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Bending, Twisting and Turning: Protein Modeling and Visualization from a Gauge-Invariance Viewpoint
Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics.
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. , p. 68
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 921
Keyword [en]
protein folding, discrete frenet frame, solitons, protein visualization
National Category
Physical Sciences
Research subject
Physics and Astronomy specializing in Theoretical Physics
Identifiers
URN: urn:nbn:se:uu:diva-172358ISBN: 978-91-554-8338-8 (print)OAI: oai:DiVA.org:uu-172358DiVA, id: diva2:514160
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
List of papers
1. Gauge field theory of chirally folded homopolymers with applications to folded proteins
Open this publication in new window or tab >>Gauge field theory of chirally folded homopolymers with applications to folded proteins
2010 (English)In: Physical Review E, ISSN 1539-3755, Vol. 82, no 2, p. 021910-Article in journal (Refereed) Published
Abstract [en]

We combine the principle of gauge invariance with extrinsic string geometry to develop a lattice model that can be employed to theoretically describe properties of chiral, unbranched homopolymers. We find that in its low temperature phase the model is in the same universality class with proteins that are deposited in the Protein Data Bank, in the sense of the compactness index. We apply the model to analyze various statistical aspects of folded proteins. Curiously we find that it can produce results that are a very good good match to the data in the Protein Data Bank.

National Category
Physical Sciences
Identifiers
urn:nbn:se:uu:diva-134959 (URN)10.1103/PhysRevE.82.021910 (DOI)000280826800006 ()
Available from: 2010-12-07 Created: 2010-12-03 Last updated: 2012-08-01Bibliographically approved
2. Elastic energy and phase structure in a continuous spin Ising chain with applications to chiral homopolymers
Open this publication in new window or tab >>Elastic energy and phase structure in a continuous spin Ising chain with applications to chiral homopolymers
2011 (English)In: Physical Review E - Statistical, Nonlinear and Soft Matter Physics, ISSN 1539-3755, Vol. 83, no 1, p. 011126-Article in journal (Refereed) Published
Abstract [en]

We present a numerical Monte Carlo analysis of the phase structure in a continuous spin Ising chain that describes chiral homopolymers. We find that depending on the value of the Metropolis temperature, the model displays the three known nontrivial phases of polymers: At low temperatures the model is in a collapsed phase, at medium temperatures it is in a random walk phase, and at high temperatures it enters the self-avoiding random walk phase. By investigating the temperature dependence of the specific energy we confirm that the transition between the collapsed phase and the random walk phase is a phase transition, while the random walk phase and self-avoiding random walk phase are separated from each other by a crossover transition. We propose that the model can be applied to characterize the statistical properties of protein folding. For this we compare the predictions of the model to a phenomenological elastic energy formula, proposed by J. Lei and K. Huang [e-print arXiv:1002.5013; Europhys. Lett. 88, 68004 (2009)] to describe folded proteins.

National Category
Physical Sciences
Identifiers
urn:nbn:se:uu:diva-147775 (URN)10.1103/PhysRevE.83.011126 (DOI)000286761300003 ()
Available from: 2011-03-01 Created: 2011-02-28 Last updated: 2012-08-01Bibliographically approved
3. 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, p. 061908-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
4. Protein loops, solitons and side-chain visualization with applications to the left-handed helix region
Open this publication in new window or tab >>Protein loops, solitons and side-chain visualization with applications to the left-handed helix region
2012 (English)In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, ISSN 1539-3755, E-ISSN 1550-2376, Vol. 85, no 6, p. 061909-Article in journal (Refereed) Published
Abstract [en]

Folded proteins have a modular assembly. They are constructed from regular secondary structures like alpha helices and beta strands that are joined together by loops. Here we develop a visualization technique that is adapted to describe this modular structure. In complement to the widely employed Ramachandran plot that is based on toroidal geometry, our approach utilizes the geometry of a two sphere. Unlike the more conventional approaches that describe only a given peptide unit, ours is capable of describing the entire backbone environment including the neighboring peptide units. It maps the positions of each atom to the surface of the two-sphere exactly how these atoms are seen by an observer who is located at the position of the central C-alpha atom. At each level of side-chain atoms we observe a strong correlation between the positioning of the atom and the underlying local secondary structure with very little if any variation between the different amino acids. As a concrete example we analyze the left-handed helix region of nonglycyl amino acids. This region corresponds to an isolated and highly localized residue independent sector in the direction of the C-beta carbons on the two-sphere. We show that the residue independent localization extends to C gamma and C-delta carbons and to side-chain oxygen and nitrogen atoms in the case of asparagine and aspartic acid. When we extend the analysis to the side-chain atoms of the neighboring residues, we observe that left-handed beta turns display a regular and largely amino acid independent structure that can extend to seven consecutive residues. This collective pattern is due to the presence of a backbone soliton. We show how one can use our visualization techniques to analyze and classify the different solitons in terms of selection rules that we describe in detail.

National Category
Condensed Matter Physics Structural Biology
Identifiers
urn:nbn:se:uu:diva-172355 (URN)10.1103/PhysRevE.85.061909 (DOI)000305128000007 ()
Available from: 2012-04-05 Created: 2012-04-05 Last updated: 2017-12-07Bibliographically approved
5. Backbone covalent bond dynamical symmetry breaking and side-chain geometry of folded proteins
Open this publication in new window or tab >>Backbone covalent bond dynamical symmetry breaking and side-chain geometry of folded proteins
(English)Article in journal (Other academic) Submitted
National Category
Condensed Matter Physics Structural Biology
Identifiers
urn:nbn:se:uu:diva-172357 (URN)
Available from: 2012-04-05 Created: 2012-04-05 Last updated: 2012-08-01Bibliographically approved

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