Oncotarget

Research Papers:

2L-PCA: a two-level principal component analyzer for quantitative drug design and its applications

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Oncotarget. 2017; 8:70564-70578. https://doi.org/10.18632/oncotarget.19757

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Qi-Shi Du _, Shu-Qing Wang, Neng-Zhong Xie, Qing-Yan Wang, Ri-Bo Huang and Kuo-Chen Chou

Abstract

Qi-Shi Du1,4,*, Shu-Qing Wang2,*, Neng-Zhong Xie1,*, Qing-Yan Wang1, Ri-Bo Huang1 and Kuo-Chen Chou3,4

1State Key Laboratory of China for Biomass Energy Enzyme Technology, National Engineering Research Center of China for Non-Food Biorefinery, Guangxi Academy of Sciences, Nanning 530007, China

2School of Pharmacy, Tianjin Medical University, Tianjin 300070, China

3Center for Informational Biology, University of Electronic Science and Technology of China, Chengdu 610054, China

4Gordon Life Science Institute, Boston, MA 02478, USA

*These authors have contributed equally to this work

Correspondence to:

Qi-Shi Du, email: [email protected]

Kuo-Chen Chou, email: [email protected]

Keywords: drug design, PCA, molecular fragments, physicochemical properties, peptides

Received: April 26, 2017     Accepted: June 30, 2017     Published: August 01, 2017

ABSTRACT

A two-level principal component predictor (2L-PCA) was proposed based on the principal component analysis (PCA) approach. It can be used to quantitatively analyze various compounds and peptides about their functions or potentials to become useful drugs. One level is for dealing with the physicochemical properties of drug molecules, while the other level is for dealing with their structural fragments. The predictor has the self-learning and feedback features to automatically improve its accuracy. It is anticipated that 2L-PCA will become a very useful tool for timely providing various useful clues during the process of drug development.


INTRODUCTION

With the fast developments of computer-aided drug design (CADD) [15], currently a number of drug design approaches are developed, and several computer software packs [6, 7] are available that can speed up the discovery of new chemical and biological drugs in more efficient and economical procedure. However, so far we still have no perfect theories, ideal technologies, and faultless software tools that can guarantee complete success of the designed drugs due to the complicity of the interactions between medicinal drugs and their biological targets [8, 9]. The factors and parameters that may affect the bioactivities of drugs are not only from the structure of drug itself, but also from its biological target, coenzymes, and interaction environment [10, 11].

Principal component analysis (PCA) [1214] is a useful tool that has been widely used in chemistry, biology, environment, and many fields of social science. The PCA approach has also been used in drug design for many years. Traditionally PCA is a single level and one direction prediction and analysis technique, described as the following equation

k=1K(akxi,k)=wi(1)

where {xi,k} are the physicochemical parameters of the i-th molecule,{ak} are the coefficients of molecular parameters, and wi is bioactivity of the i-th molecule [15, 16]. The bioactivity wi could be logarithm of IC50,i (pIC50,i=-logIC50,i), or binding free energy ΔG°i between drug and receptor.

After the coefficients {ak} of parameters are solved from the linear equations Eq.1 in a training set of drug candidates, the parameter coefficients {ak} can be used to predict the bioactivities of the designed or newly synthesized drug compounds,

wipred=k=1K(akxi,k)(2)

where K is the total number of molecular parameters. Currently hundreds even thousands of molecular parameters are available for drug design [17, 18]. However for certain drug-receptor interaction system, these parameters are not equally important; actually too many parameters may cause the over correlation problem [19, 20]. In PCA technique only the principle components are selected to describe the bioactivities of drug molecules, and to predict the bioactivities of drug candidates.

In the present study, an improved principal component analysis method, the so-called two-level principal component analysis (2L-PCA), is proposed to deal with the extreme complexity and huge amount of parameters in drug design and discovery. In the 2L-PCA predictor, the 1st level is to deal with the physicochemical properties of drug molecules, and the 2nd level is to deal with the fragments of molecular structures. The proposed two-level model can not only significantly enhance the prediction power, but also yield more useful information for in-depth analysis.

According to Chou’s 5-step rule [21] that has been widely used by many investigators (see, e.g., [2237]), to develop a really useful statistical predictor, one should consider the following five procedures: (1) benchmark dataset; (2) sample representation; (3) operation algorithm; (4) cross validation; (5) web-server. Below, let us describe how to deal with them one-by-one. However, to comply with the Journal’s rubric style, they are not exactly following the aforementioned order.

