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.

Generalized tube model of biased reptation for gel electrophoresis of DNA.

A theoretical analysis of the reptational motion of DNA in a gel that includes the effects of molecular fluctuations has been used to explain the main features found in experiments involving periodic inversion of the electric field. The resonance-like decrease of the electrophoretic mobility as a function of pulse duration is related to transient "undershoots" in the orientation of the molecule, in agreement with recent experimental data. These features arise from a delicate interplay of internal and center of mass motion of the molecules under pulsed field conditions, and are important for the separation of DNA molecules in the size range 0.2 to 10 million base pairs.

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.

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.

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.

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.

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.

Theory of band broadening for DNA gel electrophoresis and sequencing.

The problem of (thermal) band broadening during DNA gel electrophoresis is studied analytically and numerically using the reptation model. It is shown that the orientation of the end-to-end vector of the molecules leads to increased diffusion and even to molecular size-independent band broadening. Thus, the Einstein relation between mobility and diffusion holds for only very small molecules. Also, "self-trapped" molecules are predicted to give fuzzy bands, as observed experimentally. Finally, megabase molecules are discussed and the consequences of our predictions for the improvement of sequencing are considered.

Electric field gradients and band sharpening in DNA gel electrophoresis.

A mathematical study of the effect of non-uniform electric fields on the width of DNA electrophoretic bands is presented. Using a simple model, we show that field gradients sharpen these bands during an experiment if the corresponding gradient of electrophoretic velocity is large enough. This is in agreement with experimental results indicating that narrower bands form when pulsed field electrophoresis is carried out in the presence of field gradients. Moreover, it is shown that there is in fact an optimal experimental duration that maximizes separation. Finally, gradients are also predicted to reduce the relative mobilities of the DNA fragments, which is a serious drawback of this technique.

The biased reptation model of DNA gel electrophoresis: mobility vs molecular size and gel concentration.

The biased reptation model provides a good framework for interpreting the results of continuous field DNA electrophoresis experiments performed in agarose gels. Here we discuss the main features of the mobility-molecular size and mobility-gel concentration diagrams as obtained from new extensive computer simulations of the model. Our aim is to suggest a global and coherent picture of this widely used yet poorly understood experimental technique, and to point out the areas where a systematic experimental study is still needed.

Electrophoretic resolution versus fluctuations of the lateral dimensions of a capillary.

Because the local electrical resistance is inversely proportional to the local cross-section of a capillary, the intensity of the electric field varies along the migration path if the inner diameter of the capillary is not constant. Therefore, fluctuations of the lateral dimensions of a capillary can directly affect the net elution time as well as the peak width, and, hence the final resolution. In this article, we develop the theoretical framework for the study of such effects. We then examine the simple case where both the mobility and the diffusion coefficient are field-independent; in particular, we demonstrate that resolution can be severely reduced if the inner walls are not flat, and that optimal resolution is always obtained for perfectly flat walls. Generalized to ultrathin gels, our results clearly indicate that both random and systematic variations of the gel thickness can greatly affect the performance of the separation process. Acceptable degrees of flatness are estimated for both geometries. This study thus provides a quantitative understanding of the type of quality control one requires to obtain optimal results with capillaries and ultrathin gels.

Ogston gel electrophoretic sieving: how is the fractional volume available to a particle related to its mobility and diffusion coefficient(s)?

The Ogston-Morris-Rodbard-Chrambach model (OMRCM) of gel electrophoresis assumes that the mobility mu of charged particles is proportional to the fractional volume (f) of the gel that is available to them. If the gel is random, as described by Ogston, the (semi-log) Ferguson plot is the method of choice for analyzing experimental data since it permits an estimate of the gel's mean pore size to be made. However, the Ferguson plot is rarely linear; this is usually "explained" by the deformation of the anisotropy of the particle, the nonrandom or variable architecture or the gel, or the onset of some other migration mechanism. Many authors have refined this model, but the original assumption that mu varied; is directly proportional to f has not been seriously examined. Also, the model says nothing of the effect of the field intensity, the connectivity of the gel pores, nor anything about the diffusion coefficient. We have developed a Monte-Carlo computer simulation algorithm to study the electrophoretic sieving of simple particles in gels. In this brief communication, we report important preliminary results which indicate that the basic assumptions of the OMRCM are wrong. We use a two-dimensional periodic gel since the OMRCM becomes trivial in this case. Our results show that the relationship between f and mu is not the one assumed by the OMRCM. Moreover, we find that the Einstein relation between the diffusion coefficient and the mobility is not valid. This is due to the fact that the particles do not have a uniform probability of visiting the various sites that are available to them. We thus conclude that the Ferguson plot is intrinsically nonlinear; the curvature of the plot is, in fact, related to the intensity of the electric field as well as to the degree of randomness of the gel fibers.