Current Flow in a Cylindrical Nanopore with an Object-Implications for Virus Sensing.

Interest is growing in nanopores as real-time, low-cost, label-free virus size sensors. To optimize their performance, we evaluate how external electric field and ion concentrations and pore wall charges influence currents and object (disk) radius-current relationship using simulations. The physics was described using the Poisson-Nernst-Planck and Navier-Stokes equations. In a charged cylindrical nanopore with a charged disk, elevated external electric field produces higher (and polarity independent) ion concentrations and greater ion current (largely migratory). Elevated external ion concentrations also lead to higher concentrations (mainly away from the pore wall), greater axial electric field especially in the disk-pore wall space, and finally larger current. At low concentrations, current is disk radius independent. The current rises as concentrations increase. Interestingly, the rise is greater for larger disks (except when the pore is blocked mechanically). Smaller cross-sectional area for current flow or volume exclusion of electrolyte by object thus cannot be universally accepted as explanations of current blockage. Ion current rises when pore wall charge density increases, but its direction is independent of charge sign. Current-disk radius relationship is also independent of pore wall charge sign. If the pore wall and disk charges have the same sign, larger current with bigger disk is due to higher counter-ion accumulation in the object-pore wall space. However, if their signs are opposite, it is largely due to elevated axial electric field in the object-pore wall space. Finally in uncharged nanopores, current diminishes when disk radius increases making them better sensors of virus size.

Axial forces at disk surfaces in a cylindrical nanopore.

Understanding the physics of object translocation in nanopores is critical for using nanopores as sensors of molecular properties and as object size and shape sensors. Based on Poisson-Nernst-Planck and Navier-Stokes simulations we dissect three axial pressures and forces at disk edges (upper, lower and rim) - Coulomb, dielectric and fluidic. Axial Coulomb and dielectric rim forces are small and cancel each other. Upper and lower axial forces are largely controlled by the external axial electric field and interestingly by the pore wall charges that determine the amplitude and direction of axial combined force. Axial total Coulomb force (sum of its upper and lower edge components) makes the greatest contribution, but the axial total dielectric force (calculated using Maxwell stress tensor), which opposes it is surprisingly large. External ion concentration alters Coulomb and axial dielectric forces but influences only their amplitude. Axial total fluidic force is near zero (its upper and lower disk edge components are significant but cancel each other) regardless of external electric field, but pore wall charges and external fluidic pressure can alter it. Modest changes of external electric field or concentration produce axial forces comparable to those produced by large external fluidic pressures. Axial forces depend little on disk's axial position. Finally, mean axial pressures (calculated to compare forces acting on disks of different radius) are greater for larger disks.

Effect of ion concentration, solution and membrane permittivity on electric energy storage and capacitance.

Bio-membranes as capacitors store electric energy, but their permittivity is low whereas the permittivity of surrounding solution is high. To evaluate the effective capacitance of the membrane/solution system and determine the electric energy stored within the membrane and in the solution, we estimated their electric variables using Poisson-Nernst-Planck simulations. We calculated membrane and solution capacitances from stored electric energy. The effective capacitance was calculated by fitting a six-capacitance model to charges (fixed and ion) and associated potentials, because it cannot be considered as a result of membrane and solution capacitance in series. The electric energy stored within the membrane (typically much smaller than that in the solution), depends on the membrane permittivity, but also on the external electric field, surface charge density, water permittivity and ion concentration. The effect on capacitances is more specific. Solution capacitance rises with greater solution permittivity or ion concentration, but the membrane capacitance (much smaller than solution capacitance) is only influenced by its permittivity. Interestingly, the effective capacitance is independent of membrane or solution permittivity, but rises as the ion concentration increases and surface charge becomes positive. Experimental estimates of membrane capacitance are thus not necessarily a reliable index of its surface area.

Spatial profiles of potential, ion concentration and flux in short unipolar and bipolar nanopores.

