Computational Chemistry Comparison of Stable/Nonstable Protein Mutants Classification Models Based on 3D and Topological Indices ´ LEZ-DI´AZ,1,2 YUNIERKIS PE ´ REZ-CASTILLO,1 GIANNI PODDA,2 EUGENIO URIARTE1 HUMBERTO GONZA 1

Faculty of Pharmacy, University of Santiago de Compostela, Santiago de Compostela 15782, Spain 2 Dipartimento Farmaco Chimico Tecnologico, Universita´ Degli Studi di Cagliari, Cagliari 09124, Italy Received 15 December 2006; Accepted 19 January 2007 DOI 10.1002/jcc.20700 Published online 20 April 2007 in Wiley InterScience (www.interscience.wiley.com).

Abstract: In principle, there are different protein structural parameters that can be used in computational chemistry studies to classify protein mutants according to thermal stability including: sequence, connectivity, and 3D descriptors. Connectivity parameters (called topological indices, TIs) are simpler than 3D parameters being then less computationally expensive. However, TIs ignore important aspects of protein structure and hence are expected to be inaccurate. In any case, a comparison of 3D and TIs has not been reported with respect to the power of discrimination of proteins according to stability. In this study, we compare both classes of indices in this sense by the first time. The best model found, based on 3D spectral moments correctly classified 507 out of 525 (96.6%) proteins while TIs model correctly classified 404 out of 525 (77.0%) proteins. We have shown that, in fact, 3D descriptor models gave more accurate results than TIs but interestingly, TIs give acceptable results in a timely way in spite of their simplicity. q 2007 Wiley Periodicals, Inc.

J Comput Chem 28: 1990–1995, 2007

Key words: protein structure; topological indices; spectral moments; protein stability

Introduction In the post-genomic era, there is an increasing necessity for quick computational chemistry methods to predict accurately proteins properties. In this sense, quantitative structure activity relationship (QSAR) is a widely covered field, with more than 1600 molecular descriptors introduced up to now.1–3 Most of the molecular descriptors have been applied to small molecules. Nevertheless, the QSAR studies for DNA and protein sequences may be classified as an emerging field.4–7 One of the most promising applications of QSAR to proteins relates to the prediction of thermal stability, which is an essential issue in protein science.8–11 Amongst all molecular descriptors, the 3D indices (3DIs) give a more detailed description of protein structure, while connectivity indices, also called topological indices (TIs)1 serve for less precise but fast calculations. 3DIs can be calculated from 3D X-ray determined crystal structures or from NMR-fitted protein model,12,13 whereas TIs are graph invariants of different kinds of proteins or DNA graph representations. The branch of mathematical chemistry devoted to the development of new graph representations in order to encode proteins and

DNA with TIs has become an intense research area with interesting works, for instance, those of Liao et al,14–17 Randic et al.,7,18–21 or the author’s own group.22 Nevertheless, they are not QSAR studies that compare the real ability of both classes of indices, 3DIs and TIs, to classify protein mutants according to stability.8 Here, we selected as 3DIs the spatial descriptors called 3D spectral moments because they have been widely used to describe 3D and 2D structural features in different contexts such as polymer sciences, solid phase chemistry, and theoretic chemistry.23–27 Among the most frequently used TIs, one can cite the Wiener’s, Balaban’s, Connectivity, and others indices.1 In this study, we combine both 3DIs and TIs in order to build models to classify single-point mutated proteins according to their thermal stability. The work may help to yield clarity regarding the

Correspondence to: H. Gonza´lez-Dı´az; e-mail: [email protected] This article contains Supplementary Material available at http://www. interscience.wiley.com/jpages/0192-8651/suppmat

q 2007 Wiley Periodicals, Inc.

Comparison of Stable/Nonstable Protein Mutants Classification Models Based on 3D and TIs

real extent of the ratio of accuracy to speed (3DIs vs. TIs) in protein QSAR studies.