RESULTS AND DISCUSSION

As an example to show the advantage of 2L-PCA, we applied it for predicting the binding affinity of epitope-peptides with class I MHC molecules HLA-A*0201 [38, 39]. HLA-A*0201 is one of the most frequent class I alleles found in many different species and populations, which plays a critical role for antigen presentation in both viral antigens [40] and tumor antigens from a variety of cancers [4144], and is expressed in approximately 50% of Caucasians population [45].

The epitope-peptides consist of nine amino acids [38, 39]. In the 2L-PCA study for the epitope-peptides, the nine side chains of the nine amino acids are the nine fragments. Eight physicochemical properties are used as the descriptors of the 20 natural amino acids. Four of them are the HMLP parameters [15, 16], describing the lipophilic character, hydrophilic character, surface area with lipophilic potential, and surface area with hydrophilic potential, respectively. The fifth property is the volume of amino acid side chains. The remaining three properties are the secondary structural potency indices of amino acids: the α-potency, β-potency, and coil-potency [46]. Listed in Table 1 are the eight physicochemical parameters of 20 amino acids used in this study.

Table 1: Eight physicochemical parametersa of 20 natural amino acid side chains

A.A.

Lip

Hyd

SL2)

SH2)

Pα

Pβ

Pc

V(Å3)

Leu (L)

1.2906

0.0000

84.5476

0.0000

1.21

1.30

0.68

166.7

Ile (I)

1.1046

0.0000

88.6055

0.0000

1.08

1.60

0.66

166.7

Val (V)

0.5324

0.0000

77.8108

0.0000

1.06

1.70

0.62

140.0

Phe (F)

0.4412

-0.1195

105.7054

11.2472

1.13

1.38

0.71

189.9

Met (M)

1.0768

-0.3068

70.3631

23.2299

1.45

1.05

0.58

162.9

Trp (W)

0.8364

-0.4310

133.6980

14.8820

1.08

1.37

0.75

227.8

Ala (A)

0.1744

0.0000

34.7760

0.0000

1.42

0.83

0.70

88.6

Cys (C)

0.2479

-0.2402

23.5563

30.4540

0.70

1.19

1.18

108.5

Gly (G)

0.0208

0.0000

3.7616

0.0000

0.57

0.75

1.50

60.1

Tyr (Y)

0.4534

-0.5896

80.9646

42.7160

0.69

1.47

1.06

193.6

Thr (T)

1.4265

-0.4369

46.7285

16.0490

0.83

1.19

1.07

116.1

Ser (S)

0.2346

-0.6040

26.0681

15.9613

0.77

0.75

1.32

89.0

His (H)

0.8124

-0.7766

82.1701

13.8631

1.00

0.87

1.06

153.2

Gln (Q)

1.0036

-0.7211

70.0876

17.8662

1.11

1.10

0.86

143.9

Lys (K)

1.4600

-0.6229

97.7144

8.0786

1.16

0.74

0.98

168.7

Asn (N)

0.6396

-0.7211

50.5075

17.7804

0.67

0.89

1.35

117.7

Glu (E)

1.0315

-0.9298

57.1582

25.5726

1.51

0.37

0.84

138.4

Asp (D)

0.6058

-0.9298

37.4173

25.2736

1.01

0.54

1.20

111.1

Arg (R)

1.2424

-1.4797

90.8008

35.3095

0.98

0.93

1.04

173.4

Pro (P)

0.3226

0.0000

69.2297

0.0000

0.57

0.55

1.59

122.7

Lip: lipophilic index; Hyd: hydrophilic index; SL: lipophilic surface area; SH: hydrophilic surface area; Pα: potency of α-helix; Pβ: potency of β-band; Pc: potency of loop; V: volume of side chains.

In this study the HMLP parameters were used to describe the lipophilicity and hydrophilicity of molecular fragments. In peptides the HMLP parameters of the 20 natural amino acid side chains are available from literatures. However, the HMLP parameters of common chemical molecular fragments have to be derived using complicated calculations. In such cases other hydrophobic parameters can be used, e.g., the atom-based hydrophobic parameters in [47].

To reduce computational time, the cross validation in this study was performed via the independent dataset test [48], as described as follows. The sequences and experimental binding affinities of the 90 peptides were used as the training dataset to train the model, while those of the 40 peptides taken from [49] as the independent dataset to test the model. Actually, such 40 peptides had also been compiled in a series of publications [41, 42, 5055]. The logarithms (pIC50) of IC50 were used as the bioactivity, because they are related to the changes in the free binding energy [55, 56]. Listed in Table 2 are the sequences and the experimental pIC50 of the peptides used in the training set. The binding strength of the 90 training peptides and 40 testing peptides covers the low, intermediate, and high affinity. The following two criteria were applied in the choice of the testing peptides: (1) the range of binding affinities in the testing dataset should not exceed the range of affinities in the training set; (2) the amino acid at each position in the testing dataset should also be present at that position in the training set of peptides. These two conditions make the 130 peptides to be the ideal benchmark dataset for 2L-PCA method.

