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DNA separation techniques have drawn attention because of their uses in applications such as gene analysis and manipulation. There have been many studies utilizing micro-fabricated devices for faster and more efficient separations than traditional methods using gel electrophoresis. Although many experimental studies have presented various new devices and methods, computational studies have played a pivotal role in this development by identifying separation mechanisms and by finding optimal designs for efficient separation conditions. The simulation of DNA separation methods in micro-fabricated devices requires the correct capture of the dynamics and the structure of a single polymer molecule that is being affected by an applied flow field or an electric field in complex geometries. In this work, we summarize the polymer models (the bead-spring model, the bead-rod model, the slender-body model, and the touching-bead model) and the methods, focusing on Brownian dynamics simulation, used to calculate inhomogeneous fields taking into consideration complex boundaries (the finite element method, the boundary element method, the lattice-Boltzmann method, and the dissipative particle dynamics simulation). The worm-like chain model (adapted from the bead-spring model) combined with the finite element method has been most commonly used but other models have shown more efficient and accurate results. We also review the applications of these simulation approaches in various separation methods and devices: gel electrophoresis, post arrays, capillary electrophoresis, microchannel flows, entropic traps, nanopores, and rotational flows. As more complicated geometries are involved in new devices, more rigorous models (such as incorporating the hydrodynamic interactions of DNA with solid boundaries) that can correctly capture the dynamic behaviors of DNA in such devices are needed.

Gene analysis is one of the essential tasks for advances in biotechnology. Gene analysis would not be possible without DNA manipulation techniques. With the advent of lab-on-a chip technology in the early 2000s, the manipulation of DNA molecules in micro-fabricated microfluidic devices began to flourish [

The mobility of DNA molecules is an important transport property in DNA separation techniques. DNA molecules tend to have size-independent similar mobility in free solution because the overall charge to mass ratio does not change much with molecular weight. This leads to difficulties in separating longer molecules [

As mentioned earlier, for DNA separation to be feasible, size-dependent dynamics or mobility must be caused by interactions with solid boundaries in the flow system. Therefore, single polymer dynamics and inhomogeneous force field calculations must be calculated simultaneously and self-consistently [

The time and length scales for DNA separations are typically in similar or larger ranges of a single DNA molecule in a free space (length scales of 10 - 100 μm and relaxation times of 0.01 - 1 s). These scales are also larger than the base-pair molecular level so molecular dynamic simulation is not suitable. Indeed, the sequence of base-pairs does not affect the physical properties of DNA. Additionally, DNA separations are usually performed in a dilute concentration of DNA solution, which leads to an assumption that interaction with other DNA molecules can be neglected in modeling. In those situations, Brownian dynamics (BD) simulation of a coarse-grained single polymer model is used for DNA separation simulation [

The most common polymer model for DNA separation is the “bead-spring” model. Each “bead” represents a sub-chain larger than a Kuhn length, b_{k} (a shortest polymer segment length which is not bent or stretched by thermal fluctuation. DNA has b_{k}~0.1 nm which is much larger than that of typical polymer), and the “springs” lie between these beads. These springs are used to maintain the conformational entropy inside a sub-chain (represented by the beads). This is shown in

The force balance on the i-th bead in a bead-spring chain model is given by Equation (1):

Here, m is the mass of the bead, r_{i} is the position vector of the bead, t is the time, F is the total net hydrodynamic force acting on the bead, and ζ is the drag coefficient. Stokes flow condition is usually applicable to microscale flows, hence, to DNA separations, too. When using Stokes flow condition, inertial effect is considered negligible (overdamping system). Thus, the left hand side of Equation (1) can be assumed to be 0. Electric fields are used in gel electrophoresis, a common method of DNA separation. Thus, along with considering flow field, electric field (non-hydrodynamic force) is also evaluated to give an equation of motion:

Here, U(r_{i}) is the unperturbed fluid velocity at the bead position, μ is the electrophoretic mobility, E(r_{i}) is the electric force at the bead position, _{i}) or E(r_{i}) becomes 0. The evaluation of U(r_{i}) or E(r_{i}) with consideration to the micro-fabricated structure of the device is one of the most important parts in DNA separation simulations. This is discussed further in Section 3. The drag coefficient, ζ, is related to the bead diffusivity, D_{i}. For typical electrophoresis conditions, DNA, which is a negatively charged molecule, is always surrounded by counter ions. This cancels the hydrodynamic interactions (HI) in strong ionic concentration [