Materials and Methods 3D Stochastic Spectral Moments

Stochastic spectral moments, represented as SR
k(O), have been extensively used in QSAR studies of small sized and macromolecules.28 The approach used here involved MM29–31 to codify information about the 3D molecular structure of proteins and constitutes a generalization of other the so-called MARCHINSIDE (Markovian Chemicals In Silico Design) approach.32 The SR
k for proteins are straightforward to calculate as the trace (Tr) of the k-order natural power of the matrix 1P, SR


k ðOÞ ¼ Trðk Þ ¼ Tr

h
k i Xn 1 k  pii ¼ i¼1

1991

function ij is applied in such a way that a spatial electrostatic interaction takes place only between neighboring aminoacids (ij ¼ 1). Otherwise, the electrostatic interaction is nullified (ij ¼ 0).30–33 The SR
k(O) values can be calculated for different protein regions. These regions (O) called ‘‘orbits’’ are defined with respect to the distance from the AA -carbon to the protein centre of mass d(j) and the largest of these distances dmax(j). In the present study, five values were used by default for the parameter orbit ¼ 0, 1, 2, 3, 4, considering a with a ratio r ¼ 100d(j)/dmax(j) ranging between the following limits: 0  orbit0 < 25  orbit1 < 50  orbit2 < 75  orbit3 < 100% or all of the AAs together, orbit4. As the orbits are related to the position of the AA with respect to the centre of the protein, these are named as orbit0 ¼ core, orbit1 ¼ inner, orbit2 ¼ middle, and orbit3 ¼ outer. The distribution of orbits for a 3D kinase structure is represented in Figure 1.

(1) Topological Indices

where 1P is the Markov electrostatic interaction matrix built as a square table of order n, with n the total number of aminoacids (AAs) in the protein. The elements of this matrix are the AAs pairwise electrostatic interaction probabilities 1pij, 1

ij
’ij pij ¼ Pn k¼1 ik
’ik

(2)

Conversely, connectivity indices have been extensively used in QSAR.34–41 Almost all molecular descriptors, including topological indices, can also be written in the form of vector-matrixvector notation. That is the case with the Wiener index (W), which was the first molecular descriptor defined in a chemical context,

where ’ij is the electrostatic potential of interaction between two aminoacids ai and aj. For the sake of simplicity, a truncation

1 W ¼ ðu:D:uT Þ 2

Figure 1. Spatial distribution of different orbits used in the definition of the descriptors for the protein with PDB code 2C5X. We have labeled the core orbit as (i), inner orbit (ii), middle orbit (iii) and outer orbit (iv). Note that (i) and (iii) are pale while (ii) and (iv) are dark.

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(3)

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Gonza´lez-Dı´az et al. • Vol. 28, No. 12 • Journal of Computational Chemistry

where u is a unitary vector and D the topological distance matrix. Some of most used connectivity indices include first (M1) and second (M2) Zagreb Indices, Harary number (H), valence connectivity index ( ), Balaban index (J), and Molecular topological index (MTI) can also be represented as vector-matrixvector product (see section 5.1 in electronic supplementary material).42 Calculation of Molecular Descriptors

Molecular descriptors used in this work include 3D spectral moments and TIs. Three-dimensional spectral moments are calculated using the MARCH-INSIDE approach. For each mutated protein we calculated five kth-order spectral moments for the protein as a whole as well as for specific orbits or regions of the proteins. In this work, we used spectral moments from order zero (SR
0) to the fifth order (SR
5) for each orbit.8 TIs are calculated as implemented in the Chem3D Ultra software.43 Proteins Used in This Study

Protein mutants used in this work were built as in previously published paper by Zhou and Zhou.10 In the electronic supplementary material, are depicted all pdb codes for proteins that were used along with the AA residue to be mutated, its number within the protein sequence, the AA introduced upon mutation and other useful information. All three-dimensional protein structures are extracted from the Protein Data Bank database.44 After mutation, protein structures were energy minimized using HyperChem package.45 Statistical Analysis

Linear discrimination analysis (LDA)46,47 is often preferred in QSAR as a technique appropriate for classification problems. In this work, we choose LDA in order to seek for a linear discriminant function, which can classify single-point mutated proteins according to their thermal stability. The criteria used for deciding whether a protein is classified as stable or unstable was the same previously reported.8 A dummy variable (stab) is selected to encode the studied property: stab ¼ 1 for stable mutants and stab ¼ 1 if otherwise. All independent variables are standardized prior to model construction. All statistical calculations were carried out with the package STATISTICA 6.0.48 The quality of LDA analysis was determined by examining Wilk’s  also known as U-statistic (U), Fisher ratio (F), and the p-level (p). We also inspected the ratios of good classification. Additionally we tested the model stability and robustness by a re-substitution methodology of cases in four prediction series.

Results First, the data is split at random in two parts, a training series (t) used for model construction, and a prediction one (v) for model validation (see electronic supplementary material for further details on series composition). Afterwards, LDA was used to seek models that discriminate between stable/unstable protein mutants and having one, two or three molecular descriptors.