Table 2: Amino acid sequences and experimental and predicted bioactivities of 90 MHC-I peptides in the training set

No.

Peptide
sequence

Expt
pIC50

Pred
pIC50

pIC50
Diff

No.

Peptide
sequence

Expt
pIC50

Pred
pIC50

pIC50
Diff

1

VALVGLFVL

5.148

5.7543

-0.6063

46

VVMGTLVAL

7.174

7.3163

-0.1423

2

GTLVALVGL

5.342

5.9368

-0.5948

47

YLEPGPVTI

7.187

7.1654

0.0216

3

LQTTIHDII

5.501

5.8143

-0.3133

48

GLSRYVARL

7.248

7.4620

-0.2131

4

SLHVGTQCA

5.842

6.1580

-0.3160

49

LLAQFTSAI

7.301

7.4302

-0.1292

5

ALPYWNFAT

5.869

6.6416

-0.7726

50

VLLDYQGML

7.328

7.5911

-0.2631

6

SLNFMGYVI

5.881

5.9560

-0.0750

51

YLEPGPVTV

7.342

7.4078

-0.0658

7

NLQSLTNLL

6.000

6.6992

-0.6992

52

ILSPFMPLL

7.3470

7.1400

0.2070

8

FVTWHRYHL

6.025

5.7230

0.3020

53

YLSPGPVTA

7.383

7.5610

-0.1780

9

DPKVKQWPL

6.176

5.7407

0.4354

54

IIDQVPFSV

7.398

7.6528

-0.2548

10

ITSQVPFSV

6.196

6.5888

-0.3928

55

SVYDFFVWL

7.444

7.3654

0.0786

11

ALAKAAAAI

6.211

6.2433

-0.0323

56

ITWQVPFSV

7.463

7.4417

0.0213

12

GLGQVPLIV

6.301

6.5651

-0.2641

57

ITYQVPFSV

7.480

7.6613

-0.1813

13

MLDLQPETT

6.335

6.8570

-0.5220

58

GLYSSTVPV

7.481

7.6303

-0.1493

14

LLSSNLSWL

6.342

6.3502

-0.0082

59

VMGTLVALV

7.553

7.2369

0.3161

15

GLACHQLCA

6.380

6.0594

0.3206

60

LLLCLIFLL

7.585

7.1406

0.4444

16

LIGNESFAL

6.415

7.0559

-0.6409

61

SLDDYNHLV

7.585

7.1764

0.4086

17

ALAKAAAAV

6.419

6.4857

-0.0667

62

VLIQRNPQL

7.644

6.9473

0.6967

18

LLAVGATKV

6.477

6.5115

-0.0344

63

SLYADSPSV

7.658

7.7106

-0.0526

19

ALAKAAAAL

6.511

6.2262

0.2848

64

ILSQVPFSV

7.699

7.6472

0.0518

20

WILRGTSFV

6.556

6.9084

-0.3524

65

IMDQVPFSV

7.719

8.0305

-0.3115

21

IISCTCPTV

6.580

6.6649

-0.0849

66

QLFEDNYAL

7.764

7.4713

0.2927

22

FLGGTPVCL

6.623

6.8756

-0.2526

67

ALMDKSLHV

7.770

7.5250

0.2450

23

ALIHHNTHL

6.623

6.7908

-0.1677

68

YAIDLPVSV

7.796

7.6075

0.1885

24

NLSWLSLDV

6.639

6.0466

0.5924

69

FVWLHYYSV

7.824

8.1149

-0.2909

25

YMIMVKCWM

6.663

6.6427

0.02035

70

MLGTHTMEV

7.845

7.3180

0.5270

26

VLQAGFFLL

6.682

7.0412

-0.3592

71

LLFGYPVYV

7.886

8.0253

-0.1393

27

GTLGIVCPI

6.714

6.5233

0.1907

72

ILKEPVHGV

7.921

7.5915

0.3295

28

VILGVLLLI

6.785

7.4728

-0.6878

73

YLMPGPVTV

7.932

7.9139

0.0181

29

VTWHRYHLL

6.793

6.5597

0.2333

74

WLDQVPFSV

7.939

7.9514

-0.0124

30

PLLPIFFCL

6.796

7.5217

-0.7257

75

KTWGQYWQV

7.955

7.6934

0.2616

31

TLGIVCPIC

6.815

5.9499

0.8651

76

ALMPLYACI

8.000

7.4383

0.5617

32

CLTSTVQLV

6.832

7.1061

-0.2741

77

YLAPGPVTA

8.032

7.6408

0.3912

33

ILLLCLIFL

6.845

6.7815

0.0635

78

YLYPGPVTV

8.051

8.3112

-0.2602

34

FAFRDLCIV

6.886

6.6689

0.2171

79

LLMGTLGIV

8.097

7.6769

0.4201

35

FLEPGPVTA

6.898

7.4940

-0.5960

80

YLWPGPVTV

8.125

8.0916

0.0334

36

ALAKAAAAA

6.947

6.8081

0.1389

81

FLLTRILTI

8.149

7.8796

0.2694

37

LMAVVLASL

6.954

7.4908

-0.5368

82

GLLGWSPQA

8.237

8.2184

0.0185

38

YVITTQHWL

6.983

6.3410

0.6420

83

ILYQVPFSV

8.310

8.7197

