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Dynamics of Discrete Curves with Applications to Protein Structure
Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics.
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In order to perform a specific function, a protein needs to fold into the proper structure. Prediction the protein structure from its amino acid sequence has still been unsolved problem. The main focus of this thesis is to develop new approach on the protein structure modeling by means of differential geometry and integrable theory. The start point is to simplify a protein backbone as a piecewise linear polygonal chain, with vertices recognized as the central alpha carbons of the amino acids. Frenet frame and equations from differential geometry are used to describe the geometric shape of the protein linear chain. Within the framework of integrable theory, we also develop a general geometrical approach, to systematically derive Hamiltonian energy functions for piecewise linear polygonal chains. These theoretical studies is expected to provide a solid basis for the general description of curves in three space dimensions. An efficient algorithm of loop closure has been proposed.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2013. , 41 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1054
Keyword [en]
Frenet equations, integrable model, folded proteins, discrete curves
National Category
Physical Sciences Biophysics
Research subject
Physics; Physical Biology
Identifiers
URN: urn:nbn:se:uu:diva-199987ISBN: 978-91-554-8694-5 (print)OAI: oai:DiVA.org:uu-199987DiVA: diva2:621923
Public defence
2013-09-02, Å10132, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Supervisors
Available from: 2013-06-11 Created: 2013-05-17 Last updated: 2013-08-30Bibliographically approved
List of papers
1. Discrete Frenet frame, inflection point solitons, and curve visualization with applications to folded proteins
Open this publication in new window or tab >>Discrete Frenet frame, inflection point solitons, and curve visualization with applications to folded proteins
2011 (English)In: Physical Review E, ISSN 1539-3755, Vol. 83, no 6, 061908- p.Article in journal (Refereed) Published
Abstract [en]

We develop a transfer matrix formalism to visualize the framing of discrete piecewise linear curves in three-dimensional space. Our approach is based on the concept of an intrinsically discrete curve. This enables us to more effectively describe curves that in the limit where the length of line segments vanishes approach fractal structures in lieu of continuous curves. We verify that in the case of differentiable curves the continuum limit of our discrete equation reproduces the generalized Frenet equation. In particular, we draw attention to the conceptual similarity between inflection points where the curvature vanishes and topologically stable solitons. As an application we consider folded proteins, their Hausdorff dimension is known to be fractal. We explain how to employ the orientation of C-beta carbons of amino acids along a protein backbone to introduce a preferred framing along the backbone. By analyzing the experimentally resolved fold geometries in the Protein Data Bank we observe that this C-beta framing relates intimately to the discrete Frenet framing. We also explain how inflection points (a.k.a. soliton centers) can be located in the loops and clarify their distinctive role in determining the loop structure of folded proteins.

National Category
Natural Sciences
Identifiers
urn:nbn:se:uu:diva-155912 (URN)10.1103/PhysRevE.83.061908 (DOI)000291703800005 ()
Available from: 2011-07-05 Created: 2011-07-04 Last updated: 2013-08-30Bibliographically approved
2. Energy functions for stringlike continuous curves, discrete chains, and space-filling one dimensional structures
Open this publication in new window or tab >>Energy functions for stringlike continuous curves, discrete chains, and space-filling one dimensional structures
2013 (English)In: Physical Review D, ISSN 1550-7998, E-ISSN 1550-2368, Vol. 87, no 10, 105011- p.Article in journal (Refereed) Published
Abstract [en]

The theory of string-like continuous curves and discrete chains have numerous important physical applications. Here we develop a general geometrical approach, to systematically derive Hamiltonian energy functions for these objects. In the case of continuous curves, we demand that the energy function must be invariant under local frame rotations, and it should also transform covariantly under reparametrizations of the curve. This leads us to consider energy functions that are constructed from the conserved quantities in the hierarchy of the integrable nonlinear Schrödinger equation. We point out the existence of a Weyl transformation that we utilize to introduce a dual hierarchy to the standard nonlinear Schrödinger equation hierarchy. We propose that the dual hierarchy is also integrable, and we confirm this to the first nontrivial order. In the discrete case the requirement of reparametrization invariance is void. But the demand of invariance under local frame rotations prevails, and we utilize it to introduce a discrete variant of the Zakharov-Shabat recursion relation. We use this relation to derive frame-independent quantities that we propose are the essentially unique and as such natural candidates for constructing energy functions for piecewise linear polygonal chains. We also investigate the discrete version of the Weyl duality transformation. We confirm that in the continuum limit the discrete energy functions go over to their continuum counterparts, including the perfect derivative contributions.

Place, publisher, year, edition, pages
APS, 2013
National Category
Physical Sciences
Research subject
Physics
Identifiers
urn:nbn:se:uu:diva-199941 (URN)10.1103/PhysRevD.87.105011 (DOI)000319117300008 ()
Available from: 2013-05-17 Created: 2013-05-17 Last updated: 2017-12-06Bibliographically approved
3. On bifurcations in framed curves and chains.
Open this publication in new window or tab >>On bifurcations in framed curves and chains.
(English)Manuscript (preprint) (Other academic)
Abstract [en]

A closed framed curve can be characterized by the value of its self-linking number. However, in general the self-linking number is not uniquely determined. In particular, it depends on the way how the curve has been framed. Moreover, the self-linking number can also change, when a bifurcation called perestroika takes place. Here we devise a simple Hamiltonian energy function to study perestroikas during the time evolution of a closed, piecewise linear polygonal chain. We analyze several examples to follow the progress of the discrete Frenet framing during the time evolution of the chain, and we observe how self-linking number changing perestroikas occur whenever there is an inflection point along the chain.

National Category
Physical Sciences
Research subject
Physics
Identifiers
urn:nbn:se:uu:diva-199983 (URN)
Available from: 2013-05-17 Created: 2013-05-17 Last updated: 2013-08-30
4. On Coarse Grained Representations And the Problem of Protein Backbone Reconstruction
Open this publication in new window or tab >>On Coarse Grained Representations And the Problem of Protein Backbone Reconstruction
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Crystallographic protein structures reveal that generically, only two of the Ramachandran angles are flexible. The third Ramachandran angle, all the backbone bond angles, and also all the covalent bond lengths are quite rigid, displaying only insignificant deviations from their optimal values. This empirical observation is among the rationale for the construction of coarse grained force fields where only a subset of the full set of atomic coordinates is utilized as dynamically active variables. The present article addresses in a systematic manner the question, to what extent the various angles andbond lengths can be replaced by their optimal values. In the case of bond lengths, it is found that the optimal values are in practice sufficient. But a coarse graining where a subset of angular variables is replaced by optimal values, commonly yields geometrically incorrect protein structures. There appears to be an inherent numerical instability, which seems to reflect the presence of a positive Liapunov exponent in the iterative reconstruction algorithm. Besides the full and complete set of individual atomic angles, essentially only one numerically stable coarse grained subset of angular variables is found. It consists of variable virtual Cα backbone bond and torsion angles. In combination with fixed, constant valued virtual bond lengths these two angles reproduce the original structure with high precision. The present observations impose strong limitations on the subset of backbone coordinates that can be utilized, even in principle, for the development of coarse grained force fields.

National Category
Physical Sciences
Research subject
Physical Biology
Identifiers
urn:nbn:se:uu:diva-199985 (URN)
Available from: 2013-05-17 Created: 2013-05-17 Last updated: 2013-08-30

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