1 Introduction

In a chiral environment, stereoisomers may differ in their pharmacokinetic, pharmacodynamic and toxicological properties because there would be selective absorption, protein binding (ligand-target interaction), transport, metabolism etc. The difference in the bioactivity of enantiomers of various therapeutic agents has been studied and reported [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16] (Table 1). The role of chirality in agrochemicals is documented by Jeschke [17] and in the monograph by Kurihara and Miyamoto [18]. Most of these studies of chiral distinction by biological systems, de novo synthesis of enantiopure drugs and chiral separation of racemates picked up mostly after the 1992 US-FDA regulation [19]. It is well known that the thalidomide controversy [20] is a striking incident that made everyone to realize the importance in administering enantiopure drugs. It may be noted that it took more than three decades for US-FDA to come up with regulation on using stereoisomeric drugs.

Table 1 Importance of chirality in various therapeutic agents
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As there is steady increase in the synthesis and marketing of chiral drugs it is essential to develop structure–activity modelling applicable to stereoisomers. Application of conventional structure activity modelling approach to bioactivity of chiral molecules is not possible because the molecular descriptors and or the physicochemical properties that would be used as predictors are not be able to differentiate the stereoisomers (enantiomers and diastereomers). Hence, it is necessary to develop stereochemical descriptors that could encode the three-dimensional disposition of various atoms or groups present in the chiral molecules. According to the US-FDA recommendations [19], the pharmacokinetic, pharmacological, and toxicological profiles of all the enantiomers and the racemic mixtures must be characterized prior to drug approval.

There are several attempts in the direction of formulating new molecular descriptors for enantiomers and diastereomers [21,22,23,24,25,26,27,28,29,30]. The attempts in developing different types of chiral descriptors and their application in modelling bioactivity of chiral molecules had been reviewed by Natarajan et al. [31, 32]. In an earlier attempt in developing chirality descriptors Natarajan et al. [33] developed relative chirality indices (RCI) and illustrated their application in modelling mosquito repellent activity of diastereomers of SS220 and picaridin. This approach is similar to the Physicochemical Atomic Stereo-descriptors (PAS) developed by Zhang and Aires-de-Sousa [27] where PAS is based on twenty-one physicochemical properties of the ligands attached to a chiral center. There are two major differences between the two approaches:

  1. 1.

    Calculation of RCI used a topological approach and proposed possible extensions of using shape or size of the substituents and electro topological (E-state) indices for the substituents attached to the chiral center. On the other hand, calculation of PASs exclusively uses physicochemical properties.

  2. 2.

    RCI follows R/S configuration based on Cahn Ingold and Prelog priority rules [34, 35] (CIP-R/S) and PASs used CIP-like R/S assignment of priority for the ligands.

2 Modified RCI

Of the various chirality measures the relative chirality indices developed by Natarajan et al., [33] is conceptually simple and easy to compute for a large set of molecules in a batch process. Moreover, a series of indices based on different concepts of molecular topology and chemistry namely, simple connectivity [36], valence connectivity [37, 38], Balaban J index [39], bond order connectivity, information contents [40], over all connectivity [41] or shape [42] could be computed. When the activities of a diverse set of molecules are to be modelled by QSSAR approach a diverse set of descriptors that encode different structural aspects the molecules are necessary. In this paper we propose to modify the relative chirality indices proposed by us in 2007 [33]. The two assumptions or concepts that are modified to calculate the new chirality indices are:

  1. 1.

    The previous approach used diminishing importance for atoms in a group attached to the chiral carbon.

The groups/atoms a, b, c, and d are assigned valence delta values of atoms (δv) according to the method of Hall and Kier [35, 36]. When the group has more than one atom, δv for the group a, b, or c is calculated considering the relative proximities of the atoms to the chiral center and decreasing importance with increasing topological distance (through bond) was assigned while calculating the contribution of atoms other than hydrogen in a group. The valence group delta value for any group (δiv) attached to a chiral carbon is calculated according to Eq. 1.

$$\delta_{i}^{v} = \delta_{n1}^{v} + \left( {\frac{{\delta_{n2}^{v} }}{2}} \right) + \left( {\frac{{\delta_{n3}^{v} }}{4}} \right) + \left( {\frac{{\delta_{n4}^{v} }}{8}} \right) + \ldots$$
(1)

where n1 is the atom attached directly to the chiral center (nearest neighbor), n2 is separated by one atom, n3 by two atoms, etc.

