estimate of the nonpolar solvation energy basedon the solvent-accessible surface area (SA ± this acronymonly used in this paragraph); and (4) entropy estimatedusing normal-mode analysis. The resulting approaches(MM-PBSA [33,34], MM-GBSA [35]) do not requireparameter optimization, at least in the final stage.The second stream includes the Linear Response (LR,

a.k.a. Linear Interaction Energy) method [36 ± 38] and itsextended version (ELR) [39 ± 41]. The LR approach calcu-lates binding free energy �Gb as the linear combination ofthe differences � of the MD or Monte Carlo (MC) van derWaals and electrostatic energies [36 ± 38], to which the ELRapproach adds the solvent-accessible surface areas [39 ± 41](SASA):

�Gb � �� EvdW� � � �� Eel� � � �� SASA� � � �� 1�

The protein-solvent and solvent-solvent interactions areabsorbed in the adjustable parameters �, �, and �. Thescaling coefficient �, originally assumed to be equal to 1³2based on the linear response of the surroundings to electricfields [36 ± 38], was later shown to vary with the ligandnature and ligand surroundings represented by protein orwater [42 ± 44].Application of a similarmethod to hydrationof drugs and heterocycles required, in addition to thequantities present in Eq. 1, the use of H-bond donor andacceptor counts [45]. Continuing studies of more diverseligands may reveal the need for further empirical correc-tions.Single-structure-based partitioning approaches replace

the ensemble averages by a single configuration, usuallyobtained by a direct geometry optimization of the receptor-ligand complex. The first attempts in this category [46, 47]weremade twodecades ago. The simplest approach uses justthe force-field interaction energies [48]. VALIDATE [49],one of the most elaborate methods based on singlestructures considers, in addition to electrostatic and vander Waals interactions, the number of rotatable bondsfrozen in the receptor-ligand interaction, lipophilicity of theligand, surface complementarity, and ligand strain energy.The Free Energy Force Field approach [50, 51], althoughusing anMD simulation, finally selects single configurationsto calculate free energy contributions that areweighted to fitexperimental data. Conformational entropy is treated viagroup contributions and hydration through a specificsolvation model. Free energy contributions are developedfor individual types of interactions and entropies for the

QSAR Comb. Sci. 2004, 23 ¹ 2004 WILEY-VCH Verlag GmbH&Co. KGaA, Weinheim 755

Simulation-Based Predictions of Binding Affinities of Matrix Metalloproteinase Inhibitors

� ��

whole binding site and whole ligands. In contrast, theCOMBINE analysis [52] fragments the receptor into aminoacids and the ligands into parts and parametrizes the pair-wise interactions among them. As a consequence, theapproach needs to optimize several dozens of adjustableparameters for all the pairwise interactions by the PLSmethod. Simplified single-structure-based partitioning ap-proaches are called scoring functions and categorized asforce field-based methods [53, 54], empirical free energyscoring functions [55 ± 57], and knowledge-based scoringfunctions [58 ± 60]. They aremainly used in high-throughputvirtual screening, in connection with fast docking proce-dures.LR and ELR methods appear particularly useful as the

second-pass methods for computational lead optimization,providing more precise estimates of the binding freeenergies of more diverse hits obtained by docking andscoring in virtual screening. Semi-empirical character of theapproaches would be reduced if the scaling parameters �, �,and especially � for electrostatic interactions adopt pre-defined values [42] for different proteins and ligands.The aim of this study is to evaluate the ability of several

LR-based methods to predict inhibitory potency of hydrox-amate MMP inhibitors, in particular the approaches basedon MC with continuum electrostatics and on MD withexplicit water using time-averaged structures for computa-tion of the energy averages. For this purpose, 28 inhibitors[61] of MMP-9 were selected. MMP-9 plays a prominentrole in cancer metastasis. The crystal structure of MMP-9complexed with reverse hydroxamate inhibitor [62] revealsthe details of the binding mode.

2 Methods

2.1 Data Set

Inhibitory potencies of a series of 28 hydroxamate deriva-tives [61] towards MMP-9 were characterized by theinhibition constants Ki (Table 1). A structurally diverse setof 21 inhibitors with activities ranging from 0.08 to 349 nMi.e. ~3.5 log units were selected as a training set. Thepredictive ability of the model was evaluated by predictingthe activities of randomly selected test set compounds,which were not included in the development of the model.The test set consisted of compounds (3), (6), (15), (18), (19),(22), and (28), which are shown in Table 1. To eliminate asubjective input in selection of the test set, predictions werealso performed for 200 randomly selected test sets.

