Field-driven polyelectrolyte-polymer collisions in nanochannels.

Even though dilute (unentangled) polymer solutions cannot act as gel-like sieving media, it has been shown that they can be used to separate DNA molecules in capillary electrophoresis. The separation then comes from sporadic, independent DNA-polymer collisions. We study polymer-polymer collisions in nanochannels (i.e., channels that are smaller than the normal size of the polymers), a situation where a polyelectrolyte is forced to migrate "through" isolated uncharged molecules during electrophoresis. We use Langevin dynamics simulations to explore the nature of these collisions and their effect on the net motion of the two polymer chains. We identify several types of collisions, including some that are unique to nanochannels. When the uncharged polymer is much larger than the polyelectrolyte, the system is reminiscent of gel electrophoresis, and we propose a modified empirical reptation model to explain the data, with an orientation factor that depends on the tube diameter. We also observe that the duration of a collision is a non-monotonic function of the polymer size ratio when the two chains are of comparable size, a surprising resonance-like phenomenon, which, combined with the asymmetric nature of molecular conformations during collision, suggests possible ratchet-like mechanisms that could be used to sort polyelectrolytes in nanodevices.

Estimating the asymptotic characteristic time scales for diffusion-controlled drug release systems using partially sampled data.

Drug release experiments and numerical simulations only give access to partial release data (i.e., within a finite time range t∈[0,tf]). In this article, we propose fitting-based procedures to estimate the asymptotic time scales of the release process, namely the global relaxation time τ∗ and the longest (or terminal) relaxation time τ0, from partially sampled data of diffusion-controlled drug release systems. We test these procedures on both synthetic and experimental data using, as an example, the well-known Weibull function. Our results show that the Weibull function must be used with great care because the values of the fitting parameters can vary significantly depending on the ratio tf/τ0. Beyond their practical simplicity, the usefulness of our procedures is evidenced by the fact that: (1) the initial loading profile does not need to be known and (2) the chosen fitting function does not require any physical basis. These two advantages allow us to determine the diffusion coefficient of the molecules directly from the characteristic time τ0.

Diffusivity interfaces in lattice Monte Carlo simulations: Modeling inhomogeneous delivery and release systems.

Lattice Monte Carlo (LMC) simulations are widely used to investigate diffusion-controlled problems such as drug-release systems. The presence of an inhomogeneous diffusivity environment raises subtle questions about the interpretation of stochastic dynamics in the overdamped limit, an issue sometimes referred to as the "Ito-Stratonovich-isothermal dilemma." We propose a LMC formalism that includes the different stochastic interpretations in order to model the diffusion of particles in a space-dependent diffusivity landscape. Using as an example a simple inhomogeneous one-dimensional system with a diffusivity interface and different boundary conditions, we demonstrate that we can properly reproduce the steady state and dynamic properties of these systems and that these properties do depend on the choice of calculus. In particular, we argue that the version of the LMC algorithm that uses Ito calculus, which is commonly used to model drug delivery systems, should be replaced by the isothermal version for most applications. Our LMC methodology provides an efficient alternative to Langevin simulations for a wide class of space-dependent diffusion problems.

Hydrodynamic chromatography and field flow fractionation in finite aspect ratio channels.

Hydrodynamic chromatography (HC) and field-flow fractionation (FFF) separation methods are often performed in 3D rectangular channels, though ideal retention theory assumes 2D systems. Devices are commonly designed with large aspect ratios; however, it can be unavoidable or desirable to design rectangular channels with small or even near-unity aspect ratios. To assess the significance of finite-aspect ratio effects and interpret experimental retention results, an ideal, analytical retention theory is needed. We derive a series solution for the ideal retention ratio of HC and FFF rectangular channels. Rather than limiting devices' ability to resolve samples, our theory predicts that retention curves for normal-mode FFF are well approximated by the infinite plate solution and that the performance of HC is actually improved. These findings suggest that FFF devices need not be designed with large aspect ratios and that rectangular HC channels are optimal when the aspect ratio is unity.

Can slip walls improve field-flow fractionation or hydrodynamic chromatography?

One way to potentially modify the performance of field-flow fractionation (FFF) would be to move the position of the maximum flow velocity away from the mid-point of the channel, for example by using walls with non-zero slip lengths. In this short communication, we extend the ideal theory of FFF to include the effects of two slip walls. Our calculations demonstrate that while the hydrodynamic chromatography limit of FFF (weak fields) is not improved by engineering devices with slip-walls, the performance of Normal-Mode FFF can be enhanced by having slip at the depletion wall in moderate fields. We also introduce a new regime, which we call Slip-Mode FFF, where a large external field (typical of Normal-Mode FFF) and a large slip at the accumulation wall lead to sharp separations characterized by an elution order that is similar to that of hydrodynamic chromatography.

