Protein Quadratic Indices of the “Macromolecular Pseudograph’s α-Carbon Atom Adjacency Matrix”. 1. Prediction of Arc Repressor Alanine-mutant’s Stability
Abstract
:Introduction
Computational Methods
Arc Dimer Structure and Melting Temperature of a Complete Set of A-Substitution Mutants
- i)
- Between side chain in the same subunit (R16-D20, D20-R23, N29-E36, E36-R31, E36-R40, E43-K46, E43-K47) and; those between side chains in different subunits (E28-R50, R40-S44, R40-F48).
- ii)
- Between a side chain and main-chain atom intersubunit (W14-N34, N34-R13) and; those between a side chain and main-chain atom intrasubunits (E17-E17, S32-S35, S44-R40).
Protein | Classb | P% (P)c | P% (H)c | Scored | tm(Obs)e | tm(Pred)f | Resg | tm(Pred)h | Resg |
---|---|---|---|---|---|---|---|---|---|
1PA8-st6a | H | 4.31 | 95.69 | 1.47 | 74.1 | (55.1) i | 19.0 | 56.86 | 17.2 |
2SA35-st6 | H | 5.25 | 94.75 | 1.36 | 63.4 | 62.4 | 1.0 | 69.1 | -5.7 |
*3NA34-st11 | H | 59.40 | 40.60 | -0.23 | 63.0 | 61.2 | 1.8 | 52.6 | 10.4 |
4NA11-st6a | H | 40.89 | 59.11 | 0.13 | 62.1 | 54.5 | 7.6 | 49.95 | 12.1 |
5QA39-st11 | H | 9.25 | 90.75 | 1.07 | 61.4 | 59.7 | 1.7 | 62.7 | -1.3 |
*6GA52-st11 | H | 86.94 | 13.06 | -0.98 | 60.9 | 60.0 | 0.9 | 57.5 | 3.4 |
7KA6-st6a | H | 8.75 | 91.25 | 1.10 | 59.6 | 55.0 | 4.6 | 60.83 | -1.2 |
8RA16-st6 | H | 0.43 | 99.57 | 2.61 | 59.5 | 56.3 | 3.2 | 57.6 | 1.9 |
9VA25-st6 | H | 11.48 | 88.52 | 0.95 | 59.3 | 57.3 | 2.0 | 56.4 | 2.9 |
10MA4-st6 | H | 12.49 | 87.51 | 0.90 | 59.2 | 58.1 | 1.1 | 60.1 | -0.9 |
11Arc-st6a | H | 9.11 | 90.89 | 1.08 | 59 | 54.7 | 4.3 | 57.88 | 1.1 |
12EA27-st6 | H | 5.42 | 94.58 | 1.35 | 58.8 | 58.1 | 0.7 | 56.5 | 2.3 |
13KA2-st6 | H | 2.09 | 97.91 | 1.83 | 58.7 | 58.2 | 0.5 | 59.2 | -0.5 |
14QA9-st6 | H | 14.28 | 85.72 | 0.83 | 58.4 | 57.5 | 0.9 | 55.3 | 3.1 |
15GA3-st6 | H | 6.12 | 93.88 | 1.29 | 58.1 | 60.3 | -2.2 | 57.3 | 0.8 |
16MA1-st6a | H | 12.84 | 87.16 | 0.89 | 58 | 55.0 | 3.0 | 59.41 | -1.4 |
*17Arc-st11 | H | 88.80 | 11.20 | -1.06 | 57.9 | 59.0 | -1.1 | 52.4 | 5.5 |
18SA5-st6 | H | 8.09 | 91.91 | 1.14 | 57.5 | 58.2 | -0.7 | 58.8 | -1.3 |
19RA13-st6 | H | 2.28 | 97.72 | 1.79 | 57.3 | 57.7 | -0.4 | 53.9 | 3.4 |
20KA46-st11 | H | 8.04 | 91.96 | 1.14 | 57.1 | 55.9 | 1.2 | 56.1 | 1.0 |
21EA17-st6a | H | 4.58 | 95.42 | 1.43 | 57 | 55.8 | 1.2 | 56.90 | 0.1 |
22VA18-st6 | H | 6.25 | 93.75 | 1.28 | 56.9 | 58.1 | -1.2 | 55.4 | 1.5 |
23RA23-st11 | H | 18.53 | 81.47 | 0.67 | 56.7 | 57.7 | -1.0 | 51.8 | 4.9 |
24KA24-st11 | H | 29.57 | 70.43 | 0.38 | 56.3 | 57.9 | -1.6 | 49.3 | 7.0 |
25EA43-st6 | H | 2.04 | 97.96 | 1.84 | 56.1 | 57.6 | -1.5 | 54.7 | 1.4 |
26EA28-s11a | H | 47.66 | 52.34 | 0.00 | 55.7 | 56.2 | -0.5 | 50.19 | 5.5 |
27MA7-st6 | H | 8.75 | 91.25 | 1.10 | 55.5 | 58.4 | -2.9 | 60.8 | -5.3 |
28DA20-st6 | H | 2.68 | 97.32 | 1.71 | 55.3 | 57.7 | -2.4 | 49.6 | 5.7 |