2.2 Construction of Initial Inhibitor/Enzyme Complexes

Three-dimensional structures of ligands were constructedusing the SYBYL6.91 suite of programs [63] running underIrix 6.5. Energy minimizations were performed using theTripos force field [64] with a distance-dependent dielectrics

and the Powell conjugate gradient algorithm with a con-vergence criterion of 0.001 kcal/(mol ä). For severalligands, partial atomic charges calculated by the GasteigerH¸ckel method [65] and by DFT/B3LYP 6-316** [66] werein good qualitative agreement, therefore the faster formermethodwas used to calculate the charges for all ligands. Thecoordinates of the MMP-9 active site were taken from therecently reported x-ray crystal structure of theMMP-9 in thecomplex with an inhibitor [62] deposited in the ProteinDatabank (PDB code: 1GKC). The inhibitors were dockedinto the active site of MMP-9 using the FlexX program [22]that considers ligand conformational flexibility by anincremental fragment placing technique [55, 67]. Confor-mations of the ligands in the active site were selected fromthe top 30 poses generated by FlexX using the distance lessthan 2.5 ä between the hydroxamate oxygens and the zincatom of the receptor as the primary criterion and the FlexXranking as the secondary criterion [68]. In simulations, thecoordinate bonds between zinc and the ligands weredescribed using the nonbonded approach.

2.3 Molecular Dynamics Simulations

MD Simulations were performed using SYBYL 6.91 underisothermal/isobaric (NPT) conditions, with the Tripos forcefield using nonbonded zinc parameters [13]. For eachsystem, a pair of simulations was performed; one with theligand surrounded by two layers ofwater (TIP3P)molecules[64], the other with the ligand bound to the protein with acap comprising two layers of water molecules surroundingthe complex. The simulations were carried out at 300 K for200 ps. The time step of the simulations was 1 fs with a cutoffof 12 A¬ for nonbonded interactions. The nonbonded pairswere updated every 25 fs. All residues within 4 A¬ of anyligand atom were allowed to move freely and the remainingpart of the protein was kept frozen. The time averagestructures of the ligand bound with the protein and the freeligands were collected at 5, 10, 25, 50, 100, and 200 ps,respectively. These structures were briefly minimized, torelieve the worst conflicts, using the Tripos force field, usinga distance-dependent dielectrics and the Powell conjugategradient algorithm with a convergence criterion of0.001 kcal/(mol ä). The van der Waals and electrostaticenergies were calculated for the bound ligand (protein/solvent) and free ligand (solvent). The polar, non polar andtotal solvent accessible surface area (SASA) terms werecalculated using the ProsSat option in the homologymoduleof the Insight II modeling package [69].

2.4 Surface-Generalized Born Continuum Solvation Model(SGBCSM) Calculations

The energy calculations were carried out using the Liaison[70] suite of programs that incorporates SGBCSM [71, 72].All charges were treated using the OPLS-AA force field[73]. Twomethods were employed for obtaining the average

756 ¹ 2004 WILEY-VCH Verlag GmbH&Co. KGaA, Weinheim QSAR Comb. Sci. 2004, 23


� ��


Table1.

Structures,inh

ibitorypo

tencies[61],and

thesimulationresultsof

thestud

iedMMP-9

inhibitors.T

heLR

term

s(kcal/m

ol)forvande

rWallsan

delectrostaticinteractions

andforthecavity

form

ationwereob

tained

usingminim

izationan

dHMC,b

othwithacontinuu

msolven

tmod

el,a

ndMD

(5ps)withex

plicitsolven

t.The

surfacearea

term

sPSA

SAan

dNPSA

SAarein

ä2 .

No.

XR

1R

2K

i(nM

)�vd

W�ele

�cav

Min.

HMC

MD

Min.

HMC

MD

Min.

