Methods based on conformational variables

Diamond, R.

doi:10.1107/97809553602060000561

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 3.3, pp. 378-379 | 1 | 2 |

Section 3.3.2.2.1. Methods based on conformational variables

R. Diamond^a ^*

^aMRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England
Correspondence e-mail: rd10@cam.ac.uk

3.3.2.2.1. Methods based on conformational variables

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Suppose that t represents a vector from the current position of an atom in the model to a target position then (see Section 3.3.1.1.3), to first order, the observational equations are $[t_{IA} = D_{IpA} \theta_{p} + v_{IA}]$ in which $[\boldtheta]$ represents changes to conformational variables which may include dihedral angles, bond angles, bond lengths, and parameters determining overall position and orientation of the molecule as a whole. If every such parameter is included the model acquires 3n degrees of freedom for n atoms, in which case the methods of the next section are more appropriate, but if bond lengths and some or all bond angles are being treated as constants then the above equation becomes the basis of the treatment. $[D_{IPA} = {\partial r_{IA}\over \partial \theta_{P}} = \varepsilon_{Ijk} n_{jP} (r_{kA} - r_{kP})]$ in which $[{\bf n}_{P}]$ is a unit vector defining the axis of rotation for an angular variable $[\theta_{P}]$ , $[{\bf r}_{A}]$ and $[{\bf r}_{P}]$ are position vectors of the atom A and the site of the parameter P, and $[{\bf v}_{A}]$ represents a residual vector.

$[\sigma = v_{ia} v_{ia}]$ is minimized by $[{\boldtheta} = {\bi M}^{-1} {\bf V}]$ in which $[\eqalign{ M_{PQ} &= D_{iPa}D_{iQa}\cr V_{P} &= D_{iPa}t_{ia}.}]$ More generally, if σ represents any scalar quantity which is to be minimized, e.g. an energy, then $[\eqalign{ V_{P} &= -{1\over 2} {\partial \sigma\over \partial \theta_{P}}\cr M_{PQ} &= {1\over 2} {\partial^{2} \sigma\over \partial \theta_{P} \partial \theta_{Q}}.}]$

It is beyond the scope of this chapter to review the methods available for evaluating $[\boldtheta]$ from these equations. Difficulties may arise from two sources:

(i) Inversion of M may be difficult if M is large or ill conditioned and impossible if M is singular.
(ii) Successful evaluation of $[{\bi M}^{-1}{\bf V}]$ will not minimize σ in one step if t is not linearly dependent on $[\boldtheta]$ or, equivalently, $[\partial^{2}\sigma / \partial \theta_{P}\partial \theta_{Q}]$ is not constant, and substantial changes $[\boldtheta]$ are involved. Iteration is then necessary.

Difficulties of the first kind may be overcome by gradient methods, for example the conjugate gradient method without searches if M is available or with searches if it is not available, or they may be overcome by methods based on eigenvalue decompositions. If non-linearity is serious less dependence should be placed on M and gradient methods using searches are more valuable. In this connection Diamond (1966) introduced a sliding filter technique which produced rapid convergence in extreme conditions of non-linearity. These topics have been reviewed elsewhere (Diamond, 1981, 1984b) and are the subject of many textbooks (Walsh, 1975; Gill et al., 1981; Luenberger, 1984).

Warme et al. (1972) have developed a similar system using dihedral angles as variables and a variety of alternative optimization algorithms.

The modelling of flexible rings or lengths of chain with two or more fixed parts is sometimes held to be a difficulty in methods using conformational variables, although a simple two-stage solution does exist. The principle involved is the sectioning of the space of the variables into two orthogonal subspaces of which the first is used to satisfy the constraints and the second is used to perform the optimization subject to those constraints.

