Stochastic line integrals and stream functions as metrics of irreversibility and heat transfer.

Stochastic line integrals are presented as a useful metric for quantitatively characterizing irreversibility and detailed balance violation in noise-driven dynamical systems. A particular realization is the stochastic area, recently studied in coupled electrical circuits. Here we provide a general framework for understanding properties of stochastic line integrals and clarify their implementation for experiments and simulations. For two-dimensional systems, stochastic line integrals can be expressed in terms of a stream function, the sign of which determines the orientation of nonequilibrium steady-state probability currents. Theoretical results are supported by numerical studies of an overdamped two-dimensional mass-spring system driven out of equilibrium. Additionally, the stream function permits analytical understanding of the scaling dependence of stochastic area growth rate on key parameters such as the noise strength for both linear and nonlinear springs.

Experimental metrics for detection of detailed balance violation.

We report on the measurement of detailed balance violation in a coupled, noise-driven linear electronic circuit consisting of two nominally identical RC elements that are coupled via a variable capacitance. The state variables are the time-dependent voltages across each of the two primary capacitors, and the system is driven by independent noise sources in series with each of the resistances. From the recorded time histories of these two voltages, we quantify violations of detailed balance by three methods: (1) explicit construction of the probability current density, (2) constructing the time-dependent stochastic area, and (3) constructing statistical fluctuation loops. In comparing the three methods, we find that the stochastic area is relatively simple to implement and computationally inexpensive and provides a highly sensitive means for detecting violations of detailed balance.

Fluctuation loops in noise-driven linear dynamical systems.

Understanding the spatiotemporal structure of most probable fluctuation pathways to rarely occurring states is a central problem in the study of noise-driven, nonequilibrium dynamical systems. When the underlying system does not possess detailed balance, the optimal fluctuation pathway to a particular state and relaxation pathway from that state may combine to form a looplike structure in the system phase space called a fluctuation loop. Here, fluctuation loops are studied in a linear circuit model consisting of coupled RC elements, where each element is driven by its own independent noise source. Using a stochastic Hamiltonian approach, we determine the optimal fluctuation pathways, and analytically construct corresponding fluctuation loops. To quantitatively characterize fluctuation loops, we study the time-dependent area tensor that is swept out by individual stochastic trajectories in the system phase space. At long times, the area tensor scales linearly with time, with a coefficient that precisely vanishes when the system satisfies detailed balance.

Steering most probable escape paths by varying relative noise intensities.

We demonstrate the possibility to systematically steer the most probable escape paths (MPEPs) by adjusting relative noise intensities in dynamical systems that exhibit noise-induced escape from a metastable point via a saddle point. With the use of a geometric minimum action approach, an asymptotic theory is developed that is broadly applicable to fast-slow systems and shows the important role played by the nullcline associated with the fast variable in locating the MPEPs. A two-dimensional quadratic system is presented which permits analytical determination of both the MPEPs and associated action values. Analytical predictions agree with computed MPEPs, and both are numerically confirmed by constructing prehistory distributions directly from the underlying stochastic differential equation.

Aggregation according to classical kinetics: from nucleation to coarsening.

A previous paper [Y. Farjoun and J. C. Neu, Phys. Rev. E 78, 051402 (2008)] presents a simple kinetic model of the initial creation transient, starting from pure monomer. During this transient the majority of clusters are created, and the distribution of cluster sizes that emerges from it is predicted to be discontinuous at the largest cluster size. It is well known that the further evolution according to the Lifshitz-Slyozov model of coarsening preserves this discontinuity. The result is at odds with the original proposal of Lifshitz and Slyozov, that the physical late-stage coarsening distribution is the smooth one. The current paper presents an analytic-numerical solution of the Lifshitz-Slyozov equations, starting from the discontinuous creation distribution. Of course, this analysis selects the discontinuous late-stage coarsening distribution, but there is much more. It resolves the intermediate stages between the creation transient and late-state coarsening and provides specific scales of time and cluster size that characterize the onset of coarsening.

Myxobacteria gliding motility requires cytoskeleton rotation powered by proton motive force.

Myxococcus xanthus is a Gram-negative bacterium that glides over surfaces without the aid of flagella. Two motility systems are used for locomotion: social-motility, powered by the retraction of type IV pili, and adventurous (A)-motility, powered by unknown mechanism(s). We have shown that AgmU, an A-motility protein, is part of a multiprotein complex that spans the inner membrane and periplasm of M. xanthus. In this paper, we present evidence that periplasmic AgmU decorates a looped continuous helix that rotates clockwise as cells glide forward, reversing its rotation when cells reverse polarity. Inhibitor studies showed that the AgmU helix rotation is driven by proton motive force (PMF) and depends on actin-like MreB cytoskeletal filaments. The AgmU motility complex was found to interact with MotAB homologs. Our data are consistent with a mechanochemical model in which PMF-driven motors, similar to bacterial flagella stator complexes, run along an endless looped helical track, driving rotation of the track; deformation of the cell surface by the AgmU-associated proteins creates pressure waves in the slime, pushing cells forward.

Activating function of needle electrodes in anisotropic tissue.