HMC

MD

c

PSA

SANPA

SA

1X1

CH

2CH

35.05

�34

.84

�31

.70

�10

9.1

�64

.86

�73

.85

�19

1.0

�2.18

6�2.35

5�11

8.6

�26

6.5

2a

X1

CH

2CH

334

9�29

.65

�26

.60

�47

.56

�72

.85

�89

.78

�16

9.8

�2.11

1�2.07

6�96

.24

�20

5.0

3b

X1

CH

32.03

�35

.62

�36

.32

�10

4.4

�65

.94

�42

.49

�17

6.8

�2.12

1�2.11

3�13

5.2

�32

3.0

4X1

6.57

�43

.27

�39

.87

�10

3.9

�18

.78

�22

.90

�16

4.8

�2.52

2�2.65

1�12

7.3

�20

9.5

5X1

4.56

�39

.34

�42

.57

�63

.68

�68

.48

�51

.49

�11

9.2

�2.47

9�2.61

7�12

9.4

�39

3.6

6b

X1

6.47

�44

.51

�41

.99

�11

3.8

�64

.82

�72

.75

�19

5.5

�2.65

5�2.66

8�13

4.0

�42

3.5

7X1

(CH

2)2N

(CH

2CH

3)2

16.7

�42

.95

�40

.25

�96

.41

�45

.07

�57

.49

�19

6.7

�2.58

�2.45

6�12

3.0

�32

2.7

8X1

3.42

�28

.92

�40

.48

�12

7.3

�83

.37

�85

.91

�22

6.3

�2.36

�2.31

6�66

.70

�37

4.8

9X1

(CH

2)2O

CH

2CH

36.36

�41

.51

�36

.80

�10

0.4

�51

.70

�78

.77

�16

1.2

�2.40

8�2.26

5�14

3.8

�23

8.6


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Table1.

(con

t.)

No.

XR

1R

2K

i(nM

)�vd

W�ele

�cav

Min.

HMC

MD

Min.

HMC

MD

Min.

HMC

MD

c

PSA

SANPA

SA

10X1

CH

2CH

342

.7�29

.29

�26

.21

�81

.55

�72

.29

�61

.81

�15

8.4

�1.88

4�1.90

8�12

9.1

�17

4.4

11X1

CH

2CH

341

.8�33

.72

�32

.23

�88

.61

�73

.14

�52

.40

�17

4.2

�2.12

5�2.15

7�10

8.3

�27

7.1

12X1

CH

2CH

351

.9�29

.47

�26

.58

�83

.31

�66

.45

�67

.42

�17

4.5

�1.99

7�2.07

9�11

3.2

�17

3.3

13X1

CH

2CH

380

.1�29

.08

�27

.22

�88

.12

�63

.95

�64

.18

�19

6.4

�2.02

6�2.13

2�10

8.2

�18

3.1

14X1

CH

2CH

332

.1�32

.28

�31

.61

�95

.55

�38

.87

�52

.01

�16

3.6

�1.89

9�2.15

7�94

.32

�20

4.2

15b

X1

CH

2CH

327

8�29

.62

�30

.93

�86

.79

�75

.11

�45

.87

�17

1.8

�1.93

7�1.97

9�13

3.8

�18

0.0

16X1

CH

2CH

395

.6�31

.16

�28

.90

�78

.29

�66

.43

�73

.22

�19

8.9

�2.02

8�2.08

3�14

1.5

�20

1.8

17X1

CH

2CH

359

.2�28

.90

�29

.28

�73

.76

�69

.53

�65

.83

�16

8.9

�1.92

6�1.97

8�15

0.5

�15

6.9

18b

X1

CH

2CH

314

.2�47

.01

�40

.86

�11

3.3

�35

.99

�40

.47

�18

8.1

�2.75

8�2.64

5�14

3.3

�33

5.9

19b

X1

CH

2CH

30.31

�43

.64

�41

.43

�13

7.2

�62

.87

�77

.17

�23

4.7

�2.68

�2.69

4�11

3.4

�30

4.4

20X1

CH

2CH

30.08

�47

.11

�44

.47

�12

1.7

�62

.03

�70

.28

�21

3.7

�2.66

1�2.65

8�10

9.4

�48

6.8

21X1

CH

2CH

30.6

�45

.60

�45

.89

�11

5.9

�64

.23

�72

.33

�19

4.2

�2.62

4�2.70

2�12

8.9

�30

0.5


� ��


Table1.

(con

t.)

No.