-0.4097

39

LLCLIFLLV

6.996

7.5015

-0.5055

84

GILTVILGV

8.347

7.8414

0.5056

40

ITAQVPFSV

7.020

6.6685

0.3515

85

NMVPFFPPV

8.398

8.0854

0.3126

41

YLEPGPVTL

7.058

7.1483

-0.0903

86

ILDQVPFSV

8.481

7.6904

0.7906

42

YTDQVPFSV

7.066

7.0742

-0.0082

87

YLFPGPVTA

8.495

8.3473

0.1477

43

NLYVSLLLL

7.114

6.9769

0.1371

88

YLDQVPFSV

8.638

8.1326

0.5054

44

ILHNGAYSL

7.127

7.3493

-0.2223

89

ILFQVPFSV

8.699

8.4335

0.2655

45

SIISAVVGI

7.159

7.3048

-0.1458

90

ILWQVPFSV

8.770

8.5002

0.2698

Statistical indices:

R=0.887132 R2= 0.787003 RES=0.366873 SEE=0.038672.

The iterative 2L-PCA technique described in Method section is used for the binding affinity study of peptides based on the sequences and experimental data listed in Table 2. The initial coefficient values {bl(0)}  of fragment parameters were assigned to 1, implying that all fragment parameters are equally important. Shown in Figure 1 are the curves of correlation coefficients R vs iterations, where the curve Ra is for the iteration of coefficients {ak}, and the curve Rb is for the iterations of coefficients {bl}. The average fitting error Q between the calculated bioactivities and the experimental bioactivities of peptides are shown in Figure 2, where Qa is for {ak}iteration and Qb for {bl} iteration. It has been observed that, after 10 to 12 iterations, the iterative result converged smoothly. The converged prediction coefficient sets {ak(n)} and {bl(n)}  are given in Table 3. In the iterative solution precedure the correlation coefficien increases from the first value RA(1)=0.4167 to the converged value RA(98)=0.8871, and the prediction residue decreases from the first value QA (1)=0.7223 to the converged value QA(98)=0.0387.

Table 3: Prediction coefficients of eight physicochemical properties and nine residue positions obtained from the training set of MHC-I peptides

No.

Property

Coefficient

No.

Position

Coefficient

{ak}

(Residue)

{bkl}

1

Lip

-0.02445

1

R1

2.53268

2

Hyd

0.19258

2

R2

8.36712

3

SL

-0.00212

3

R3

3.06856

4

SH

0.00348

4

R4

-4.89559

5

Pα

0.15367

5

R5

3.12686

6

Pβ

0.07823

6

R6

2.45367

7

Pc

0.19764

7

R7

1.24669

8

Vol

0.00366

8

R8

-3.50416

--

--

--

9

R9

-3.79249

The correlation coefficients between experimental and predicted bioactivities increase with the iterations.

Figure 1: The correlation coefficients between experimental and predicted bioactivities increase with the iterations. Ra is the correlation coefficient in the iterative procedure for {ak(n)}  of the physicochemical properties, and Rb is the correlation coefficient in the iterative procedure for {bl(n)}  of the molecular fragments.

The residue between predicted bioactivities and experimental bioactivities in the iterative procedure.

Figure 2: The residue between predicted bioactivities and experimental bioactivities in the iterative procedure. The Q is the average square root of the summation of squared differences between predicted bioactivities and experimental bioactivities. Qa is for {ak(n)}  iteration and Qb is for {bl(n)}  iteration.

