Group theory and biomolecular conformation: I. Mathematical and computational models

Fourier Transform on the Homogeneous Space of 3D Positions and Orientations for Exact Solutions to Linear PDEs

Fokker–Planck PDEs (including diffusions) for stable Lévy processes (including Wiener processes) on the joint space of positions and orientations play a major role in mechanics, robotics, image analysis, directional statistics and probability theory. Exact analytic designs and solutions are known in the 2D case, where they have been obtained using Fourier transform on SE(2). Here, we extend these approaches to 3D using Fourier transform on the Lie group SE(3) of rigid body motions. More precisely, we define the homogeneous space of 3D positions and orientations R3⋊S2:=SE(3)/({0}×SO(2)) as the quotient in SE(3). In our construction, two group elements are equivalent if they are equal up to a rotation around the reference axis. On this quotient, we design a specific Fourier transform. We apply this Fourier transform to derive new exact solutions to Fokker–Planck PDEs of α-stable Lévy processes on R3⋊S2. This reduces classical analysis computations and provides an explicit algebraic spectral decomposition of the solutions. We compare the exact probability kernel for α=1 (the diffusion kernel) to the kernel for α=12 (the Poisson kernel). We set up stochastic differential equations (SDEs) for the Lévy processes on the quotient and derive corresponding Monte-Carlo methods. We verified that the exact probability kernels arise as the limit of the Monte-Carlo approximations.

Hybrid simulation approach incorporating microscopic interaction along with rigid body degrees of freedom for stacking between base pairs

Stacking interaction between the aromatic heterocyclic bases plays an important role in the double helical structures of nucleic acids. Considering the base as rigid body, there are total of 18 degrees of freedom of a dinucleotide step. Some of these parameters show sequence preferences, indicating that the detailed atomic interactions are important in the stacking. Large variants of non-canonical base pairs have been seen in the crystallographic structures of RNA. However, their stacking preferences are not thoroughly deciphered yet from experimental results. The current theoretical approaches use either the rigid body degrees of freedom where the atomic information are lost or computationally expensive all atom simulations. We have used a hybrid simulation approach incorporating Monte-Carlo Metropolis sampling in the hyperspace of 18 stacking parameters where the interaction energies using AMBER-parm99bsc0 and CHARMM-36 force-fields were calculated from atomic positions. We have also performed stacking energy calculations for structures from Monte-Carlo ensemble by Dispersion corrected density functional theory. The available experimental data with Watson-Crick base pairs are compared to establish the validity of the method. Stacking interaction involving A:U and G:C base pairs with non-canonical G:U base pairs also were calculated and showed that these structures were also sequence dependent. This approach could be useful to generate multiscale modeling of nucleic acids in terms of coarse-grained parameters where the atomic interactions are preserved. This method would also be useful to predict structure and dynamics of different base pair steps containing non Watson-Crick base pairs, as found often in the non-coding RNA structures. © 2015 Wiley Periodicals, Inc. Biopolymers 105: 212-226, 2016.

Optimization on the space of rigid and flexible motions: an alternative manifold optimization approach

In this paper we consider the problem of minimization of a cost function that depends on the location and poses of one or more rigid bodies, or bodies that consist of rigid parts hinged together. We present a unified setting for formulating this problem as an optimization on an appropriately defined manifold for which efficient manifold optimizations can be developed. This setting is based on a Lie group representation of the rigid movements of a body that is different from what is commonly used for this purpose. We illustrate this approach by using the steepest descent algorithm on the manifold of the search space and specify conditions for its convergence.

