Theory

Molecular Shape

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 can not 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. 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:

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

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.

Shape from Hermite Representation

So far, discussion has focused on a multi-center expansion of molecular shape based on a Gaussian representation of atoms. An interesting alternative is a single-center expansion of the shape into a complete set of functions. One possibility is to use a product of three Hermite functions, one per space dimension and expand the shape into the resulting complete set of functions.

Hermites are a class of functions known as “special” functions. They are special because linear combinations of these functions can reproduce any other reasonably behaved function. Just as the Discrete Fourier Transform shows that any periodic function can be represented by a sum of the trigonometric functions {\(sin(n\varphi), cos(n\varphi)\)}, Hermite functions can be used to represent any localized, non-periodic function such as molecular shape. The “special” in special functions refers to the fact that each function in one of these classes is orthogonal to every other function. This means that the integral of the product of any two such functions is 1 if they are the same function, or zero if they are not. This makes much of the mathematics of finding the coefficients of the representational linear sum much simpler: a coefficient is then just the integral of the product of each particular special function with the function being represented.

There are many well-known special functions such as sine and cosine, Hermite polynomials, Legendre polynomials, and Laguerre polynomials. So why work with Hermite functions? The art of special functions is to choose one that fits the purpose. For instance, for a periodic function, sine and cosine make sense. Hermites make sense for shape because of their form. The Hermite function of order n (any non-negative integer) is defined as follows:

\[{\text{H}}_{n}(x) = H_{n}(x) e^{-x^2/2}\]

The Hermite polynomial is defined as:

\[H_{n}(x) = (-1)^n e^{x^2}\left(\frac{d}{dx}\right)^n e^{-x^2}.\]

The first few Hermite polynomials are:

\[H_{0}(x)=1, \quad H_{1}(x)=2x, \quad H_{2}(x) = 4x^2-2, \quad H_{3}(x)=8x^3-12x.\]

Hermite functions can be thought of as generalizations of Gaussian function to a complete set: this is why we like them as candidates for representing molecular shape! Since its formation, OpenEye has worked with molecular shape as a sum Gaussian ([Grant-1995], [Grant-1996], [Grant-1997]) and Hermites represent an extension of that work. The difference is that Hermites are all centered at the origin, while the “classical” OpenEye representation of molecular shape is to place a Gaussian at each non-hydrogen atom.

So, are there any advantages to using a single-centered Hermite representation rather than a multi-centered set of Gaussians? This is a similar question to the one we first posed at the beginning of OpenEye: Does representing shape by Gaussians give us any advantages over the canonical “sum of spheres” representation? We felt that it did. Shape became a smooth function that seemed more physical than sharp spherical functions, and they allowed a more efficient molecular overlay of shapes. We postulated that this might make a better dielectric function for Poisson-Boltzmann, which has been largely verified in ZAP TK ([Grant-2001]). The concept seemed very general, so we embedded it into an OpenEye toolkit, Shape TK, so that we and our customers could explore different uses. This led to applications in crystallography (AFITT), bioisostere replacement (BROOD), docking (FRED), pose prediction using ligand information (POSIT), and shape-based alignment (ROCS).

So what advantages might Hermites represent? For one, they are very compact (i.e., they have few coefficients). Second, the more coefficients we include (i.e., higher order functions), the more exact the match to a sum of atom-centered Gaussians; conversely, the fewer functions, the more “coarse” the representation becomes, while retaining the smooth properties we like about Gaussians in the first place. Third, the Fourier Transform of a Hermite function is the same function! If we imagine wanting to generate Fourier representations of shape, Hermites make it easy. Fourth, the overlap of two shapes represented by Hermite functions is just the sum of the product of the coefficients: it’s that easy! It’s not difficult to imagine ways in which we could apply Hermites to the same problems we currently tackle with atom-centered Gaussians.

