Abstract

STOCHASTIC-BASED DESCRIPTORS FOR MODELING BIOLOGICAL PROPERTIES OF PEPTIDES: MODELING ANGIOTENSIN-CONVERTING ENZYME INHIBITION OF DIPEPTIDES.

Ronal Ramos^a*, Humberto González Díaz^b, , Alexander Duran^a, Maykel Perez^{c, b}

^a Chemistry Department, Chemistry and Pharmacy Faculty, Central University of Las Villas, Santa Clara Villa Clara, Cuba. PC 54830.

^b Department of Drug Design, Chemical Bioactive Center, Central University of Las Villas. Cuba. Santa Clara Villa Clara, Cuba. PC 54830.

Abstract: A stochastic based molecular descriptor was introduced to predict inhibition properties of a set of 58 dipeptides over Angiotensin Converting Enzyme. For this purpose several model (equations) were found with total and locally defined descriptors over the side chain of each amino acid position. In all cases very good regressions models explaining more than 75% of total variance and having a very good results in cross-validation procedures employed (less than 15% of the variance is lost when they are used to predict an external compound).

Key Words: QSAR, Dipeptides, ACE inhibition, Markov Theory, Stochastic Process.

Quantitative Structure Activity Relationship (QSAR) techniques have become indispensable for in all aspects of search into the molecular interpretation of biological properties ⁽¹⁾. Several kinds of methodologies yielding different molecular descriptors have been defined and widely employed in order to study physico-chemical and biological properties ^(2-4). A lot of research has been made in the specific field of connectivity indices ^(5-6).

On the other hand, several problems arise when QSAR studies are applied to peptides and proteins. The peptide comprises a great variety of biologically active linear and cyclic biopolymers with diverse functions, which can be divided according to their functions into different classes such as antibiotics, enzyme inhibitors and substrates, hormones and regulatory factors, peptide alkaloids, toxins and sweeteners. Most of those bioactive peptides consist of a larger number of amino acid residues but there are also a small drugs consisting of up to three amino acid residues. Despite these difficulties, there have been successful reports that describe their molecular structure in quantitative way in the last decades ^(7-8).

Special attention has been paid to Angiotensin and related enzymes with peptidic derivatives and non-peptidic as well. Krovat ⁽⁹⁾ reported Chemical feature based pharmacophore models elaborated for angiotensin II receptor subtype 1 (AT(1)) antagonists using both a quantitative and a qualitative approach (Catalyst HypoGen and HipHop algorithms, respectively). Literature also reports studies on this specific compounds employing conformational and chemometric combined analysis ⁽¹⁰⁾, steric parameters ⁽¹¹⁾, hologram-based methodologies. Several references has been found also dealing with QSAR studies of ACE inhibitor employing minimum analogue peptide sets (MAPS) of nine dipeptides, and from a set of 58 dipeptides inhibiting Angiotensin Converting Enzyme, descriptor encoding 3-D information^(12-14) and molecular holographic distances vector ⁽¹⁵⁾

Despite the extension of the methodologies employed in the above-mentioned studies, they are still limited in that they either not allow a straightforward interpretation of the resulting QSAR, as in the z-scales, in terms of physico-quemical factors important for biological activity. They may also require previous procedures, e.g. structural alignment in CoMFA, ⁽⁴⁾ or identification of a putative bioactive conformation derived from several methods as in CoMFA and classical 3-D QSAR ⁽¹⁶⁾. All these facts can introduce uncertainties into QSAR models especially in dealing with a structurally diverse dataset of highly flexible compounds such as peptides and proteins. That is why it is still reasonable and meaningful to construct a set of descriptors which posses’ good prediction, speed, and case of use without 3-d structural information.

The use of stochastic matrix formalism as source of simple molecular descriptors has not appeared in the literature before 2002. Our research group recently developed a new methodology (MARCH INSIDE) based on a Markov chain formalism to codify molecular structure towards virtual screening, and rational drug discovery that has been applied to rational design of fluckicidal drugs, ⁽¹⁷⁾. These ideas have been extended to the study of Proteins structure property relationships ⁽¹⁸⁾. Thus, taking into consideration all aspects highlighted in this introductory section, the present paper has very specific aims. Primordially, the present work deals with the QSAR study of a set of 58 dipeptides and their ACE-Inhibition activity applying MARCH-INSIDE methodology in order to proof their applicability on the prediction of biological properties of amino acids. The results were compared with previous reports and the contribution of the amino acid residues on either position of the dipeptide was also analyzed reaching to very interesting conclusions.