Conformational Modeling of Continuum Structures in Robotics and Structural Biology: A Review

Hyper-redundant (or snakelike) manipulators have many more degrees of freedom than are required to position and orient an object in space. They have been employed in a variety of applications ranging from search-and-rescue to minimally invasive surgical procedures, and recently they even have been proposed as solutions to problems in maintaining civil infrastructure and the repair of satellites. The kinematic and dynamic properties of snakelike robots are captured naturally using a continuum backbone curve equipped with a naturally evolving set of reference frames, stiffness properties, and mass density. When the snakelike robot has a continuum architecture, the backbone curve corresponds with the physical device itself. Interestingly, these same modeling ideas can be used to describe conformational shapes of DNA molecules and filamentous protein structures in solution and in cells. This paper reviews several classes of snakelike robots: (1) hyper-redundant manipulators guided by backbone curves; (2) flexible steerable needles; and (3) concentric tube continuum robots. It is then shown how the same mathematical modeling methods used in these robotics contexts can be used to model molecules such as DNA. All of these problems are treated in the context of a common mathematical framework based on the differential geometry of curves, continuum mechanics, and variational calculus. Both coordinate-dependent Euler-Lagrange formulations and coordinate-free Euler-Poincaré approaches are reviewed.

A symmetry model for genetic coding via a wallpaper group composed of the traditional four bases and an imaginary base E: towards category theory-like systematization of molecular/genetic biology

Background

Previously, we suggested prototypal models that describe some clinical states based on group postulates. Here, we demonstrate a group/category theory-like model for molecular/genetic biology as an alternative application of our previous model. Specifically, we focus on deoxyribonucleic acid (DNA) base sequences.

Results

We construct a wallpaper pattern based on a five-letter cruciform motif with letters C, A, T, G, and E. Whereas the first four letters represent the standard DNA bases, the fifth is introduced for ease in formulating group operations that reproduce insertions and deletions of DNA base sequences. A basic group Z₅ = {r, u, d, l, n} of operations is defined for the wallpaper pattern, with which a sequence of points can be generated corresponding to changes of a base in a DNA sequence by following the orbit of a point of the pattern under operations in group Z₅. Other manipulations of DNA sequence can be treated using a vector-like notation ‘D_j’ corresponding to a DNA sequence but based on the five-letter base set; also, ‘D_j’s are expressed graphically. Insertions and deletions of a series of letters ‘E’ are admitted to assist in describing DNA recombination. Likewise, a vector-like notation R_j can be constructed for sequences of ribonucleic acid (RNA). The wallpaper group B = {Z₅^×∞, ●} (an ∞-fold Cartesian product of Z₅) acts on D_j (or R_j) yielding changes to D_j (or R_j) denoted by ‘D_j◦B_(j→k) = D_k’ (or ‘R_j◦B_(j→k) = R_k’). Based on the operations of this group, two types of groups—a modulo 5 linear group and a rotational group over the Gaussian plane, acting on the five bases—are linked as parts of the wallpaper group for broader applications. As a result, changes, insertions/deletions and DNA (RNA) recombination (partial/total conversion) are described. As an exploratory study, a notation for the canonical “central dogma” via a category theory-like way is presented for future developments.

Conclusions

Despite the large incompleteness of our methodology, there is fertile ground to consider a symmetry model for genetic coding based on our specific wallpaper group. A more integrated formulation containing “central dogma” for future molecular/genetic biology remains to be explored.

A New Approach to Rigid Body Minimization with Application to Molecular Docking

Our work is motivated by energy minimization in the space of rigid affine transformations of macromolecules, an essential step in computational protein-protein docking. We introduce a novel representation of rigid body motion that leads to a natural formulation of the energy minimization problem as an optimization on the [Formula: see text] manifold, rather than the commonly used SE(3). The new representation avoids the complications associated with optimization on the SE(3) manifold and provides additional flexibilities for optimization not available in that formulation. The approach is applicable to general rigid body minimization problems. Our computational results for a local optimization algorithm developed based on the new approach show that it is about an order of magnitude faster than a state of art local minimization algorithms for computational protein-protein docking.

