A unified approach to conformational statistics of classical polymer and polypeptide models

Kim, Jin Seob; Chirikjian, Gregory S.

doi:10.1016/j.polymer.2005.09.012

Cited by 4 publications

(4 citation statements)

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“…If the frames of reference g i and g i +1 are attached at the C α atoms of residues i and i + 1 in a polypeptide, then the function f i,i +1 ( g i,i +1 ) would be the six-dimensional generalization of a Ramachandran map that could include small bond angle bending, warping of the peptide plane and even bond stretching. If one chooses not to model these effects, then the classical Ramachandran map [64] can be reflected by appropriately defining f i,i +1 ( g i,i +1 ), as has been done in [44]. This is consistent with the Flory isolated pair model [33], which has been challenged in recent years [61].…”

Section: Computing Bounds On the Entropy Of The Unfolded Ensemblementioning

confidence: 85%

“…The case of statistical distributions when semi-flexible polymers have internal joints and rigid bends has also been addressed using these methods [87, 88]. And it has been shown that this method can be applied to more general polymers including unfolded polypeptide chains [44]. Similar tools can be used to analyze large amounts of geometric data in the protein data bank [5] such as statistics of helix-helix crossing angle [50] and the relative pose (position and orientation) between alpha carbons in proteins [14, 49].…”

Section: Introductionmentioning

confidence: 99%

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Modeling Loop Entropy

Chirikjian

2011

Computer Methods, Part C

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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.

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Section: Computing Bounds On the Entropy Of The Unfolded Ensemblementioning

confidence: 85%

Section: Introductionmentioning

confidence: 99%

Modeling Loop Entropy

Chirikjian

2011

Computer Methods, Part C

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“…Similar problems arise in the study of chainlike biological macromolecules that undergo thermal fluctuations in solution. See, for example, Zhou and Chirikjian (2006) and Kim and Chirikjian (2005). As another example, consider a non-holonomic mobile robot that executes an open loop trajectory.…”

Section: Introductionmentioning

confidence: 99%

Nonparametric Second-order Theory of Error Propagation on Motion Groups

Wang

Chirikjian

2008

The International Journal of Robotics Research

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Error propagation on the Euclidean motion group arises in a number of areas such as in dead reckoning errors in mobile robot navigation and joint errors that accumulate from the base to the distal end of kinematic chains such as manipulators and biological macromolecules. We address error propagation in rigid-body poses in a coordinate-free way. In this paper we show how errors propagated by convolution on the Euclidean motion group, SE(3), can be approximated to second order using the theory of Lie algebras and Lie groups. We then show how errors that are small (but not so small that linearization is valid) can be propagated by a recursive formula derived here. This formula takes into account errors to second-order, whereas prior efforts only considered the first-order case. Our formulation is nonparametric in the sense that it will work for probability density functions of any form (not only Gaussians). Numerical tests demonstrate the accuracy of this second-order theory in the context of a manipulator arm and a flexible needle with bevel tip.

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“…This has been applied to polymer chains. 45,46 One of the most popular methods for numerical inverse kinematics relies on Jacobian iterations to update joint angles. Many variations on Jacobian-based iterative methods for solving serial chain inverse kinematics problems have been proposed in the literature.…”

Section: An Inverse Kinematic Solution Using the Jacobianmentioning

confidence: 99%

Inverse kinematic solutions of 6-D.O.F. biopolymer segments

Kim¹,

Chirikjian²

2016

Robotica

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SUMMARYWe present two methods to find all the possible conformations of short six degree-of-freedom segments of biopolymers which satisfy end constraints in position and orientation. One of our methods is motivated by inverse kinematic solution techniques which have been developed for “general” 6R serial robotic manipulators. However, conventional robot kinematics methods are not directly applicable to the geometry of polymers, which can be treated as a degenerate case where all the “link lengths” are zero. Here, we propose a method which extends the elimination method of Kohli and Osvatic. This method can be applied directly to the geometry of biopolymers. We also propose a heuristic method based on a Lie-group-theoretic description. In this method, we utilize inverse iterations of the Jacobian matrix to obtain all conformations which satisfy end constraints. This can be easily implemented for both the general 6R manipulator and polymers. Although the extended elimination method is computationally faster than the Jacobian method, in cases where some of the joint angles are 180° (i.e., where the elimination method fails), we combine these two methods effectively to obtain the full set of inverse kinematic solutions. We demonstrate our approach with several numerical examples.

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A unified approach to conformational statistics of classical polymer and polypeptide models

Cited by 4 publications

References 39 publications

Modeling Loop Entropy

Modeling Loop Entropy

Nonparametric Second-order Theory of Error Propagation on Motion Groups

Inverse kinematic solutions of 6-D.O.F. biopolymer segments

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