Here, k_{B} is the Boltzmann constant, T is the absolute temperature, η is the solvent viscosity, and a is the bead radius. The bead radius, a, is typically chosen to match the experimental diffusivity data [

The Brownian force for a free-draining bead is evaluated at each time step from the fluctuation dissipation theorem, which must satisfy the following conditions:

Here,

Here, w is a random vector, of which average is 0 and variance is 1, evaluated by any random vector generator algorithm [

The net spring force is the sum of the spring forces between adjacent beads:

Here, the sub-index i,i + 1 represents the force between the i-th and the i + 1 th beads. For the beads at both ends (i = 1 and i = N), only one of these spring forces exists. There are various models used to describe the spring force, which is closely related to polymer conformation. The simplest spring force model is the Gaussian chain model also known as the Hookean spring model [

Note here that the persistence length for WLC model is the half length of b_{k}. Underhill and Doyle examined the nonlinearity of the extension-force relation further to propose a correction method by incorporating the “effective” persistence length [

The excluded volume force is the sum of each excluded volume force between each bead:

Streek et al. used a force derived from a truncated Leonard-Jones potential equation [

Here, _{j} with the nearest boundary position [

Numerical integration of Equation (2) is required to get the new bead position at a new time step t + Δt. An explicit Euler scheme requires a very small Δt to prevent numerical instability attributed to new spring lengths exceeding l or new bead positions overlapping the solid boundaries of the model. Although an implicit Euler scheme can be used to avoid spring overstretch, the new position must be solved using Newton-Raphson iterations. This also results in long computational times. Therefore, Jendrejack et al. devised a semi-implicit scheme where an implicit Euler scheme is applied only to the integration of the term related to the spring force and the rest of the terms are integrated by an explicit Euler scheme [

As mentioned earlier, Equation (3) can be only used when HIs are neglected. This assumes that DNA undergoing gel electrophoresis is uniformly negatively charged and the Debye length is smaller than the persistence length of DNA. With these conditions, HIs are assumed to be screened due to counterion movement [

Inclusion of HIs for the bead-spring model is described by Jendrejack et al. [

Here, Ω is the HI tensor. For HIs with beads to be evaluated, the Oseen-burger tensor or Rotne-Prager tensor is used [

Here, B is the decomposed tensor of

While the bead-spring model is the most widely used model in DNA separation simulations, other polymer models can be applied to simulation of DNA. Below we discuss bead-rod model, slender-body model, and touching-bead model.

1) Bead-rod model: As shown in _{k}, which leads to a less coarse-grained model than when using the bead-spring model. Compared to when using the bead-spring model, penetration between chains is not allowed. Constraint forces are assigned to maintain a constant rod length between beads and prevents an overstretch of the chain [_{k} because bending within the rods is neglected [_{k} [

2) Slender-body model: As shown in _{k} can be simulated as single slender-bodies [

3) Touching-bead model: As shown in _{k} and can allow for bending within the model. This aspect makes this model more accurate than the bead-rod model. This flexibility within b_{k} enables us to calculate rotational diffusivity more accurately [_{k}), the actual effective persistence length becomes smaller than 0.5b_{k}, which results in inaccurate prediction of DNA stretch [

In summary, the bead-spring models, more specifically the WLC model, have been widely used in simulations of DNA separations due to their efficiency. However, too much coarse-graining, in other words not enough beads, may result in an inaccurate description of dynamics and crossing of polymer chains. The bead-rod model can prevent the overstretch issue and the slender-body model can include HI more accurately. However, connector rigidity can cause limitations in the length scale of confinement. The touching-bead model can simulate DNA properties on a more realistic scale, but at the cost of a high computational load. Therefore, this model is mainly used in the study of DNA structure in nano-confinement.

As explained earlier, DNA separation simulations require local flow or force values , as in U(r_{i}) and E(r_{i}) in Equations (2) and (12), for polymer motion in the flow or force field of the separation device. If the geometry of the separation device is simple, such as a straight microchannel, its force or flow values at each position can be solved analytically. However, advances in DNA separation methods utilize DNA flows in complex geometries which induce nonlinear force or flow fields. These must be solved numerically. Therefore, DNA separation simulations require a proper combination of DNA dynamics predictions and field calculations.