XR

1R

2K

i(nM

)�vd

W�ele

�cav

Min.

HMC

MD

Min.

HMC

MD

Min.

HMC

MD

c

PSA

SANPA

SA

22b

X1

CH

2CH

30.1

�46

.71

�43

.61

�13

5.8

�70

.64

�71

.19

�23

2.1

�2.71

6�2.79

1�15

7.7

�44

3.3

23X1

CH

2CH

338

.3�30

.46

�26

.62

�62

.62

�76

.94

�97

.71

�18

5.3

�2.20

6�2.22

3�10

4.1

�20

5.0

24X2

CH

2CH

311

.0�41

.48

�31

.64

�10

2.3

�45

.27

�73

.93

�19

9.9

�2.25

1�2.28

2�13

3.7

�23

6.4

25X3

CH

2CH

33.8

�33

.83

�32

.88

�10

7.4

�70

.69

�76

.21

�20

7.6

�2.19

9�2.19

4�13

8.5

�27

0.3

26X4

CH

2CH

31.13

�39

.03

�29

.62

�10

8.2

�40

.48

�75

.90

�20

5.2

�2.26

9�2.18

4�15

3.2

�35

8.2

27X5

CH

2CH

35.20

�43

.83

�30

.91

�12

5.5

�74

.70

�74

.40

�23

1.4

�2.36

4�2.22

4�14

2.7

�27

0.6

28b

X6

CH

2CH

331

6�37

.82

�30

.58

�98

.87

�39

.73

�77

.28

�16

8.3

�2.23

9�2.20

6�12

7.8

�23

3.5

aconfigurationof

C-3

(marke

dwithan

asterisk)is

Sin

2while

inothe

rcompo

unds

itis

R;b

test

setcompo

und;

cSA

SAis

thesum

ofPSA

SAan

dNPSA

SA.


� ��

LR energies: conjugate-gradient minimization and theHybrid Monte Carlo (HMC) simulation that followed theminimization step. HMC simulation employed 5 ps ofheating, a sample target temperature of 300 K, and 5 ps ofsampling for the LR energies (total simulation time�10 ps). The time step was 0.002 ps; there were 5 MD stepsper HMC cycle, and energies were sampled every 10 steps.For all simulations, protein residues beyond 4 A¬ of theligands were frozen. The default 15 A¬ residue-based cutofffor electrostatic interactions was employed in all calcula-tions. TheSASAterm inEq. 1was replacedby theSGBCSMcavity term.

2.5 Regression and cross-validation

The least-square fits, based on singular value decomposi-tion, of Eq. 1without the constant terms to theminimizationand HMC data were performed using the Liaison package[70]. The least square fits for cross-correlation were done byORIGIN software [74], using Eq. 1 with the constant term.Statistical analyses for the MD-based models described byEqs. 1 and 2 were performed using the ORIGIN program[74]. The robustness of the regression equations and theirpredictive abilities were probed by cross-validation. For thispurpose, the fits to the potency data are generated leavingout one ormore inhibitors from the calibration process. Theresulting equation for each fit is used topredict thepotenciesof the omitted compounds. To generate the cross-validatedroot mean squared errors (RMSE) for the 21 inhibitors ofthe training set, we used the leave-one-out (LOO) approachand the leave-several-out (LSO) approach, where 3 inhib-itors were randomly omitted and the process was repeated200 times. To suppress the influence of the test set selection,theLSOapproachwas repeated for all 28 inhibitors, where 7inhibitors were randomly omitted and the process wasrepeated for 200 times. The correlations of the LSOpredictions with the actual potencies were characterizedby the root mean square errors (RMSE).

3 Results and Discussion

Three samplingmethodswere compared for description andprediction of inhibitory potencies of hydroxamate inhibitorstoMMP-9 (Table 1) using the structure-basedLRapproach:(1) a single-structure-based technique using just the mini-mized structures of the inhibitor and the FlexX-dockedinhibitor-enzyme complex, and ensemble-based methodsutilizing (2) HMC with a continuum solvation model, and(3) MD simulations with explicit solvation. For the MDsimulations, the optimized time-averaged geometries timesof the ligands in the binding site and alone were used tocalculate the LR terms. In the LR approximation, thebinding affinity is expressed as the linear combination ofterms related to van der Waals interactions, electrostatics,and cavity formation, as shown in Eq. 1. The respectivecontributions for all inhibitors, as obtained by all three usedsampling methods are summarized in Table 1 and theircross-correlations are given in Table 2.