Results for molecular descriptors in the model, stable proteins, unstable ones, and total model accuracy rates, model statistics, and model coefficients are presented in Table1. Models Based on 3DIs

For models based only on 3DIs, the most significant variables were the fifth-order spectral moment corresponding to the outer orbit, SR
5(outer), and the total first order spectral moment, SR
1(total). Introducing a new variable does not produce a significant improvement of the model. The obtained discriminant equation is, Stab ¼ 6:69xSR
5 ðouterÞ þ 3:83xSR
1 ðtotalÞ þ 0:49 N ¼ 395; U ¼ 0:39; F ¼ 308:22;

p < 0:001

where N is the number of proteins used to find the model, U is Wilk’s lambda or U-statistic, F is Fisher ratio, and p is p-level. This model correctly classifies 207 out of 217 (95.4%) stable proteins and 174 out of 178 (97.8%) unstable mutated proteins for training series. For the prediction series, 126 out of 130 (96.9%) are correctly classified, more specifically 58/59 and 68/ 71 unstable and stable mutated proteins are well classified. Additionally a resubstitution procedure was carried out by interchanging proteins in training and prediction series.8 Results for resubstitution procedure are shown in Table2. Models Based on TIs

The best LDA model obtained using TIs without considering 3D information is, Stab ¼ 3:39xMTI þ 2:76xCC þ 0:24 N ¼ 395; U ¼ 0:80; F ¼ 49:86;

p < 0:001

where MTI is the molecular topological index and CC is cluster count. Introducing additional TIs does not significantly improve the model. This model correctly classifies 189 out of 217 (87.1%) of stable proteins and 115 out of 178 (64.6%) unstable ones in the training series, this makes an overall 77.0% of correctly predicted proteins. For prediction series, 61/71 (85.9%) and 39/59 (66.1%) of stable and unstable mutated proteins respectively are well classified. Models Based on Both 3DIs and TIs

Models based on both 3DIs and TIs do not present significant improvement in model quality with respect to the model containing only the 3DIs. For instance, the best model found with one 3DI was: Stab ¼ 2:45xSR
5 ðouterÞ þ 0:34 N ¼ 395; U ¼ 0:55;

F ¼ 322:37;

p < 0:001

This model, the best model containing only one 3DI, correctly classifies 329/395 (83.1%) for training series and 111/130

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Table 1. Summary of All Models.

Variables in the model V1a

Model statistics V2a

V3a

%Sb

%Ub

%Tb

Uc

a0d

A1d

a2d

a3d

MTI ShA ShC

94.1 92.3 64.9 64.9 64.9 64.9 55.9 55.9 55.9 55.9 15.9 0.6 95.3 90.9 61.1 87.1 95.8 86.8 95.4

71.3 62.4 56.5 48.9 48.1 48.1 48.5 48.5 48.5 48.5 95.3 100 97.7 70.8 66.6 64.6 94.5 65.4 97.8

83.8 78.8 61.1 57.7 57.3 57.3 52.5 52.5 52.5 52.5 51.8 45.5 96.4 81.9 63.6 76.9 95.2 77.1 96.5

0.54 0.87 0.93 0.92 0.90 0.90 0.96 0.96 0.96 0.96 1.00 1.00 0.38 0.51 0.81 0.80 0.35 0.80 0.36

0.36 0.22 0.21 0.21 0.22 0.22 0.20 0.20 0.20 0.20 0.20 0.20 0.51 0.38 0.24 0.24 0.56 0.24 0.55

2.50 0.82 0.56 0.62 0.71 0.71 0.44 0.44 0.43 0.43 0.10 0.11 6.84 2.50 0.76 3.30 7.31 3.30 4.55

0 0 0 0 0 0 0 0 0 0 0 0 3.91 0.70 0.64 2.63 4.34 2.63 1.10

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.09 0.00 1.36


5(outer)
1(total)

SR SR

De Re MTI W CC ShAe SODe SVDe ShCe TCe SR
5 (outer) SR
5 (outer) SR
1 (total) MTI SR
5 (outer) MTI SR
5 (outer)


1 (total) MTI MTI CC SR
1 (total) CC SR
1 (total)

Variables coefficient

SR

a

Variables in the model, V1, V2, V3. Accuracy of the model: for stable proteins (%S), unstable ones (%U) and for all (%T). c Wilks’ lambda or U-statistic (U). d Parameters for the discriminant equation: a1 for the first variable, a2 for the second, a3 for the third, and a0 for constant coefficient. e TIs used in the study but not cited above: R, radius; D, diameter; Sha, shape attribute; SOD, sum of degrees; SVD, sum of valence degrees; ShC, shape coefficient; TC, total connectivity. b

Table 2. Results obtained for Resubstitution procedure in 3D Spectral

Moments Model. Training series % Subset 1 Stable Unstable Total Subset 2 Stable Unstable Total Subset 3 Stable Unstable Total Subset 4 Stable Unstable Total Average Stable Unstable Total

Stab.

