The translocation of macromolecular chains through a narrow pore is ubiquitous and important in life processes,e.g.,injection of viral DNA,transfer of ribonucleic acid (RNA) across nuclear pores,and transport of proteins through biological membrane channels^[1]. In recent years,nanopores have been used to detect and analyze DNA bases,which is considered as a new generation technique of the DNA sequencing technique^{[2, 3]}. An electric field is imposed on the electrolyte solutions,in which a membrane with a nature or synthetic solid-state nanopore is placed inside,thus resulting in ionic currents in the nanopores. The charged ploymer translocation driven by the electricity leads to a temporary current reduction. The difference between the amplitude and the duration of the ionic current blockade is sensitively relied on the characteristics of the molecules. If the DNA molecules are immersed in the solution,four types of nucleotides,i.e.,A,T,G,and C,will produce the essential distinction for the ionic current reductions^[4]. Therefore,DNA translocation can be used to explore the diameter, the length,the volume,and the conformation of macromolecules.

The protein pore $\alpha $-hemolysin is one of the choices for detecting DNA,because it has an appropriate pore radius which can allow the passage of single-stranded DNA (ssDNA)^{[5, 6, 7]}. Kasianowicz et al.^[8] investigated the translocation of the ssDNA through the biological protein pore with the 2.6 nm diameter,and found that the passage of each DNA molecule was detectable as a transient decrease in the ionic current whose duration was proportional to the polymer length. Meller et al.^[9] reported the ssDNA sensing with an $\alpha $-hemolysin biological nanopore. They demonstrated that the ssDNA translocation velocity was independent of the DNA length if the length of the DNA was much longer than the nanopore length.

Many researchers focused on applying synthetic nanopores on DNA sequencing^{[10, 11, 12, 13, 14, 15, 16]}. Compared with natural nanopores, fabricated solid-state nanopores have the advantages in the potential of device integration and stability under a wide range of voltage biases,pH values,salinity,and temperatures. Storm et al.^[17] investigated the translocation of the double-stranded DNA (dsDNA) through a silicon nitride (SiN) nanopore,and found a scaling law of the translocation time $\tau $ to the polymer length $N$,i.e.,$\tau \sim N^{1.27}$. Several authors reported the salt dependence on the ion transport and DNA translocation through solid-state nanopores^{[18, 19, 20]}.

Recently,graphene nanopores became particularly promising in nanopore sequencing due to their great advantages in mechanical, electrical,and thermal properties^[21]. Graphene is constituted of one layer of carbon atoms,structured in the honeycomb lattice, with the thickness of approximately 0.3 nm,which is comparable to the ssDNA base pair stacking distance (0.32 nm-0.52 nm)^[22]. This superiority is significantly helpful in improving the sensitivity and resolution of the technique. Several research groups have achieved the dsDNA translocation through graphene nanopores^{[23, 24, 25]}. They found that the current blockades associated with the dsDNA translocation in folded and unfolded conformations.

There has been much development in the experiments of nanopore sequencing. However,some aspects of this new generation technology are still needed to be investigated. To reliably recognize the single base,a major challenge in the field is to increase the transit time per basepair/nucleotide^[4]. Therefore,numerical and theoretical investigations are beneficial to understand the transport characteristics and DNA movement in the electrolyte solution.

Many simulations have been performed by various numerical methods on polymer chain dynamics. The molecular dynamics simulation is presently too inefficient for the entire process of the DNA translocation in the mesoscale systems,since the large number of the particles has to be involved. Alternatively,the dissipative particle dynamics (DPD) becomes a powerful simulation technique so as to investigate the mesoscopic scale dynamics of the polymer,and it has already been widely used to study the behaviors of macromolecules in a diversity of circumstances^{[26, 27, 28, 29, 30]}. Smiatek et al.^[31] investigated the polyelectrolyte electrophoresis in narrow microchannels for different salt concentrations and surface slip conditions by DPD simulations,and found that the effective polyelectrolyte mobility was closely related to the slip length and the thickness of the electric double layer in the system.

