Generalization of distance to higher dimensional objects

Abstract

The measurement of distance between two objects is generalized to the case where the objects are no longer points but are one-dimensional. Additional concepts such as nonextensibility, curvature constraints, and noncrossing become central to the notion of distance. Analytical and numerical results are given for some specific examples, and applications to biopolymers are discussed.T he distance, as conventionally defined between two zerodimensional objects (points) A and B at positions r A and r B , is the minimal arclength travelled in the transformation from A to B. A transformation r(t) between A and B is a vector function that may be parametrized by a scalar variable t: 0 Յ t Յ T, r(0) ϭ r A , r(T) ϭ r B , and the distance travelled is a functional of r(t). The (minimal) transformation r*(t) is an object of dimension one higher than A or B; i.e., it yields a distance that is one-dimensional. The distance D* is found through the variation of the functional (1):Here, ẋ ϭ dx/dt, ṙ ϭ dr/dt, and we have let g v ϭ ␦ v (Euclidean metric). The boundary conditions mentioned above are present at the end points of the integral. The Einstein summation convention will be used where convenient (e.g., eq. 1b); however, all the analysis here deals with spatial coordinates, ϭ 1, 2, 3, on a Euclidean metric. Generalizations to dimension higher than 3, as well as non-Euclidean metrics, are straightforward to incorporate into the formalism. On a Euclidean metric, the minimal distance becomes the diagonal of a hyper-cube. However, formulated as above, the solutions minimizing D are infinitely degenerate, because particles moving at various speeds but tracing the same trajectory over the total time T all give the same distance. To circumvent this problem, what is typically done is to let one of the space variables (e.g., x) become the independent variable. However, for higher dimensional objects or zero dimensional objects on a manifold with nontrivial topology, there is no guarantee that the dependent variables (y, z) constitute single valued functions of x. Alternatively, one can study the ''time'' trajectory of the parametric curve defined above, but under a gauge that fixes the speed to a constant, v o , for example. One can either fix the gauge from the outset with Lagrange multipliers, or choose a gauge that may simplify the problem after finding the extremum equations. The latter is often simpler in practice.To be specific, the effective Lagrangian appearing in the above problem is ͌ ṙ 2 , and the Euler-Lagrange (EL) equations arewith v the unit vector in the direction of the velocity. The boundary conditions are r*͑0͒ ϭ r A and r*͑T͒ ϭ r B.[3]Because the derivative of a unit vector is always orthogonal to that vector, Eq. 2 says that the direction of the velocity cannot change, and therefore straight line motion results. Applying the boundary conditions gives v ϭ (r B Ϫ r A )/͉r B Ϫ r A ͉. However, any function v(t) ϭ ͉v o (t)͉v satisfying the boundary conditions is a solution, so long as ͐ 0 In what follows, we generalize...

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

“…Such constraints may become particularly important for collapsed or semi-collapsed proteins, and knotted proteins where they restrict stereochemically-allowed folding pathways. These effects may in principle be treated by extending the present formalism to include non-zero chain thickness, and by extending the minimal folding pathway to the partition function of pathways, with each pathway having weight proportional to the exponent of the distance [72]. Such a treatment is an interesting and important topic of future work.…”

“…The beads of the chain follow straight trajectories from initial to final positions. This is an approximation to the actual Euclidean distance D of the transformation, where straight line transformations of the beads are generally preceded or proceeded by non-extensive local rotations to preserve the link length connecting the beads as a rigid constraint [72,73]. The instances of self-crossing along with their times are recorded.…”

“…The approximate transformation is carried out in a way to minimize deviations from the true transformation (D), such that link lengths are kept as constant as possible, given that all beads must follow straight-line motion. We thus only allow deviations from constant link length when rotations would be necessary to preserve it; this only occurs for a small fraction of the total trajectory, typically either at the beginning or the end of the transformation [72,73].…”

“…The minimal transformation inevitably involves curvilinear motion if bond, angle, or stereochemical constraints are involved [72,76]. Such curvilinear transformations as a result of bond constraints were developed in [72][73][74][75].…”

Minimal distance transformations between links and polymers: principles and examples

“…The standard variational definition of distance can be generalized to higher dimensional objects such as strings or membranes. In a previous paper [1], one of us has introduced the formalism for this calculation. Consider first zero-dimensional objects (points).…”