In situations involving large potentials or surface charges, the Poisson Boltzman(PB) equation has shortcomings because it neglects ion-ion interactions and steric effects. This has been widely recognized by the electrochemistry community, leading to the development of various alternative models resulting in different sets "modified PB equations", which have had at least qualitative success in predicting equilibrium ion distributions. On the other hand, the literature is scarce in terms of descriptions of concentration dynamics in these regimes. Here, adapting strategies developed to modify the PB equation, we propose a simple modification of the widely used Poisson-Nernst-Planck (PNP) equations for ionic transport, which at least qualitatively accounts for steric effects. We analyze numerical solutions of these MPNP equations on the model problem of the charging of a simple electrolyte cell, and compare the outcome to that of the standard PNP equations. Finally, we repeat the asymptotic analysis of Bazant, Thornton, and Ajdari(2004} for this new system of equations to further document the interest and limits of validity of the simpler equivalent electrical circuit models introduced in Part I for such problems.

Steric effects in the dynamics of electrolytes at large applied voltages: II. Modified Poisson-Nernst-Planck equations

Biological nanopores are emerging as powerful tools for single-molecule analysis and sequencing. Here, we engineered the two-component pleurotolysin (PlyAB) toxin to assemble into 7.2 × 10.5 nm cylindrical nanopores with a low level of electrical noise in lipid bilayers, and we addressed the nanofluidic properties of the nanopore by continuum simulations. Surprisingly, proteins such as human albumin (66.5 kDa) and human transferrin (76-81 kDa) did not enter the nanopore. We found that the precise engineering of the inner surface charge of the PlyAB induced electro-osmotic vortices that allowed the electrophoretic capture of the proteins. Once inside the nanopore, two human plasma proteins could be distinguished by the characteristics of their current blockades. This fundamental understanding of the nanofluidic properties of nanopores provides a practical method to promote the capture and analysis of folded proteins by nanopores.

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Electro-Osmotic Vortices Promote the Capture of Folded Proteins by PlyAB Nanopores.

Many strategies have been pursued to trap and monitor single proteins over time to detect the molecular mechanisms of these essential nanomachines. Single-protein sensing with nanopores is particularly attractive because it allows label-free high-bandwidth detection on the basis of ion currents. Here we present the nanopore electro-osmotic trap (NEOtrap) that allows trapping and observing single proteins for hours with submillisecond time resolution. The NEOtrap is formed by docking a DNA-origami sphere onto a passivated solid-state nanopore, which seals off a nanocavity of a user-defined size and creates an electro-osmotic flow that traps nearby particles irrespective of their charge. We demonstrate the NEOtrap’s ability to sensitively distinguish proteins on the basis of size and shape, and discriminate between nucleotide-dependent protein conformations, as exemplified by the chaperone protein Hsp90. Given the experimental simplicity and capacity for label-free single-protein detection over the broad bio-relevant time range, the NEOtrap opens new avenues to study the molecular kinetics underlying protein function. A trap, formed by a DNA-origami sphere docked onto a solid-state nanopore, allows the hydrodynamic trapping and label-free observation of single proteins, enabling nucleotide-dependent protein conformation to be discriminated on the timescale of submilliseconds to hours.

Nanopore electro-osmotic trap for the label-free study of single proteins and their conformations

The interaction between nanoparticles dispersed in a fluid and nanopores is governed by the interplay of hydrodynamical, electrical, and chemical effects. We developed a theory for particle capture in nanopores and derived analytical expressions for the capture rate under the concurrent action of electrical forces, fluid advection, and Brownian motion. Our approach naturally splits the average capture time in two terms, an approaching time due to the migration of particles from the bulk to the pore mouth and an entrance time associated with a free-energy barrier at the pore entrance. Within this theoretical framework, we described the standard experimental condition where a particle concentration is driven into the pore by an applied voltage, with specific focus on different capture mechanisms: under pure electrophoretic force, in the presence of a competition between electrophoresis and electroosmosis, and finally under dielectrophoretic reorientation of dipolar particles. Our theory predicts that dielectrophoresis is able to induce capture for both positive and negative voltages. We performed a dedicated experiment involving a biological nanopore (α-hemolysin) and a rigid dipolar dumbbell (realized with a β-hairpin peptide) that confirms the theoretically proposed capture mechanism.

