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, which 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 does reproduce the generalized Frenet equation. As an application we consider folded proteins, their Hausdorff dimension is known to be fractal. We explain how to employ the orientation of C β 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 β framing relates intimately to the discrete Frenet framing. We also explain how inflection points can be located in the loops, and clarify their distinctive rôle in determining the loop structure of foldel proteins. I. I: INTRODUCTIONThe visualization of a three dimensional discrete framed curve is an important and widely studied topic in computer graphics, from the association of ribbons and tubes to the determination of camera gaze directions along trajectories. Potential applications range from aircraft and robot kinematics to stereo reconstruction and virtual reality [1], [2].We are interested in addressing the problem of characterizing the physical laws that govern protein folding. For this we develop a technique for framing a general discrete and piecewise linear curve in a manner that will eventually enable us to combine the geometric problem of framing with an appropriate physical principle for frame determination. Our ultimate goal is to have an approach, where instead of purely geometric considerations the frames along a curve are determined directly from the properties of an underlying physical system. As a consequence we expect that our formalism and our results will find wide applicability well beyond the protein folding problem.The classical theory of continuous curves in three dimensional space employs the Frenet equation [1], [2] to determine a moving coordinate frame along a sufficiently differentiable space curve. However, if the curve has inflection points and/or straight segments or if it fails to be at least three times continuously differentiable, the Frenet frame becomes either discontinuous or may not even exist. In such cases there can be good reasons to consider the option to introduce an alternative framing such as Bishop's parallel transport frame [3], a geodetic reference frame or some possibly hybrid variants [1], [2].In this article we derive a discrete version of the Frenet equation that introduces a framing along an intrinsically discrete and piecewise linear curve in R 3 . We develop the general formalism for the visualization of such a curve without any underlying assumption that it approaches a continuous space curve in the...

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.

The enterobacteria lambda phage is a paradigm temperate bacteriophage. Its lysogenic and lytic life cycles echo competition between the DNA binding λ-repressor (CI) and CRO proteins. Here we scrutinize the structure, stability and folding pathways of the λ-repressor protein, that controls the transition from the lysogenic to the lytic state. We first investigate the super-secondary helix-loophelix composition of its backbone. We use a discrete Frenet framing to resolve the backbone spectrum in terms of bond and torsion angles. Instead of four, there appears to be seven individual loops. We model the putative loops using an explicit soliton Ansatz. It is based on the standard soliton profile of the continuum nonlinear Schrödinger equation. The accuracy of the Ansatz far exceeds the B-factor fluctuation distance accuracy of the experimentally determined protein configuration. We then investigate the folding pathways and dynamics of the λ-repressor protein. We introduce a coarse-grained energy function to model the backbone in terms of the Cα atoms and the side-chains in terms of the relative orientation of the C β atoms. We describe the folding dynamics in terms of relaxation dynamics, and find that the folded configuration can be reached from a very generic initial configuration. We conclude that folding is dominated by the temporal ordering of soliton formation. In particular, the third soliton should appear before the first and second. Otherwise, the DNA binding turn does not acquire its correct structure. We confirm the stability of the folded configuration by repeated heating and cooling simulations. I: INTRODUCTIONThe transition between the lysogenic and the lytic state in bacteriophage λ infected E. coli cell is the paradigm genetic switch mechanism. It is described in numerous molecular biology textbooks and review articles [1]- [7]. The interplay between the lysogeny maintaining λ-repressor (CI) protein and the CRO regulator protein that controls the transition to the lytic state is a simple model for more complex regulatory networks, including those that can lead to cancer in humans.In the present article we describe the physical properties of the λ-repressor protein, that controls the lysogenic-to-lytic transition. We investigate in detail the stability of its native conformation, the dynamics of the folding process, and the landscape of folding pathways. We find that the folded configuration displays a structure which is unique among all known protein structures. We also conclude that the folding pathways are entirely dominated by the loop regions. In particular, the temporal ordering of loop formation appears to be the decisive factor for the protein's ability to reach its native fold. If solitons form in a wrong order the protein may misfold.Full crystallographic information of the experimental λ-repressor structure that we use in our investigation is available in Protein Data Bank (PDB) [8] under the code 1LMB. This structure is a homo-dimer with 92 residues in each of the two monomers. It maintain...

Protein collapse from a random chain to the native state involves a dynamical phase transition. During the process, new scales and collective variables become excited while old ones recede and fade away. The presence of different phases and many scales causes formidable computational bottle-necks in approaches that are based on full atomic scale scrutiny. Here we propose a way to describe the folding and unfolding processes effectively, using only the biologically relevant time and distance scales. We merge a coarse grained Landau theory that models the static collapsed protein in the low-temperature limit with a Glauber protocol that describes finite-temperature relaxation dynamics in a statistical system which is out of thermal equilibrium. As an example we inspect the collapse of a HP35 chicken villin headpiece subdomain, a paradigm specimen in protein folding studies. We simulate the folding and unfolding process by repeated heating and cooling cycles between a given low-temperature, i.e. bad solvent, environment where the protein is collapsed and various different high-temperature, i.e. good solvent, environments. We find that as long as the high temperature value stays below a value in the range that separates the random walk phase from the self-avoiding walk phase, we consistently recover the native state upon cooling. But, when heated to sufficiently high temperatures, the native state practically never recurs. Our result confirms Anfinsen's thermodynamical hypothesis and estimates a temperature range for its validity, in the case of villin.

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.

Topological methods are indispensable in theoretical studies of particle physics, condensed matter physics, and gravity. These powerful techniques have also been applied to biological physics. For example, knowledge of DNA topology is pivotal to the understanding as to how living cells function. Here, the biophysical repertoire of topological methods is extended, with the aim to understand and characterize the global structure of a folded protein. For this, the elementary concept of winding number of a vector field on a plane is utilized to introduce a topological quantity called the folding index of a crystallographic protein. It is observed that in the case of high resolution protein crystals, the folding index, when evaluated over the entire length of the crystallized protein backbone, has a very clear and strong propensity towards integer values. The observation proposes that the way how a protein folds into its biologically active conformation is a structural self-organization process with a topological facet that relates to the concept of solitons. It is proposed that the folding index has a potential to become a useful tool for the global, topological characterization of the folding pathways.

Folded proteins have a modular assembly. They are constructed from regular secondary structures like α-helices and β-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 twosphere. Unlike the more conventional approaches that only describe 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 Cα 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 non-glycyl amino acids. This region corresponds to an isolated and highly localized residue independent sector in the direction of the C β carbons on the two-sphere. We show that the residue independent localization extends to Cγ and C δ 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 β-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.

We investigate the fine structure of the sp3 hybridized covalent bond geometry that governs the tetrahedral architecture around the central Cα carbon of a protein backbone, and for this we develop new visualization techniques to analyze high resolution X-ray structures in Protein Data Bank. We observe that there is a correlation between the deformations of the ideal tetrahedral symmetry and the local secondary structure of the protein. We propose a universal coarse grained energy function to describe the ensuing side-chain geometry in terms of the C β carbon orientations. The energy function can model the side-chain geometry with a sub-atomic precision. As an example we construct the Cα-C β structure of HP35 chicken villin headpiece. We obtain a configuration that deviates less than 0.4Ȧ in root-mean-square distance from the experimental X-ray structure.