Computers Chem. Vol. 17, No. 3, PP. 283-286, 1993 Printed in Great Britain. All rights reserved
0097-8485/93 $6.00 + 0.00 Copyright 0 1993 Pergamon Press Ltd
MOLECULAR CONNECTIVITY MODEL FOR DETERMINATION OF T, RELAXATION
TIMES OF cc-CARBONS OF AMINO ACIDS AND CYCLIC DIPEPTIDES
LIONELLO POGLIANI Dipartimento di Chimica, Universit$ della Calabria, 87030 Rende (CS), Italy
(Received I December 1992; in revised form IO February 1993)
Abstract-A molecular connectivity model for the calculation of the relaxation times of amino acids and cyclic dipeptides is presented. A power type regression shows the best fit between experimental and calculated data. The simple molecular connectivity index and the simple sum-delta index are the most appropriate indexes in describing the relaxation times of the investigated compounds. The given computational method shows that it is possible to model tumbling rates of solute molecules in a relatively easy and direct way.
INTRODUCTION
Recently a molecular connectivity model has been employed to encode the isoelectric point and the electronic structure of natural amino acids (Pogliani, 1992a, b). In this paper we will develop a compu- tational method to derive the longitudinal relaxation time of the 13Cc1 of amino acids and small peptides, T, (Car), in the context of a QSAR theory (quantitat- ive structure-activity relationships) (Kier & Hall, 1986). Longitudinal relaxation times of the #-carbons of amino acids and cyclic dipeptides are particularly well-suited for a QSAR study, as they are related to the overall molecular motion in solution (Allerhand et al., 1971; Deslauriers et al., 1975). They thus represent a useful dynamical and structural molecular parameter. The finding of a common QSAR based computational method for the T, values of these two classes of compounds could provide us some interest- ing clues about the derivation of a molecular connec- tivity model adequate to predict the relaxation times of a wide range of pharmaceutical and biochemical compounds.
METHOD
The simple one-bond molecular connectivity index of the first-order, IX,
‘X = z(6,&j)-i (1)
and the corresponding one-bond valence molecular connectivity index, IX “, where the valence 6’ replaces 6, are used in this study (Kier & Hall, 1986); the subscripts i andj denote two adjacent atoms forming a bond, other than hydrogen. The following sum- delta index (Pogliani, 1992a) is also used here:
D =ZSi (2)
together with the corresponding D' index, where S’ replaces 6.
Summations run over the nonhydrogen atoms of the molecule. Index 6 is a count of nonhydrogen a-bond electrons contributed by atom i, ai = oi - h, while the valence delta, 6’, is a count of all nonhydro- gen valence electrons contributed by atom i, 6: = dj + pi+ ni, where u, p and n are o, p and Ione-pair electrons respectively and h is the number of bonded hydrogens on atom i (Kier & Hall, 1986). Index 6 can also be formulated as the number of bonded neighbours other than hydrogen.
Even if the arithmetic operations performed in the calculation of the different connectivity indexes do not require any sophisticated mathematics, a GW- BASIC program, available by request, was developed to derive the connectivity indexes of the amino acids. Power curve fitting was done with the aid of the curve fitting program of the STAT PAC program of the Hewlett Packard HP-41CV calculator.
RESULTS
In Table 1 the 6 and 6’ values of the investigated amino acids expressed as rectangular (2 3 n) matrices have been collected: n signifies the number of atoms other than hydrogen in the main backbone. While the zero values in the second row of these matrices mean no connection, c in the matrices of cyclic side-chains denotes adjacency due to ring closure. To describe an amino acid in a peptidic moiety the right-hand side of these matrices (the 2.3 matrix head) should be replaced by
6: 3,3,1 6’: 3,4,6
2,P,O 4,P,O.
CAC 17,3--E 283
284 LIONELLO PCGLIANI
Due to the peptidic bond, p denotes adjacency be- tween the 6 value at its top and the 5 value at its left.
In Table 2 experimental T,(Ccc) values in eight amino acids (Pogliani et al., 1984) and in Seven nongIycine residues of diketopiperazines (Fig. 1) (Deslauriers et al., 1975) together with their corre- sponding connectivity indexes have been collected. The connectivity indexes of the diketopiperazines, c(AA,-AA,), are the sum of the corresponding (‘X/D)i or (‘Xv/D’), (i = 1,2) values resulting from the corrected 6 matrices.