In the present study,the translocation process of a homopolymer through a nanopore is focused on,and the electrokinetic transport dynamics of the polymer is investigated by implementing the electric fields into the DPD simulations,in which the coarse-grained worm-like chain (WLC) is used to model the homopolymer molecules. Our purpose is to visualize the conformation and dynamic properties of the polymer,and investigate the effects on the flow structure in the solution. The rest of the article is organized as follows. The theoretical model and simulation method are presented in Section 2. Section 3 focuses on the numerical results. Finally,we conclude with a brief summary in Section 4.

The DPD method was first proposed by Hoogerbrugge and Koelman^[32] to take the advantages of the mesh-free algorithm in molecular dynamics (MD) simulations^[33] and the large time-scale in the lattice-gas automata (LGA)^[34]. Instead of considering the atoms/molecules themselves,the particles in the DPD are coarse-grained,which represent a cluster of molecules. Thereafter,the DPD method becomes a widely used coarse-grained and momentum-conserving mesoscopic simulation method. It has been proved to be an effective tool for the computer simulation of polymer solutions. With a simplified model for polymer chains,e.g., bead-spring,we can formulate and compare a variety of realistic conservative interbead forces^[35].

In the DPD simulation,the time evolution of a typical DPD particle $i$ with unit mass follows the Newton laws of motion given by

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where $t$,$r_i$,and $v_i $ denote the time,the position,and the velocity vectors,respectively. $F^{\rm i}$ is the interparticle force,and $F^{\rm e}$ is the external force applied on each particle. In the present study,the externally applied forces include the external electric force and the spring force. $F^{\rm i} $ contains the conservative,dissipative,and random forces,i.e.,

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where the sum runs over all other particles within a certain cut-off radius $r_{\rm c} $. The conservative force $F_{ij}^{\rm C} $ between the particle $i$ and the particle $j$ is responsible for the thermodynamic behavior of the system. The conservative force is usually chosen to be a soft repulsion as follows:

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where $a_{ij} $ is a coefficient measuring the interaction between the particle $i$ and the particle $j.$ $ r_{ij} =r_i-r_j $,and $r_{ij}=\left| {r_{ij} } \right|$. $e_{ij} =r_{ij}/\left| {r_{ij} } \right|$ is the unit vector. The dissipative and random forces are given by

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respectively,where $\gamma $ is the dissipative parameter,${v_{ij}} = {v_i} - {v_j}$ is the velocity difference,$\sigma $ is the amplitude of the white noise,$\theta _{ij} $ is a white noise function with zero mean and unit variance. The weighting functions of the dissipative force and the stochastic force are related by the fluctuation-dissipation theorem as follows^[36]:

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where $k_{\rm B} $ is the Boltzmann constant,and $T$ is the equilibrium temperature. The weight function $\omega ^{\rm R}$ is given by

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The DPD simulations are conducted in a cubic box with the dimensions $L_x $,$L_y $,and $L_z $. The origin of the coordinates is set at the center of the box. The nanopore,as presented in Figs. 1 and 2, is modelled as a cylindrical pore with the length $l_{\rm {pore}} $ and the radius $r$. The periodic boundary conditions are imposed in all directions. A homogeneous polyelectrolyte with $N $ beads is considered,in which each bead has a negative unit charge $e$. The solvent contains positive counterions for the polyelectrolyte and additional positively and negatively charged salt ions. All charges are monovalent,and the system as a whole is electro-neutral. In order to simulate the homopolymer translocation through nanopores,a uniform electric field $E$ is used,parallel to the $x$-direction. In this system,the charged particles are subject not only to the electric force but also to the long-range electrostatic interactions. The electrostatic interactions are calculated by the standard Ewald sum method^{[37, 38, 39]}. The wall of the nanopore contains immobilized DPD particles. The multi-layers of the frozen DPD particles on the wall together with the bounce-back reflections are adopted to ensure that the no-slip condition of the fluid is satisfied.