The predicted pIC50 of the 40 queried peptides in the testing set are given in Table 4, which were predicted using the coefficients {ak(n)}  of properties and {bl(n)}  of fragments based on the eight physicochemical parameters and the nine fragments (amino acid side chains). The diversity of the peptides in the training set is very important for the prediction power of TLPC, especially for the residue positions at which we want to make prediction. It is expected that, with more experimental data available, the predictive power of 2L-PCA will be further improved. Actually, 30 prediction servers for human MHC-I peptide molecules were evaluated in a review article [57]. Among the 30 existing servers, 16 were ranked as the first class that provided the most accurate prediction results for MHC-I peptide molecules with the correlation coefficients ranging from r = 0.55 to r = 0.87. It has been shown in this study that the prediction correlation coefficient yielded by our 2L-PCA method is r = 0.868, being ranked around the very top of the first class.

Table 4: Amino acid sequences and experimental and predicted bioactivities of 40 MHC-I peptides in the testing set

No.

Sequence

Expt
pIC50

Pred
pIC50

pIC50
Diff

No.

Sequence

Expt
pIC50

Pred
pIC50

pIC50
Diff

1

LLGCAANWI

5.301

5.1708

0.1302

21

ITFQVPFSV

7.179

7.3750

-0.1960

2

SAANDPIFV

5.342

4.8592

0.4828

22

FTDQVPFSV

7.212

6.8379

0.3741

3

TTAEEAAGI

5.380

5.4678

-0.0878

23

RLMKQDFSV

7.342

7.5681

-0.2261

4

LTVILGVLL

5.580

5.3216

0.2584

24

KLHLYSHPI

7.352

6.6450

0.7070

5

HLLVGSSGL

5.792

6.4811

-0.6891

25

ITMQVPFSV

7.398

7.2641

0.1340

6

GIGILTVIL

6.000

5.7321

0.2679

26

KIFGSLAFL

7.478

6.7818

0.6962

7

TVILGVLLL

6.072

5.4662

0.6058

27

ALVGLFVLL

7.585

7.3852

0.1998

8

WTDQVPFSV

6.145

6.8930

-0.7480

28

YLSPGPVTV

7.642

7.2387

0.4033

9

AIAKAAAAV

6.176

6.4480

-0.2720

29

GLYSSTVPV

7.699

7.6303

0.0687

10

ILTVILGVL

6.419

7.0160

-0.5970

30

YLYPGPVTA

7.772

8.6335

-0.8615

11

AVAKAAAAV

6.495

5.9131

0.5819

31

YLAPGPVTV

7.818

7.3184

0.4996

12

ILDEAYVMA

6.623

7.4445

-0.8215

32

VVLGVVFGI

7.845

7.4509

0.3941

13

LLWFHISCL

6.682

6.3594

0.3226

33

MMWYWGPSL

7.921

7.4007

0.5203

14

TLDSQVMSL

6.793

7.2566

-0.4636

34

ILAQVPFSV

7.939

7.7270

0.2120

15

HLYQGCQVV

6.832

7.6799

-0.8479

35

FLLSLGIHL

8.053

8.1578

-0.1048

16

QLFHLCLII

6.886

7.6475

-0.7615

36

ILMQVPFSV

8.125

8.3225

-0.1975

17

ITDQVPFSV

6.947

6.6320

0.3150

37

YLFPGPVTV

8.237

8.0249

0.2121

18

ALCRWGLLL

7.000

7.2766

-0.2766

38

YLMPGPVTA

8.367

8.2363

0.1307

19

NLGNLNVSI

7.119

7.0974

0.02160

39

YLWPGPVTA

8.495

8.4140

0.0810

20

HLYSHPIIL

7.131

7.5663

-0.4353

40

FLDQVPFSV

8.658

7.8964

0.7616

Statistical indices:

R=0.867872 R2= 0.753202 RES=0.469728 SEE=0.074271.

2L-PCA neither needs knowing the exact comformations of the peptides nor needs aligning the peptides according to a template. The two steps are necessary but quite difficult for CoMFA [58, 59] and CoMSIA [60, 61] owing to that there are numerous possible conformations for peptides and that the experimental crystal structure for serving as a template is often not available. 2L-PCA method provides an alternate way for design of the chemical drugs and peptide drugs.

The eigenvalues and contributions of physicochemical properties and amino acid positions in peptides are summarized in Table 5 and shown in Figure 3. In Table 5 the eigenvalues are normalized. The eigenvalue portion of the first three property eigenvectors is almost 100%, and the eigenvalue portion of the first eigenvector alone is larger than 99%. Most contributions are made by the three properties: side chain volume (Vol), lipophilic surface area (SL), and hydrophilic surface area (SH), as shown in Figure 3b and Table 5. The contributions of other 5 properties seem very small. The eigenvalue of the first peptide position eigenvector is larger than 98%. In Table 5 the contributions of the nine amino acid positions are different. However the differences are not big, implying that all positions are almost equally important. The detailed computation results are given in Supplementary Information 1.

