From quantum similarity measures to quantum analogy functors: tools for QShAR, quantitative shape-activity relations

Abstract

An analogy can be regarded as a relation between two families of similarities, and the mathematical tool of functor has been proposed for certain chemical applications where not just one type of similarity, but entire families of similarities, hence, analogies, are playing important roles. Some of the relevant background important in molecular applications, especially, in studies of chemical and biochemical activity problems are discussed, where several factors of various levels of similarities are determining the overall chemical processes. Such analogies are highly relevant in the interpretation of complex biochemical actions, as well as in pharmaceutical drug design. The Quantum Similarity Measures and alternative measures based on quantum chemistry are combined into a Quantum Analogy Functor model, with special emphasis on the 3D molecular shapes represented by the electron density clouds, and the associated Quantitative Shape-Activity Relations, QShAR. Some aspects of the Quantum Analogy Functor approach, with focus on Molecular Shapes and QShAR are discussed in this contribution.

Appendices

Appendix

1.1 Brief introduction to functors representing analogies

Here only three brief mathematical examples of the simplest type of Functor is given, for more background the reader is advised to consult the Category Theory description of Functors in Ref. [64].

A functor can be regarded as a transformation operating on two levels:

• (1) mapping (interrelating) one family of sets to another family of sets,

• (2) transforming the mappings (interrelations, in our case, similarity comparisons) among sets in the first family of sets to the mappings among the second family of sets.

The actual, individual mappings interrelating the sets within the first family can be regarded as expressions of similarities, and the actual, individual mappings interrelating the sets within the second family can also be regarded as expressions of similarities, consequently, the complete Functor then can be regarded as an iterated similarity, or a higher level of similarity, expressing, in one sense, the similarity of similarities that can be regarded as an analogy.

As an illustration, consider two families of sets, A and B, each family containing three sets, U, V, W, and X, Y, Z, respectively:

$$A \, = \, \left\{ {U, \, V, \, W} \right\}$$

(1)

$$B \, = \, \left\{ {X, \, Y, \, Z} \right\}$$

(2)

Among the sets U, V, and W of the first family A of sets, consider the following series of mappings, actually, several such mappings between each pair of sets, which mappings may actually represent similarity comparisons:

$$f_{AUVi} \left( {i = 1, \ldots \, n_{UV} } \right),$$

(3)

$$f_{AUWi} \left( {i = 1, \ldots \, n_{UW} } \right),$$

(4)

$$f_{AVWi} \left( {i = 1, \ldots \, n_{VW} } \right),$$

(5)

In a similar (actually, analogous) way, consider among the sets X, Y, and Z of the second family B of sets, the following series of mappings, which also may represent actual similarity comparisons:

$$f_{BXYi} \left( {i = 1, \ldots \, n_{XY} } \right),$$

(6)

$$f_{BXZi} \left( {i = 1, \ldots \, n_{XZ} } \right),$$

(7)

$$f_{BYZi} \left( {i = 1, \ldots \, n_{YZ} } \right),$$

(8)

We may denote the collection of all mappings within family A of sets by G_A, and the collection of all mappings within family B of sets by G_B,

$$G_{A} = \, \{ f_{AUVi} \left( {i = 1, \ldots \, n_{UV} } \right), f_{AUWi} \left( {i = 1, \ldots \, n_{UW} } \right), f_{AVWi} \left( {i = 1, \ldots \, n_{VW} } \right)\} ,$$

(9)

$$G_{B} = \left\{ { f_{BXYi} \left( {i = 1, \ldots \, n_{XY} } \right), f_{BXZi} \left( {i = 1, \ldots \, n_{XZ} } \right), f_{BYZi} \left( {i = 1, \ldots \, n_{YZ} } \right)} \right\}.$$

(10)

Then, the Functor F can be considered as the mapping that acts, on the one hand, between the two families of sets A and B, and also acts, on the other hand between the two families of mappings G_A and G_B:

$${\mathbf{F}}: \{ \, A\left\{ {U, \, V, \, W} \right\}, G_{A} \} \to \{ \, B\left\{ {X, \, Y, \, Z} \right\}, G_{B} \}$$

(11)

Example 1: Functorial analogies of electron densities of molecular functional groups

As a simple example for possible applications of the above Functor model to a Quantum Analogy problem, with the QShAR approach in mind, consider several quantum chemistry representations of functional groups.

