Shape Theory

What do we mean by shape? The word is often used without consideration of precise meaning but in this document we shall be very clear as to the definition of shape. Two entities will have the same shape if their volumes exactly correspond. The more the volumes differ, the more the shapes will differ. We will give a precise mathematical exposition below, but it is worth noting even at this most basic level shape is defined as a relative quantity, depending on references to other shapes. In this we differ from approaches that attempt to provide absolute, canonical, shapes by which to categorize molecules.

What do we mean by volume? A volume is any scalar field. This means a function that has a single number, or scalar, value at each point in space. The special case for the common understanding of volume is a specific scalar field that has a value of one inside an object and zero outside. The volume of a scalar field is:

\[V \mbox{(volume)} = \int f(x,y,z) dv\]

The volume function, f, is also referred to as the characteristic function. When the characteristic function corresponds to the common definition of a volume field this integral corresponds to what is commonly expected by volume. However, we are not restricted to such simple functions and can still calculate a V. In general the volume of a scalar field is a contraction of the information represented by that characteristic function. It is more precisely referred to as the zeroth-order contraction, or moment. We will discuss other moments and their uses later, but one immediate observation is that two objects cannot have the same shape if their volumes are not the same. The converse is obviously not true. Rather, two objects can have the same volume and not have the same shape. Volume is typical, therefore, of most contractions of information.

We can now write down a precise definition of shape similarity. Consider the integral:

\[S_1 = \int |f(x,y,z) - g(x,y,z)| dV\]

where f and g are different characteristic functions. If this integral is zero then f and g are actually the same function and therefore correspond to the same shape. The larger the integral, the more different the shapes defined by f and g. It defines a metric quantity between the two fields f and g. The word metric is used loosely to mean shape, but here we mean the precise mathematical definition: i.e. a distance that is 1) always positive, 2) zero if and only if two entities are identical and 3) that obeys the triangle inequality. The triangle inequality states that if entity A is distance x from entity B and B is distance y from entity C then the distance between A and C is bounded by |x-y| and |x+y|. The type of comparison shown in S₁ is referred to as an L₁ metric. Another metric is the S₂ metric:

\[S_2=\sqrt{\int [f(x,y,z)-g(x,y,z)]^2 dV}\]

Multiplying the terms in the integral out gives:

\[S_2^2 = \int f(x,y,z)^2dV + \int g(x,y,z)^2dV - 2\int f(x,y,z)g(x,y,z)dV\]

This is the fundamental equation for shape comparison. We rewrite it as:

\[S_{f,g} = I_f + I_g - 2O_{f,g}\]

The I terms are the self-volume overlaps of each entity (for our purposes - molecule), while the O term is the overlap between the two functions. They constitute the three terms we need to compare the shapes of two fields. The I terms are independent of orientation but not O. Finding the orientation that maximizes O, and hence minimizes S_{f,g}, is equivalent to finding the best overlay between the two objects (a quantity that has its own, distinct metric properties). We also note here that the quantity referred to as a Tanimoto coefficient may be derived by recombining I’s and O so:

\[Tanimoto_{f,g} = \frac{O_{f,g}}{I_f+I_g-O_{f,g}} \label{Tanimoto}\]

Tanimoto coefficients will be familiar to those who use them for bitvector fingerprint comparison. An alternative measure is the Tversky coefficient, also mostly used for similarity between bitvector fingerprints. Similarly to the Tanimoto coefficient above, we can define a shape Tversky measure. The base equation for the Tversky coefficient is:

\[Tversky_{f,g} = \frac{O_{f,g}}{\alpha I_f+\beta I_g} \label{tversky}\]

Normally, alpha + beta = 1, and for our current use, alpha is chosen to be 0.95. Since this introduces an asymmetry, the Tversky calculation depends on which molecule’s self-overlap has the alpha pre-factor. ROCS calculates two Tversky values, one with the query molecule with alpha as the pre-factor and a second with the database molecule with alpha as the pre-factor. Also, note that since shape is a field property, instead of a simple scalar like a bitvector, shape Tversky can be larger than 1.0 since the overlap O_{f,g} can be larger than a molecule’s self-overlap, I_f.

The OpenEye Shape Toolkit is a set of calculational objects designed to facilitate the calculation of these field-metric quantities. ROCS is an application built on top of the Shape toolkit.

Shape Characteristics and the Use of Gaussians

Molecules are traditionally viewed as a set of fused spheres, sometimes referred to as the CPK model. The common view of molecular volume is then of a characteristic function that is one (1) inside at least one sphere and zero (0) outside. How do we calculate the volume of such a seemingly simple function? The volume of a single sphere is (4pi r^3)/3 but the complication for two fused spheres is that we have to account for the shared volume and not count it twice. For more than two atoms, there are triple intersections that must be added back in if we have removed the three pairs of intersections. The general formula for N spheres that explicitly calculates the volume of every level of overlap and its correct contribution is:

\[V = 1 - \int \prod_i^N (1 - f_i)dv\]

This is easy to write, not so easy to solve because the analytic formulae for overlaps of increasing order are highly non-trivial (although they have been derived to arbitrary order). It is fair to say that this has hindered the development of shape comparison in many ways. Attempts to use analytic formulae led to very slow programs and approximate methods, for instance using grids of points that are turned in or out by each sphere, do not give smooth gradients required for minimization. Brian Masek (AstraZeneca) was the first to attempt to optimize overlaps of molecules using the analytic approach [Masek-1993]. His program would take several minutes per minimization. In addition it would often suffer from a common problem when using functions that vary sharply (such as solid spheres): it would often get stuck in local minima. Nevertheless, Brian did have encouraging success using this method to find similarities not obvious from chemical structure.

