Improved algorithms for symmetry analysis: structure preserving permutations

Abstract

We propose an improved algorithm for calculating Avnir’s continuous symmetry and chirality measures of molecules. These measures evaluate the deviation of a given structure from symmetry by calculating the distance between the structure and its nearest symmetric counterpart. Our new algorithm utilizes structural properties of the given molecule to increase the accuracy of the calculation and dramatically reduce the running time by up to tens orders of magnitude. Consequently, a wide variety of molecules of medium size with ca. 100 atoms and even more can be analyzed within seconds. Numerical evidence of the algorithm’s efficiency is presented for several families of molecules such as helicenes, porphyrins, dendrimers building blocks, fullerene and more. The ease and efficiency of the calculation make the continuous symmetry and chirality measures promising descriptors for integration in quantitative structure–activity relationship tools, as well as chemical databases and molecular visualization software.

Appendix: Proof of formulas (3) and (4)

For the sake of completeness, we present a rigorous proof of formulas (3) and (4). Given a permutation $\pi$ of $\{1,2,\ldots ,N\}$ satisfying $\pi ^{n}=id$, we look at the minimum of $\mathop {\sum }\limits _{k=1}^N {\left| {\mathbf{Q}_k -\mathbf{P}_k } \right| ^{2}}$ over all the $\pi$-symmetric structures $\left\{ {\mathbf{P}_k } \right\}$. Let $C_1 ,C_2 ,\ldots ,C_f$ be the cycles of $\pi$, as in Eq. (8). Given the symmetry operation T, we have

$$\begin{aligned} \min \sum _{k=1}^N {\left| {\mathbf{Q}_k -\mathbf{P}_k } \right| ^{2}} =\min \sum _{j=1}^f {\sum _{k\in C_j } {\left| {\mathbf{Q}_k -\mathbf{P}_k } \right| ^{2}} } \end{aligned}$$

(13)

where the minimum is over all the structures $\left\{ {\mathbf{P}_k } \right\}$ satisfying Eq. (2).

For each cycle $C_j$ and $k_0 \in C_j$, we have

$$\begin{aligned} C_j =\{\pi (k_0 ),\pi ^{2}(k_0 ),\ldots ,\pi ^{l}(k_0 )=k_0 \} \end{aligned}$$

(14)

where l is a divisor of n. We therefore have (as T is an isometry)

$$\begin{aligned} \sum _{k\in C_j } {\left| {\mathbf{Q}_k -\mathbf{P}_k } \right| ^{2}}= & {} \sum _{u=1}^l {\left| {\mathbf{Q}_{\pi ^{u}(k_0 )} -\mathbf{P}_{\pi ^{u}(k_0 )} } \right| ^{2}} \nonumber \\= & {} \sum _{u=1}^l {\left| {\mathbf{Q}_{\pi ^{u}(k_0 )} -T^{u}{} \mathbf{P}_{k_0 } } \right| ^{2}} =\sum _{u=1}^l {\left| {T^{-u}{} \mathbf{Q}_{\pi ^{u}(k_0 )} -\mathbf{P}_{k_0 } } \right| ^{2}} \end{aligned}$$

(15)

Therefore, the minimum in (13) can be found by minimizing the expression in (15) separately for each cycle, and summing over j.

It is a well known and easy to prove fact that given vectors $\mathbf{x}_1 ,\ldots ,\mathbf{x}_m$, the vector $\mathbf{x}$ which minimizes the sum $\sum _{k=1}^m {\left| {\mathbf{x}_k -\mathbf{x}} \right| ^{2}}$ is the average, $\mathbf{x}=\frac{1}{m}\mathop {\sum }\limits _{k=1}^m {\mathbf{x}_k }$. The minimum value is

$$\begin{aligned} \sum _{k=1}^m {\left| {\mathbf{x}_k -\mathbf{x}} \right| ^{2}}= & {} \sum _{k=1}^m {\left| {\mathbf{x}_k } \right| ^{2}} +m\left| \mathbf{x} \right| ^{2}-2\mathbf{x}\cdot \sum _{k=1}^m {\mathbf{x}_k } \nonumber \\= & {} \sum _{k=1}^m {\left| {\mathbf{x}_k } \right| ^{2}} -m\left| \mathbf{x} \right| ^{2}=\sum _{k=1}^m {\left| {\mathbf{x}_k } \right| ^{2}} -\frac{\sum _{k=1}^m {\mathbf{x}_k } \cdot \sum _{k=1}^m {\mathbf{x}_k } }{m} \nonumber \\= & {} \frac{1}{2m}\left( {2m\sum _{k=1}^m {\left| {\mathbf{x}_k } \right| ^{2}} -2\sum _{k=1}^m {\mathbf{x}_k } \sum _{k=1}^m {\mathbf{x}_k } } \right) \nonumber \\= & {} \frac{1}{2m}\sum _{k_1 =1}^m {\sum _{k_2 =1}^m {\left| {\mathbf{x}_{k_1 } -\mathbf{x}_{k_2 } } \right| ^{2}} } \end{aligned}$$

(16)

Therefore, the minimum of (15) is attained when

$$\begin{aligned} \mathbf{P}_{k_0 } =\frac{1}{l}\sum _{u=1}^l {T^{-u}{} \mathbf{Q}_{\pi ^{u}(k_0 )} } \end{aligned}$$

(17)

Since $\pi ^{l}(k_0 )=k_0$, and by (2), the summand in Eq. (17) is periodic in u with period l. Since l divides n, we conclude that

$$\begin{aligned} \mathbf{P}_{k_0 } =\frac{1}{n}\sum _{u=1}^n {T^{-u}{} \mathbf{Q}_{\pi ^{u}(k_0 )} } \end{aligned}$$

(18)

This proves Eq. (4).

By Eq. (16), the minimum value of Eq. (15) is

$$\begin{aligned} \min \sum _{k\in C_j } {\left| {\mathbf{Q}_k -\mathbf{P}_k } \right| ^{2}}= & {} \min \sum _{u=0}^{l-1} {\left| {T^{-u}{} \mathbf{Q}_{\pi ^{u}(k_0 )} -\mathbf{P}_{k_0 } } \right| ^{2}} \nonumber \\= & {} \frac{1}{2l}\sum _{u,v=0}^{l-1} {\left| {T^{-u}{} \mathbf{Q}_{\pi ^{u}(k_0 )} -T^{-v}{} \mathbf{Q}_{\pi ^{v}(k_0 )} } \right| ^{2}} \nonumber \\= & {} \frac{1}{2l}\sum _{u,v=0}^{l-1} {\left| {T^{v-u}{} \mathbf{Q}_{\pi ^{u}(k_0 )} -\mathbf{Q}_{\pi ^{v}(k_0 )} } \right| ^{2}} \nonumber \\= & {} \frac{1}{2l}\sum _{u,v=0}^{l-1} {\left| {T^{v-u}{} \mathbf{Q}_{\pi ^{u}(k_0 )} -\mathbf{Q}_{\pi ^{v-u}(\pi ^{u}(k_0 ))} } \right| ^{2}} \nonumber \\= & {} \frac{1}{2l}\sum _{i=1}^l {\sum _{k\in C_j } {\left| {T^{i}{} \mathbf{Q}_k -\mathbf{Q}_{\pi ^{i}(k)} } \right| ^{2}} } \nonumber \\= & {} \frac{1}{2n}\sum _{i=1}^n {\sum _{k\in C_j } {\left| {T^{i}{} \mathbf{Q}_k -\mathbf{Q}_{\pi ^{i}(k)} } \right| ^{2}} } \end{aligned}$$

(19)

Summing over j, we get (3).