Unstab.

Prediction series %

Stab.

Unstab.

95.4 97.8 96.5

207 4

10 174

95.8 98.3 96.9

68 1

3 58

95.8 98.3 97.0

207 3

9 175

94.4 98.3 96.2

68 1

4 57

95.8 98.3 97.0

206 3

9 176

94.5 98.3 96.2

69 1

4 56

95.4 97.8 96.5

206 4

10 174

95.8 100.0 97.7

69 0

3 58

95.6 98.0 96.7

207 4

10 175

95.1 98.7 96.7

69 1

4 57

(85.4%) for the prediction one. The best combined model is that containing the fifth order spectral moment corresponding to the outer orbit and the molecular topological index (MTI): Stab ¼ 2:44xSR
5 ðouterÞ  0:65xMTI þ 0:36 N ¼ 395; U ¼ 0:52; F ¼ 179:34;

p < 0:001

This combined model (3DIs and TIs) correctly classifies 320/395 (81.0%) of mutated proteins in the training series and 109/130 (83.8%) for the prediction one.

Discussion The best model found includes only two 3DIs. It means that we can discriminate between the two groups only by calculating two 3DIs: SR
5(outer) and SR
1(total). From Table 2, one can note the high robustness and predictability of the model. However, the calculation of these 3DIs presupposes knowing of 3D structure and may become a time-consuming calculation task for large protein databases.49,50 As a consequence, we investigated the ability of TIs to classify proteins according to thermal stability. The results obtained for models based only on TIs have shown that this family of descriptors, the simplest one, can

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encode in some way protein 3D structure related information. Although the rates of good classification are not as high as in the case of 3D spectral moments, as average 77.0% of accuracy, the classification rate for TIs model can be considered as acceptable. This makes the TIs-based model a fast alternative to the more accurate, but complex and higher time-consuming 3D model. This result presented here for the first time for proteins had been demonstrated in the past for small molecules.51–54 An interesting result has been obtained for models combining TIs along with 3DIs. In this case a model containing only SR
5(outer) shows an average of 83.1% of accuracy. In contrast, when a TI is introduced in the model, a small decreasing in the classification rate is obtained (81.0%). This result shows that on adding topological information to a model already containing 3D information does not necessarily improve model accuracy. It is straightforward to realize that the addition of TIs does not provide any additional information to that contained in 3DIs. It is a fact that almost all TIs can be derived by similar vector-Matrix-vector multiplication of different matrices and contain essentially connectivity information.42,55,56 The obtained results confirm the capability of 3DIs for encoding proteins structural properties related to the thermal stability of protein mutants.57 Specifically, the present work confirm the success of 3D spectral moments for protein QSAR studies.58 As expected, the 3DIs perform better than TIs in the prediction of protein thermal stability. Nevertheless, 3DI-based methodology is also the more computationally expensive. For this reason, we can confirm in this work that TI models could be useful for processing large databases of proteins to obtain a raw but fast result,59 which can be later refined with 3DIs.

Acknowledgments H. Gonzalez-Dı´az acknowledges contract/grant sponsorship from the Program Isidro Parga Pondal of the ‘‘Direccio´n Xeral de Investigacio´n y Desenvolvemento’’ of ‘‘Xunta de Galicia’’ and scholarship funding from the same institution for a one-year post-doctoral position in the Dipartimento Farmaco Chimico Tecnologico of the University of Cagliari, Italy, 2007. He also acknowledges two contracts as guest professor in the Department of Organic Chemistry, Faculty of Pharmacy, University of Santiago de Compostela, Spain in 2006. The authors thank the Xunta de Galicia (projects PXIB20304PR and BTF20302PR) and the Ministerio de Sanidad y Consumo (project PI061457) for partial financial support. The authors sincerely thank Prof. Gernot Frenking and the two unknown referees for their editing.

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