Table 5: Eigenvalues and contributions of physicochemical properties and amino acid positions in training set of peptides

Physicochemical properties

Positions (fragments)

No.

Eigenvaluea

Property

Contribution

No.

Eigenvaluea

Property

Contribution

1

0.99118

Lip

0.00004

1

0.98873

Residue-1

0.12126

2

0.00641

Hyd

0.00000

2

0.00300

Residue-2

0.12043

3

0.00240

SL

0.20595

3

0.00249

Residue-3

0.1130

4

0.00005

SH

0.00534

4

0.00213

Residue-4

0.09871

5

0.00004

Pα

0.00004

5

0.00106

Residue-5

0.1046

6

0.00003

Pβ

0.00005

6

0.00094

Residue-6

0.10804

7

0.00002

Pc

0.00002

7

0.0007

Residue-7

0.11931

8

0.00001

Vol

0.78857

8

0.00058

Residue-8

0.10186

--

--

--

--

9

0.00037

Residue-9

0.11278

a Eigenvalues are normalized.

b The positions of peptides are equal to the fragments of molecules.

Eigenvalues and contributions of properties and peptide positions.

Figure 3: Eigenvalues and contributions of properties and peptide positions. (a) The eigenvalues of property eigenvectors. (b) The contributions of properties to the eigenvalues. The volumes (Vol) and hydrophobic surface areas (SL) of amino acid side chains make the largest contributions. (c) The eigenvalues of peptide position eigenvectors. (d) The contributions of peptide positions to the eigenvalues. The contributions of all nine amino acid positions are almost equally important.

We are often facing two kinds of challenges in theoretical prediction for drug design: one is over-correlation problem, and the other is lack of information and explanation for the predicted results. The over-correlation problem is caused by large amount of parameters used in the prediction model, which may yield quite good correlation results in self-consistency test [62, 63], but very poor predicted results in independent dataset test owing to the high dimensional disaster [19] or “curse of dimensionality” problem. To solve this problem, the pseudo amino acid composition (PseAAC) was introduced [64]. Ever since then, the concept of PseAAC or the general PseAAC [21] has been widely used in drug development and biomedicine [65, 66] and nearly all the areas of computational proteomics (see, e.g., [67] as well as a long list of references cited in [68, 69]). Actually, the physicochemical properties used here can be regarded as some optimal pseudo components [70]. It is through such a PseAAC approach to remove the trivial parameters (or reduce the feature vector’s dimension) and grasp the key ones. Besides, the traditional prediction methods fail to provide a good explanation for the predicted results; i.e., how do the physicochemical properties and the structural changes affect the bioactivities? In contrast to that, the proposed “2L-PCA” method can provide more information about the impact of the physicochemical properties and molecular fragments to the bioactivities of drug candidates.

MATERIALS AND METHODS

In practical drug design and development, usually the basic structure of drug candidates keep constant, only small modifications are made on several fragments. The structure parameters of the entire molecules cannot clearly describe the detailed characters of the small changes at individual fragments or substitutes. In the 2L-PCA model the molecular structures are separated into several fragments, and are described by a set of fragment parameters. An example of molecular structure and its fragments is shown in Figure 4A. The idea of molecular fragments also can be applied to the peptide drugs, in which each side chain of an amino acid is a fragment, as shown in Figure 4B.

Illustration of molecular fragments.

Figure 4: Illustration of molecular fragments. (A) The structural fragments in neuraminidase (NA) of influenza virus A inhibitors. The molecular structure is divided into 4 fragments according to the substitutes being investigated. The fragments F1, F2 and F3 are three substituent groups, and the fragment F4 is the remaining part of the molecular parent. (B) In short peptides each side chain of amino acid residue is a fragment.

General 3D equation of 2L-PCA

In the 2L-PCA prediction model the bioactivity wi of molecule i is the summation of contributions Δgi,l from all molecular fragments; i.e.,

l=1LblΔgi,l=wi  (3)

where Δgi,l is the contribution of fragment l to the bioactivity wi of molecule i, bl is the prediction coefficient of fragment l, and L is the total number of molecular fragments. The contribution Δgi,l of fragment l is the summation of the contributions from all physicochemical properties of fragment l, namely

Δgi,l=k=1Kakxi,l,k                        (4)

where xi,l,k is the physicochemical property k of fragment l in molecule i, ak is the prediction coefficient of physicochemical property k, and K is the total number of physicochemical properties.