The algebra of the method may be outlined as follows, and is given in more detail by Diamond (1971, 1980a,b). Parametric shifts $[{\boldtheta}_{1}]$ which satisfy the constraints are solutions of $[{\bf V}_{1} = {\bi M}_{1} {\boldtheta}_{1}]$ in which $[{\bf V}_{1}]$ and $[{\bi M}_{1}]$ depend only on the target vectors, $[{\bf t}_{1}]$ , of the atoms with constrained positions and on the corresponding derivatives. We then find a partitioned orthogonal matrix $[({\bi A}_{1}{\bi B}_{1})]$ satisfying $[({\bi A}_{1}{\bi B}_{1}) = ({\bi E}_{\lambda}{\bi E}_{0}) \pmatrix{{\bi R}_{\lambda} &{\bi 0}\cr {\bi 0} &{\bi R}_{0}\cr}]$ in which $[{\bi E}_{\lambda}]$ are the eigenvectors of $[{\bi M}_{1}]$ having positive eigenvalues, $[{\bi E}_{0}]$ are those having zero eigenvalues, and $[{\bi R}_{\lambda}]$ and $[{\bi R}_{0}]$ are arbitrary orthogonal matrices. Then $[\openup-3pt\displaylines{\eqalign{\pmatrix{{\bi A}_{1}^{T}\cr {\bi B}_{1}^{T}\cr} {\bf V}_{1} &= \pmatrix{{\bi A}_{1}^{T}\cr {\bi B}_{1}^{T}\cr} {\bi M}_{1}({\bi A}_{1}{\bi B}_{1}) \pmatrix{{\bi A}_{1}^{T}\cr {\bi B}_{1}^{T}\cr} {\boldtheta}_{1}\cr \noalign{\vskip3pt} &= \pmatrix{{\bi A}_{1}^{T}{\bi M}_{1}{\bi A}_{1} &{\bi 0}\cr {\bi 0} &{\bi 0}\cr} \pmatrix{{\bi A}_{1}^{T}\cr {\bi B}_{1}^{T}\cr} {\boldtheta}_{1}\cr} \cr {\bi A}_{1}^{T}{\boldtheta}_{1} = ({\bi A}_{1}^{T}{\bi M}_{1}{\bi A}_{1})^{-1} {\bi A}_{1}^{T}{\bf V}_{1}}]$ in which the matrix to be inverted is positive definite. $[{\bi A}_{1}]$ , however, is rectangular so that multiplying on the left by $[{\bi A}_{1}]$ does not necessarily serve to determine $[{\boldtheta}_{1}]$ , but we may write $[{\boldtheta}_{1} = ({\bi A}_{1}{\bi B}_{1}) \pmatrix{{\boldvarphi}_{1}\cr {\boldpsi}_{1}\cr}]$ giving $[\displaylines{\pmatrix{{\bi A}_{1}^{T}\cr {\bi B}_{1}^{T}\cr} {\bf V}_{1} = \pmatrix{{\bi A}_{1}^{T}{\bi M}_{1}{\bi A}_{1} &{\bi 0}\cr {\bi 0} &{\bi 0}\cr} \pmatrix{{\boldvarphi}_{1}\cr {\boldpsi}_{1}\cr}\cr \noalign{\vskip3pt} {\boldvarphi}_{1} = ({\bi A}_{1}^{T}{\bi M}_{1}{\bi A}_{1})^{-1} {\bi A}_{1}^{T}{\bf V}_{1}}]$ and $[{\boldpsi}_{1}]$ is indeterminate and free to adopt any value. We therefore adopt $[{\boldtheta}_{1} = {\bi A}_{1}{\boldvarphi}_{1} = {\bi A}_{1}({\bi A}_{1}^{T}{\bi M}_{1}{\bi A}_{1})^{-1} {\bi A}_{1}^{T}{\bf V},]$ which is the smallest vector of parametric shifts which will satisfy the constraints, and allow $[{\boldpsi}_{1}]$ to be determined by the remaining observational equations in which the target vectors, t, are now modified to $[{\bf t}_{2}]$ according to $[{\bf t}_{2} = {\bf t} - {\bi D}_{2}{\boldtheta}_{1},]$ $[{\bi D}_{2}]$ and $[{\bf t}_{2}]$ being the derivatives and effective target vectors for the unconstrained atoms. We then solve $[{\bf V}_{2} = {\bi M}_{2}{\boldtheta}_{2}]$ in which $[{\boldtheta}_{2}]$ is required to be of the form $[{\boldtheta}_{2} = {\bi B}_{1}{\boldpsi}_{1}]$ giving $[{\boldtheta}_{2} = {\bi B}_{1} ({\bi B}_{1}^{T}{\bi M}_{2}{\bi B}_{1})^{-1} {\bi B}_{1}^{T}{\bf V}_{2}]$ and apply the total shifts $[{\boldTheta} = {\boldtheta}_{1} + {\boldtheta}_{2}]$ to obtain a structure which is optimized within the restrictions imposed by the constraints.