The approach of applying decreasing importance to the atoms in a group as one moves away from the chiral center is modified in the current approach instead equal importance is given to all the vertices (atoms) because the atoms away from the chiral carbon are more flexible is expected to play a role in binding to an active site in a protein.

  1. 2.

    While calculating the weights of each of the groups attached to a chiral carbon, we included the chiral carbon as part of the group so that the molecular connectivity remains unaltered, and this also facilitate us to calculate a large pool of chirality indices for a molecule. The modification facilitated in calculating large number of descriptors for the groups attached to the chiral center and thus a large set of new chirality measures could be computed. Moreover, software available to compute various types of topology-based molecular descriptors, based shape indices, overall connectivity indices and triplet indices could be calculated for each of the four groups bound to the chiral carbon and thus, a pool of chirality descriptors based on different molecular characteristics such as connectivity, nature of chemical bonds, shape indices are available.

Calculations of RCI based on the earlier method [33] and the modified approach suggested in this paper are illustrated using tyrosine a model compound in Example-1 of illustrations of Sect. 2.2.

2.1 Calculation of relative chirality indices for the R and S isomers

The steps followed in calculating the relative chirality indices (RCI) for the R and the S isomers denoted as RRCI and SRCI, respectively are given below:

  1. 1.

    The four groups attached to the chiral carbon are assigned priority following the Cahn, Ingold and Prelog (CIP) system as a, b, c, and d, where a is the highest priority and d is the lowest priority group.

  2. 2.

    SMILES notation [43, 44] is written for each of the four groups and in order to maintain the connectivity of the atom (vertex) connected to the chiral carbon chiral carbon is also included to the groups.

  3. 3.

    Various molecular descriptors such as connectivity indices [36], valence connectivity indices [37, 38], Balaban J index [39], information theoretic indices [40], overall connectivity indices [41] and shape indices [42] are calculated for each of the four substituents (a, b, c, and d) using the SMILES notations as the input. The computer program POLLY [45], TRIPLET (the computer program TRIPLET computes the triplet indices proposed by Balaban et al. [46, 47]) and INDCAL [48] were used to compute the molecular descriptors for each of the four substituents.

  4. 4.

    The descriptors thus calculated for each substituent a, b, c and d are taken as the weight of that substituent namely, δa, δb, δc, and δd.

  5. 5.

    Relative Chirality Indices (RCI) for are then calculated for the R and S isomers using the following formulae:

    $$\begin{aligned} {}_{{}}^{R} RCI & = \delta_{a} + \left\{ {\delta_{a} + \left( {\delta_{a} \times \delta_{b} } \right)} \right\} + \left\{ {\delta_{a} + \left( {\delta_{a} \times \delta_{b} } \right) + \left( {\delta_{a} \times \delta_{b} \times \delta_{c} } \right)} \right\} + \\ & \quad \left( {\delta_{a} \times \delta_{b} \times \delta_{c} \times \delta_{d} } \right) \\ \end{aligned}$$
    (2)
    $$\begin{aligned} {}_{{}}^{S} RCI & = \delta_{a} + \left\{ {\delta_{a} + \left( {\delta_{a} \times \delta_{c} } \right)} \right\} + \left\{ {\delta_{a} + \left( {\delta_{a} \times \delta_{c} } \right) + \left( {\delta_{a} \times \delta_{b} \times \delta_{c} } \right)} \right\} + \\ & \quad \left( {\delta_{a} \times \delta_{b} \times \delta_{c} \times \delta_{d} } \right) \\ \end{aligned}$$
    (3)

where δa, δb, δc and δd are the weights assigned to the four substituents attached the chiral center and the priority order (assigned as per CIP rules) of the four substituents are a > b > c > d

Information regarding the least priority group (d) was also included in the new index by the fourth term. It becomes zero when the fourth substituent is a hydrogen atom otherwise, it contributes to the RCI for the chiral center. From the above equations (Eq. 2 and 3) the difference between the values of RRCI and SRCI is:

$${}_{{}}^{R} RCI - {}_{{}}^{S} RCI = 2\delta_{a} (\delta_{b} - \delta_{c} )$$
(4)

Several other approaches on chirality descriptors did not consider compounds with more than one chiral center, which was not the case in the calculation of RCI. A root means square method to obtain RCI for molecules containing more than one chiral center i.e., for diastereomers was adopted.