Operational-modes of field-flow fractionation in microfluidic channels.

Through a careful consideration of the retention ratio for field-flow fractionation (FFF), we show that a single unified ideal retention theory can predict a wide range of separation behaviours including hydrodynamic chromatography, normal-mode FFF and steric-mode FFF by introducing the concept of a device retention parameter. We determine the critical device retention parameter above which normal-mode does not exist and there is no clear distinction between hydrodynamic chromatography and steric-mode FFF. Numerical analysis of the elution order as a function of particle size quantitatively predicts the transitions between these regimes. The resulting map of the operational-modes shows each of the regions and their connectivity, and so may guide future device design. By extending this analysis to account for the variation of stress over particle surfaces, a hitherto unreported regime called Faxén-mode FFF is predicted, which has the same elution order as normal-mode FFF. This mode arises when particle sizes approach the channel height, as can occur when microfluidic devices are utilized for FFF. The transition from steric-mode to Faxén-mode FFF is numerically mapped and approximations for each transition are presented.

Molecular Dynamics simulation of a polymer chain translocating through a nanoscopic pore: hydrodynamic interactions versus pore radius.

The detection of linear polymers translocating through a nanoscopic pore is a promising idea for the development of new DNA analysis techniques. However, the physics of constrained macromolecules and the fluid that surrounds them at the nanoscopic scale is still not well understood. In fact, many theoretical models of polymer translocation neglect both excluded-volume and hydrodynamic effects. We use Molecular Dynamics simulations with explicit solvent to study the impact of hydrodynamic interactions on the translocation time of a polymer. The translocation time tau that we examine is the unbiased (no charge on the chain and no driving force) escape time of a polymer that is initially placed halfway through a pore perforated in a monolayer wall. In particular, we look at the effect of increasing the pore radius when only a small number of fluid particles can be located in the pore as the polymer undergoes translocation, and we compare our results to the theoretical predictions of Chuang et al. (Phys. Rev. E 65, 011802 (2001)). We observe that the scaling of the translocation time varies from tau approximately N 11/5 to tau approximately N 9/5 as the pore size increases (N is the number of monomers that goes up to 31 monomers). However, the scaling of the polymer relaxation time remains consistent with the 9/5 power law for all pore radii.

Tethered polyelectrolytes under the action of an electrical field: a molecular-dynamics study.

For a polyelectrolyte undergoing electrophoretic motion, it is predicted (D. Long, J.L. Viovy, A. Ajdari, Phys. Rev. Lett. 76, 3858 (1996); D. Long, A. Ajdari, Electrophoresis 17, 1161 (1996)) that the mechanical force necessary to stall the molecule is substantially smaller than the sum of electrical forces applied on all monomers. In fact, it should be proportional to its hydrodynamic friction coefficient and therefore to the size of its conformation. In our work we examine this prediction using coarse-grained molecular-dynamics simulations in which we explicitly include the polymer, the solvent, the counterions and salt. The electrophoretic mobility of polyelectrolytes is evaluated, the mechanical force necessary to keep the molecules tethered is measured and the resulting anisotropic polymer conformations are observed and quantified. Our results corroborate Long et al.'s prediction.

Electric and hydrodynamic stretching of DNA-polymer conjugates in free-solution electrophoresis.

The conjugation of an uncharged polymer to DNA fragments makes it possible to separate DNA by free-solution electrophoresis. This end-labeled free-solution electrophoresis method has been shown to successfully separate ssDNA with single monomer resolution up to about 110 bases. It is the aim of this paper to investigate in more detail the coupled hydrodynamic and electrophoretic deformation of the ssDNA-label conjugate at fields below 400 V/cm. Our model is an extension of the theoretical approach originally developed by Stigter and Bustamante [Biophys. J. 75, 1197 (1998)] to investigate the problems of a tethered chain stretching in a hydrodynamic flow and of the electrophoretic stretch of a tethered polyelectrolyte. These two separate models are now used together since the charged DNA is "tethered" to the uncharged polymer (and vice versa), and the resulting self-consistent model is used to predict the deformation and the electrophoretic velocity for the hybrid molecule. Our theoretical and experimental results are in good qualitative agreement.

Molecular-dynamics simulations with explicit hydrodynamics II: on the collision of polymers with molecular obstacles.