During release of vesicular content the resistance of the fusion pore sometimes changes rapidly and repeatedly. However, it is not clear why the pore 'flickers'. Engineered nanopores often rectify, but how different factors influence the rectification requires clarification. To better understand the ionic 'causes' of pore conductivity and its changes we simulated ion transport through a short nanopore using Poisson-Nernst-Planck equations, coupling it to the transport of water using Navier-Stokes equations. We extracted the potential, concentration, and ion flux profiles. In uniformly charged nanopores the voltage bias determines which counter-ion flux dominates, and it is carried by the counter-ions of the highest concentration. In unipolar nanopores this simple rule breaks down. The dominant counter-ion in the charged half is from the adjacent compartment, but the bias determines what counter-ion flux is dominant--the same ion (regular bias), or a different and smaller (reverse bias), and this difference determines the level of rectification. In bipolar nanopores the dominant counter-ions in each half are from the adjacent compartments, and the total ion concentration dips in the middle near the wall. With regular bias the total ion concentration peaks in the pore center; the ions that carry the current through the nanopore start as counter-ions and their fluxes are large. With reverse bias the total concentration dips near the wall and in the center, both dominant ion fluxes through the nanopore start as co-ions and are very small, whereas those starting as counter-ions do not go through.

Temperature dependence of vesicular dynamics at excitatory synapses of rat hippocampus.

How vesicular dynamics parameters depend on temperature and how temperature affects the parameter change during prolonged high frequency stimulation was determined by fitting a model of vesicular storage and release to the amplitudes of the excitatory post-synaptic currents (EPSC) recorded from CA1 neurons in rat hippocampal slices. The temperature ranged from low (13 °C) to higher and more physiological temperature (34 °C). Fitting the model of vesicular storage and release to the EPSC amplitudes during a single pair of brief high-low frequency stimulation trains yields the estimates of all parameters of the vesicular dynamics, and with good precision. Both fractional release and replenishment rate decrease as the temperature rises. Change of the underlying 'basic' parameters (release coupling, replenishment coupling and readily releasable pool size), which the model-fitting also yields is complex. The replenishment coupling between the readily releasable pool (RRP) and resting pool increases with temperature (which renders the replenishment rate higher), but this is more than counterbalanced by greater RRP size (which renders the replenishment rate lower). Finally, during long, high frequency patterned stimulation that leads to significant synaptic depression, the replenishment rate decreases markedly and rapidly at low temperatures (<22 °C), but at high temperatures (>28 °C) the replenishment rate rises with stimulation, making synapses better able to maintain synaptic efficacy.

Is replenishment of the readily releasable pool associated with vesicular movement?

At the excitatory synapse of rat hippocampus the short-term synaptic depression observed during long high-frequency stimulation is associated with slower replenishment of the readily-releasable pool. Given that the replenishment rate is also not [Ca(++)]o sensitive this puts into question a widely held notion that the vesicles-constrained by the cytoskeleton and rendered free from such constraints by Ca(++) entry that renders them more mobile-are important in the replenishment of the readily-releasable pool. This raises a question-Is vesicular replenishment of the readily releasable pool associated with significant movement? To answer this question we evaluated how okadaic acid and staurosporine (compounds known to affect vesicular mobility) influence the replenishment rate. We used patterned stimulation on the Schaffer collateral fiber pathway and recorded the excitatory post-synaptic currents (EPSCs) from rat CA1 neurons, in the absence and presence of these drugs. The parameters of a circuit model with two vesicular pools were estimated by minimizing the squared difference between the ESPC amplitudes and simulated model output. [Ca(2+)]o did not influence the progressive decrease of the replenishment rate during long, high frequency stimulation. Okadaic acid did not significantly affect any parameters of the vesicular storage and release system, including the replenishment rate. Staurosporine reduced the replenishment coupling, but not the replenishment rate, and this is owing to the fact that it also reduces the ability of the readily releasable pool to contain quanta. Moreover, these compounds were ineffective in influencing how the replenishment rate decreases during long, high frequency stimulation. In conclusion at the excitatory synapses of rat hippocampus the replenishment of the readily releasable pool does not appear to be associated with a significant vesicular movement, and during long high frequency stimulation [Ca(++)]o does not influence the progressive decrease of vesicular replenishment.

