Autophagy is a major catabolic pathway in eukaryotic cells whereby the lack of amino acids induces the formation of autophagosomes, double-bilayer membrane vesicles that mediate delivery of cytosolic proteins and organelles for lysosomal degradation. The biogenesis and turnover of autophagosomes in mammalian cells as well as the molecular mechanisms underlying induction of autophagy and trafficking of these vesicles are poorly understood. Here we utilized different autophagic markers to determine the involvement of microtubules in the autophagic process. We show that autophagosomes associate with microtubules and concentrate near the microtubule-organizing center. Moreover, we demonstrate that autophagosomes, but not phagophores, move along these tracks en route for degradation. Disruption of microtubules leads to a significant reduction in the number of mature autophagosomes but does not affect their life span or their fusion with lysosomes. We propose that microtubules serve to deliver only mature autophagosomes for degradation, thus providing a spatial barrier between phagophores and lysosomes.

In recent years there has been a resurgence of interest in Bohmian mechanics as a numerical tool because of its local dynamics, which suggest the possibility of significant computational advantages for the simulation of large quantum systems. However, closer inspection of the Bohmian formulation reveals that the nonlocality of quantum mechanics has not disappeared -it has simply been swept under the rug into the quantum force. In this paper we present a new formulation of Bohmian mechanics in which the quantum action, S, is taken to be complex. This leads to a single equation for complex S, and ultimately complex x and p but there is a reward for this complexificationa significantly higher degree of localization. The quantum force in the new approach vanishes for Gaussian wavepacket dynamics, and its effect on barrier tunneling processes is orders of magnitude lower than that of the classical force. We demonstrate tunneling probabilities that are in virtually perfect agreement with the exact quantum mechanics down to 10 −7 calculated from strictly localized quantum trajectories that do not communicate with their neighbors. The new formulation may have significant implications for fundamental quantum mechanics, ranging from the interpretation of nonlocality to measures of quantum complexity. PACS numbers:Ever since the advent of Quantum Mechanics, there has been a quest for a trajectory-based formulation of quantum theory that is exact. In the 1950's, David Bohm, building on earlier work by Madelung[1] and de Broglie [2], developed an exact formulation of quantum mechanics in which trajectories evolve in the presence of the usual Newtonian force plus an additional quantum force [3]. In recent years there has been a resurgence of interest in Bohmian mechanics (BM) as a numerical tool because of its local dynamics, which suggests the possibility of significant computational advantages for the simulation of large quantum systems [4,5,6,7,8,9,10,11]. However, closer inspection of the Bohmian formulation reveals that the non-locality of quantum mechanics has not disappeared -it has simply been swept under the rug into the quantum force. Particularly disturbing is the fact that for simple cases such as Gaussian wave packet dynamics of the free particle or the harmonic oscillator, where classical-quantum correspondence should be perfect, the quantum force is not only non-vanishing but is the same magnitude as the classical force [12].In this paper we present a new formulation of BM in which the quantum phase,S, is taken to be complex. This leads to a single equation for the complex phase, as opposed to coupled equations for real phase and real amplitude in the conventional BM. Complex phase leads to equations of motion for trajectories with complex x and p but there is a reward for this complexification -a significantly higher degree of localization than in conventional BM. We demonstrate tunneling probabilities that are in virtually perfect agreement with the exact quantum mechanics down to 10 −7 calculated from strict...

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We consider RCMS, a method for integrating differential equations of the form y = [ A + A 1 (t)]y with highly oscillatory solution. It is shown analytically and numerically that RCMS can accurately integrate problems using stepsizes determined only by the characteristic scales of A 1 (t), typically much larger than the solution "wavelength". In fact, for a given t grid the error decays with, or is independent of, increasing solution oscillation. RCMS consists of two basic steps, a transformation which we call the right correction and solution of the right correction equation using a Magnus series. With suitable methods of approximating the highly oscillatory integrals appearing therein, RCMS has high order of accuracy with little computational work. Moreover, RCMS respects evolution on a Lie group. We illustrate with application to the 1D Schrödinger equation and to Frenet-Serret equations. The concept of right correction integral series schemes is suggested and right correction Neumann schemes are discussed. Asymptotic analysis for a large class of ODEs is included which gives certain numerical integrators converging to exact asymptotic behaviour.

The dynamics of two electrons in a 2-dimensional quantum dot molecule in the presence of a time-dependent electromagnetic field is calculated from first principles. We show that carefully selected microwave pulses can exclusively populate a single state of the first excitation band and that the transition time can be further decreased by optimal pulse control. Finally we demonstrate that an oscillating charge localized state may be created by multiple transitions using a sequence of pulses.

We present a new semiclassical method that yields an approximation to the quantum mechanical wavefunction at a fixed, predetermined position. In the approach, a hierarchy of ODEs are solved along a trajectory with zero velocity. The new approximation is local, both literally and from a quantum mechanical point of view, in the sense that neighboring trajectories do not communicate with each other. The approach is readily extended to imaginary time propagation and is particularly useful for the calculation of quantities where only local information is required. We present two applications: the calculation of tunneling probabilities and the calculation of low energy eigenvalues. In both applications we obtain excellent agrement with the exact quantum mechanics, with a single trajectory propagation.Comment: 16 pages, 7 figure

Based on a novel point of view on 1-dimensional Gaussian quadrature, we present a new approach to the computation of d-dimensional cubature formulae. It is well known that the nodes of 1-dimensional Gaussian quadrature can be computed as eigenvalues of the so-called Jacobi matrix. The d-dimensional analog is that cubature nodes can be obtained from the eigenvalues of certain mutually commuting matrices. These are obtained by extending (adding rows and columns to) certain noncommuting matrices A 1 , . . . , A d , related to the coordinate operators x 1 , . . . , x d , in R d . We prove a correspondence between cubature formulae and "commuting extensions" of A 1 , . . . , A d , satisfying a compatibility condition which, in appropriate coordinates, constrains certain blocks in the extended matrices to be zero. Thus, the problem of finding cubature formulae can be transformed to the problem of computing (and then simultaneously diagonalizing) commuting extensions. We give a general discussion of existence and of the expected size of commuting extensions and describe our attempts at computing them, as well as examples of cubature formulae obtained using the new approach.

In their comment, Sanz and Miret-Artés (SMA) describe previous trajectory-based formalisms based on the quantum Hamilton-Jacobi (QHJ) formalism. In this reply, we highlight our unique contributions: the identification of the smallness of the quantum force in the complex QHJ and its solution using complex trajectories. SMA also raise the question of how the term locality should be used in quantum mechanics. We suggest that at least certain aspects of nonlocality can depend on the method used to solve the problem.