We present an analytical solution for the electrical potential and activating function (AF) established by cylindrical needle electrodes in anisotropic tissue. We compare this activating function to (1) AF computed assuming line-source electrodes and (2) AF computed using a finite element program. The results show that when the fiber is two needle diameters away from the electrodes, the maximum of the AF for needle electrodes is 1.43-times larger than for line-source electrodes, which results in lower thresholds for stimulation and electroporation. Therefore, for fibers that are close to the stimulating electrodes, one would benefit from using the formula that accounts for the electrodes' geometry.

Exhaustion of nucleation in a closed system.

We determine the distribution of cluster sizes that emerges from an initial phase of homogeneous aggregation with conserved total particle density. The physical ingredients behind the predictions are essentially classical: Supercritical nuclei are created at the Zeldovich rate, and before the depletion of monomers is significant, the characteristic cluster size is so large that the clusters undergo diffusion-limited growth. Mathematically, the distribution of cluster sizes satisfies an advection partial differential equation (PDE) in "size space." During this creation phase, clusters are nucleated and then grow much larger than the critical size, so nucleation of supercritical clusters at the Zeldovich rate is represented by an effective boundary condition at zero size. The advection PDE subject to the effective boundary condition constitutes a "creation signaling problem" for the evolving distribution of cluster sizes during the creation era. Dominant balance arguments applied to the advection signaling problem show that the characteristic time and cluster size of the creation era are exponentially large in the initial free-energy barrier against nucleation, G*. Specifically, the characteristic time is proportional to e(2/5)G*/kBT} and the characteristic number of monomers in a cluster is proportional to e(3/5)G*/kBT. The exponentially large characteristic time and cluster size give a posteriori validation of the mathematical signaling problem. In a short note, Marchenko [JETP Lett. 64, 66 (1996)] obtained these exponentials and the numerical prefactors 2/5 and 3/5. Our work adds the actual solution of the kinetic model implied by these scalings, and the basis for connection to subsequent stages of the aggregation process after the creation era.

Singular perturbation analysis of the pore creation transient.

Electroporation, in which electric pulses create transient pores in the cell membrane, is an important technique for drug and DNA delivery. Electroporation kinetics is mathematically described by an advection-diffusion boundary value problem. This study uses singular perturbation to derive a reduced description of the pore creation transient in the form of a single integrodifferential equation for the transmembrane voltage Vt. The number of pores and the distribution of their radii are computed from Vt. The analysis contains two nonstandard features: the use of the voltage deviation to peel away the strong exponential dependence of pore creation upon the transmembrane potential, and the autonomous approximation of the pore evolution. Comparing the predictions of the reduced equation with the simulations of the original problem demonstrates that this analysis allows one to predict with good accuracy the number and distribution of pores as a function of the electric pulse strength.

Model of creation and evolution of stable electropores for DNA delivery.

Electroporation, in which electric pulses create transient pores in the cell membrane, is becoming an important technique for gene therapy. To enable entry of supercoiled DNA into cells, the pores should have sufficiently large radii (>10 nm), remain open long enough for the DNA chain to enter the cell (milliseconds), and should not cause membrane rupture. This study presents a model that can predict such macropores. The distinctive features of this model are the coupling of individual pores through membrane tension and the electrical force on the pores, which is applicable to pores of any size. The model is used to explore the process of pore creation and evolution and to determine the number and size of pores as a function of the pulse magnitude and duration. Next, our electroporation model is combined with a heuristic model of DNA uptake and used to predict the dependence of DNA uptake on pulsing parameters. Finally, the model is used to examine the mechanism of a two-pulse protocol, which was proposed specifically for gene delivery. The comparison between experimental results and the model suggests that this model is well-suited for the investigation of electroporation-mediated DNA delivery.

Electrical energy required to form large conducting pores.

This study computes the contribution of the externally induced transmembrane potential to the energy of large, highly conductive pores. This work was undertaken because the pore energy formulas existing in the literature predict qualitatively different behavior of large pores: the original formula proposed by Abidor et al. in 1979 implies that the electrical force expanding the pore increases linearly with pore radius, while later extensions of this formula imply that this force decreases to zero for large pores. Starting from the Maxwell stress tensors, our study derives the formula for the mechanical work required to deform a dielectric body in an ionic solution with steady-state electric current. This formula is related to a boundary value problem (BVP) governing electric potentials and fields in a proximity of a pore. Computer simulations yield estimates of the electrical energy for pores of two different shapes: cylindrical and toroidal. In both cases, the energy increases linearly for pore radii above approximately 20 nm, implying that the electrical force expanding the pore asymptotes to a constant value for large pores. This result is different from either of the two energy formulas mentioned above. Our study traces the source of this disagreement to approximations made by previous studies, which are suitable only for small pores. Therefore, this study provides a better understanding of the energy of large pores, which is needed for designing pulsing protocols for DNA delivery.

Modeling postshock evolution of large electropores.

The Smoluchowski equation (SE), which describes the evolution of pores created by electric shocks, cannot be applied to modeling large and long-lived pores for two reasons: (1) it does not predict pores of radius above 20 nm without also predicting membrane rupture; (2) it does not predict postshock growth of pores. This study proposes a model in which pores are coupled by membrane tension, resulting in a nonlinear generalization of SE. The predictions of the model are explored using examples of homogeneous (all pore radii r are equal) and heterogeneous (0