It may happen that $[{\bi B}_{1}^{T} {\bi M}_{2} {\bi B}_{1}]$ is itself singular because there are insufficient data in the vector $[{\bf t}_{2}]$ to control the structure and the parametric shifts contained in $[{\boldtheta}_{2}]$ fully. In this event the same process may be applied again, basing the solution for $[{\boldtheta}_{2}]$ on $[\openup3pt\pmatrix{{\bi A}_{2}^{T}\cr {\bi B}_{2}^{T}\cr} {\bi B}_{1}^{T}{\bi M}_{2}{\bi B}_{1} ({\bi A}_{2}{\bi B}_{2})]$ so that the vectors in $[{\bi B}_{2}]$ represent the degrees of freedom which remain uncommitted. This method of application of constraints by subspace sectioning may be nested to any depth and is completely general.

A valid matrix $[{\bi A}_{1}]$ may be found from $[{\bi M}_{1}]$ by using the fact that the columns of $[{\bi M}_{1}]$ are all linear combinations of the columns of $[{\bi E}_{\lambda}]$ and are void of any contribution from $[{\bi E}_{0}]$ . It follows that $[{\bi A}_{1}]$ may be found by using the columns of $[{\bi M}_{1}]$ as priming vectors in the Gram–Schmidt process [Section 3.3.1.2.3 (i)] until the normalizing step involves division by zero. $[{\bi A}_{1}]$ is then complete if all the columns of $[{\bi M}_{1}]$ have been tried. $[({\bi A}_{1}{\bi B}_{1})]$ may then be completed by using arbitrary vectors as primers.

Manipulation of a ring of n atoms may be achieved by treating it as a chain of [(n + 2)] atoms [having [(n + 1)] bond lengths, n bond angles and [(n - 1)] dihedral angles] in which atom 1 is required to coincide with atom and atom 2 with . $[{\bf t}_{1}]$ then contains two vectors, namely the lack-of-closure vectors at these points, and is typically zero. $[{\bi A}_{1}]$ is then found to have five columns corresponding to the five degrees of freedom of two points of fixed separation; $[\boldtheta_{1}]$ contains only zeros if the ring is initially closed, and contains ring-closure corrections if, through non-linearity or otherwise, the ring has opened. $[{\bi B}_{1}]$ contains [(p - 5)] columns if the chain of [(n + 2)] points has p variable parameters. It follows, if bond lengths and bond angles are treated as constants, that the seven-membered ring is the smallest ring which is flexible, that the six-membered ring (if it can be closed with the given bond angles) has no flexibility (though it may have discrete alternatives) and that it may be impossible to close a five-membered ring. Therefore some variation of bond angles and/or bond lengths is essential for the modelling of flexible five- and six-membered rings. Treating the ring as a chain of [(n + 1)] atoms is less satisfactory as there is then no control over the bond angle at the point of ring closure.

A useful concept for the modelling of flexible five-membered rings with near-constant bond angles is the concept of the pseudorotation angle P, and amplitude $[\theta_{m}]$ , for which the jth dihedral angle is given by $[\theta_{j} = \theta_{m} \cos \left(P + {4\pi j\over 5}\right).]$ This formulation has the property $[\textstyle{\sum}_{j = 0}^{4} \theta_{j} = 0]$ , which is not exactly true; nevertheless, $[\theta_{j}]$ values measured from observed conformations comply with this formulation within a degree or so (Altona & Sundaralingam, 1972).

Software specialized to the handling of condensed ring systems has been developed by van der Lieth et al. (1984) (Section 3.3.3.3.1) and by Cohen et al. (1981) (Section 3.3.3.3.2).

References

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International Tables for Crystallography (2006). Vol. B. ch. 3.3, pp. 378-379