Rigid Body Energy Minimization on Manifolds for Molecular Docking

Virtually all docking methods include some local continuous minimization of an energy/scoring function in order to remove steric clashes and obtain more reliable energy values. In this paper, we describe an efficient rigid-body optimization algorithm that, compared to the most widely used algorithms, converges approximately an order of magnitude faster to conformations with equal or slightly lower energy. The space of rigid body transformations is a nonlinear manifold, namely, a space which locally resembles a Euclidean space. We use a canonical parametrization of the manifold, called the exponential parametrization, to map the Euclidean tangent space of the manifold onto the manifold itself. Thus, we locally transform the rigid body optimization to an optimization over a Euclidean space where basic optimization algorithms are applicable. Compared to commonly used methods, this formulation substantially reduces the dimension of the search space. As a result, it requires far fewer costly function and gradient evaluations and leads to a more efficient algorithm. We have selected the LBFGS quasi-Newton method for local optimization since it uses only gradient information to obtain second order information about the energy function and avoids the far more costly direct Hessian evaluations. Two applications, one in protein-protein docking, and the other in protein-small molecular interactions, as part of macromolecular docking protocols are presented. The code is available to the community under open source license, and with minimal effort can be incorporated into any molecular modeling package.

Multiscale modeling of double-helical DNA and RNA: a unification through Lie groups

Several different mechanical models of double-helical nucleic-acid structures that have been presented in the literature are reviewed here together with a new analysis method that provides a reconciliation between these disparate models. In all cases, terminology and basic results from the theory of Lie groups are used to describe rigid-body motions in a coordinate-free way, and when necessary, coordinates are introduced in a way in which simple equations result. We consider double-helical DNAs and RNAs which, in their unstressed referential state, have backbones that are either straight, slightly precurved, or bent by the action of a protein or other bound molecule. At the coarsest level, we consider worm-like chains with anisotropic bending stiffness. Then, we show how bi-rod models converge to this for sufficiently long filament lengths. At a finer level, we examine elastic networks of rigid bases and show how these relate to the coarser models. Finally, we show how results from molecular dynamics simulation at full atomic resolution (which is the finest scale considered here) and AFM experimental measurements (which is at the coarsest scale) relate to these models.

Mathematical aspects of molecular replacement. I. Algebraic properties of motion spaces

Molecular replacement (MR) is a well established method for phasing of X-ray diffraction patterns for crystals composed of biological macromolecules of known chemical structure but unknown conformation. In MR, the starting point is known structural domains that are presumed to be similar in shape to those in the macromolecular structure which is to be determined. A search is then performed over positions and orientations of the known domains within a model of the crystallographic asymmetric unit so as to best match a computed diffraction pattern with experimental data. Unlike continuous rigid-body motions in Euclidean space and the discrete crystallographic space groups, the set of motions over which molecular replacement searches are performed does not form a group under the operation of composition, which is shown here to lack the associative property. However, the set of rigid-body motions in the asymmetric unit forms another mathematical structure called a quasigroup, which can be identified with right-coset spaces of the full group of rigid-body motions with respect to the chiral space group of the macromolecular crystal. The algebraic properties of this space of motions are articulated here.

Modeling loop entropy

Proteins fold from a highly disordered state into a highly ordered one. Traditionally, the folding problem has been stated as one of predicting "the" tertiary structure from sequential information. However, new evidence suggests that the ensemble of unfolded forms may not be as disordered as once believed, and that the native form of many proteins may not be described by a single conformation, but rather an ensemble of its own. Quantifying the relative disorder in the folded and unfolded ensembles as an entropy difference may therefore shed light on the folding process. One issue that clouds discussions of "entropy" is that many different kinds of entropy can be defined: entropy associated with overall translational and rotational Brownian motion, configurational entropy, vibrational entropy, conformational entropy computed in internal or Cartesian coordinates (which can even be different from each other), conformational entropy computed on a lattice, each of the above with different solvation and solvent models, thermodynamic entropy measured experimentally, etc. The focus of this work is the conformational entropy of coil/loop regions in proteins. New mathematical modeling tools for the approximation of changes in conformational entropy during transition from unfolded to folded ensembles are introduced. In particular, models for computing lower and upper bounds on entropy for polymer models of polypeptide coils both with and without end constraints are presented. The methods reviewed here include kinematics (the mathematics of rigid-body motions), classical statistical mechanics, and information theory.