Fig. 1 Schematic representation of simulation model for polymer chain (bigger spheres) and membrane with narrow pore, where smaller spheres represent positive or negative ions, counterions, and solvent, and solid lines denote nanochannel

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Fig. 2 Membrane with nanopore of radius r = 1, where electric field is applied in x-direction

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Many biological polymers have an intrinsic elasticity,which causes them to remain rigid at small length scales while display significant flexibility over large length scales. One of the theories,which are able to account for this characteristic behavior and have been successful used to describe the elastic properties of a variety of biomolecules,such as ssDNA,dsDNA,RNA,and polypeptide chains,is the worm-like chain (WLC) model[35, 40-41]. It is adopted in the present study to imitate the ssDNA. In the WCL model,a polymer is discretized into $N$ beads connected by $( {N-1} )$ springs,where the spring force law for a chain segment can be expressed by

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where $\lambda _{\rm p}$ is the persistence length measuring the chain stiffness,and $l_{\rm m}$ is the maximum permissible length of one chain segment. The subscripts $i$ and $j$ denote the neighbored beads. If the total length of the chain is $L$ and the number of the beads in the chain is $N$,then $l_{\rm m} =L/\left( {N-1} \right)$. $\lambda _{\rm p}$ is set to be 0.75 nm in the simulations^{[42, 43]}. The persistence length of dsDNA has been widely known to be 50 nm^[44]. For ssDNA,the persistence length is between 0.75 nm and 3 nm^{[42, 43]}. Therefore,dsDNA should be much stiffer than ssDNA due to the hydrogen bonding between the base pairs.

In the DPD simulations,all of the parameters are nondimensionlized to the DPD units. The length,mass,and energy are scaled by the cutoff radius $r_{\rm c} $,the mass of the particles,and $k_{\rm B} T$,respectively. The parameter $a_{\rm {ll}} $ describes the repulsive interaction between liquid particles and liquid particles, while $a_{\rm {sl}} $ stands for the repulsive interaction between solid particles and liquid particles,$E$ is the uniform electric field,and $n$ is the number of the salt ions. All computational quantities are listed in Table 1.

Table 1 Computational parameters used in present simulations, where all quantities are given in DPD units

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To investigate the statistics of the macroscopic physical variables, the coordinates of all particles are recorded every time interval $\Delta t$. The spatial location vector and the velocity vector of the particles are transformed to the cylindrical coordinates in the $rx$-plane,which is partitioned into the gridded cells with the interval range $(\Delta r,\Delta x)$.

For the cell at ($r$,$x)$ containing $N_{\rm w} $ fluid particles,the transient local fluid velocity vector can be calculated by

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where $r_i (t_j )$ is the location vector of the fluid particle $i$ at the time $t_j $,and $\Delta t$ is the time interval.

The chain structure changes of the polymer during the translocation process are investigated by considering the static properties such as the MSR of gyration $\langle {R_{\rm g}^2} \rangle $. For a chain of $N$ beads,the MSR is defined by

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where $r_i $ is the position vector of each bead,$r_{\rm {cm}} $ is the position vector of the center of the mass of the chain,and $\langle \dot \rangle $ denotes for the time averaging.

The number density of the anions and cations in the cells are obtained by

where $N_{\rm c} $ and $V_{\rm c}$ are the number of the ions and the volume of the corresponding cell,respectively.

At the time $t$,the average total ionic currents are computed by $I=eN_i /\Delta t$,where $N_i $ stands for the number of the anions or cations across the nanopore during the time interval $\Delta t$, and $e$ is the amount of the charge per electron.

In the present study,focusing on the translocation process of the polymer through a nanopore,the electrokinetic transport of a charged polymer chain of the length $N=20$ in the electrolyte solutions is simulated,where the dimensions of the system are $L_x \times L_y \times L_z =14\times 5\times 5$. Figure 3 shows a translocation event for an external electric field $E=1.5$ and the number of the salt ions $n=400$. There are mainly three phases in the translocation process of the polymer molecule,i.e., approach,capture,and migration^[45].

Fig. 3 Sequential snapshots of typical translocation process of polymer passing through nanopore, where E = 1.5 and r = 1

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At the beginning of the simulation,the homopolymer is placed in front of the nanopore,and then accelerated by the external electric field (see Fig. 3(a)). Since the polymer is negatively charged,the chain is driven to approach the mouth of the nanopore. When the front bead of the polymer is pulled into the pore,it is forced to go forward,and pulls the connected neighbors into the nanopore (see Figs. 3(b) and 3(c)). Eventually,the translocation process is fulfilled when the end of the chain traverses the narrow pore (see Fig. 3(d)).