29IA51-st11 | P | 93.91 | 6.09 | -1.39 | 50.9 | 40.4 | 10.5 | 47.7 | 3.2 |
30GA49-st11a | P | 91.79 | 8.21 | -1.23 | 48.7 | 47.0 | 1.7 | 40.71 | 8.0 |
*31LA19-st6 | P | 9.99 | 90.01 | 1.03 | 48.3 | 45.4 | 2.9 | 51.8 | -3.5 |
32GA30-st11 | P | 52.78 | 47.22 | -0.10 | 47.9 | 42.5 | 5.4 | 56.1 | -8.2 |
33RA50-st11 | P | 62.68 | 37.32 | -0.30 | 47.9 | 44.5 | 3.4 | 49.5 | -1.6 |
*34KA47-st11 | P | 20.15 | 79.85 | 0.62 | 47.2 | 50.0 | -2.8 | 40.7 | 6.5 |
35PA15-st11a | P | 66.88 | 33.12 | -0.39 | 46.6 | 38.4 | 8.2 | 55.56 | -9.0 |
36SA44-st11 | P | 99.90 | 0.10 | -3.42 | 46.3 | 44.3 | 2.0 | 37.0 | 9.3 |
37NA29-st11 | P | 80.97 | 19.03 | -0.76 | 45.3 | 47.7 | -2.4 | 49.6 | -4.3 |
38VA33-st11 | P | 94.46 | 5.54 | -1.43 | 44.1 | 41.5 | 2.6 | 49.8 | -5.7 |
39EA48-st11 | P | 82.37 | 17.63 | -0.80 | 43.2 | 42.3 | 0.9 | 44.7 | -1.5 |
40LA12-st11 | P | 97.37 | 2.63 | -1.81 | 42.3 | 44.3 | -2.0 | 43.2 | -0.9 |
*41FA10-st6a | P | 31.24 | 68.76 | 0.34 | 40.6 | 45.8 | -5.2 | 49.41 | -8.8 |
42LA21-st11 | P | 90.68 | 9.32 | -1.16 | 39.6 | 39.9 | -0.3 | 46.7 | -7.1 |
*43RA31-st11 | P | 15.18 | 84.82 | 0.79 | 37.1 | 41.6 | -4.5 | 45.8 | -8.7 |
44MA42-st11 | P | 84.06 | 15.94 | -0.86 | 35.6 | 37.5 | -1.9 | 35.6 | 0.0 |
45SA32-st11a | P | 90.07 | 9.93 | -1.13 | 33.5 | 34.2 | -0.7 | 61.35 | -27.8 |
46YA38-st11 | P | 90.77 | 9.23 | -1.17 | 33.0 | 40.6 | -7.6 | 36.4 | -3.4 |
47WA14-st11 | P | 97.38 | 2.62 | -1.82 | 31.5 | 38.8 | -7.3 | 36.6 | -5.1 |
48RA40-st11 | P | 98.44 | 1.56 | -2.08 | 31.2 | 30.2 | 1.0 | 40.6 | -9.4 |
49VA22-st11 | P | 83.85 | 16.15 | -0.85 | <20 | ||||
50EA36-st11a | P | 69.58 | 30.42 | -0.45 | <20 | ||||
51IA37-st11 | P | 91.53 | 8.47 | -1.21 | <20 | ||||
52VA41-st11 | P | 95.81 | 4.19 | -1.58 | <20 | ||||
53FA45-st11 | P | 99.52 | 0.48 | -2.66 | <20 |
Protein Quadratic Indices of the “Macromolecular Pseudograph’s α-Carbon Atom Adjacency Matrix”
Amino Acids | z1 | z2 | z3 | |
---|---|---|---|---|
Ala | A | 0.07 | -1.73 | 0.09 |
Val | V | -2.69 | -2.53 | -1.29 |
Leu | L | -4.19 | -1.03 | -0.98 |
Ile | I | -4.44 | -1.68 | -1.03 |
Pro | P | -1.22 | 0.88 | 2.23 |
Phe | F | -4.92 | 1.30 | 0.45 |
Trp | W | -4.75 | 3.65 | 0.85 |
Met | M | -2.49 | -0.27 | -0.41 |
Lys | K | 2.84 | 1.41 | -3.14 |
Arg | R | 2.88 | 2.52 | -3.44 |
His | H | 2.41 | 1.74 | 1.11 |
Gly | G | 2.23 | -5.36 | 0.30 |
Ser | S | 1.96 | -1.63 | 0.57 |
Thr | T | 0.92 | -2.09 | -1.40 |
Cys | C | 0.71 | -0.97 | 4.13 |
Tyr | Y | -1.39 | 2.32 | 0.01 |
Asn | N | 3.22 | 1.45 | 0.84 |
Gln | Q | 2.18 | 0.53 | -1.14 |
Asp | D | 3.64 | 1.13 | 2.36 |
Glu | E | 3.08 | 0.39 | -0.07 |
= 1 if i = j and the amino acid i has a hydrogen bond between its side chain and
its main-chain atom
= 0 otherwise
= 1/2 kaij if vi or vj are vertices (amino-acid) contained within FR but not both
= 0 otherwise