Inserting the Eq.4 into Eq.3 we get the general equation of 2L-PCA prediction model as given by

l=1Lbl(k=1Kakxi,l,k)=wi(i=1,2,,N)                                (5)

where N is the total number of molecular samples. Eq.5 can be expressed in vector and matrix form as given below

XN,L,KBLAK=WN                         (6)

where XN,L,K is the three dimensional (3D) data matrix of molecular parameters, WN is the bioactivity column vector of molecular samples, BL is the coefficient vector of fragments, and AK is the coefficient vector of physicochemical properties.

2D equations of properties and fragments

The general three-dimensional 2L-PCA equation of Eq.6 can be reduced to two 2D equations with the following algebra operations,

XN,L,KAK=HN,L                                  (7)

where HN,L is the 2D data matrix of molecular fragments. Substituting HN,L into Eq.6, we obtain the following fragment 2D equation

HN,LBL=WN                               (8)

Likewise, the property 2D equation can also be expressed as

XN,L,KBL=FN,K(9)

and

FN,KAK=WN(10)

where FN,K is the 2D data matrix of physicochemical properties.

Algebra solutions of property and fragment 2D equations

The fragment 2D equation Eq.8 and the property 2D equation Eq.10 can be solved using the standard algebra method. Both sides of the fragment 2D equation of Eq.8 are multiplied with the transposed matrix HtN,L from left, it follows that

HN,LtHN,L=UL,L(11)

and

HN,LtWN=SL(12)

Thus, we get the following symmetrically square matrix equation of fragments

UL,LBL=SL(13)

Since the fragment square matrix equation of Eq.13 is multiplied by its inverse matrix U-1L,L, the prediction coefficients BL for the fragments are obtained, as given below

BL=UL,L1SL(14)

where the inverse matrix U-1L,L can be obtained by solving the eigen equation [71] [48] of UL,L, namely the equation

UL,LΨL,L=αLΨL,L(15)

meaning

UL,L1=1αLΨL,L(16)

where ΨL,L is the eigenvectors and αL is the eigenvalues of fragment square matrix UL,L[72, 73].

Similarly, left-multiplying both sides of property 2D equation of Eq.10 with FtN,K, we have

FN,KtFN,K=VK,K  (17)

and

FN,KtWN=TK   (18)

From Eqs.17-18, we get the following square matrix equation of properties

VK,KAK=TK   (19)

Multiplying Equation Eq.19 with the inverse matrix V-1K,K, will give the solution of property prediction coefficients AK; i.e.

AK=VK,K1TK(20)

Thus, the inverse matrix V-1K,K is obtained by solving the eigen equation of property square matrix VK,K:

VK,KΦK,K=βKΦK,K(21)

and

VK,K1=1βKΦK,K(22)

where ΦK,K is the eigen-vectors and βk is the eigen-values of the property square matrix VK,K.

Iterative solution of 2L-PCA equations

In the training dataset for drug candidates the two prediction coefficients set AK and BL in the 2L-PCA general equation Eq.6 are solved in an iterative procedure [74, 75]. Firstly the initial fragment coefficients B(0)L are assigned to 1 {bi=1, i=1,2…,L}, implying all fragments are equally important. The initial B(0)L are used in the property 2D equations Eq (9) and (10), thus the first solution of property coefficients A(1)K is obtained by solving the eigen-equations Eq.17-20. Then the property coefficients A(1)K are used in the fragment equations Eqs.7-8, and the first solution of fragment coefficients B(1)L are obtained by solving eigen-equations Eq.11-14. In the next iterative cycle the B(1)L is used to find the A(2)K. Above iterative procedure is repeated for n times, until to reaching a threshold value ε; i.e.,

| Q(n+1)Q(n) | =| 1Ni=1N(wiexptwi(n+1))21Ni=1N(wiexptwi(n))2 | ε(23)

The bioactivities of designed drugs and newly synthesized drug candidates are predicted using the converged coefficients {ak(n)} and {bl(n)}  as given below

wipred=l=1Lbl(n)(k=1Kak(n)xi,l,k)  (24)

Illustrated in Figure 5 is the iterative solution procedure for the 2L-PCA predictor.

The iterative algebra solution procedure of the solution of 2L-PCA prediction model for the two sets of coefficients {ak(n)}  and {bl(n)} , where N is the number of molecular samples, L is the number of fragments in molecules, L′ is the principal number of fragments, K is the number of physicochemical properties, and K′ is the principal number of properties.