3.1 Analysis of Minimization and HMC Data

A summary of fits of the training set data to Eq. 1 for acontinuum-solvation-based minimization and HMC sam-plings is shown in Table 3. The best statistical indices wereobtainedusing all threedescriptors, without a constant term,for minimization (r2� 0.622) and HMC (r2� 0.646). Theerror terms of the regression coefficients for these modelsare quite high, reflecting the cross-correlations (Table 2)between the van der Waals terms and the cavity termsobtained by minimization (r2� 0.723) and HMC (r2�0.795). The error terms improved when the van der Waalsterms and the cavity terms were not kept in the samemodel.The statistical significance of all the descriptors (Table 3)was evaluated by the F test. For each descriptor, except forminimization and theHMCmodelwith all three descriptors,the probability�F ratio was� 0.0001, implying a low like-lihood of a random occurrence of the descriptor. The beststatistically significant model for minimization (r2� 0.570)


Table 2. Inter-correlations between the LR tems used in the MD models for the training set molecules. The dependencies arecharacterized by r2 (r is the correlation coefficient), with the values above 0.5 shown in bold.

Descriptor Pairs Minimization HMC MD ± Simulation Period [ps]

5 10 25 50 100 200

SASAa vs. Vdw 0.723 0.795 0.148 0.394 0.514 0.043 0.229 0.165SASAa vs. Ele 0.057 0.073 0.059 0.193 0.345 0.029 0.030 0.430Vdw vs. Ele 0.226 0.046 0.878 0.522 0.609 0.191 0.073 0.054SASA vs. PSASA NA NA 0.384 0.026 0.288 0.404 0.266 0.253SASA vs. NPSASA NA NA 0.956 0.935 0.961 0.955 0.953 0.971Vdw vs. PSASA NA NA 0.199 0.023 0.203 0.002 0.099 0.015Vdw vs. NPSASA NA NA 0.155 0.321 0.470 0.039 0.190 0.326Ele vs. PSASA NA NA 0.086 0.003 0.091 0.030 0.000 0.515Ele vs. NPSASA NA NA 0.072 0.151 0.333 0.016 0.049 0.326PSASA vs. NPSASA NA NA 0.390 0.001 0.155 0.438 0.118 0.123

a Instead of SASA, the cavity-related term is used.


� ��

and HMC (r2� 0.593) contains only cavity and van derWaals terms respectively. The predictive abilities of themodels are characterized by the RMSE for the LOO andLSO procedures (Table 3). Interestingly, the conformationspace search provided by the HMCmethod did not result ina significantly better fit than simple minimization (Table 3).This fact may be largely due to the comparatively high levelof accuracy of the starting bound conformations. Bothminimization and HMC provide the van der Waals andcavity terms of similar magnitudes, in contrast to widelydiffering electrostatic energy terms (Table 1).

3.2 Analysis of MD Simulation Data

Toassesswell-knownproblemswith coordinationbindingofligands to zinc during the simulations [13], all five coordi-

nation bonds of catalytic zinc were monitored and theresults for one of the ligands in 2 ps intervals are shown inFigure 1. The distances of coordinating ligand atoms fromzinc are reasonable: hydroxamate oxygens oscillate around1.85 ä and 1.90 ä, histidine nitrogens around 1.95 ä [35].Carefully constructed initial geometries and identical zincbinding groups in all ligands may have contributed tosimulation of the correct coordination sphere. The time-averaged geometries were used to calculate pertinentensemble averages after 5, 10, 25, 50, 100, and 200 ps. Therespective conformations for the most active ligand (20,Table 1) after different simulation times are illustrated inFigure 2. In addition to using SASA in Eq. 1 basedcorrelations, the influence of the SASA×s polar and non-polar components (PSASA and NPSASA, respectively), aswell as that of the constant term were examined:


Table 3. Statistically significant correlations (Eq. 1) of inhibitory potencies with the LR terms obtained from Minimization and HMCsimulations of 21 compounds in the training set. The regression coefficients characterize the contribution of the LR terms for van derWaals interactions (�), electrostatic interactions (�), and the solvent accessible surface area (�).