The first application that has already intrigued us is the representation of protein shapes. Consider that an arbitrary molecular shape can be expanded into the following combination of Hermite functions:

\[f(x,y,z) = \sum_{l,m,n}^{\infty}\,f_{lmn}\,{\text{H}}_l(\lambda_x x)\, {\text{H}}_{m}(\lambda_y y)\, {\text{H}}_n(\lambda_z z).\]

The (infinite) set of coefficients for \(f_{lmn}\) forms the Hermite representation. Due to reflection properties of Hermite polynomials, we can assume without loss of generality that the coefficients \(\lambda_x, \lambda_y, \lambda_z\) are all positive. In practice we take a finite sum in the above equation by truncating it with the following condition on l, m, n:

\[l+m+n\le {\text{NPolyMax}},\]

where NPolyMax is a resolution parameter, and can be varied from 0 (very inaccurate expansion) to infinity (exact Hermite expansion). The recommended value varies depending on the size of the molecule.

Below is an example of the Hermite representation of the protein DHFR.

Hard and soft shape gaussians

Exact shape versus Hermite Representation

The left figure represents the VIDA view of the exact shape of the protein. The middle figure shows the Hermite representation with NPolyMax = 5. The right figure shows the Hermite representation with NPolyMax = 30. Remarkably, we can encode the main features of the protein shape using Hermite representation with only 56 floating point numbers (middle figure).

The advantages of representing proteins by Hermites include:

  • smooth representations that capture any level of detail, from the atomistic (many coefficients, as in the picture on the right) to the very coarse (few coefficients, as in the middle picture),
  • the ability to store these representations in a compact manner,
  • the ability to transform these representations easily,
  • very fast overlap calculations between proteins.

Of course, the mathematics of Hermites is more complex than that of a set of atom-centered Gaussians. We have to know how to make, rotate, translate, and scale such representations. Therefore, we are releasing this toolkit with no immediate application or goal—rather, in the spirit of OpenEye, to make potentially useful tools for our customers.

Color Features

In addition to shape-alignments Shape TK, optionally, considers chemistry alignment, known as ‘color’. User-specified definitions of chemistry can be included in the superposition and similarity analysis process to facilitate the identification of those compounds that are similar both in shape and chemistry.

Color atoms are described as Gaussians and usually displayed as colored spheres in visualizations. The Gaussian for a color atom is relatively hard with a steep gradient. Figure: Hard vs. Soft Gaussians illustrates hard vs fuzzy Gaussians. Both Gaussians in the figure represent the same volume as the sphere. However, the hard Gaussian, with the steep gradient, reaches a probability of zero (0) within the radius of the sphere. The color features are either matched, if they fall within the sphere radius, or not matched. In the case of the fuzzy Gaussian there are areas outside the volume of the sphere (the area under the curve indicated by the two arrows) where the Gaussian probability is greater than zero. This would allow color features to match even when they align well outside the sphere representing the color atom. That situation would lead to less precise alignments and, for that reason, the ‘hard’ Gaussian is employed.

Hard and soft shape gaussians

Hard vs. Soft Gaussians

A sphere described by two different Gaussian functions. The ‘hard’ Gaussian (dashed) is the one employed by Shape TK to approximate a color atom sphere.

Shape TK comes pre-loaded with two color force fields, Implicit Mills Dean (default) and Explicit Mills Dean. These are described in associated color force field files (*.cff). The desired force field file can be supplied to the OEColorForceField::Init method. Further information on editing color force field files is given in the below section Color File Format.

The color force field is used to measure chemical similarity between the query and the database molecule and to refine shape-based overlays. The color force field file describes:

  • Color atom types

  • The functional groups to which the color atoms should be applied.

    Shape TK uses only the heavy atoms of molecules; hydrogens are ignored.

  • Whether the interaction between color atoms is attractive or repulsive. Interactions between color atoms of the same type are always attractive. The weight term describes the strength of the interaction relative to the shape gradients and the range term affects the range of the interaction.