  1. 6.

    If more than one chiral center is present then RRCI and SRCI are calculated for each chiral center. RCI for the diastereomers are calculated by taking root-mean square of the RCI for all centers.

    $$RCI = \sqrt{\frac{1}{N}} \mathop \sum \limits_{i = 1}^{N} RCI_{i}^{2}$$
    (5)

For instance, if a molecule has two chiral centers and they are the 2nd and the 3rd carbon atoms of the molecule then the molecule will have four diastereomers namely, (2R, 3R), (2S, 3R), (2R, 3S) and (2S, 3S). RCI for each of these diastereomers 2R3RRCI, 2S3RRCI, 2R3SRCI and 2S3SRCI may be calculated as below

$${}_{{}}^{2R3R} RCI^{{}} \sqrt {\frac{{{}_{{}}^{{}} \left( {{}_{{}}^{2R} RCI^{2} } \right) + \left( {{}_{{}}^{3R} RCI^{2} } \right)}}{2}}$$
(6)
$${}_{{}}^{2S3R} RCI\sqrt {\frac{{{}_{{}}^{{}} \left( {{}_{{}}^{2S} RCI^{2} } \right) + \left( {{}_{{}}^{3R} RCI^{2} } \right)}}{2}}$$
(7)

Similarly, RCI is calculated for the other two stereoisomers.

  1. 7.

    Based on the type of molecular descriptor assigned to the substituents the RCI are denoted as RRCIV0, RRCIB1, SRCISUM-R, RRCIOPM etc., where the subscript indicates the type of molecular descriptor used in assigning the weights (δ) to the four substituents attached to the chiral carbon.

  2. 8.

    To calculate the new chirality descriptors, δa, δb, δc and δd may calculated using any of the available computer programs such as POLLY and TRIPLET with SMILES code as input and then the delta values (δa, δb, δc and δd) be converted to RRCI and SRCI using MS. EXCEL or any other spreadsheet. A single software incorporating both the stages of calculation can be developed to compute the new chirality descriptorsfor a large set of molecules in a batch mode.

2.2 Illustrations for the calculation of relative chirality indices

Tyrosine is taken as the example with one chiral center and the calculation of RRCI and SRCI are demonstrated according the previous convention [33] and the modified approach presented here (Fig. 1).

Fig. 1
figure 1

Tyrosine with substituents a, b, c, and d have the order of priority a > b > c > d according to CIP rule. (Degrees of each vertex as per valence connectivity are shown according to the previous convention [27] in A in and as per the modification suggested in this paper in B. The lowest priority group d (H) is not shown.)

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Example 1 According to earlier convention [33] which uses diminishing importance of atoms as one moves away from a chiral center, the δ values for the four substituents a, b, c, and d are:

$$\delta^{{\text{v}}}_{{\text{a}}} = { 3}$$
$$\delta v_{b} = 4 + \frac{5}{2} + \frac{6}{2} = { 9}.{5}$$
$$\delta v_{c} = 2 + \frac{4}{2} + \left( {2 \times \frac{4}{4}} \right) + \left( {2 \times \frac{4}{8}} \right) + \frac{4}{16} + \frac{5}{32} = { 7}.{4}0{62}$$
$$\delta^{{\text{v}}}_{{\text{d}}} = \, 0$$

RRCIv and SRCIv for tyrosine could be obtained by substituting the above δ values for the four substituents a, b, c, and d in Eq. 2 and Eq. 3, respectively.

Values of RRCIv and SRCIv calculated as per the old convention suggested by the authors are:

$$^{R} RCI_{v} = { 28}0.0{767};\,\quad{\text{ and}}^{S} RCI_{v} = { 267}.{5139}$$

In the revised convention proposed in this paper, δ values for the groups a, b, c and d could be calculated for different orders and types of connectivity indices, Balaban J index, triplet indices etc. The weights of the four groups (δ values) may therefore be based on any formulation for computing molecular descriptors. Incorporating these δa, δb, δc, and δd in the equations Eqs. 2 and 3 a pool of RRCI and SRCI could be calculated for a pair of enantiomers and each class of RCI thus computed encodes a particular feature of the groups attached to the chiral center. Calculation of four types of RRCI and SRCI for tyrosine enantiomers based on zero-order valence connectivity index (0χv), first order Randić connectivity index (1χs), Balaban J index (J) and first-order kappa index (1κ) are illustrated.