Method Regression Coefficients r2 SD F RMSE

� � � � LOO LSO

Minimization ± 3.553 0.060 ± 0.537 0.632 3481 0.525 0.654Minimization 0.168 0.011 0.031 0.006 ± ± 0.510 0.656 1562 0.579 0.745Minimization ± ± 2.865 0.570 � 1.559 1.287 0.570 0.615 25.22 0.531 0.633Minimization 0.106 0.022 ± ± � 4.164 0.806 0.549 0.630 23.13 0.534 0.645Minimization 0.0727 0.0475 0.0156 0.0095 1.962 0.950 ± 0.622 0.562 1223 0.574 0.606HMC ± ± 3.502 0.058 ± 0.551 0.632 3586 0.503 0.669HMC 0.173 0.015 0.030 0.007 ± ± 0.546 0.583 1233 0.654 0.723HMC ± ± 3.019 0.608 � 1.110 1.390 0.565 0.618 24.70 0.542 0.640HMC 0.109 0.021 ± ± ± 4.307 0.712 0.593 0.599 27.66 0.523 0.541HMC 0.0476 0.0350 0.0147 0.0070 2.352 0.640 ± 0.646 0.531 1368 0.485 0.599

Figure 1. Coordination distances between catalytic zinc andhydroxamate oxygens (CO ± black, OH ± blue) ligand (20) andhistidine nitrogens (His401 ± cyan, His405 ± red, and His411 ±green) in 2 ps intervals throughout 200 ps MD simulation.Structure of the ligand is given in Table 1.

Figure 2. Time-averaged conformations of the most active com-pound (20, Table 1) for the MD simulations lasting 5 (blue), 10(cyan), 25 (green), 50 (magenta), 100 (orange), and 200 (red) ps.Conformations of catalytic zinc (sphere) and coordinating histi-dines (thin lines) are also shown with the same color-coding.


� ��

�Gb � �� EvdW� � � �� Eel� � � �� PSASA� �� NPSASA� � � �� 2�

A summary of the cross-correlation analysis of independentvariables in Eqs. 1 and 2 is given in Table 2. The results aremostly satisfactory; the only concern are the dependenciesof the van der Waals and electrostatic LR terms for the 5 ±25 ps collection periods. Other inter-correlated variables donot appear in the same equation.The fits of Eqs. 1 and 2 to the MD data obtained for

varying collection times are summarized in Tables 4 and 5respectively. The most interesting aspect is the apparentunimportance of the electrostatic and van der Waals LRterms for most data collection periods. The electrostatic LRterms are only significant for the shortest, 5 ps simulation, inthe fit based on Eq. 1; however, they are inter-correlatedwith the van der Waals terms for this data collection period(Table 2). The van der Waals terms were significant for the5 ps and 10 ps simulations. Separation of SASA into its polarand nonpolar component helped to obtain slightly bettercorrelations, except the 50 and 200 ps simulations. The bestmodel contained just the van derWaals (vdW), PSASA, andNPSASA terms and was obtained using the data from a10 ps simulation (Table 5). The correlation for the 21compounds is characterized by r2�0.903 and SD� 0.310,reflecting a good agreement between actual and calculatedvalues. For each descriptor, the probability�F ratio was�

0.0001, which implies a low likelihood of a randomlyoccurring significant descriptor. The inter-correlations be-tween the vdW/PSASA, vdW/NPSASA and PSASA/NPSASA terms were weak, as indicated by the r2 values of0.023, 0.321, and 0.001, respectively.The results are in accordance with previous findings that

the actual values of the coefficients � [36 ± 38, 75] and � [76,77] depend on the force field and other conditions of thesimulation. Dominance of the surface-area-related LRterms, clearly seen in Tables 4 and 5, is probably reflectingthe effect of burial of the inhibitor in the binding site. Thisphenomenon was described previously in the analysis ofbinding energies of several ligand-protein complexes [78].Our observations confirm also other conclusions of thatstudy [78], namely the importance of explicit solvation,especially thewatermolecules inside the binding site, aswellas the role of both polar andnonpolar components of SASA.Interestingly, the quality of correlations is not improving

with the increase in simulation time. Consequently, thesimulation times of 5 ± 10 ps seem to be sufficient for thebinding energy analyses, if the starting geometries of thecomplexes are comparatively good.The coordinate bonds between the hydroxamate oxygens

and the catalytic zinc center are crucial for ligand binding.Due to the absence of a proper force field, the zinccoordinating interactions were treated in a simplified wayusing a nonbonded model in the MD simulations. The


Table 4. The best correlations (Eq. 1) of inhibitory potencies with the LR terms obtained from MD simulations of 21 compounds in thetraining set for shown data collection periods. The regression coefficients characterize the contribution of the LR terms for van der Waalsinteractions (�), electrostatic interactions (�), and the solvent accessible surface area (�).