The conceptual breakthrough in shape comparison came in 1995 in a paper by Andrew Grant (AstraZeneca) and Barry Pickup (University of Sheffield) ([Grant-1995], [Grant-1996], [Grant-1997]). They showed that if one let go of the concept of the characteristic function being binary, and instead use a sum of continuous functions, i.e. a Gaussian, that the solid-sphere volume, could be recovered to high accuracy (typically ~0.1%). A sphere has one defining parameter, its radius, whereas a Gaussian has two defining parameters, its prefactor, p, and its width, w:

\[pe^{-wx^2}\]

Grant and Pickup found that by fixing p to 2.7 and setting w for each atom such that the volume integral for each atom agreed with its solid-sphere volume, they achieved remarkable precision. In addition, the overlap terms between any two atoms, and hence any higher-order overlaps, are all Gaussian functions themselves because of the Gaussian Contraction formula (shown here for one spatial variable):

\[\int e^{a(x-x_i)^2}e^{b(x-x_i)^2} = \int e^{(a+b)(x-x_i)^2}\]

i.e. two atomic-Gaussians overlap to produce another Gaussian. Likewise, a three atomic-Gaussian overlap is that of an overlap-Gaussian with an atomic-Gaussian, hence another Gaussian. The simplicity of these formulae and the formula for the volume of each individual Gaussian leads to very efficient algorithms for the calculation of the volume of a molecule so represented (the OpenEye method calculates several thousand volumes per second while calculating intersections up to sixth order).

In addition to simple calculation of molecular volume, which is the zeroth-order moment of the characteristic function, the ease of evaluation of intersections allows for accurate calculation of high-order moments: called the steric multipoles. For instance, if the product formulae for atomic and intersection Gaussians yields n Gaussians, the first order moments are:

\[ \begin{align}\begin{aligned}M_{1,x} = \sum_{i=1}^n \int xe^{a_i|(x-x_i)^2+(y-y_i)^2+(z-z_i)^2|}\\M_{1,y} = \sum_{i=1}^n \int ye^{a_i|(x-x_i)^2+(y-y_i)^2+(z-z_i)^2|}\\M_{1,z} = \sum_{i=1}^n \int ze^{a_i|(x-x_i)^2+(y-y_i)^2+(z-z_i)^2|}\end{aligned}\end{align} \]

These integrals are easy to solve and their sum can be set to zero by an appropriate choice of origin: the center of mass for the sum of Gaussians. Second-order moments are found from integrals of the type:

\[M_{2,PQ} = \sum_{i=1}^n \int PQe^{a_i|(x-x_i)^2+(y-y_i)^2+(z-z_i)^2|}\]

where P and Q are chosen from (x,y,z), e.g. x², xy etc.

These moments can be thought of as a symmetric 3*3 matrix which we refer to as the mass matrix. Rotating or translating the molecule will change the moments and the transform that sets the first-order moments to zero and diagonalizes the mass-matrix puts the molecule into its inertial frame. By convention we assign the x-axis to the largest eigenvector of the mass matrix, y-axis to the median eigenvector, z-axis to the smallest. Note that this orientation is still not uniquely defined: 180 degree rotations around any axis also diagonalize the mass-matrix. The eight (2³) possible transforms that can be generated by combinations of such rotations actually lead to four unique inertial orientations.

If a molecule is aligned to its inertial frame, all higher-order steric multipoles become invariant, ignoring certain sign-changes from the four-fold degeneracy of the inertial frame. As such they, as well as the second-order moments, are shape descriptors. They are still contractions of the information contained in the characteristic field, i.e. two molecules can have the same steric moments and yet have different shapes. (Moments are complete in that if we calculate them to infinite order they do exactly define volume but this is seldom a practical approach!) Nevertheless, they do contain useful information and can be used as a rapid, approximate, filter for shape similarity.

The same advantages that allow for the calculation of molecular volume carry over to the calculation of molecular volume overlap. The overlap of volumes are Gaussian contractions, easily tabulated and efficiently retrieved. Andy re-wrote Brian’s program and obtained an order-of-magnitude improvement in performance as well as another remarkable observation: if the starting orientation of each molecule is that given from the inertial frame then very few “false” minima are produced. The smoothness of the Gaussian characteristic function is enough to overcome the problems with convergence in Brian’s program. The four possible “inertial” starting points were enough to find the best, global, overlay between two molecules. This observation and the Gaussian approach are the basis of the OpenEye Shape Toolkit and ROCS program for rapid shape overlay.

But note, despite the algorithmic advantages, a correlation with common perception has been lost. Because the pre-factor of each atomic Gaussian is not unity, the characteristic function does not correspond to the inside/outside description with which we are most comfortable. In the Gaussian model all points in space are to some degree inside and to some degree outside. That is, the Gaussian model typically shows about 0.1% error with respect to the solid sphere model due to the fact that is includes a portion of all points in space inside the volume.