Pervious investigations indicated that the polymer chain passed through the nanopore in various folded structures^[46]. The present simulations successfully reproduce three typical conformations (see Fig. 4). For instance,Fig. 4(a) shows an event of single-file translocation,and Fig. 4(b) shows a fully double-folded structure of the polymer chain. In Fig. 4(c),the leading head in the solution shows a double-folded structure and the remaining part is unfolded,which is considered as a partially folded conformation.

Fig. 4 Polymer molecule in nanopore with different conformations, where E = 1.5 and r = 1

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The polymer translocation through a narrow pore is a complex phenomenon. In order to study the dynamics of the polymers in the translocation process,the MSR of gyration of the chains $\left\langle {R_{\rm g}^2 } \right\rangle $ during the migration is analyzed and presented in Figs. 5 and 6,together with its $x$-component $\left\langle {R_{{\rm g},x}^2 } \right\rangle $. In addition,to track the trajectory of the polymer,the variation of the $x$-component of the center-of-mass $X_{\rm com}$ with time is also plotted. In this figure,we observe that the polymer undergoes a large conformational deformation during the translocation. Initially,there is only a slight change in $\left\langle {R_{\rm g}^2 } \right\rangle $ during the approach process. When the polymer is subsequently captured,$\left\langle {R_{\rm g}^2 } \right\rangle $ and $\left\langle {R_{{\rm g},x}^2 } \right\rangle $ increase gradually when several beads come into the pore,which is likely a squeezing effect due to the presence of the nanopore walls^[47],resulting in an elongation of the polymer in the axial direction of the channel. Then,$\left\langle {R_{\rm g}^2 } \right\rangle $ increases remarkably until it reaches the maximum, implying that the polymer molecule is extended and stretched by the electric field during the translocation. The value of $\left\langle {R_{{\rm g},x}^2 } \right\rangle $ is dominant in comparison with the other two components of $\left\langle {R_{\rm g}^2 } \right\rangle $,and most of the polymer monomers reside in the nanochannel at this moment. After the polymer exits from the pore, the MSR of gyration decreases dramatically.

Fig. 5 Evolutions of MSR of gyration of polymer in single-file conformation, where center-of-mass of polymer is along x-direction during migration

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Fig. 6 Variations of MSR of gyration of polymer during translocation

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Aside from the single-file translocation event,we also examine the variations of the radius of gyration with time for the other two different translocation events,i.e.,the fully double-folded conformation in Fig. 4(b) and the partially folded conformation in Fig. 4(c). Figure 6 shows that the maximum values of $\left\langle {R_{\rm g}^2 } \right\rangle $ are 2.81$r_{\rm c} ^2$ and 2.02$r_{\rm c} ^2$ for the chain in the single-file and partially folded conformations, respectively. In comparison,the equilibrated radius of gyration of the polymer chain in the solution is approximately 0.868$r_{\rm c} ^2$. However,for the double-folded translocation event,no apparent increase in $\left\langle {R_{\rm g}^2 } \right\rangle $ is observed during the translocation process. This may be ascribed to that the polymer we considered is not long enough due to the limitation of the computational resources. Furthermore,the slight difference between the MSR of gyration before and after the translocation of the polymer can be ignored.

The effect of the translocation of a charged polymer chain is considered on the ionic currents through the pore in this subsection. A constant electric field ($E=1.5$) is used to induce the ionic transport in the solution through a nanopore. The coordinates of all charged particles in the simulations are recorded to calculate the ionic currents. At the beginning of the simulation, the system is allowed to evolve until the current has reached the steady-state value.