We present a study of the dynamics of single polymers colliding with molecular obstacles using Molecular-dynamics simulations. In concert with these simulations we present a generalized polymer-obstacle collision model which is applicable to a number of collision scenarios. The work focusses on three specific problems: i) a polymer driven by an external force colliding with a fixed microscopic post; ii) a polymer driven by a (plug-like) fluid flow colliding with a fixed microscopic post; and iii) a polymer driven by an external force colliding with a free polymer. In all three cases, we present a study of the length-dependent dynamics of the polymers involved. The simulation results are compared with calculations based on our generalized collision model. The generalized model yields analytical results in the first two instances (cases i) and ii)), while in the polymer-polymer collision example (case iii)) we obtain a series solution for the system dynamics. For the case of a polymer-polymer collision we find that a distinct V-shaped state exists as seen in experimental systems, though normally associated with collisions with multiple polymers. We suggest that this V-shaped state occurs due to an effective hydrodynamic counter flow generated by a net translational motion of the two-chain system.

Molecular dynamics study of tethered polymers in shear flow.

Single macromolecules can now be isolated and characterized experimentally using techniques such as optical tweezers and videomicroscopy. An interesting and important single-molecule problem is that of the dynamics of a polymer chain tethered to a solid surface and subjected to a shear flow. An experimental study of such a system was reported by Doyle et al. (Phys. Rev. Lett. 84, 4769 (2000)), and their results showed a surprising recirculating motion of the DNA chain. We explore this problem using molecular dynamics computer simulations with explicit hydrodynamic interactions. The dynamical properties of a Freely Jointed Chain (FJC) with Finitely Extensible Nonlinear Elastic (FENE) links are examined in similar conditions (i.e., confined between two surfaces and in the presence of a Poiseuille flow). We see the remarkable cyclic polymer motion observed experimentally, and we show that a simple cross-correlation function can be used to measure the corresponding period of motion. We also propose a new empirical equation relating the magnitude of the shear flow to the amount of chain deformation, an equation that appears to apply for both weak and strong flows. Finally, we report on packing effects near the molecularly flat wall, an associated chain-sticking phenomenon, and the impact of the chain hydrodynamic drag on the local fluid flow.

Exact lattice calculations of dispersion coefficients in the presence of external fields and obstacles.

We present a study of the field-dependent dispersion coefficient of point-like particles in various 2D overdamped systems with obstructions (periodic, percolating, and trapping distributions of obstacles). These calculations profit from the synthesis of a newly proposed Monte Carlo algorithm--the first such algorithm that correctly reproduces the free dispersion coefficient in the presence of finite external fields--and an asymptotically exact calculation technique. The resulting method efficiently produces algebraic and numerical results without the need to actually perform Monte Carlo simulations. When compared to such simulations, our exact method features a negligible computational cost and exponentially small errors. Utilizing the power of this numerical method, we engage in comprehensive parametric analysis of several model systems, revealing very subtle effects that would otherwise be swamped by statistical errors or incur prohibitive computational costs. The unified framework presented here serves as a template for further applications of lattice random-walk models of biased diffusion.

Molecular-dynamics simulations with explicit hydrodynamics I: on the friction coefficients of deformed polymers.

We implement large-scale Molecular-Dynamics (MD) simulations which incorporate hydrodynamic interactions via the inclusion of explicit Lennard-Jones solvent to examine the behaviour of polymer chains in sieving media. We begin by examining the friction coefficients of polymers in long-lived states responsible for inducing length-dependent mobility, i.e., allowing separation of polymers (or polyelectrolytes) by molecular weight. In particular, the conformations we examine occur in devices which utilize arrays of molecular obstacles or dilute solutions of polymers. We compare the results from our MD simulations with expressions from macroscopic hydrodynamics for four specific cases: i) a random coil excluded-volume Zimm polymer, ii) a rigid polymer moving perpendicular to its major axis iii) a rigid polymer moving parallel to its major axis and iv) a rigid polymer, folded at different points along its contour. We also examine the behaviour of the friction coefficient of a fully flexible molecule pulled by its middle monomer as a function of an applied force F and show that there are several distinct frictional regimes.

An exactly solvable Ogston model of gel electrophoresis: VIII. Nonconducting gel fibers, curved field lines, and the Nernst-Einstein relation.

In this article, we examine the low-field electrophoretic migration of infinitely small analytes in dilute sieving media made of nonconducting gel fibers. Using an Ogston obstruction model, we show that the electrophoretic mobility is not affected by the presence of curved field lines. In other words, the Nernst-Einstein relation between the mobility and the diffusion coefficient is valid regardless of the electrical properties of the gel fibers. Although this finding may greatly simplify the development of obstruction models of electrophoretic sieving, it also represents a critical test for any analytical or computational approach.

Capillary electrophoretic separation of uncharged polymers using polyelectrolyte engines. Theoretical model.