In this context, a quantum chemical functional group has been defined as a quasi-autonomous molecular fragment, using the criterion that there exists some closed molecular isodensity surface that contains all the nuclei of the functional group but none of the other nuclei of the molecule, a condition similar to those of two, weakly interacting, but still “autonomous” molecules near to each other, before a reaction between them would start going in earnest. For background on quantum chemical functional groups see [102], and for applications [28, 103,104,105,106,107,108,109].

Take, for example, three amino acids, Mol1, Mol2, Mol3, and the most relevant functional groups, -NH₂ and -COOH from each (these groups, without known exceptions in stable conformations, do satisfy the above mentioned “quantum chemical functional group” condition). Then, assign the various computed or otherwise obtained sets of local shape descriptors of these functional groups in the following way:

$$U \, = \, \left\{ { {\text{Set}}\;{\text{of}}\;{\text{shape}}\;{\text{descriptors}}\;{\text{of}} - {\text{NH}}_{2} \;{\text{in}}\;{\text{Mol}}\;1} \right\},$$

(12)

$$V \, = \, \left\{ { {\text{Set}}\;{\text{of}}\;{\text{shape}}\;{\text{descriptors}}\;{\text{of}} - {\text{NH}}_{2} \;{\text{ in}}\;{\text{Mol}}\;{2 }} \right\},$$

(13)

$$W \, = \, \left\{ { {\text{Set}}\;{\text{of}}\;{\text{shape}}\;{\text{descriptors}}\;{\text{of}} - {\text{NH}}_{2} \;{\text{ in}}\;{\text{Mol3}}} \right\},$$

(14)

$$X = \left\{ { {\text{Set}}\;{\text{of}}\;{\text{shape}}\;{\text{descriptors}}\;{\text{of}} - {\text{COOH}}\;{\text{in}}\;{\text{Mol}}1} \right\},$$

(15)

$$Y = \, \left\{ { {\text{Set}}\;{\text{of}}\;{\text{shape}}\;{\text{descriptors}}\;{\text{of}} - {\text{COOH}}\;{\text{in}}\;{\text{Mol}}2} \right\},$$

(16)

$$Z = \left\{ { {\text{Set}}\;{\text{of}}\;{\text{shape}}\;{\text{descriptors}}\;{\text{of}} - {\text{COOH }}\;{\text{in}}\;{\text{Mol}}3} \right\}.$$

(17)

The actual f mappings of Eqs. (3)—(8) are taken as functions expressing various local similarity measures, such as Quantum Similarity Measures or various alternative similarity measures.

With these choices, a Functor F satisfying the conditions of Eqs. (1–11) does represent a description of a higher level of similarity among the relations for the –NH₂ functional groups and the –COOH functional groups for the selected three amino acids, that is, an analogy between trends of shape variations for the two most important functional groups of the selected amino acids.

Example 2: Functorial analogies of protein structures

As motivated in part by the detailed studies of Balasubramanian and co-workers on systems of proteins, the Functor model appears suitable to handle the multitude of similarities and analogies in families of proteins and their reaction partners.

In this protein context too, a functor can be regarded as a transformation operating on two levels:

• (a) Mapping (interrelating) one family of sets of proteins and their reaction partners to another family of such sets of proteins and their reaction partners,

• (b) Transforming the similarity comparisons between the proteins themselves, on the one hand, and also the similarity comparisons of their reaction partners, among sets in the first family of sets to similarity comparisons among the second family of sets.

The actual, individual comparisons interrelating the sets within the first family can be regarded as expressions of similarities, and the actual, individual transformations interrelating the sets within the second family can also be regarded as expressions of similarities, consequently, the complete Functor based on proteins, their structures, their similarities, and the same type of properties for their reaction partners, then can be regarded as an iterated similarity, or a higher level of similarity, expressing, in one sense, the similarity of similarities, that can be regarded as an analogy.