Synaptic activity slows vesicular replenishment at excitatory synapses of rat hippocampus.

Short-term synaptic depression mainly reflects the depletion of the readily releasable pool (RRP) of quanta. Its dynamics, and especially the replenishment rate of the RRP, are still not well characterized in spite of decades of investigation. Main reason is that the vesicular storage and release system is treated as time-independent. If it is time-dependent all parameters thus estimated become problematic. Indeed the reports about how prolonged stimulation affects the dynamics are contradictory. To study this, we used patterned stimulation on the Schaeffer collateral fiber pathway and model-fitting of the excitatory post-synaptic currents (EPSC) recorded from CA1 neurons in rat hippocampal slices. The parameters of a vesicular storage and release model with two pools were estimated by minimizing the squared difference between the ESPC amplitudes and simulated model output. This yields the 'basic' parameters (release coupling, replenishment coupling and RRP size) that underlie the 'derived' and commonly used parameters (fractional release and replenishment rate). The fractional release increases when [Ca(++)]o is raised, whereas the replenishment rate is [Ca(++)]o independent. Fractional release rises because release coupling increases, and the RRP becomes less able to contain quanta. During prolonged stimulation, the fractional release remains generally unaltered, whereas the replenishment rate decreases down to ~10 % of its initial value with a decay time of ~15 s, and this decrease in the replenishment rate significantly contributes to synaptic depression. In conclusion, the fractional release is [Ca(++)]o-dependent and stimulation-independent, whereas the replenishment rate is [Ca(++)]o-independent and stimulation-dependent.

Recovery of vesicular storage and release parameters after high frequency stimulation in rat hippocampus.

The replenishment rates estimated from the recovery of synaptic efficacy following synaptic depression are known to be widely scattered. Given the importance of the replenishment during stimulation, especially if it is prolonged, it is important to better understand what influences the recovery of the synaptic efficacy following stimulation. We fit a two-pool model of vesicular secretion to the changes of the excitatory post-synaptic currents recorded in CA1 neurons of rat hippocampal slices to determine how the model parameters change during, and following, long stimulation. The replenishment rate at the end of stimulation inducing synaptic depression differs greatly from that at the beginning of stimulation. It decreases progressively and rapidly (by ~75 % and with a time constant of <10 s) during stimulation, and this is followed by a similarly fast recovery (time constant of ~10 s), but to a steady-state that is approximately twice as large as its pre-stimulation value. Both [Ca(++)]o and the duration of long stimulation influence the recovery of the replenishment rate. Its new steady-state is significantly higher, if either [Ca(++)]o is higher or stimulation longer, but the recovery of the replenishment rate becomes clearly slower if [Ca(++)]o is higher, and faster if stimulation is longer. Many factors thus influence the recovery of the replenishment rate and of the synaptic efficacy, but the stimulation induced [Ca(++)]i accumulation cannot explain the change of the replenishment rate during recovery. Finally, okadaic acid, which speeds up vesicular trafficking, does not alter the recovery of the replenishment rate. The vesicular replenishment of the RRP following stimulation is thus not likely to be associated with significant vesicular movement.

Interfacial interactions of glutamate, water and ions with carbon nanopore evaluated by molecular dynamics simulations.