MATERIALS AND METHODS

Markovian Chemicals “in Silico” Design (MARCH-INSIDE). The MARCH-INSIDE methodology uses MCH ^(19-20) to codify information about molecular structure. This procedure considers as states of the MCH the external electron layers of any atom core in the molecule ⁽²¹⁾ (valence shell). The method uses as a source of molecular descriptors the matrix ¹P, which has the elements p_ij. This matrix is called the 1-step electron-transition stochastic matrix. ¹P is built as a squared table of order n, where n represents the number of atoms in the molecule. The elements (¹p_ij) of the 1-step electron-transition stochastic matrix are the transition probabilities:

Where c_j is the electronegativity of the atom a_j, which is bonded to the atom a_i. ^(17,18-21) The elements of ¹P (¹p_ij) are defined to codify information about the electron-withdrawing strength of atoms to withdraw electrons from their neighbors in the molecule. The MARCH-INSIDE molecular descriptors used in the previous paper ⁽³⁵⁾ are defined as:

These molecular descriptors are the traces of the kth-step-electron-transition stochastic matrices (^kP). These matrices are the successive powers of ¹P. The trace (Tr) ^(22-25) is the sum of the main diagonal elements (^kp_ii) of ¹P. The construction of the ¹P matrix for Nitrilo-acetyl Fluoride is shown in Figure 1. It can be observed that the p_ij values are proportional to the electronegativity of the atom a_j (the atom that attracts the electrons of a_i). Conversely, the p_ij values are inversely related with the electronegativity of the atoms that “compete” with a_j to withdraw electrons from a_i. In equation (2), S_m represents a specific group of atoms in the molecule. When S_m contains all the atoms in the molecule the term ^SRp_k(S) becomes a global molecular indices and we write only ^SRp_k.

Fig. 1 construction of the ¹P matrix for Nitrilo-acetyl Fluoride Markovian Electronic Delocalization Entropies (MEDEs).

The MCH theory permits us to explore the potential of the ¹P matrix as a source of simple molecular descriptors that a clear physical meaning and are based on the use of the physical informational concept of entropy in this stochastic theory. These molecular descriptors, which we shall define here as kth-MEDEs, have the use of the entropy concept in common with those reported previously ^(2,26). Thus, we can expect that the use of entropy in the generalization of MARCH-INSIDE may not only clarify the physical meaning of this method but further improve its effectiveness in QSTR studies.

In equation (3) we can carry out the sum in the dominator over all the atoms in the molecule (n) instead of the sum only up to d + 1. We call this new probability the 0-step absolute Markovian probability of electronic delocalization for atom j (^Ap₀(j)), note the use of A (absolute) rather than SR (self-return). In fact, these probabilities for each atom are normalized, i.e. ^Ap₀(j) values obey the normalization condition because their sum up to n is equal to 1.

By following the MCH theory formulation ^{(19, 24)} the physical meaning of ^Ap₀(j) is very clear. Under the approximation that electronegativity describes the electron withdrawing strength of atoms in the molecule, ⁽²¹⁾ then p₀(j) are the probabilities with which a specific atom j attracts any electrons in the molecule in 0 steps. The calculation of any ^Ap_k(j) value, which is the probability with which a specific atom j attracts any electrons in the molecule in k steps, is straightforward:

Where ^AP_k are 1 × n vectors whose elements ^Ap_k(j) are the found probabilities, ^AP₀ is a 1 × n vector whose elements are the ^Ap₀(j) probabilities for the n atoms in the molecule and ^kP are the kth natural powers of the ¹P matrix. It is now very simple to calculate the kth-steps Markovian Electron Delocalization Entropies, which represents the entropy involved in the attraction of electrons at least k-steps (bonds) away from by any atom j in the molecule (Q_k (j)). The sum of the Q_k (j) for all n atoms in the molecule gives the kth MEDEs used here as total molecular descriptors:

We use natural logarithmic expressions instead of base 2 counterparts, as use to be in Shannon´s expression, and we multiply by k_B(Boltzmann`s constant). This change only implies a change of scale from bits (more commonly used in information theory) to kJ/K (more familiar to the chemist) ⁽²⁷⁾ and, in any case, the physical meaning remains the same.