  1. (a).

    For tyrosine calculation of weights of a, b, c and d (δa, δb, δc, and δd) based on zero-order valence connectivity index (0χv) are:

    $$\delta^{{{\text{v}}0}}_{{\text{a}}} \, = \,{1}.{5774}; \, \delta^{{{\text{v}}0}}_{{\text{b}}} \, = \,{2}.{3555}; \, \delta^{{{\text{v}}0}}_{{\text{c}}} \, = \,{5}.{4637}.$$

Substituting these values in Eqs. 2 and 3, RRCIv0 and SRCIv0 could be calculated and the values are:

$$^{R} RCI_{v0} = { 32}.{463};\,\,^{S} RCI_{v0} = { 43}.{829}$$
  1. (b)

    The above calculation could be extended to calculate a diverse set of RRCI and.SRCI for a pair of enantiomers. Three more such examples are illustrated in Table 2

Table 2 Substituents and their weights (δi) calculated on different principles of connectivity
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Using first order simple connectivity index 1χ (S1)


RRCIS1 = 13.956 ; SRCIS1 = 19.144

Using Balaban index J


RRCIJ = 18.204; SRCIJ = 17.822

Using shape index 1κ


RRCI1κ = 13.111; SRCI1κ = 17.222

Example 2 In this example 2,3-dibromopentane (Fig. 2) with two chiral centers is taken as the example (Table 3).

Fig. 2
figure 2

Applying sequence rule to 2,3-dibromopentane for the two chiral centers

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Table 3 Substituents attached to each of the chiral centers and their weights (δi) calculated based on zero order valence connectivity and bond order connectivity
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First calculate RCI for each position.

For position 1; RCI are:


RRCIvo = 71.016


SRCIvo = =51.794

For position 2; RCI are:


RRCIvo = 72.144


SRCIvo = 61.297

RCI for the diastereomers are calculated using root mean square (See Eqs. 5 and 6)

$$\begin{aligned} {\text{RCI for 2}}R,{3}R & = \sqrt {\frac{{{}_{{}}^{{}} \left( {{}_{{}}^{2R} RCI^{2} } \right) + \left( {{}_{{}}^{3R} RCI^{2} } \right)}}{2}} \\ & = \sqrt[{}]{{\frac{{(71.016_{{}}^{2} + 72.144_{{}}^{2} }}{2}}} = \, 71.582 \\ \end{aligned}$$

By similar calculation one can calculate RCI for other diastereomers

$$\begin{gathered} RCI \, for\left( {2S,3R} \right) = {62}.{799} \hfill \\ RCI \, for\left( {2R,3S} \right) = {66}.{335} \hfill \\ RCI \, for\left( {2S,3S} \right) = {56}.{745} \hfill \\ \end{gathered}$$

2.3 Calculation of RCI for diastereomers with one, two and three chiral centers

A large number (≈ 150) of chirality indices were calculated for the stereoisomers of different chiral compounds. In order to check the differentiation of the diastereomers by the chirality indices, they were computed for three classes of chiral chemicals namely,

  1. 1.

    Compounds with one chiral center: naturally occurring α-amino acids were taken as examples. Among the 21 α-amino acids is optically inactive, threonine and isoleucine contain two chiral carbons. Hence, these three were dropped and the chirality indices are calculated for the other 18 α-amino acids. In addition to these 18 compounds, 2-butanol, lactic acid, 2-chlorobutane, mandelic acid and glyceraldehyde were also used for calculation.

  2. 2.

    Compounds with two chiral centers (4 diastereomers) threonine, isoleucine, 2,3-dibromopentane, 2-bromo-3-chloropentane, 2,3-dichloropentane, tartaric acid and 2-bromo-3-methylpentanoic acid were considered

  3. 3.

    Compound with three chiral centers (8 diastereomers) a pentose sugar was taken as the typical example.