Previous investigations showed that different current signatures were observed in the electronic measurements of the polymer translocation through nanopores,which were ascribed to the passage of the polymer molecules through the pore with different folding conformations^{[23, 24, 25]}. Figure 7 presents the level of the total ionic current during the translocation. The thermal noise in the electrical current is found to partially affect the numerical measurements. When a molecule enters the pore,the current drops from its equilibrium value for the open pore to the blocked level. This characteristic blockade of the ionic current can be observed when the polymer resides in the nanopore,whereas the current recovers its original value when the polymer escapes from the exit of the nanopore. The current signatures indicate that the magnitude of the current reduction crucially depends on the ways of the polymer translocation. The first valley of the curve represents the translocation of the single-file event,the second blockade shows partially folded polymer translocation with two different current blockades,and the third one with longer duration and deeper valley corresponds to the event of the fully-folded translocation. These theoretical results are in qualitative agreement with pervious experimental observations^[21].

Fig. 7 Current reductions of three translocation events through nanopore, where curve shows translocation events of single-file, partially folded, and fully folded polymer molecules from left to right, respectively

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The distribution of the ions is of great importance on the transport properties of the polymer molecules. Because of the blockage of the ionic flux by the membrane,the ions may accumulate and form a concentration polarization layer (CPL) in its vicinity when an external electric field is used^[48]. The ionic number density distributions of the anions and cations in the solution is plotted in Fig. 8 for the salt concentration when $n=100$ and $E=3$,where $x=0$ corresponds to the location of the neutral membrance. Aiming at examining the role of the polymer on the transport properties of the electrolyte solution,both simulations,namely,with polymer and without polymer,are performed. As shown in Fig. 8,the anions and cations form the CPLs on the opposite sides of the sheet. These results are in good agreement with the MD simulation of Hu et al. on the ionic transport through a nanopore^[48]. The profiles without polymer (not shown here) are also analyzed, and no remarkable difference is observed.

Fig. 8 Number density profile of cations and anions in x-direction, where E = 3 and r = 1

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The CPLs are able to induce the electric pressure gradients near the membrane,especially in the regions adjacent to the nanopore. Therefore,the alterations of the electric field are expected to affect the ionic transport in the solution. If the applied electric pressure is sufficiently strong,the viscous resistance and thermal fluctuations may be overcome,and the vortices may be generated in the fluid system. This phenomenon was addressed by the molecular simulations of Hu et al.^[48],and later confirmed by the nanoscale simulations based on the coupled Nernst-Planck and hydrodynamic equations^[49]. The flow streamlines for the nanopore with the radius $r=1$ are shown in Fig. 9 for the simulations with polymer and without polymer. Comparing Fig. 9(a) with Fig. 9(b),it can be found that the movement of the macromolecular chain might have the effect of eliminating the structure of the vortices and the fluid velocity in the electrolyte system. To seek the physics behind this phenomenon,the consideration of the distributions of the ions is only slightly different between the cases with polymer and without polymer,the reason might lie in that the moving polymer driven by the electric field will disturb the formation of the nanoscale vortices and flow structures.

Fig. 9 Flow fields of electrolyte solution, where n = 100, E = −3, and r = 1, and rectangles (grey near x = 0) denotes neutral membrance

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The mesoscopic DPD simulations of the electric-induced homopolymer translocation through a nanopore are carried out in the present study,taking account of the electrostatic interactions. The results show that the polymer chain,which is modeled as a WLC,undergoes conformation changes during the translocation process. Being stretched in the nanopore,the leading head bases pass through the nanopore pulling the connected neighbors. Three types of polymer structures,i.e.,single-file,partially folded,and fully folded, are obtained,which have been observed in the previous experiments. Moreover,the simulations reveal that the ionic current blockades strongly correlate with the local conformation of the polymer inside the pore. We further investigate the transport properties of an electrolyte through a nanopore. The applied electrical field is found to force the anions and cations to move through the pore in the opposite directions. Due to the partial blockage of the ionic current by the membrane,a CPL is observed to be closed to the membrane. The CPL is able to drive a flow,as long as the forces are large enough to overcome the effects of the viscosity and Brownian fluctuations. The results confirm that the formation of vortices can be affected by the polymer movement in the polyelectrolyte of the solution. The present study addresses that the existence of polymer molecules might play the role of eliminating the fluid vortices. This study shows that the DPD is a powerful tool for the theoretical investigation of mesoscopic phenomena. We hope that these results could provide useful information for the design of advanced nanopore devices for DNA sequencing.