The color features described in the Implicit and Explicit Mills Dean color force field files include:

Donor:Functional groups that can act as H-bond donors e.g. acid-OH
Acceptor:Functional groups that can act as H-bond acceptors e.g. carboxylate
Anion:Functional groups with either localized or delocalized negative charge e.g. tetrazole
Cation:Functional groups with either localized or delocalized positive charge e.g. guanidinium
Hydrophobe:Terminal or non-terminal aromatic or aliphatic groups e.g. phenyl
Rings:Rings of defined size e.g. 4-7 atoms

A custom force field file can include other features that you define e.g. positive, negative, carbonyl_linker, metal_binder. For each color atom type a set of SMARTS is used to define the specific functional groups to which the color atom will be applied. The Implicit and Explicit Mills Dean force fields differ in these functional group definitions. For example, the Explicit Mills Dean force field allows a primary amine to be an acceptor as well as a donor whereas it is a donor only in the Implicit Mills Dean force field.

The color force field can also be used for post-shape scoring either alone, e.g. ColorTanimoto and Color Tversky, or in combination with shape scores, e.g. TanimotoCombo and TverskyCombo.

Color File Format

As an alternative to the built-in force fields, the user can define a new color force field using the format described in this section. The following is a simplified example of a color force field specification.

DEFINE hetero [#7,#8,#15,#16]
DEFINE notNearHetero [!#1;!$($hetero);!$(\*[$hetero])]
#
#
TYPE donor
TYPE acceptor
TYPE rings
TYPE positive
TYPE negative
TYPE structural
#
#
PATTERN donor [$hetero;H]
PATTERN acceptor [#8&!$(\*~N~[OD1]),#7&H0;!$([D4]);!$([D3]-\*=,:[$hetero])]
PATTERN rings [R]~1~[R]~[R]~[R]1
PATTERN rings [R]~1~[R]~[R]~[R]~[R]1
PATTERN rings [R]~1~[R]~[R]~[R]~[R]~[R]1
PATTERN rings [R]~1~[R]~[R]~[R]~[R]~[R]~[R]1
PATTERN positive [+,$([N;!$(\*-\*=O)])]
PATTERN negative [-]
PATTERN negative [OD1+0]-[!#7D3]~[OD1+0]
PATTERN negative [OD1+0]-[!#7D4](~[OD1+0])~[OD1+0]
PATTERN structural [$notNearHetero]
#
#
INTERACTION donor donor attractive gaussian weight=1.0 radius=1.0
INTERACTION acceptor acceptor attractive gaussian weight=1.0 radius=1.0
INTERACTION rings rings attractive gaussian weight=1.0 radius=1.0
INTERACTION positive positive attractive gaussian weight=1.0 radius=1.0
INTERACTION negative negative attractive gaussian weight=1.0 radius=1.0
INTERACTION structural structural attractive gaussian weight=1.0 radius=1.0

There are four basic keywords in a cff file: DEFINE, TYPE, PATTERN, and INTERACTION. The TYPE field can be any user-defined term. TYPES can be any user-specified string such as “donor”, “acceptor”, “lipophilic anion” etc. The PATTERN keyword is used to associate SMARTS patterns with these types. There is no restriction on the number of patterns that can be associated with a user defined type. The position in Cartesian space of the PATTERN is taken as the average of the coordinates of the atoms that match the SMARTS pattern. If the desired location of the PATTERN is on a single atom of a larger SMARTS pattern recursive SMARTS (written as ‘[$(SMARTS)]’ can be used to this effect. Only the first atom in a recursive SMARTS pattern ‘matches’ the molecule, and the rest of the SMARTS pattern defines an environment. By writing a SMARTS pattern in recursive notation the location of the PATTERN will be taken as the atomic position of the first matching atom in the pattern. In order to simplify both reading and writing SMARTS, intermediate SMARTS can be associated with words using the DEFINE keyword. Once defined, these words can then be used as atom primitives in subsequent SMARTS patterns with the $ prefix (see “DEFINE hetero” and “PATTERN donor” above).

Interactions between types are associated with the INTERACTION keyword. Two user-defined types must be listed, and whether their interaction is attractive or repulsive. The height and radius can be modified by keywords WEIGHT and RADIUS. At present, the only alternative to a Gaussian decay is invoked by the DISCRETE keyword. A discrete interaction contributes all of WEIGHT if the inter-type distance is less than RADIUS, or zero. Since it is not differentiable it makes no contribution to optimization (i.e. because the gradient of a DISCRETE function is 0 or infinite).