Pentapeptide Structure (sequence) Macromolecular ‘Pseudograph’ (Gm) of the α-Carbon Atoms (Polypeptide’s backbone) Amino acid residue (side chain R) Here, we consider only covalent interaction (peptidic bond), but non-covalent interaction (hydrogen-bond and salt bridge interaction) can be taken into consideration (within a chain as well as between chains) | Macromolecular Vector: Xm = [V K W A A] ℜ5 In the definition of the Xm, as macromolecular vector, the one letter symbol of the amino acids indicates the corresponding side-chain amino-acid property, e.g., z1-values. That is to say, if we write V it means z1(V), z1-values or some amino acid property, which characterizes each side chain in the polypeptide. Therefore, if we use the canonical bases of R5, the coordinates of any vector Xm coincide with the components of that macromolecular vector [mX]t = [-2.69 2.84 -4.75 0.07 0.07] [mX]t = transposed of [mX] and it means the vector of the coordinates of Xm in the canonical basis of R5 (an 1x5 matrix) [mX]: vector of coordinates of Xm in the canonical basis of R5 (an 5x1matrix) | |||||
Total (whole molecule) protein quadratic indices of zero, first and second order are a quadratic maps; qk(xm): ℜn→ ℜ such that, q0(V, K, W, A, A) = (V2+K2+W2+A2+A2) = 37.874 q1(V, K, W, A, A) = (2VK+KW+2WA+2AA) = -42.9144 q2(V, K, W, A, A) = (A2+V2+2K2+2W2+2A2+2WV+2AW) = 93.7946 | ||||||
If the peptide is partitioned into each (5) amino acid, the matrix Mk(Gm) can be partitioned into 5 local matrices MkL(Gm), L = 1,... 5. The kth power of the matrix M(Gm) is exactly the sum of the kth power of the local (5) matrices: | ||||||
The zero, first and second powers of the local (amino-acid) matrix | ||||||
and the total (whole-molecule) quadratic indices are the sum of the macromolecular quadratic indices of the 5 amino-acids, qk(xm) = | ||||||
Amino Acid (AA) | q0L(xm, AA) | q1L(xm, AA) | q2L(xm, AA) | q3L(xm, AA) | q4L(xm, AA) | |
Val (V) | 7.2361 | -7.6396 | 20.0136 | -15.4675 | 52.6164 | |
Lys (K) | 8.0656 | -21.1296 | 16.33 | -55.5504 | 41.1232 | |
Trp (W) | 22.5625 | -13.8225 | 57.57 | -41.4675 | 172.71 | |
Ala (A) | 0.0049 | -0.3276 | 0.2086 | -1.176 | 0.8197 | |
Ala (A) | 0.0049 | 0.0049 | -0.3276 | 0.2086 | -1.176 | |
Pentapeptide | 37.874 | -42.9144 | 93.7946 | -113.453 | 266.0933 |
TOMOCOMD Software
- Draw the macromolecular pseudographs for each protein of the data set, using the software’s drawing mode. This procedure is carried out by a selection of the active aminoacid symbol belonging to ‘natural’ aminoacid code. Here, we consider only covalent interaction (peptidic bond) and hydrogen-bond interaction (within a chain as well as between chains). Afterward, we draw the mutants by changing an AA for alanine and considering that this change only affect the possibility of this region of the protein to form polar interaction (because we suppressed the hydrogen interaction if the former AA had it).