The finite element method (FEM) is a numerical method for solving differential equation within a boundary. This method discretizes the domain of the problem into smaller sub-domains, called finite elements or meshes, as shown in

rectangles, as with structured microchannels, the finite difference method can be used [

As mentioned earlier, FEM can be used for electric field calculations with DNA electrophoresis simulations. The electric field of potential is denoted by Φ. The governing Laplace equation, in the fluid domain, Ω, is shown below:

The boundary where the electric potential is explicitly applied, given as Φ = Φ_{given}, is

Boundary element method (BEM) is a numerical method used to solve “linear” partial differential equation in a boundary. In this method, the fundamental solution of the linear differential equation (Green’s function) must be available first. Compared to FEM, discretization is only required on boundaries, which results in fewer mesh points and more efficient calculations. Instead of the interpolation used in FEM, the boundary integral equation is used in BEM to evaluate flow or electric potential values at the positions of interest. The surface integrals of the Green’s function and its derivative are utilized for this [^{T} much more efficiently. This method can be applied to complex geometries and hydrodynamic interaction is considered as much level as Stokeian dynamics simulation [

The lattice-Boltzmann Method (LBM) is a numerical method for the simulation of fluid using the discrete Boltzmann equation instead of conservative momentum balance equations like the Navier-Stokes equation [

As in LBM, mesoscale models can accurately represent the hydrodynamic properties of a flow system and they are not as expensive as atomic models in terms of computation load. Dissipative Particle Dynamics (DPD) is a simulation technique for fluid which utilizes the dynamic simulation of coarse-grained particles on a mesoscale. Mesoscale methods are intermediate methods between atomic scale and microscale [

As in LBM, DPD is suitable for the calculation of flow fields in complex geometries including HIs. Another similarity is that electric force fields must be calculated explicitly. Additionally, the original DPD technique has a low Schmidt number, which is the ratio between kinematic viscosity to diffusivity. This causes slower momentum transfer when compared to mass transfer. This can be a major problem when simulating fluids within complex geometries [

DPD was applied to DNA separation simulations in microfluidic devices that utilized electrophoresis and structured microchannels to examine the HI effects [

In summary, inhomogeneous electric field considering complex geometry can be calculated either by FEM or BEM. BEM is more efficient but there are many available popular commercial tools for FEM. If flow field considering complex geometry can be calculated by FEM, LBM [

In this section, we summarize the simulations of popular DNA separation methods.

Gel electrophoresis is one of the most popular DNA separation tools. It is still widely used in many DNA related experiments [

Duke and Viovy adapted a MC simulation for studying DNA motion in gel electrophoresis [

FEM | BEM | LBM | DPD | |
---|---|---|---|---|

Advantage | Many commercial tools are available. Both electric field and flow filed can be calculated. | Mathematically accurate (using Green’s function). Both electric field and flow field can be calculated. | HI inclusion is intrinsic. Both linear and nonlinear (inertial effect) PDEs are possible. | HI inclusion is intrinsic. Both linear and nonlinear (inertial effect) PDEs are possible. |

Disadvantage | Inclusion of HI is difficult. | Limited to linear PDE. Green’s function must be available. | Limited to flow field calculation. | Limited to flow field calculation. Simulation parameters must be tuned. |

[

Azuma and Takayama performed a BD simulation of DNA in a constant electric field gel electrophoresis. They modeled DNA as a bead-spring model and the gel structure as immobilized bars, simulated as lines of beads, in a 3D periodic box. They tracked the evolution of the radius of the longer principal axis and the velocity of the center-of-mass and found that those values show periodic behaviors in relatively strong fields. This was inferred as the “elongation-contraction” mechanism in DNA. The period of the elongation-contraction mechanism was also found to be proportional to DNA length. They used this finding to explain why long DNA strands cannot be separated under a constant electric field gel electrophoresis [

Although gel electrophoresis is a very common method, its limitations were described previously in this paper: time consuming procedures, inconsistency of random gel structure, and difficulty in the separation of relatively long DNA chains [

With advances in post array devices, simulation studies have been used to both identify separation mechanisms and to explore optimal array designs. Saville and Sevick performed a BD simulation of a bead-spring model flowing around an obstacle [

Randall and Doyle incorporated an analytical expression for the inhomogeneous electric field around a circular object for more accurate DNA motion.

