Numerical Solution of Differential Equations by the Parker-Sochacki Method

A tutorial is presented which demonstrates the theory and usage of the Parker-Sochacki method of numerically solving systems of differential equations. Solutions are demonstrated for the case of projectile motion in air, and for the classical Newtonian N-body problem with mutual gravitational attraction.Comment: Added in July 2010: This tutorial has been posted since 1998 on a university web site, but has now been cited and praised in one or more refereed journals. I am therefore submitting it to the Cornell arXiv so that it may be read in response to its citations. See "Spiking neural network simulation: numerical integration with the Parker-Sochacki method:" J. Comput Neurosci, Robert D. Stewart & Wyeth Bair and http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2717378

Mapping of functionalized regions on carbon nanotubes by scanning tunneling microscopy

Scanning tunneling microscopy (STM) gives us the opportunity to map the surface of functionalized carbon nanotubes in an energy resolved manner and with atomic precision. But this potential is largely untapped, mainly due to sample stability issues which inhibit reliable measurements. Here we present a simple and straightforward solution that makes away with this difficulty, by incorporating the functionalized multiwalled carbon nanotubes (MWCNT) into a few layer graphene - nanotube composite. This enabled us to measure energy resolved tunneling conductance maps on the nanotubes, which shed light on the level of doping, charge transfer between tube and functional groups and the dependence of defect creation or functionalization on crystallographic orientation.Comment: Keywords: functionalization, carbon nanotubes, few layer graphene, STM, CITS, ST

MyosinVIIa Interacts with Twinfilin-2 at the Tips of Mechanosensory Stereocilia in the Inner Ear

In vertebrates hearing is dependent upon the microvilli-like mechanosensory stereocilia and their length gradation. The staircase-like organization of the stereocilia bundle is dynamically maintained by variable actin turnover rates. Two unconventional myosins were previously implicated in stereocilia length regulation but the mechanisms of their action remain unknown. MyosinXVa is expressed in stereocilia tips at levels proportional to stereocilia length and its absence produces staircase-like bundles of very short stereocilia. MyosinVIIa localizes to the tips of the shorter stereocilia within bundles, and when absent, the stereocilia are abnormally long. We show here that myosinVIIa interacts with twinfilin-2, an actin binding protein, which inhibits actin polymerization at the barbed end of the filament, and that twinfilin localization in stereocilia overlaps with myosinVIIa. Exogenous expression of myosinVIIa in fibroblasts results in a reduced number of filopodia and promotes accumulation of twinfilin-2 at the filopodia tips. We hypothesize that the newly described interaction between myosinVIIa and twinfilin-2 is responsible for the establishment and maintenance of slower rates of actin turnover in shorter stereocilia, and that interplay between complexes of myosinVIIa/twinfilin-2 and myosinXVa/whirlin is responsible for stereocilia length gradation within the bundle staircase

Lipid-dependent gating of a voltage-gated potassium channel

Recent studies hypothesized that phospholipids stabilize two voltage-sensing arginine residues of certain voltage-gated potassium channels in activated conformations. It remains unclear how lipids directly affect these channels. Here, by examining the conformations of the KvAP in different lipids, we showed that without voltage change, the voltage-sensor domains switched from the activated to the resting state when their surrounding lipids were changed from phospholipids to nonphospholipids. Such lipid-determined conformational change was coupled to the ion-conducting pore, suggesting that parallel to voltage gating, the channel is gated by its annular lipids. Our measurements recognized that the energetic cost of lipid-dependent gating approaches that of voltage gating, but kinetically it appears much slower. Our data support that a channel and its surrounding lipids together constitute a functional unit, and natural nonphospholipids such as cholesterol should exert strong effects on voltage-gated channels. Our first observation of lipid-dependent gating may have general implications to other membrane proteins

Voltage-Dependent Gating in a “Voltage Sensor-Less” Ion Channel

An unusual mechanism of ion channel regulation generates voltage-dependent gating in the absence of a canonical voltage-sensing domain

I–II Loop Structural Determinants in the Gating and Surface Expression of Low Voltage-Activated Calcium Channels