2.3.1 RCI for compounds with one chiral center

Relative chirality indices for the R and S isomers (enantiomers) for the 23 compounds containing one chiral carbon are presented in Table 1A, 1b and 1 of Supporting Information-1 (SI-1). The chirality indices are able to differentiate the enantiomers because they differ in the numerical values of the RRCI and SRCI isomers. In case of simple connectivity and graph theoretic based indices, where the nature of the atoms is not considered, same RCI values are obtained for compounds that have identical molecular graphs. For example, asparagine, aspartic acid and leucine have the same set of RRCI and SRCI values (degenerate values) calculated from graph theoretic and simple connectivity-based indices. These three compounds have the same hydrogen-suppressed molecular graph (Fig. 3) and this explains the reason for degenerate values for these compounds. The same is true for serine and cystine. However, RCI calculated from other substituent weights based on valence connectivity are different for these compounds.

Fig. 3
figure 3

Molecular graph of asparagine, aspartic acid and leucine

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2.3.2 RCI for compounds with two chiral centers

It is well known that the number of stereoisomers possible for a compound with n chiral centers is 2n. Hence, compounds with two chiral centers will have 22 = 4 stereoisomers. Two pairs of enantiomers or 4 diastereomers namely, (S, S), (S, R), (R, S) and (S, S). Structures of the eight examples used in the study along with their stereoisomers are shown in Figure RCI values calculated for the four stereoisomers of each of the eight compounds are listed in Tables 2A to 2C of Supporting Information-1 (SI-1).

In the case of tartaric acid and 2,3-dichlorobutane each of the two chiral carbons are attached to similar set of substituents. Hence, the two stereoisomers (2R, 3S) and (2S, 3R) have a plane of symmetry and they will be optically inactive forms. They are considered to be reduced one inactive form called the meso isomers or meso form. RRCI and SRCI for the meso isomers have the same values. For example, in the case of tartaric acid and 2,3-dichloropentane the meso isomers with configuration 2R3S and 2S3R have the same RCI values irrespective of the approach used to compute them (please refer to Table 2A in Supporting Information-1 (SI-1).

2.3.3 Molecule with three chiral centers

Pentose sugar with three chiral centers was considered. The compound has eight stereoisomers (23 = 8). The eight stereoisomers have the configurations (2S,3S,4S), (2S,3S,4R), (2S,3R,4S) (2S,3R,4R), (2R,3R,4R), (2R,3R,4S), (2R,3S,4R), (2R,3S,4S). RCI were calculated for the eight stereoisomers (diastereomers) using the root mean square scheme discussed in the previous section. The values of RCI for the eight diastereomers are given in Table 3A and B of Supporting Information-1 (SI-1).

3 Chiral descriptors based on cartesian coordinates

Diudea and Ursu developed a descriptor based on the internal cartesian coordinates of the molecule (Å). The chirality of an ordered quadruple of atoms numbered 1, 2, 3, 4 is measured in terms of their (x, y, z) Cartesian coordinates, adopting some geometrical constraints, by the sign of the following determinant [49, 50].

$$C_{1234} = sign \times \left[ A \right]$$
(7)

where

$$A = \left| {\begin{array}{*{20}c} {x_{1} } & {x_{2} } & {x_{3} } \\ {y_{1} } & {y_{2} } & {y_{3} } \\ {z_{1} } & {z_{2} } & {z_{3} } \\ \end{array} } \right|$$

The value of the determinant is discriminant for each molecule. The sign of the resulting determinant should be sensitive to chirality. Furthermore, most bioactive molecules are chiral. Cluj centrality descriptor [49, 51] allows the finding of the graph center (e.g., the vertex having the largest Ci value) and provides an ordering of graph vertices according to their centrality best describes bioactivity (it focuses on heteroatoms where the active chemical center of the molecule is. Hence, Cluj centrality index was added when bioactivity of organic chiral compounds are considered.