Molecular dynamics simulations were used to assess the transport of glutamate, water and ions (Na(+) and Cl(-)) in a single wall carbon nanopore. The spatial profiles of Na(+) and Cl(-) ions are largely determined by the pore wall charges. Co-ions are repelled whereas the counter-ions are attracted by the pore charges, but this 'rule' breaks down when the water concentration is set to a level significantly below that in the physiological bulk solution. In such cases water is less able to counteract the ion-wall interactions (electrostatic or non-electrostatic), co-ions are layered near the counter-ions attracted by the wall charges and are thus layered as counter-ions. Glutamate is concentrated near the pore wall even at physiological water concentration, and irrespective of whether the pore wall is neutral or charged (positively or negatively), and its peak levels are up to 40 times above mean values. The glutamate is thus always layered as a counter-ion. Layering of water near the wall is independent of charges on the pore wall, but its peak levels near the wall are 'only' 6-8 times above the pore mean values. However, if the mean concentration of water is significantly below the level in the physiological bulk solution, its layering is enhanced, whereas its concentration in the pore center diminishes to very low levels. Reasons for such a 'paradoxical' behavior of molecules (glutamate and water) are that the non-electrostatic interactions are (except at very short distances) attractive, and electrostatic interactions (between the charged atoms of the glutamate or water and the pore wall) are also attractive overall. Repulsive interactions (between equally charged atoms) exist, and they order the molecules near the wall, whereas in the pore center the glutamate (and water) angles are largely randomly distributed, except in the presence of an external electric field. Diffusion of molecules and ions is complex. The translational diffusion is in general both inhomogeneous and anisotropic. Non-electrostatic interactions (ion-wall, glutamate-wall or water-wall) powerfully influence diffusion. In the neutral nanopore the effective axial diffusion constants of glutamate, water and Na(+) and Cl(-) ions are all <10% of their values in the bulk, and the electrostatic interactions can reduce them further. Diffusion of molecules and ions is further reduced if the water concentration in the pore is low. Glutamate(-) is slowed more than water, and ions are reduced the most especially co-ions. In conclusion the interfacial interactions influence the spatial distribution of glutamate, water and ions, and regulate powerfully, in a complex manner and over a very wide range their transport through nanosize pores.

Molecular dynamics simulations of glutamate diffusion in synaptic cleft.

Diffusion of transmitters in the synaptic cleft critically influences synaptic efficacy by affecting both the amplitude and the time course of quantal events, but the value of the diffusion constant is speculative. In this study, we use molecular dynamics simulations to determine how the spatial confinement and membrane charges affect the diffusion constants of glutamate- and water as well as general properties of their diffusion. The synaptic cleft is represented as the space enclosed by two single-wall carbon sheets. Both water and especially glutamate are concentrated near the pore wall, where the concentration of glutamate can reach 30-50 times the mean value and the concentration of water can reach 2-8 times the mean value. Such spatial profiles of glutamate contradict the classical notions of diffusion on which both continuous and Monte Carlo simulations are built. The layering of glutamate- and water molecules suggests that the interfacial glutamate-cleft wall (or water-cleft wall) interactions may critically regulate their diffusion in the cleft. Indeed, the effective longitudinal diffusion constant of glutamate is steeply dependent on the cleft width, but only when the cleft is very narrow (< 5 nm). Therefore, even for a cleft as narrow as at the glutamatergic synapse in the central nervous system, the effective diffusion constant of glutamate will not be much lower than free diffusion in the bulk solution due to confinement. The effective diffusion constant of water is considerably less sensitive to cleft width over the same range of cleft widths than is glutamate, but is also higher than that of glutamate. Finally, the layering of glutamate and water and their effective diffusion constants are largely independent of how the cleft wall is charged. In conclusion, in the confined space of the synaptic cleft, glutamate is layered near the wall. Consequently, its diffusion constant becomes dependent on the cleft width. However, the diffusion of glutamate is slower than its free diffusion in water only if the cleft is very narrow. If the width of the cleft is consistent with that determined by morphometric studies in the central nervous system, glutamate diffusion should not be slowed by confinement and is thus likely to be similar to that in free solution.

Mechanisms shaping fast excitatory postsynaptic currents in the central nervous system.

How different factors contribute to determine the time course of the basic element of fast glutamate-mediated excitatory postsynaptic currents (mEPSCs) in the central nervous system has been a focus of interest of neurobiologists for some years. In spite of intensive investigations, these mechanisms are not well understood. In this review, basic hypotheses are summarized, and a new hypothesis is proposed, which holds that desensitization of AMPA receptors plays a major role in shaping the time course of fast mEPSCs. According to the new hypothesis, desensitization shortens the time course of mEPSCs largely by reducing the buffering of glutamate molecules by AMPA receptors. The hypothesis accounts for numerous findings on fast mEPSCs and is expected to be equally fruitful as a framework for further experimental and theoretical investigations.