The intracellular loops that interlink the four transmembrane domains of Ca2+- and Na+-channels (Cav, Nav) have critical roles in numerous forms of channel regulation. In particular, the intracellular loop that joins repeats I and II (I–II loop) in high voltage-activated (HVA) Ca2+ channels possesses the binding site for Cavβ subunits and plays significant roles in channel function, including trafficking the α1 subunits of HVA channels to the plasma membrane and channel gating. Although there is considerable divergence in the primary sequence of the I–II loop of Cav1/Cav2 HVA channels and Cav3 LVA/T-type channels, evidence for a regulatory role of the I–II loop in T-channel function has recently emerged for Cav3.2 channels. In order to provide a comprehensive view of the role this intracellular region may play in the gating and surface expression in Cav3 channels, we have performed a structure-function analysis of the I–II loop in Cav3.1 and Cav3.3 channels using selective deletion mutants. Here we show the first 60 amino acids of the loop (post IS6) are involved in Cav3.1 and Cav3.3 channel gating and kinetics, which establishes a conserved property of this locus for all Cav3 channels. In contrast to findings in Cav3.2, deletion of the central region of the I–II loop in Cav3.1 and Cav3.3 yielded a modest increase (+30%) and a reduction (−30%) in current density and surface expression, respectively. These experiments enrich our understanding of the structural determinants involved in Cav3 function by highlighting the unique role played by the intracellular I–II loop in Cav3.2 channel trafficking, and illustrating the prominent role of the gating brake in setting the slow and distinctive slow activation kinetics of Cav3.3

Optoelectronic and Excitonic Properties of Oligoacenes: Substantial Improvements from Range-Separated Time-Dependent Density Functional Theory

The optoelectronic and excitonic properties in a series of linear acenes (naphthalene up to heptacene) are investigated using range-separated methods within time-dependent density functional theory (TDDFT). In these rather simple systems, it is well-known that TDDFT methods using conventional hybrid functionals surprisingly fail in describing the low-lying La and Lb valence states, resulting in large, growing errors for the La state and an incorrect energetic ordering as a function of molecular size. In this work, we demonstrate that the range-separated formalism largely eliminates both of these errors and also provides a consistent description of excitonic properties in these systems. We further demonstrate that re-optimizing the percentage of Hartree-Fock exchange in conventional hybrids to match wavefunction-based benchmark calculations still yields serious errors, and a full 100% Hartree-Fock range separation is essential for simultaneously describing both of the La and Lb transitions. Based on an analysis of electron-hole transition density matrices, we finally show that conventional hybrid functionals overdelocalize excitons and underestimate quasiparticle energy gaps in the acene systems. The results of our present study emphasize the importance of both a range-separated and asymptotically-correct contribution of exchange in TDDFT for investigating optoelectronic and excitonic properties, even for these simple valence excitations.Comment: Accepted by the Journal of Chemical Theory and Computatio

Fluorescence-Tracking of Activation Gating in Human ERG Channels Reveals Rapid S4 Movement and Slow Pore Opening

Background: hERG channels are physiologically important ion channels which mediate cardiac repolarization as a result of their unusual gating properties. These are very slow activation compared with other mammalian voltage-gated potassium channels, and extremely rapid inactivation. The mechanism of slow activation is not well understood and is investigated here using fluorescence as a direct measure of S4 movement and pore opening. Methods and Findings: Tetramethylrhodamine-5-maleimide (TMRM) fluorescence at E519 has been used to track S4 voltage sensor movement, and channel opening and closing in hERG channels. Endogenous cysteines (C445 and C449) in the S1–S2 linker bound TMRM, which caused a 10 mV hyperpolarization of the VK of activation to 227.562.0 mV, and showed voltage-dependent fluorescence signals. Substitution of S1–S2 linker cysteines with valines allowed unobstructed recording of S3–S4 linker E519C and L520C emission signals. Depolarization of E519C channels caused rapid initial fluorescence quenching, fit with a double Boltzmann relationship, F-VON, with VK,1 = 237.861.7 mV, and VK,2 = 43.567.9 mV. The first phase, VK,1, was,20 mV negative to the conductance-voltage relationship measured from ionic tail currents (G-VK = 218.361.2 mV), and relatively unchanged in a non-inactivating E519C:S620T mutant (V K = 234.461.5 mV), suggesting the fast initial fluorescence quenching tracked S4 voltage sensor movement. The second phase of rapid quenching was absent in the S620T mutant. The E519C fluorescence upon repolarizatio