We recently demonstrated that the molecular mass distribution of an uncharged polymer sample can be analyzed using free-solution capillary electrophoresis of DNA-polymer conjugates. In these conjugates, the DNA is providing the electromotive force while the uncharged polydisperse polymer chains of the sample retard the DNA engine with different amounts of hydrodynamic drag. Here we present a theoretical model of this new analytical method. We show that for the most favourable, diffusion-limited electrophoresis conditions, there is actually an optimal DNA size to achieve the separation of a given polymer sample. Moreover, we demonstrate that the effective friction coefficient of the polymer chains is related to the stiffness of the two polymers of the conjugate, thus offering a method to estimate the persistence length of the uncharged polymer through mobility measurements. Finally, we compare some of our predictions with available experimental results.

Diffusion coefficient of DNA molecules during free solution electrophoresis.

The free-draining properties of DNA normally make it impossible to separate nucleic acids by free-flow electrophoresis. However, little is known, either theoretically or experimentally, about the diffusion coefficient of DNA molecules during free-flow electrophoresis. In fact, many authors simply assume that the Nernst-Einstein relation between the mobility and the diffusion coefficient still holds under such conditions. In this paper, we present an experimental study of the diffusion coefficient of both ssDNA and dsDNA molecules during free-flow electrophoresis. Our results unequivocally show that a simplistic use of Nernst-Einstein's relation fails, and that the electric field actually has no effect on the thermal diffusion process. Finally, we compare the dependence of the diffusion coefficient upon DNA molecular size to results obtained previously by other groups and to Zimm's theory.

Molar mass profiling of synthetic polymers by free-solution capillary electrophoresis of DNA-polymer conjugates.

The molar mass distribution of a polymer sample is a critical determinant of its material properties and is generally analyzed by gel permeation chromatography or more recently, by MALDI-TOF mass spectrometry. We describe here a novel method for the determination of the degree of polymerization of polydisperse, uncharged, water-soluble polymers (e.g., poly(ethylene glycol) (PEG)), based upon single-monomer resolution of DNA-polymer conjugates by free-solution capillary electrophoresis. This is accomplished by end-on covalent conjugation of a polydisperse, uncharged polymer sample (PEG) to a monodisperse, fluorescently labeled DNA oligomer, followed by electrophoretic analysis. The monodisperse, charged DNA "engine" confers to each conjugate an equal amount of electromotive force, while the varying contour lengths of the uncharged, polydisperse polymers engender different amounts of hydrodynamic drag. The balance of electromotive and hydrodynamic forces enables rapid, high-resolution separation of the DNA-polymer conjugates as a function of the size of the uncharged PEG tail. This provides a profile of the molar mass distribution of the original polymer sample that can be detected by laser-induced fluorescence through excitation of the dye-labeled DNA. We call this method free solution conjugate electrophoresis (FSCE). Theory-based analysis of the resulting electrophoresis data allows precise calculation of the degree of polymerization of the PEG portion of each conjugate molecule. Knowledge of the molecular mass of the uncharged polymer's repeat unit allows for direct calculation of the molar mass averages as well as sample polydispersity index. The results of these analyses are strikingly reminiscent of MALDI-TOF spectra taken of the same PEG samples. PEG samples of 3.4-, 5-, and 20-kDa nominal average molar mass were analyzed by FSCE and MALDI-TOF; the values of the molar mass averages, Mw and Mn, typically agree to within 5%. Measurements and molar mass calculations are performed without any internal standards or calibration. Moreover, when DNA-polymer conjugate analysis is performed in a chip-based electrophoresis system, separation is complete in less than 13 min. FSCE offers an alternative to MALDI-TOF for the characterization of uncharged, water-soluble polymers that can be uniquely conjugated to DNA.

An exactly solvable Ogston model of gel electrophoresis. VI. Towards a theory for macromolecules.

In this article, we present a generalized version of our lattice model of low-field gel electrophoresis that allows us to treat the case of macromolecules such as short linear or circular oligomers and semi-flexible rods. We show that free-solution electrophoresis problems can be seen as random walks in the conformational space of the analyte. For sufficiently small molecules, our mathematical approach provides exact mobilities. In a quenched gel-like environment, however, both conformational and positional degrees of freedom must be used, but exact solutions can also be obtained. As an example, we then investigate several two-dimensional model gels, as well as a simple channel system where we see evidence of entropic effects that cannot be captured by the traditional Ogston concept of free volume.

Gel electrophoretic mobility of single-stranded DNA: the two reptation field-dependent factors.

The reptation model is the dominant theory in understanding the electrophoretic separation of single-stranded DNA molecules in gels or entangled polymer solutions. Recently, we showed that the Ogston and reptation regimes are separated by an entropic trapping regime at low field intensities. Here, we report the first comparison of the field-dependent part of the DNA mobility for both small and long reptating molecules. We show that both mobilities increase linearly with field intensity, with the mobility of the longer (comigrating) fragments increasing faster than that of the smaller ones. We compare our results to the predictions of the biased reptation model.