DISCUSSION
Table 2. TI(CLZ) (in s), ‘X, D, ‘.I” and D’ in amino acids and diketopiperazines (DP)
AA/DP T, ‘X D ‘A’” D’
G&J 6.0 2.270 8 I.190 20 Pro 4.3 3.805 16 2.767 28 Ser 3.5 3.181 Ala Tbr :::
2.643 3.553
Ile 2.2 4.091 L.eu 1.9 4.036
:E”-Gly) 0.9 I .4 4.681 5.592 c(Phe-Gly) 0.64 1.254 c(Phe-Val) 0.51 8.575 c(Trp-‘31~) 0.38 8.737 c(Leu-Trp) 0.31 IO.541 c(Tyr-Gly) + DMSO 0.30 9.380 cCTrp_TrpJ 0.18 13.686
I2 1.774 28 IO 1.627 22 14 2.219 30 I6 3.076 28 I6 3.021 28 18 3.366 32 24 4.613 40 32 5.314 54 38 6.224 M) 40 6.308 66 48 7.701 74 40 1.536 69.3 64 9.396 100
In Table 2, the sixth diketopiperazine is considered together with a DMSO solvent molecule, as its anomalous T, value has been explained (Deslauriers et al., 1975) on the basis that this compound is tumbling in solution with one solvent molecule (DMSO) hydrogen bonded to the OH group of Tyr. For a sulphur atom bonded to an oxygen atom, Kier and Hall (1986) have postulated at 6’ = 1.33. Thus DMSO can be denoted by the following 6 matrices and indexes:
6= 1,391
( >
‘X = 1.732; a’= 1,1.33,1
O,l,O D=6 ( > 0, 6 ,O
IX” = 2.088 D’=9.3
In Table 2, connectivity indexes for c(Tyr-Gly) include these DMSO values.
Table I, The (2. n) S and d’ matrices of the inwstiaated amino acids
AA 6 Matrix 6’ Matrix
GUY
ser
Pro
Val
Tbr
Leu
Ile
I-Ys
Phe
TY~
Trp
( ::~:E::::) ( :;:z:Y)
( :::::::::, > ( lE:$)
( E%) ( z:::“,:::: >
(;:5::::::) (::::%:;:)
(E::::~) (E::::: >
( b5:Z:E::, > ( ~zz:~ >
(
2222232331 Y.,.??.,. 3333342346 .,,1>,>,1 c0000c0110 .,,*,o,* > ( cooooco35o >....I,., >
( 2.2,3,2,2,3.2,3,3,1 3.3.4.3.3,4,2.X4.6 c0100c0110 ,,,,,..I, > ( c0500c0350 0.1)?,11 >
3.2,2.2,2,3,2,2,3,2,3.3,~ 4.3.3,3,3,4,4,3.4.2,3,4,6 c0000c00c0110 ,.1,,.(1,.11 ~,0,0,~,~,~,0,0,~,0,3,5,~
Calculated T,, values of Table 3, second column, which have been plotted vs the experimental ones in Fig. 2, have been obtained by the aid of the following correlation:
T, = 31.258 * ‘X -‘.g’9 (3)
r = 0.978, n = 15 and F = 285.1, with r = correlation coefficient, II = number of observations and F = variance ratio; the standard deviation was, for n = 15,s,,= 0.73; and for n = 12 (the best calculated values), s,~ = 0.12.
The following correlation also shows a good qual- ity:
T, = 230.88 . D-L” (4)
r = 0.973, n = 15 and F = 226.8, s,,==O.77 and s,~ = 0.17. The corresponding calculated T,, values arc shown in Table 3, fourth column.
Correlations with valence indexes, ‘Xv and D’ also show a good quality as exemplified by their corre- lation coefficients, respectively 0.961 and 0.976. Equations (3) and (4) are relatively easy to calculate, as ‘X and D parameters are functions of the pure branching number 6.
Consistent deviations (particularly the two outliers Pro and Ala) between calculated and experimental T,, values are due: (a) to consistent contribution to the relaxation pathway of the cc-carbon of Pro from fast internal ring-puckering motions (Pogliani et al., 1984); (b) to errors in the relaxation data which normally are estimated to be ~20%; and (c) to the influence of traces of paramagnetic impurities which shorten the T, values. It should be noticed that the
Fig. I. The diketopiperazine ring of a cyclic dipcptidc: R, and R, are the side-chains of two different amino acids.
T, relaxation times of cc-carbons 285
Table 3. Experimental, T,,,,, and calculated, T,,, relaxation limes (in s) of compounds in Table 2
T le,P T,, [equation (311 T,, [equation (5)) T,, [equation (411
6 6.2 6.2 6.5 4.3 2.2 2.2 2.0 3.5 3.2 3.2 3.2 3.1 4.6 4.6 4.4 2.2 2.5 2.5 2.5 2.2 1.9 :?I 2.0 1.9 2.0 2.0 1.4 1.5 1.5 1.6 0.9 1.0 1.0 I.0 0.64 0.62 0.61 0.60 0.51 0.44 0.44 0.45 0.38 0.43 0.42 0.41 0.31 0.30 0.29 0.30 0.3 0.37 0.36 0.41 0.18 0.18 0.17 0.18
compounds in Table 2 have been studied in different solvents and at different concentrations: AAs in D20 and DPs in deuterated DMSO, while amino acid and dipketopiperazine concentrations ranged from 0.5 to 1 M and from 0.1 to 0.3 M, respectively.