As an illustration, consider two families of sets of proteins and their reaction partners, P_A and P_B, each family containing sets of proteins and sets of their reaction partners, respectively:

$$P_{A} = \, \{ P_{AU,} P_{AV,} P_{AZ,} R_{AU,} R_{AV,} R_{AZ} \}$$

(18)

$$P_{B} = \, \{ P_{BU,} P_{BV,} P_{BZ,} R_{BU,} R_{BV,} R_{BZ} \}$$

(19)

Among the sets P_AU, P_AV, P_AZ, R_AU, R_AV, R_AZ of the first family of proteins or reaction partners of sets, consider the following series of mappings, actually, several such mappings between each pair of sets, which mappings may actually represent similarity comparisons:

$$fP_{AU,} P_{AV \, i} \left( {i = 1, \ldots \, nP_{AU,} P_{AV} } \right) \, ,$$

(20)

$$fP_{AU,} P_{AZ \, i} \left( {i = 1, \ldots \, nP_{AU,} P_{AZ} } \right),$$

(21)

$$fP_{AU,} R_{AU \, i} \left( {i = 1, \ldots \, nP_{AU,} R_{AU} } \right),$$

(22)

$$fP_{AU,} R_{AV \, i} \left( {i = 1, \ldots \, nP_{AU,} R_{AV} } \right),$$

(23)

$$fP_{AU,} R_{AZ \, i} \left( {i = 1, \ldots \, nP_{AU,} R_{AZ} } \right),$$

(24)

$$fP_{AV,} P_{AZ \, i} \left( {i = 1, \ldots \, nP_{AV,} P_{AZ} } \right),$$

(25)

$$fP_{AV,} R_{AU \, i} \left( {i = 1, \ldots \, nP_{AV,} R_{AU} } \right),$$

(26)

$$fP_{AV,} R_{AV \, i} \left( {i = 1, \ldots \, nP_{AV} R_{AV} } \right) \, ,$$

(27)

$$fP_{AV,} R_{AZ \, i} \left( {i = 1, \ldots \, nP_{AV,} R_{AZ} } \right),$$

(28)

$$fP_{AZ} R_{AU \, i} \left( {i = 1, \ldots \, nP_{AZ,} R_{AU} } \right),$$

(29)

$$fP_{AZ,} R_{AV \, i} \left( {i = 1, \ldots \, nP_{AZ} R_{AV} } \right),$$

(30)

$$fP_{AZ,} R_{AZ \, i} \left( {i = 1, \ldots \, nP_{AZ,} R_{AZ} } \right),$$

(31)

$$fRAU_{,} R_{AV \, i} \left( {i = 1, \ldots \, nRAUR_{AV} } \right),$$

(32)

$$fRAUR_{AZ \, i} \left( {i = 1, \ldots nRAU_{,} R_{AZ} } \right),$$

(33)

$$fRAVR_{AZ \, i} \left( {i = 1, \ldots \, nRAV_{,} R_{AZ} } \right) \, ,$$

(34)

Here Eqs. 20–34 are collectively referred to as mapping Family G_A. Among the “B” family of proteins or reaction partners of sets, P_BU, P_BV, P_BZ, R_BU, R_BV, R_BZ, we may consider the analogous series of mappings, actually, several such mappings between each pair of sets, which mappings may also actually represent similarity comparisons, collectively referred to as mapping Family G_B.

This completes the Protein-based actual Functor model:

$$\begin{aligned} {\mathbf{F}}: \{ \, \{ P_{AU,} P_{AV,} P_{AZ,} R_{AU,} R_{AV,} R_{AZ} \} ,{\mathbf{G}}_{{\mathbf{A}}} \} \\ & \to \{ \, \{ P_{BU,} P_{BV,} P_{BZ,} R_{BU,} R_{BV,} R_{BZ} \} ,{\mathbf{G}}_{{\mathbf{B}}} \} \\ \end{aligned}$$

(28)

The above presented, general Functor example (Eqs. (1)–(11)), in addition, the more chemically motivated actual Functor model for chemical functional groups, as well as the Functor example for protein structures, properties, and analogies [131, 132], have been motivated by the original Quantum Similarity advances.