- Compute the protein quadratic indices of the “macromolecular pseudograph’s α-carbon atom adjacency matrix”. They can be performed in the software calculation mode, in which one can select the side chain properties and the family descriptor previously to calculate the molecular indices. This software generates a table in which the rows and columns correspond to the compounds and the qk(xm), respectively.
- Find a QSPR/QSAR equation by using statistical techniques, such as multilinear regression analysis (MRA), Neural Networks (NN), Linear Discrimination Analysis (LDA), and so on. That is to say, we can find a quantitative relation between a property P and the qk(xm) having, for instance, the following appearance,P = a0q0(x) + a1q1(x) + a2q2(x) +….+ akqk(x) + c
- Test the robustness and predictive power of the QSPR/QSAR equation by using internal and external cross-validation techniques,
- Develop a structural interpretation of the obtained QSAR/QSPR model using macromolecular quadratic indices as molecular descriptors.
Statistical Analysis
Results and Discussion
Classification Model
-0.0170188.Z1q1(xm) +0.0132179.Z2q2(xm)
N = 41 λ = 0.476 D2 = 4.40 F(4,36) = 9.8965 p(F) < 0.0001
-0.008317831.Z1q1(xm) +0.006460173.Z2q2(xm)
N = 41 λ = 0.476 Rcanc = 0.72 χ2 = 27.44 Mean (+) = 0.998 Mean (-) = -1.048
Quantitative Structure-Stability Relationships (QSSP) Study
0.121(±0.048).Z1q1(xm) +8.89x10-5(±3.18x10-5).Z2q10(xm)
-1.369x10-5(±4.11x10-6).Z1q10(xm) +5.998x10-4(±2.157x10-4).Z1q7(xm)
+0.026(±0.014).Z1q2(xm) +3.99x10-5(±3.44x10-5).Z3q8(xm)
N = 41 R = 0.85 R2 = 0.72 s = 5.64 q2 = 0.55 scv = 6.24 F(8.28) = 9.0425 p < 0.0001
-9.4481x10-4.Z1q9(xm) -0.03023.Z3q3(xm) +0.01565.Z3q6(xm)
-0.0037.Z3q8(xm) +0.2131x10-3.Z3q10(xm)
tm (oC)>BKPT = 44.547 +0.0232.Z1q3(xm) -0.0159.Z1q5(xm) +3.046x10-3.Z1q7(xm)
-1.6594x10-4.Z1q9(xm) + 2.5765.Z3q3(xm) +0.0106.Z3q6(xm) -2.3478.Z3q8(xm)
+1.2647x10-4.Z3q10(xm)
N = 41 R = 0.94 R2 = 88.15 Bkpt = 51.32 p < 0.0001
Interpretation of Obtained Models
Conclusions
Acknowledgements
References and Notes
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Share and Cite
Ponce, Y.M.; Marrero, R.M.; Castro, E.A.; Ramos de Armas, R.; Díaz, H. G.; Zaldivar, V.R.; Torrens, F. Protein Quadratic Indices of the “Macromolecular Pseudograph’s α-Carbon Atom Adjacency Matrix”. 1. Prediction of Arc Repressor Alanine-mutant’s Stability. Molecules 2004, 9, 1124-1147. https://doi.org/10.3390/91201124
Ponce YM, Marrero RM, Castro EA, Ramos de Armas R, Díaz H G, Zaldivar VR, Torrens F. Protein Quadratic Indices of the “Macromolecular Pseudograph’s α-Carbon Atom Adjacency Matrix”. 1. Prediction of Arc Repressor Alanine-mutant’s Stability. Molecules. 2004; 9(12):1124-1147. https://doi.org/10.3390/91201124
Chicago/Turabian StylePonce, Yovani Marrero, Ricardo Medina Marrero, Eduardo A. Castro, Ronal Ramos de Armas, Humberto González Díaz, Vicente Romero Zaldivar, and Francisco Torrens. 2004. "Protein Quadratic Indices of the “Macromolecular Pseudograph’s α-Carbon Atom Adjacency Matrix”. 1. Prediction of Arc Repressor Alanine-mutant’s Stability" Molecules 9, no. 12: 1124-1147. https://doi.org/10.3390/91201124