They identified the trends of these mechanisms in terms of the radius of gyration of DNA, Rg, the size of the obstacle, and the electric field strength. For example, when the field is strong enough and the obstacle’s diameter is small, the dominant mechanism is hooking [

Studies on the effects of different array types have been performed systematically with the help of simulations. Patel and Shaqfeh investigated BD of a freely-jointed bead-rod chain in a sparse array of posts when they are ordered versus

randomly dispersed. They concluded that disordered arrays in strong electric fields are optimal conditions for separation [

Capillary electrophoresis (CE) separates macromolecules in a capillary when an electric field is applied to the system. CE needs less time to separate DNA and gives higher resolutions and sensitivities compared to typical gel electrophoresis. CE has mainly contributed to human genome analysis [

Kekre et al. performed a BD simulation of DNA in CE [

Studies on DNA dynamics in “straight” (this is different from the structured microchannel discussed in Section 4.5) microchannel flows have been performed for basic understanding of DNA and solid boundary interactions. It is well known that if a pressure drop is applied to a Newtonian fluid between two parallel plates, a parabolic shape velocity distribution is created at steady state. Therefore, the velocities of DNA flowing in a microchannel are dependent on its cross-sectional position (faster elution for DNA flowing near a center) and any factors affecting the cross-sectional DNA position can be a separation mechanism. Jendrejack et al. performed BD simulation considering DNA-wall HI [

Periodically constricted channels were introduced as an effective way of creating entropic traps to separate DNA chains based on their length. The mechanism used in the entropic constriction of polymer molecules was first studied by Arvanitidou et al. [

As shown in

it was shown that longer strands of DNA molecules elute faster. Initially, this was explained by Han et al. [

The first attempt to simulate the device designed by Han et al. and to prove their theory was done by Tessier et al. [

Streek et al. performed BD simulation using the bead-spring model with a Hookean spring force. HI was ignored and the electric field was calculated using FDM [

Panwar and Kumar performed BD simulation with the bead-rod model [

In earlier works, HIs were neglected in simulations of DNA separation by electrophoresis. The decision to neglect these interactions was based on the assumption that HIs are screened if the Debye length of the DNA is smaller than the scale of the device confinements. Therefore, this is a questionable assumption in the small channels. Application of DPD to the entropic trap simulation enables to investigate the HI effects. Moeendarbary et al. found that larger molecules have higher probability of hernia (kink) formation entering the smaller channel. These chain dynamics contribute to the higher mobility of longer DNA chains [

Along with investigating the HI effects on separation simulations, the effects of using short DNA fragments and the effects of different entropic trap geometries have also been studied. Laachi et al. investigated the transport of shorter, or rigid, DNA molecules through periodic arrays of narrow channels [

Microscale rotational flows, or streaming flows, with counter-rotating vortices have been known as another method for trapping particles, or DNA strands [

Watari et al. performed a BD simulation using WLC model and an analytic stream of Taylor-vortex flow. The inclusion of HIs were conducted in the same manner as in the Equations (11)-(12), excluding DNA-wall HI. They investigated the effect of vortex flow conditions on DNA conformations and positions to show the potential for trapping DNA in vortices [

It was discovered that the sequencing and detection of DNA and RNA strands can be possible by forcing them through a narrow biological nanopore using an electric field, as shown in

A BD simulation of this process was done by Tian and Smith and considered the repulsive force from the nanopore’s walls [

barrier effect. Investigation of the conformation difference before and after translocation, found that the polymer chains were not in equilibrium during the process. Izmitli et al. took HI into account in their simulation study [

Similar to the studies on DNA structure in nanoconfinement [

In this study, we have reviewed the computational studies of DNA separations in micro-fabricated devices. We focused on the dynamic simulation of double stranded DNA in geometries related to separation methods and devices. The reviewed simulation approaches can also be extended to the dynamic simulation of other biopolymers in microscale flows [

The authors gratefully acknowledge financial supports from Missouri University of Science and Technology (UMRB and OURE).

Monjezi, S., Behdani, B., Palaniappan, M.B., Jones, J.D. and Park, J. (2017) Computational Studies of DNA Separations in Micro-Fabricated Devices: Review of General Approaches and Recent Applications. Advances in Chemical Engineering and Science, 7, 362-392. https://doi.org/10.4236/aces.2017.74027

BD: Brownian dynamics simulation

BEM: Boundary element method

CE: Capillary electrophoresis

DPD: Dissipative particle dynamics

FDM: Finite difference method

FEM: Finite element method

FENE: Finite extensibility nonlinear elastic chain

HI: Hydrodynamic interaction

LBM: Lattice-Boltzmann method

MC: Monte-Carlo

WLC: Worm-like chain model

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