A set of 58 dipeptides inhibiting ACE was obtained from the report of Hellberg et al ⁽¹⁴⁾. The dipeptides and the ACE inhibition data in their study were taken from a previous compilation ⁽²⁸⁾ and synthesized as part of the development of Captopril.

The calculation of ^SRp_k or Q_k for any organic or inorganic molecule was implemented in the software MARCH-INSIDE. ⁽²⁹⁾ The chemical structure is input directly using the molecular graphics in the software draw mode. The structure can then be saved and it is then possible to select the calculation option and perform the calculation of the 10^th first values for each descriptor (total and local molecular indices). The local calculations were carried out over the side chain residues of each position on the dipeptide molecule. At this point, we have a set of electronic delocalization entropy (Θ_k) , self-return (^SRπ_k)and absolute probability (^absπ^R2_k) for each set of calculation (total and over R₁ and R₂) STATISTICA ⁽³⁰⁾ and its Multiple Regression Analysis Module of were used to obtain the mathematical relationship between ACE-inhibition and molecular structure (total and local molecular descriptors). Several statistical parameters were calculated such as correlation coefficient (R²), the root mean square error (s), the F statistic, cross validation regression coefficient (R²_cv).

Trying to obtain quantitative equation to model this properties several model arouse comprising either total or local molecular descriptors. The best ones of each set was:

Total descriptors

Local descriptor over side chain position 1

Local descriptor over side chain position 2

The statistical parameters of the above equation are shown in the following table (Table1)

Table 1- Statistical parameters of Eq. 6-8

As can be seen the three model showed good linearity (R²) explaining more than 74% of the variance in the experimental data. Even models 6 and 8 can expain more than 83% of such variance. The ρ coefficient (should be > 5 for linear models) also shows an acceptable 5.8 for model 6-7 and a very good value of 8.3 for model 8. The result of the Leave-One-Out cross validation procedure (Q²) also showed the superiority of models 6 and 8 over model 7. The ratio (1-Q²/ R²) measures the percent of variance that can be explained for the model that is lost in the prediction of a specific compound. These values were 9, 16, and 5.4% respectively for model 6, 7, and 8 showing that more than 90% of the former explained variance in the linearity test remains explained in the prediction test. The model 6 and 8 also show better values of the Fischer ratio and SD than model 7. the following table and figure shows the observed and predicted values of each model (table 2 and Fig 2)

^a observed value of pIC₅₀. ^{b, c, d} predicted Values for Eq. 8, 7, 6 respectively.

DIPEP	Obs^a	Pred^b R2	Pred^c R1	Pred^d Total	DIPEP	Obs^a	Pred^b R2	Pred^c R1	Pred^d Total
Ala-Ala	3.21	2.93	2.96	3.32	Gly-Thr	2.24	2.74	2.72	2.79
Ala-Gly	2.60	2.39	2.49	2.77	Gly-Trp	4.52	4.25	2.95	3.78
Ala-Phe	3.72	3.80	3.95	3.79	Gly-Tyr	3.68	3.75	2.90	3.35
Ala-Pro	3.64	3.55	3.43	3.46	Gly-Val	2.34	2.57	2.75	2.47
Ala-Trp	5.00	5.23	4.25	4.73	His-Gly	2.20	2.35	2.19	1.72
Ala-Tyr	4.06	4.27	4.03	4.12	His-Leu	2.49	2.81	2.79	2.39
Arg-Ala	3.34	2.78	3.63	3.19	Ile-Gly	2.92	2.41	3.30	2.91
Arg-Phe	3.64	3.30	4.21	4.13	Ile-Phe	3.03	4.09	4.46	4.06
Arg-Pro	3.74	3.27	3.69	3.99	Ile-Pro	3.89	3.76	3.86	3.98
Arg-Trp	4.80	4.94	5.03	5.10	Ile-trp	5.70	5.59	5.16	5.08
Asp-Ala	2.42	2.83	1.83	2.16	Ile-Tyr	5.43	4.58	4.60	4.45
Asp-Gly	1.85	2.36	1.88	1.82	Leu-Ala	3.51	2.93	3.00	2.82
Gln-Gly	2.13	2.36	1.85	2.76	Leu-Gly	2.06	2.39	2.22	2.33
Glu-Ala	2.00	2.81	2.14	2.22	Lys-Ala	3.42	2.85	3.46	2.82
Glu-Gly	2.00	2.36	1.70	1.81	Lys-Gly	2.49	2.37	2.53	2.41
Gly-Ala	2.70	2.69	2.53	2.39	Met-Gly	2.32	2.38	2.84	2.75
Gly-Arg	2.49	2.91	2.95	2.92	Phe-Arg	3.04	3.34	3.09	3.19
Gly-Asp	2.04	1.73	2.73	1.89	Phe-Gly	2.43	2.37	1.58	2.33
Gly-Gln	2.15	2.37	2.83	2.78	Pro-Gly	1.77	2.41	1.73	1.90
Gly-Glu	2.27	1.84	2.82	1.96	Ser-Gly	2.07	2.37	2.23	1.67
Gly-Gly	2.14	2.34	2.37	2.11	Thr-Gly	2.00	2.41	2.31	2.21
Gly-His	2.51	2.35	2.82	2.86	Trp-Gly	2.23	2.34	2.13	2.88
Gly-Ile	2.92	2.57	2.79	2.69	Tyr-Ala	2.70	2.82	3.32	3.07
Gly-Leu	2.60	2.58	2.81	3.05	Tyr-Gly	3.34	2.36	2.94	2.75
Gly-Lys	2.27	2.18	2.90	2.49	Val-Gly	2.96	2.42	3.11	3.14
Gly-Met	2.85	2.83	2.82	2.48	Val-Phe	4.28	4.17	4.17	4.30
Gly-Phe	3.20	3.30	2.87	3.06	Val-Pro	3.38	3.81	3.97	4.11
Gly-Pro	3.35	3.03	2.71	3.05	Val-Trp	5.80	5.53	4.67	5.20
Gly-Ser	2.42	2.23	2.65	2.34	Val-Tyr	4.66	4.66	4.30	4.66