Analytical Model for Particle Capture in Nanopores Elucidates Competition among Electrophoresis, Electroosmosis, and Dielectrophoresis.

We derive a continuum model, called the Poisson–Fermi equation, of biological calcium channels (of cardiac muscle, for example) designed to deal with crowded systems in which ionic species and side chains nearly fill space. The model is evaluated in three dimensions. It includes steric and correlation effects and is derived from classical hard-sphere lattice models of configurational entropy of finite size ions and solvent molecules. The maximum allowable close packing (saturation) condition is satisfied by all ionic species with different sizes and valences in a channel system, as shown theoretically and numerically. Unphysical overcrowding (“divergence”) predicted by the Gouy–Chapman diffuse model (produced by a Boltzmann distribution of point charges with large potentials) does not occur with the Fermi-like distribution. Using probability theory, we also provide an analytical description of the implicit dielectric model of ionic solutions that gives rise to a global and a local formula for the chemical potential. In this primitive model, ions are treated as hard spheres and solvent molecules are described as a dielectric medium. The Poisson–Fermi equation is a local formula dealing with different correlations at different places. The correlation effects are apparent in our numerical results. Our results show variations of dielectric permittivity from bath to channel pore described by a new dielectric function derived as an output from the Poisson–Fermi analysis. The results are consistent with existing theoretical and experimental results. The binding curve of Poisson–Fermi is shown to match Monte Carlo data and illustrates the anomalous mole fraction effect of calcium channels, an effective blockage of permeation of sodium ions by a tiny concentration (or number) of calcium ions.

Correlated Ions in a Calcium Channel Model: A Poisson–Fermi Theory B

Despite the broad success of biological nanopores as powerful instruments for the analysis of proteins and nucleic acids at the single-molecule level, a fast simulation methodology to accurately model their nanofluidic properties is currently unavailable. This limits the rational engineering of nanopore traits and makes the unambiguous interpretation of experimental results challenging. Here, we present a continuum approach that can faithfully reproduce the experimentally measured ionic conductance of the biological nanopore Cytolysin A (ClyA) over a wide range of ionic strengths and bias potentials. Our model consists of the extended Poisson-Nernst-Planck and Navier-Stokes (ePNP-NS) equations and a computationally efficient 2D-axisymmetric representation for the geometry and charge distribution of the nanopore. Importantly, the ePNP-NS equations achieve this accuracy by self-consistently considering the finite size of the ions and the influence of both the ionic strength and the nanoscopic scale of the pore on the local properties of the electrolyte. These comprise the mobility and diffusivity of the ions, and the density, viscosity and relative permittivity of the solvent. Crucially, by applying our methodology to ClyA, a biological nanopore used for single-molecule enzymology studies, we could directly quantify several nanofluidic characteristics difficult to determine experimentally. These include the ion selectivity, the ion concentration distributions, the electrostatic potential landscape, the magnitude of the electro-osmotic flow field, and the internal pressure distribution. Hence, this work provides a means to obtain fundamental new insights into the nanofluidic properties of biological nanopores and paves the way towards their rational engineering.

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Accurate modeling of a biological nanopore with an extended continuum framework.