Time [ps] Regression Coefficients r2 SD F RMSE

� � �� 10�3 � LOO LSO

5 0.0221 0.008 � 0.0114 0.0047 7.680 0.884 � 4.828 0.390 0.892 0.326 46.80 0.318 0.3685 0.00282 0.00317 ± 8.460 0.930 � 4.380 0.389 0.854 0.368 52.66 0.353 0.43410 0.0143 0.0042 ± 6.313 0.983 � 4.101 0.349 0.881 0.333 66.57 0.281 0.37825 ± ± 8.971 0.931 � 4.453 0.376 0.830 0.387 92.84 0.324 0.45850 ± ± 8.489 1.230 � 4.757 0.482 0.713 0.502 47.31 0.401 0.513100 ± ± 8.300 0.818 � 4.773 0.327 0.844 0.370 102.9 0.341 0.405200 ± ± 8.474 0.768 � 4.717 0.303 0.867 0.342 123.8 0.302 0.367

Table 5. The best correlations (Eq. 2) of inhibitory potencies with the LR terms obtained from MD simulations of 21 compounds in thetraining set for shown data collection periods. The regression coefficients characterize the contribution of the LR terms for van der Waalsinteractions (�), electrostatic interactions (�), polar solvent accessible surface area (�), and nonpolar solvent accessible surface area (�).

Time [ps] Regression Coefficients r2 SD F RMSE

�� 10�2 �� 10�2 �� 10�3 �� 10�3 � LOO LSO

5 2.530 0.007 � 1.370 0.004 ± 8.050 0.848 � 5.838 0.297 0.907 0.303 55.18 0.281 0.3345 0.00252 0.00373 ± ± 8.900 0.991 � 5.441 0.332 0.851 0.372 51.33 0.324 0.43010 1.570 0.003 ± 7.790 0.003 6.160 0.898 � 3.871 0.453 0.903 0.310 52.49 0.276 0.35425 ± ± 10.84 0.001 8.200 1.060 � 4.464 0.564 0.838 0.388 46.51 0.351 0.46650 ± ± ± 8.130 1.350 � 5.948 0.358 0.658 0.549 36.55 0.422 0.523100 ± ± 10.11 0.00 7.800 0.914 � 4.723 0.462 0.857 0.364 53.98 0.335 0.417200 ± ± ± 9.020 0.915 � 5.653 0.251 0.836 0.380 97.18 0.315 0.414


� ��

simplified treatment seems to be sufficient for the usedhomologous series. It is assumed that the coordinate bondsintroduce equal effects to all inhibitor-enzyme systems,which are implicitly embedded in the constant term in Eq. 2.The constant � yields an attractive term of about�3.871 logunits. The constant attractive term is providing a base valuefor these inhibitors that are then modulated by vdW,PSASA, and NPSASA terms. The values of the vdW termsare negative and the associated positive coefficient impliesthat a good geometrical fit between the inhibitor and thebinding site is important for the 5 and 10 ps simulations. ThevdW terms include both favorable packing contributions tobinding as well as any unfavorable steric repulsions. TheNPSASA terms are negative, implying that nonpolarinhibitor parts are buried upon binding. The associatedcoefficient is positive so that the removal of hydrophobicsurface area from the contact with water upon binding isfavorable for the binding free energy, which simply reflectsthe hydrophobic effect [79]. The PSASA terms are alsonegative, indicating that the polar solvent accessible surfacearea finds stronger interactions with the binding site thanwith the surroundingwater.The robustness of the regressionequations and their predictive abilities were probed by theLOO and LSO procedures, with satisfactory results (thevalues of RMSE in Tables 4 and 5).The RMSE values using LOO (0.281) and LSO (0.378)

were comparable to that of theRMSEof training set (0.333).The models were also evaluated by cross-validations using200 random selections of the 7-compound test sets from thetotal set of 28 compounds. The predictive RMSE for the10 ps MD time scale was lowest among all the samplingtimes and technique. The results are summarized in Table 6.The obtained RMSE are a little higher than those obtainedon the training set but still are very reasonable. The resultsindicate that obtainedmodels have good prediction abilitiesand they are all approximately equivalent in this aspect(Figure 3). All methods also significantly improve FlexXdocking.