Furthermore, a point symmetrical to another point is a point. Point (A) symmetrical to point (O) is point (A'), located along the extension of the line AO so that AO = OA'. The symmetric of point A concerning line (d) is the point (A') located on the extension of the perpendicular from (A) to (d) such that AO = OA'. The symmetry of a surface concerning an axial axis is another surface obtained by the symmetrical union of the given surface points. The symmetry of the surface ABC concerning axis (d) is the surface A'B'C' obtained by joining the symmetries of the points A'B'C,' symmetrical to the points A, B, and C. The critical property of the symmetry of the surface is the equality of the two surfaces: given and its symmetry, Area ABC = Area A'B'C'. Analytical geometry provides the coordinates of the points of the given and symmetrical surfaces. The distances between different points of the surfaces can be determined. Thus, the areas of the two surfaces can be established by observing their equality. It is observed that a surface is represented by a number that is the area of that surface. The theory of algebraic structures developed in the contemporary period has become essential for topological and differential structures. An algebraic structure is a set on which some operations have been defined: addition, multiplication, intersection, and reunion. In topology, the connection of a structure is essential. A function is bijective if its derivative is different from zero. This is valid only in the definition domain of the function and if it is convex. A matrix represents a linear application. It is valid only for assembly; it multiplies the scope of the product. Of all the square matrices, having the number of rows (m) equal to that of the columns (n), m = n has the property of symmetry and complementarity. If a matrix (A) is equal to the transposed matrix (A−): A = A−, then the matrix (A) is symmetric. The complement of a matrix (A) is denoted by Ac or (A). A multimer (A') is the complement of many sets (A) if the elements of the set (A') are not contained in set A. A determinant of the same order can be associated with the matrix. An additional minor Mij of an element Aij is the determinant obtained by deleting the line (i) and the column (J). The algebraic complement of an element (Aij) is:

$$A_{ij} = \left( { - 1} \right)^{i + j} M_{ij}$$
(8)

The determinant of matrix A is the sum of the products of all the elements of a row or column, and their algebraic complements.

$$\det A \Delta = \sum \sum a_{ij} M_{ij}$$
(9)
$$= \sum \sum a_{ij} ( - 1)^{i + j} M_{ij}$$
(10)

Thus, a chirality index can be computed using a square matrix composed of the atom's internal cartesian coordinates computed for the heteroatom and the neighbour atoms. The chiral molecule (Fig. 4) with one chiral center is considered as an example.

Fig. 4
figure 4

Example of a chiral molecule with heteroatoms (chiral carbon is marked)

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The descriptors obtained from the Cartesian coordinates for the R- and S-isomers are 16.900 and 237.09, respectively.

4 Application of the new chirality descriptors is QSAR

Waterbeemd and others [52] reported the QSAR studies of 7 pairs of enantiomers of 3-(3-hydroxyphenyl)piperidines (3HPP) for dopamine D2 and σ receptor affinities. Calculation of the new chirality descriptors for the 7 pairs of enantiomers of a congeneric set of molecules was carried out. Chirality descriptors calculated from the Cartesian for the 7 pairs of enantiomers are given in Table 4 along with the dopamine (DA) D2 and opiate σ receptor affinities.

Table 4 Structures of 3HPPs, the new chiral descriptors and the Dopamine (DA) D2 and opiate σ receptor affinities
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Fig. 5
figure 5

Cartesian coordinates of the R- and S-isomers of the example shown in Fig. 4

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Various molecular descriptors (top structural, topochemical, triplet and shape indices 1κ and 2κ were calculated as weights (δ) for the substituents (groups) attached to the chiral carbon. The list of descriptors and their brief definitions are given in Supplementary Information (SI-2). These weights were then used to compute RRCI and SRCI using Eq. 1 and 2 for all the chiral molecules in both the data sets. Illustrations for calculation of RCI is given in Supplementary Information (SI-3) (Fig. 5).

More than 200 molecular indices were calculated using SMILES for substituents as input and if two substituents have zero value for a molecule then that descriptor was deleted from the list before computing RCIs. Type of descriptors that could be calculated using POLLY, INDCAL and TRPLET are listed in Supplementary Information-2 (SI-2). In addition to the above topological indices, shape indices, proposed by Hall and Kier [42], otherwise called as kappa indices (1κ, 2κ) of the first order and the second order were also calculated for the groups. All statistical analysis were carried out using SPSS 25.0 [53].

5 Results of bioactivity modelling using RCI

The chirality indices calculated based on various molecular descriptors or the seven pairs enantiomers of 3-(3-hyroxyphenyl)piperidines. Some of the higher order connectivity-based descriptors have zero value and therefore, the final set of descriptors contains lower number of chirality descriptors for each enantiomer (R or S) than the original number of descriptors computed. The large pool of chirality descriptors computed for the data sets of 3HPPs (14 compounds) are presented as Excel worksheets in the Supplementary Information-4 (SI-4). The final data set for 3HPPs consists of 146 chirality descriptors comprising of 100 of them computed from TRIPLET indices, 44 from connectivity indices and two based on shape indices of the four substituents attached to the chiral carbon(s) in a molecule. Intercorrelation among the RCI indices calculated for the 7 pairs of enantiomers is presented in separate sheet in the Excel workbook (SI-4). The data set consists of two biological activities data for the 14 compounds.