This paper presents the design and development of an optimized aluminum microheater integrated onto a biochip for the amplification of DNA using polymerase chain reaction (PCR). A coupled 3D finite element electro-thermal simulation has been used to aid in the design of the microheater and the PCR reactor. The microheater has a special shape, designed to provide a uniform temperature throughout the PCR chamber. The microreactor is fabricated at the center of a 20 mm × 20 mm silicon chip. It has a meandered shape, a volume of 1.89 µl and occupies a square area with a side of 3.8 mm. Microchannels to transport fluid in and out of the reactor are also provided. After heater design optimization, the simulated temperature of the fluid volume within the PCR chamber is very uniform (95% of the volume has a temperature within ±0.2 °C when the average temperature is 60 °C). This result is validated by DNA melting point experiments, showing a very similar uniformity. A PCR experiment, consisting of 50 cycles of amplification is conducted to demonstrate functionality of the system; amplifications is uniform across the reactor with variation of the threshold cycle within about 0.5 units.

https://iopscience.iop.org/article/10.1088/1361-6439/aabd2c/pdf

A comprehensive methodology for design and development of an integrated microheater for on-chip DNA amplification

In recent years, the protein nanopore cytolysin A (ClyA) has become a valuable tool for the detection, characterization and quantification of biomarkers, proteins and nucleic acids at the single-molecule level. Despite this extensive experimental utilization, a comprehensive computational study of ion and water transport through ClyA is currently lacking. Such a study yields a wealth of information on the electrolytic conditions inside the pore and on the scale the electrophoretic forces that drive molecular transport. To this end we have built a computationally efficient continuum model of ClyA which, together with an extended version of Poison-Nernst-Planck-Navier-Stokes (ePNP-NS) equations, faithfully reproduces its ionic conductance over a wide range of salt concentrations. These ePNP-NS equations aim to tackle the shortcomings of the traditional PNP-NS models by self-consistently taking into account the influence of both the ionic strength and the nanoscopic scale of the pore on all relevant electrolyte properties. In this study, we give both a detailed description of our ePNP-NS model and apply it to the ClyA nanopore. This enabled us to gain a deeper insight into the influence of ionic strength and applied voltage on the ionic conductance through ClyA and a plethora of quantities difficult to assess experimentally. The latter includes the cation and anion concentrations inside the pore, the shape of the electrostatic potential landscape and the magnitude of the electro-osmotic flow. Our work shows that continuum models of biological nanopores—if the appropriate corrections are applied—can make both qualitatively and quantitatively meaningful predictions that could be valuable tool to aid in both the design and interpretation of nanopore experiments.

https://biorxiv.org/content/biorxiv/early/2020/01/09/2020.01.08.897819.full-text.pdf

Modeling of Ion and Water Transport in the Biological Nanopore ClyA

A novel method is presented for triggering a robust capillary stop valve fabricated in silicon using the thermal expansion of trapped air bubble (with a footprint of just 300 μm × 320 μm) as the actuation element. A heating element on the backside of a bubble trap chamber is utilized for thermal expansion of the air bubble. A voltage pulse of around 6 V, the capillary barrier, around 1400 Pa was easily breached. A non-dimensionalized model has been developed using equivalent circuit model to describe the complex thermal/hydraulic behavior of the system. The trapped gas bubble temperature is input as a function of time in the model. A thermal finite element-based simulation is conducted to determine the gas temperature from the experimentally measured heater temperature. The model results are validated against experiments to aid in characterizing the dynamics of the problem.

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Capillary stop valve actuation by thermo-pneumatic- pressure for lab-on-chip systems

In this study, we demonstrate a compact actuation mechanism of a silicon capillary stop microvalve, based on electrothermal expansion of a trapped air bubble in a chamber. The bubble is heated using an integrated aluminum microheater deposited on the silicon substrate above the air chamber. The heater occupies an area of 320 µm × 300 µm and has a resistance of 40 Ohms. By applying a 500 ms voltage pulse of 3 V amplitude we could generate a pressure sufficient to breach the capillary barrier pressure of valve, which is around 1000 Pa.

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Ujjal Barman

Papers

Accurate modeling of a biological nanopore with an extended continuum framework.

A comprehensive methodology for design and development of an integrated microheater for on-chip DNA amplification

Modeling of Ion and Water Transport in the Biological Nanopore ClyA

Capillary stop valve actuation by thermo-pneumatic- pressure for lab-on-chip systems

Capillary Microvalve Actuation Using Thermal Expansion of Trapped Air Bubble