3.3 Difference in Potency of Isomers (1) and (2)

Compound (1) is a potent inhibitor (Ki�5.05 nM) whilecompound (2) exhibits only moderate activity(Ki�349 nM). The compounds are structurally similar(Table 1) except that the configuration of C-3 in compound(1) is R while in compound (2) it is S. The importance of theRconfiguration at the� carbon iswell documented [80 ± 84].Binding modes after MD simulation (10 ps) of the twoinhibitors are shown in Figure 4. In case of compound (1),several hydrogen bonds can be discerned: between theoxygen atom of the phosphonamide group and the mainchain NH of Ala-189; the oxygen atom of the ethyl estergroup and the main chain of Leu-188; between thehydroxamate NH and the carboxyl oxygen of Glu-402(and also an intramolecular H-bond with the phosphona-mide oxygen). In contrast, compound (2) forms only one

hydrogen bond with the water molecule (Figure 4). Thezinc-binding group is tightly bound to zinc in both thecompounds but the rest of the ligand is flipped by 180o. Thelower activity of derivative (2) as compared to derivative (1)may be due a different mode of binding leading to fewerinteractions with the enzyme.

3.4 Comparison of Sampling Techniques

The � coefficients scaling the electrostatic contributions aremuch lower than the originally suggested values of 0.5 [85]for all the LR based methods. For the MD simulation, the �coefficient was originally negative. Examination of thecross-correlations between individual variables (Table 2)showed that this phenomenon is due to the dependency


Table 6. Cross-validated statistics for the complete set of 28compounds for all used sampling techniques using leave-several-out (LSO) technique (7-member test sets were randomly created200 times).

Sampling Techniques Time [ps] RMSE

Model A Model B

Minimization NA 0.785 NAHMC NA 0.797 NAMD 5 0.751 0.770MD 10 0.584 0.815MD 25 0.702 0.716MD 50 1.173 0.786MD 100 0.714 0.702MD 200 0.649 0.682

Figure 3. Experimental activities of the training set (Table 1)compared with those calculated by FlexX scoring function (whitepoints), and Linear Response approach with different samplingmethods: minimization (blue), hybrid Monte Carlo (red), andMolecular Dynamics (black). The identity line shown.


� ��

between van der Waals and electrostatic terms for the 5 psMD simulation. The electrostatic term was, therefore,eliminated from the correlation, without a significantdecrease in the quality of the model (Tables 4 and 5). Thedependence between calculated and experimental inhibi-tory potencies is given in Figure 3. The calculated valuesbased on the FlexX scores are also included for comparison.All second-pass methods improve the predictions obtainedby the fast scoring approach, especially for highly activeinhibitors.

4 Conclusions

A model for the prediction of binding affinities of hydrox-amate derivatives to MMP-9 was developed. The correla-tions obtained by sampling structures at shorter times (5 psand 10 ps) were better or comparable to 200 ps simulations.This fact suggests that if the initial ligand-receptor geo-metries are accurate, the shorter molecular dynamicssimulations can be used. The result obtained by MD forthis training set is better than that of minimization andHMC.However, all threemethods are comparable in case ofrandom selection of training set, as evident by the RMSEvalue. On the basis of the fits to Eq. 2, the most importantprocess for the prediction of binding affinities is the loss ofpolar and nonpolar solvent accessible surface area of theinhibitors due to their binding with the enzyme. This factor

was complemented, in some cases, by the enhancement ofvan der Waals interactions upon binding.

Acknowledgement

This work was supported in part by the NIH NCRR grants1P20RR15566 and 1P20RR16471.

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Received on July 22, 2004; Accepted on September 1, 2004



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Documents

Simulation-Based Predictions of Binding Affinities of Matrix Metalloproteinase Inhibitors