These results can be compared with previous reports about the same series. The result are summarized in the following table (table 3):

Table 3 Result obtained applying other QSAR approaches compared with MARCH-INSIDE approach.

As can be seen from the above table, our methodology gives similar result predicting this property compared with well-established descriptors such as Z, MHDV and MD-WHIM scores (even better if we take into account only eq. 6 and 8 for the comparison with Z and MD-WHIM descriptors). Even in the case of the MHDV that , at first sight, seems to give better statistical parameters our models show higher predictability for external samples in the case of our two best model (6, 8) reflected on their Q² value.

On the other hand, taking a close look to the residual of the predicted cases of each models, we found that cases like Ile-Phe has the largest values of residual and delete residual (see table 2 for residual). These results have been previously reported ⁽³²⁾ and may be due to assaying the cyclized form of the dipeptide, a very common phenomenon on peptides containing lipophilic amino acids like Phe-Ile.

Interpretation of this model becomes quite complicated just by analyzing the variable’s coefficient in the regression equation, but in all the equation can be pointed out the contribution of entropic and probability factors (self-return and absolute as well) except for equation 8 were the entropic factors are absent.

CONCLUSION

Angiotensin Converting Enzyme is the target of many anti hypertensive drugs. In this work several model were found that allow the quantitative and qualitative determination of a possible inhibition activity of several dipeptides applying several statistical techniques. Multiple regression model explain more than 75% of the experimental variance in the data (2 of them explain more than 80%) all these models have adequate statistical parameters and high stability in the cross validation procedure (LOO). All these results confirm the applicability of stochastic-based descriptor to describe biological properties of dipeptides. These methodology is been applying to other and more complicated problems of bioinformatics such as RNA-drug interactions and mutation-based protein stability.

N	R²	SD	Q²	Descriptor	Ref
58	0.769	-	-	Z scores	Collantes, 1995
58	0.878	0.35	0.75	MHDV scores	Shushen, 2001
58	0.705	0.54	-	T scores	Shushen, 2001
58	0.747	0.50	-	MD-WHIM scores	Shushen, 2001
58	0.847	0.42	0.80	^absπ^R2_k ^SRπ^R2_k Θ_k^R2 (Eq. 8)	This paper
58	0.740	0.56	0.62	^absπ^R1_k ^SRπ^R1_k Θ_k^R1(Eq. 7)	This paper
58	0.835	0.45	0.76	^SRπ_k Θ_k (Eq. 6)	This paper

Eq.	N	R²	Q²	SD	F	Variables	ρ	(1-Q²/ R²)
8	58	0.847	0.801	0.419	39.479	7	8.3	9.0
7	58	0.740	0.623	0.563	13.351	10	5.8	16.0
6	58	0.835	0.760	0.450	24.84	10	5.8	5.4