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Discrete Frenet frame, inflection point solitons, and curve visualization with applications to folded proteins
Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics.
Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics.
Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics.
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.

Place, publisher, year, edition, pages
2011. Vol. 83, no 6, 061908- p.
National Category
Natural Sciences
Identifiers
URN: urn:nbn:se:uu:diva-155912DOI: 10.1103/PhysRevE.83.061908ISI: 000291703800005OAI: oai:DiVA.org:uu-155912DiVA: diva2:429650
Available from: 2011-07-05 Created: 2011-07-04 Last updated: 2013-08-30Bibliographically approved
In thesis
1. Bending, Twisting and Turning: Protein Modeling and Visualization from a Gauge-Invariance Viewpoint
Open this publication in new window or tab >>Bending, Twisting and Turning: Protein Modeling and Visualization from a Gauge-Invariance Viewpoint
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Proteins in nature fold to one dominant native structure. Despite being a heavily studied field, predicting the native structure from the amino acid sequence and modeling the folding process can still be considered unsolved problems. In this thesis I present a new approach to this problem with methods borrowed from theoretical physics. In the first part I show how it is possible to use a discrete Frenet frame to define the discrete curvature and torsion of the main chain of the protein. This method is then extended to the side chains as well. In particular I show how to use the discrete Frenet frame to produce a statistical distribution of angles that works in similar fashion as the commonly used Ramachandran plot and side chain rotamers. The discrete Frenet frame displays a gauge symmetry, in the choice of basis vectors on the normal plane, that is reminiscent of features of Abelian-Higgs theory. In the second part of the thesis I show how this similarity with Abelian-Higgs theory can be translated into an effective energy for a protein. The loops of the proteins are shown to correspond to solitons so that the whole protein can be constructed by gluing together any number of solitons. I present results of simulating proteins by minimizing the energy, starting from a real line or straight helix, where the correct native fold is attained. Finally the model is shown to display the same phase structure as real proteins.

 

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2012. 68 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 921
Keyword
protein folding, discrete frenet frame, solitons, protein visualization
National Category
Physical Sciences
Research subject
Physics and Astronomy specializing in Theoretical Physics
Identifiers
urn:nbn:se:uu:diva-172358 (URN)978-91-554-8338-8 (ISBN)
Public defence
2012-05-25, Å80101, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Opponent
Supervisors
Available from: 2012-05-04 Created: 2012-04-05 Last updated: 2012-08-01Bibliographically approved
2. Dynamics of Discrete Curves with Applications to Protein Structure
Open this publication in new window or tab >>Dynamics of Discrete Curves with Applications to Protein Structure
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
Frenet equations, integrable model, folded proteins, discrete curves
National Category
Physical Sciences Biophysics
Research subject
Physics; Physical Biology
Identifiers
urn:nbn:se:uu:diva-199987 (URN)978-91-554-8694-5 (ISBN)
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

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