Philipp Reiter

Energetics and dynamics of global integrals modeling interaction between stiff filaments

The attractive and spacing interaction between pairs of filaments via cross-linkers, e.g. myosin oligomers connecting actin filaments, is modeled by global integral kernels for negative binding energies between two intersecting stiff and long rods in a (projected) two-dimensional situation, for simplicity. Whereas maxima of the global energy functional represent intersection angles of ‘minimal contact’ between the filaments, minima are approached for energy values tending to −∞, representing the two degenerate states of parallel and anti-parallel filament alignment. Standard differential equations of negative gradient flow for such energy functionals show convergence of solutions to one of these degenerate equilibria in finite time, thus called ‘super-stable’ states. By considering energy variations under virtual rotation or translation of one filament with respect to the other, integral kernels for the resulting local forces parallel and orthogonal to the filament are obtained. For the special modeling situation that these variations only activate ‘spring forces’ in direction of the cross-links, explicit formulas for total torque and translational forces are given and calculated for typical examples. Again, the two degenerate alignment states are locally ‘super-stable’ equilibria of the assumed over-damped dynamics, but also other stable states of orthogonal arrangement and different asymptotic behavior can occur. These phenomena become apparent if stochastic perturbations of the local force kernels are implemented as additive Gaussian noise induced by the cross-link binding process with appropriate scaling. Then global filament dynamics is described by a certain type of degenerate stochastic differential equations yielding asymptotic stationary processes around the alignment states, which have generalized, namely bimodal Gaussian distributions. Moreover, stochastic simulations reveal characteristic sliding behavior as it is observed for myosin-mediated interaction between actin filaments. Finally, the forgoing explicit and asymptotic analysis as well as numerical simulations are extended to the more realistic modeling situation with filaments of finite length.

Regularity theory for tangent-point energies: The non-degenerate sub-critical case

Abstract In this article we introduce and investigate a new two-parameter family of knot energies TP (p,q)

The Elastic Trefoil is the Doubly Covered Circle

{\operatorname{TP}^{(p,\,q)}}

Modeling repulsive forces on fibres via knot energies

that contains the tangent-point energies. These energies are obtained by decoupling the exponents in the numerator and denominator of the integrand in the original definition of the tangent-point energies. We will first characterize the curves of finite energy TP (p,q)

Harmonic analysis meets critical knots. Critical points of the Möbius energy are smooth

{\operatorname{TP}^{(p,\,q)}}

Repulsive knot energies and pseudodifferential calculus for O’Hara’s knot energy family E(α), α ∈ [2, 3)

in the sub-critical range p ∈ (q+2,2q+1) and see that those are all injective and regular curves in the Sobolev–Slobodeckiĭ space W (p-1)/q,q (ℝ/ℤ,ℝ n )

Regularity theory for the Möbius energy

{W^{\scriptstyle (p-1)/q,q}(\mathbb {R}/\mathbb {Z},\mathbb {R}^n)}

Stationary points of O’Hara’s knot energies

. We derive a formula for the first variation that turns out to be a non-degenerate elliptic operator for the special case q = 2: a fact that seems not to be the case for the original tangent-point energies. This observation allows us to prove that stationary points of TP (p,2)

DOES FINITE KNOT ENERGY LEAD TO DIFFERENTIABILITY

\operatorname{TP}^{(p,2)}

TOWARDS A REGULARITY THEORY FOR INTEGRAL MENGER CURVATURE

+ λ length, p ∈ (4,5), λ > 0, are smooth – so especially all local minimizers are smooth.