Figure 5: The iterative algebra solution procedure of the solution of 2L-PCA prediction model for the two sets of coefficients {ak(n)}  and {bl(n)} , where N is the number of molecular samples, L is the number of fragments in molecules, L′ is the principal number of fragments, K is the number of physicochemical properties, and K′ is the principal number of properties.

Principal component analysis of properties and fragments

The property eigenvectors {φk}  are orthogonal and normalized; i.e.,

φk·φj=0  (kj)   (25)

and

φk·φk=j=1Kφj,k2=1   (26)

where the term φ2j,k is the component of the j-th property in the k-th eigen-vector φk. The first K′ property eigen-vectors are the principal components whose eigen-values are larger than a threshold (e.g., ε=90% or 95%); i.e.,

k=1K'βk2k=1Kβk2ε   (27)

The total contribution γj of the j-th property to the bioactivity of molecular samples in training set is defined as the following summation,

γj=k=1K'βkφj,k2(28)

The property eigen-vectors {φk}  span an orthogonal multiple space, in which a drug molecule Pi is a vector, and its projection Ji,k on the k-th property-eigenvector φk is calculated by

Ji,k=fiΦϕk|fi||ϕk|=j=1Kfi,jφj,kj=1Kfi,j2j=1Kϕi,j2(29)

where fi is the i-th row vector of the property matrix FNK of Eq.9. In the projection Ji,k of molecular sample Pi on the k-th property-eigenvector jk the component of the r-th property is αkφ2r,k, therefore the total contribution of r-th property to the sample Pi is the summation of components from all principal property eigenvectors, namely

ξi,r=k=1K'βkJi,kφr,k2(30)

Similarly, the fragment eigenvectors ψl span an L-dimensional orthogonal space. The first L′ fragment eigenvectors are the principal components. The total contribution factor λj of the j-th fragment to the bioactivity of peptide set is given by

λj=l=1L'αlψj,l2(31)

In the same way the projection Ii,l of sample Pi on the l-th fragment-eigenvector ψl can be calculated by

Ii,l=hiΦϕl|hi||ϕl|=j=1Lhi,jψj,lj=1Lhi,j2j=1Lψi,j2(32)

where hi is the i-th row vector of the fragment matrix HNL of Eq.7. In the projection Ii,l of molecule Pi on the l-th fragment-eigenvector φl the component of the r-th fragment is αlψ2r,l, therefore the total contribution of r-th fragment to the sample Pi is the summation of components from all principal fragment eigenvectors; i.e.,

ςi,r=l=1L'αlIi,lψr,l2(33)

Web-server

As pointed out in [76], user-friendly and publicly accessible web-servers represent the future direction for developing practically more useful predictors or any computational tools. Actually, user-friendly web-servers as given in a series of recent publications [2325, 30, 32, 3436, 69, 70, 7791] will significantly enhance the impacts by attracting the broad experimental scientists [66, 92]. We will do our best to establish a web-server for 2L-PCA as soon as possible. Once it has been done, an announcement will be made thorough a publication or our webpage.

CONCLUSION

The 2L-PCA predictor proposed in this paper is a very useful tool for drug design. Its advantages can be summarized as follows. (1) With 2L-PCA, the molecular structures of drug candidates can be separated into several fragments described by physicochemical parameters of the molecular fragments, thus the small modifications on individual fragments can be clearly shown. (2) Its two prediction coefficient sets {ak}  of properties and {bl}  of fragments can be solved in an iterative procedure, which possesses self-learning ability and information feed-back function in certain degree, and hence greatly promoting the prediction power of 2L-PCA. (3) It possesses the information from both of the structures of molecular fragments and the physicochemical properties, able to significantly improve the drug candidates in both the structure and property. (4) Its elegant algebra solution procedure will be very useful for further enhancing the ability of principal component analysis (PCA).

ACKNOWLEDGMENTS

The authors are very much indebted to the eight anonymous reviewers, whose constructive comments are very helpful for strengthening the presentation of this paper. This work was supported by grants from the National Science Foundation of China (NSFC http://www.nsfc.gov.cn/) under the contract numbers 31360207, 31370716, and 31400079, Guangxi Science and Technology Development Projects (no. 14123001-19, 1425008-2-22, and 15104001-1), Guangxi President Academic Foundation under number 16449-02, and Science foundation Project of Guangxi Academy of Sciences (No.15YJ22SW01). We thank the National Supper Computing Center (NSCC www.nscc-tj.gov.cn) in Tianjin for the valuable help in the modeling calculations using TH-1A super computer.

CONFLICTS OF INTEREST

The authors declare no conflicting interest.

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