Minimal distance transformations between links and polymers: Principles and examples

The calculation of Euclidean distance between points is generalized to one-dimensional objects such as strings or polymers. Necessary and sufficient conditions for the minimal transformation between two polymer configurations are derived. Transformations consist of piecewise rotations and translations subject to Weierstrass-Erdmann corner conditions. Numerous examples are given for the special cases of one and two links. The transition to a large number of links is investigated, where the distance converges to the polymer length times the mean root square distance (MRSD) between polymer configurations, assuming curvature and non-crossing constraints can be neglected. Applications of this metric to protein folding are investigated. Potential applications are also discussed for structural alignment problems such as pharmacophore identification, and inverse kinematic problems in motor learning and control.Comment: Submitted to J. Phys.:Condens. Matte

A Trapped Fermi Gas Model of a Black Hole in a Non-Commutative Spacetime

Previously, we introduced a non-commutative spacetime, with the quantized states filled with non-interacting fermionic quasiparticles. An occupation number of one signifies the possibility of measuring an event whose coordinates are associated with the quantum numbers of the occupied state. An occupation number of zero corresponds to a hole in the Fermi sea, which in turn signifies the impossibility of measuring an event with the corresponding coordinates in the spacetime. Here, we model a black hole as a trapped ideal Fermi gas of holes in the Fermi sea. The Fermi surface of the hole gas corresponds to the event horizon of the black hole. We calculate the entropy of the Fermi gas at a sufficiently low temperature and recover the Bekenstein-Hawking value for the entropy in the thermodynamic limit. We observe, as expected, that the entropy comes from fluctuations at the Fermi surface

Proper Time and Distance Quantization and Applications to Black Holes

We introduce a non-commutative spacetime, after providing some physical motivation for the choice of noncommutativity. We then define a simple quantum mechanical system based on the noncommuting operators. We observe that the simple system has features reminiscent of black hole thermodynamics, as it gives rise to discrete area patches at quantized proper distances

Entanglement for Thee, Collapse for Me: An Interpretation of Quantum Mechanics Based on the Relativity of Information

We propose an information-centric interpretation of quantum mechanics. According to this interpretation, a quantum mechanical system, $A$, can always choose a basis in which its own state is an eigenstate; it never sees itself in a superposition. This holds true even if, from the perspective of another system $B$, $A$ is entangled with system $S$. In this case, $A$ and $S$ are in eigenstates (in their own systems) with their counterpart subsystem wave functions collapsed in the entangled degrees of freedom, but they are entangled according to $B$. This interpretation does not introduce new physics, and is consistent with the experimental predictions of quantum mechanics. At the same time, it avoids many ambiguities and potential inconsistencies associated with other interpretations. We further clarify concepts of measurement, macroscopic superposition, entanglement, and the dichotomy between the classical and quantum realms, in accordance with our interpretation

Minimal Folding Pathways for Coarse-Grained Biopolymer Fragments

ABSTRACT The minimal folding pathway or trajectory for a biopolymer can be defined as the transformation that minimizes the total distance traveled between a folded and an unfolded structure. This involves generalizing the usual Euclidean distance from points to one-dimensional objects such as a polymer. We apply this distance here to find minimal folding pathways for several candidate protein fragments, including the helix, the b-hairpin, and a nonplanar structure where chain noncrossing is important. Comparing the distances traveled with root mean-squared distance and mean root-squared distance, we show that chain noncrossing can have large effects on the kinetic proximity of apparently similar conformations. Structures that are aligned to the b-hairpin by minimizing mean root-squared distance, a quantity that closely approximates the true distance for long chains, show globally different orientation than structures aligned by minimizing root mean-squared distance

Polymer Uncrossing and Knotting in Protein Folding, and Their Role in Minimal Folding Pathways

We introduce a method for calculating the extent to which chain non-crossing is important in the most efficient, optimal trajectories or pathways for a protein to fold. This involves recording all unphysical crossing events of a ghost chain, and calculating the minimal uncrossing cost that would have been required to avoid such events. A depth-first tree search algorithm is applied to find minimal transformations to fold a, b, a=b, and knotted proteins. In all cases, the extra uncrossing/non-crossing distance is a small fraction of the total distance travelled by a ghost chain. Different structural classes may be distinguished by the amount of extra uncrossing distance, and the effectiveness of such discrimination is compared with other order parameters. It was seen that non-crossing distance over chain length provided the best discrimination between structural and kinetic classes. The scaling of non-crossing distance with chain length implies an inevitable crossover to entanglement-dominated folding mechanisms for sufficiently long chains. We further quantify the minimal folding pathways by collecting the sequence of uncrossing moves, which generally involve leg, loop, and elbowlike uncrossing moves, and rendering the collection of these moves over the unfolded ensemble as a multipletransformation ‘‘alignment’’. The consensus minimal pathway is constructed and shown schematically for representative cases of an a, b, and knotted protein. An overlap parameter is defined between pathways; we find that a proteins have minimal overlap indicating diverse folding pathways, knotted proteins are highly constrained to follow a dominan