Equation (3) can be approximated to the following equation with integer exponent
T,=32.‘X-* (5)
which is easier to memorize. In Table 3, third column, are collected the corresponding calculated T,, values, from which it follows that deviations from the orig- inal calculated values are minimal. The reciprocal of equation (5), R, x 0.032 IX*, gives an easy relation for the relaxation rates of the a-carbons.
Molecular motion and T1 for small and medium sized solute molecules, assuming overall isotropic motion, are related by (Noggle & Schirmer, 1971)
l/NT,={*~6).kZ.y::.y:,-z,, (6)
6.00 -
I/4, 0.w 9.w 2.00 3M 4.w 5.w 6.W 700
EXPERIMENTAL Ti
Fig. 2. Plot of the experimental vs the calculated relaxation times of eight amino acids and seven diketopiperazines
resulting from equation (3).
where N are the number of interacting protons, (r m6> is the vibrationally averaged sixth power of the ‘H-W internuclear distance, h is Planck’s constant divided by 2x, yc and yH are the gyromagnetic ratios of 13C and ‘H respectively and rem is the effective correlation time for overall molecular reorientation which can be related to the rotational diffusion constant D by
The found computational method for TI offers thus the possibility to encode information on the molecu- lar motions and also, by the aid of the Stokes-Einstein equation (rz = 3kTtd/4nq, where rs is the radius of the solute), on the structure of a solute molecule.
Let us check the validity of equation (4) with some sugars, a phosphoamino acid and a smooth muscle relaxant agent.
(a) Sucrose (D =47) 0.5 and 2 M in H,O solutions: <T,& = 0.64 and 0.16 s respectively (Allerhand et al., 1971), T,ea,c = 0.31 s, that is to say, a value in between.
(T,& value is the mean over the eight T, values of the monoprotonated carbons of the sucrose molecule; the scattering around this value being f 0.15 and -40.01 s for the 0.5 and 2 M solutions respectively.
(b) Sugar ring carbons of AMP (Allerhand et al., 1971) (D = 50) 1 M in H,O: <T,& =0.21 s, T,,, = 0.28 s, a rather good result.
<T,& value is, here, the mean over the four T, values of the monoprotonated carbons of the sugar ring; the scattering is f0.02 s.
(c) Glucose (D = 24) 0.3 M in D,O: (T,,,,) = 1.2 s (Lepri et al., 1987) and T,, = 1.0 s also a rather good result. (d) Cellobiose (D = 48) 0.15 M in D,O: (T,,,) =0.48 s (Lcpri et al., 1987) and T,, = 0.3 s: a satisfying result weighing the low concentration of the solution.
(T,cxp) values in (cc) and (d) are the mean values computed over the entire set of T, values of the monoprotonated carbons of the two molecules in both anomeric forms (10 and 15 carbons respect- ively); the scattering around this mean values being fO.l s for glucose and +0.09 s for cellobiose.
(e) Phosphothreonine (D = 21) 0.5 M in D,O: T,,,(Ca) = 1.3 s (Pogliani et al., 1984), T,,= 1.2s. (f) p-(o-Propyloxybenzamido)benzoate di- ethyl(2-hydroxyethyl)methyl ammonium bro- mide (D = 62) 0.2 M in DMSO: (T,& =0.20 s (Valensin et al., 1984) and T,,=O.l9s.
286 LIONELU) t’OGLlAN1
This ( TnXp> value derives from ((R,,,>)-‘, where tational method, which should be able to derive the (R,,,} is the mean among the three relaxation rates T, values of many chemical compounds by the aid of of the two aliphatic and one aromatic carbon a relation of the kind T, = A . (MCI)“. Molecular (R, = 4.63, 5.92 and 4.93 s-’ respectively), the relax- connectivity indexes (MCI) could then be used, ation values of which account for the molecular through the relaxation values, to derive dynamical rotational reorientation in solution of this highly and structural information on small and medium anisotropic molecule. sized organic molecules in solution.
The small dispersion around (Yf,,,,) values of examples (a)-(f), excepting example (ej (concerning the only a-carbon), lowers the possibility that ob- tained results may be due to chance. In fact, the encoding capability of the given equations, con- sidered as the total number of carbons of which the T, has been encoded, amounts to 4.6.
CONCLUSION
The results obtained tell us that the effort to encode longitudinal relaxation times by the aid of molecular connectivity indexes is worthwhile. The found QSAR model for the relaxation times of specific carbons of amino acids and small peptides can be considered a good starting step in the formulation of a compu-
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