5.1 Correlation of chirality indices with σ receptor affinities of 3HPP

Correlation of the σ receptor affinities of 3HPPs with each of the final set of chirality descriptors computed from connectivity-based indices of the substituents are presented in the Supplementary Information (SI-4) in the decreasing order of correlation coefficient, r values. Several of the chirality indices have good correlation with R > 0.8. The top ten chirality descriptors with highest correlation coefficient values are given in Table 5. The least correlated descriptors are also given in Table 5 to understand the spread of correlation among the RCIs with the σ receptor affinities of 3HPPs.

Table 5 Correlation of chirality indices calculated from group connectivity indices and TRIPLET indices with σ, receptor affinities of 3HPPs
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5.2 Correlation of chirality indices with D2 affinities

Correlation of chirality indices with dopamine receptors are presented in Table 6. The descriptors are arranged in the decreasing order of r values. Unlike the σ receptor affinities the D2 affinities are not highly correlated with the chirality indices. The highest and the least correlated chirality descriptors (RCI) with D2 affinities of 3HPPs are presented in Table 6.

Table 6 Correlation of chirality indices calculated from group connectivity indices and TRIPLET indices with D2 receptor affinities of 3HPPs
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5.3 Modeling biological activity using cartesian coordinates based chirality indices

The values of the new chirality descriptor based on cartesian coordinates is given for the 7 pairs of enantiomers in Table 4 along with the two biological activities namely, the affinities for dopamine D2 and σ receptors. For σ receptor affinity the new chiral descriptors correlated reasonably well (r = − 0.7780). However, for the D2 receptor affinities the correlation is very poor (r = 0.1947).

6 Conclusions

Owing to the modification introduced by us in the calculation of RCI a large pool of chirality descriptors for each compound could be computed using available computer programs. As diversity handles diversity these descriptors encode different attributes of the substituents attached to the chiral carbon. RCIs calculated can distinguish enantiomer and diastereomer. In case of molecules with two chiral centers, all the four stereoisomers are differentiated by the chirality indices. Singularity of meso form in accounted for by the chirality indices developed. The pair of enantiomers with a plane symmetry have the same set of RRCI and SRCI for all the different types indices. RRCI, SRCI were calculated using shape attributes. Protein–ligand binding is very much driven by the shape of the substituents which plays a very important role in fitting to the cavity of a protein and therefore, RCI computed using shape attributes are expected to play a significant role in QSSAR modelling. The application of the RCI is tried for a data set that contains 7 pairs of enantiomers. Several RCIs are correlated reasonably well (r ≥ 0.8) with the bioactivates of the chiral molecules. Hence, the new chirality descriptors (RCI) would be an extremely useful computational method to precisely direct the asymmetric synthesis of new chiral pharmaceuticals and agrochemicals. The approach may be extended using other types of molecular parameters and topological indices such as electrotopological state indices introduced by Kier and Hall [54].

The chirality descriptors calculated using the cartesian coordinates of the atoms in a chiral molecule are able to distinguish the enantiomers. When the new descriptors were used to model the biological activities of the 7 pairs of enantiomers of 3HPPs the correlation with the σ receptor affinities was − 0.778 while the correlation with the D2 receptor affinities was poor (r = 0.1647). Here we are trying to model two biological activities of a set of molecules with one index. The two biological activities namely, σ receptor affinities and D2 receptor affinities are not mutually correlated (r = 0.1947). This is expected when one tries to model two mutually uncorrelated biological activities with a single set of indices.

The RCI approach yields a diverse pool of descriptors based weighting scheme used that is, on different aspects such as simple atom connectivity, valence connectivity, information content, shape attributes, electro-topological state etc. In most of the other approaches only a pair of indices are generated for the enantiomers. If a set of chiral molecules has multiple observable properties which are not mutually correlated with one another, one cannot expect only one index to correlate to the various properties and this is illustrated in the case of chirality indices based on Cartesian coordinates. As different inter- and intra-molecular interactions lead to different observable properties hence, collection of indices such as RCI has a better chance of building high quality QSAR for chiral molecules.