Translocation of Diblock Copolymer through Compound Channels: A

Oct 6, 2014 - Macromolecules , 2014, 47 (20), pp 7215–7220 ... The physical mechanisms are discussed from the free energy landscape of polymer trans...
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Translocation of Diblock Copolymer through Compound Channels: A Monte Carlo Simulation Study Chao Wang,†,‡ Ying-Cai Chen,‡ Shuang Zhang,† and Meng-Bo Luo*,† †

Department of Physics, Zhejiang University, Hangzhou 310027, China Department of Physics, Taizhou University, Taizhou 318000, China

‡

ABSTRACT: The forced translocation of diblock copolymers (ANABNB) through compound channels composed of part α with length Lα near the cis side and part β with length Lβ near the trans side was investigated. The interaction between monomer A and channel α is strongly attractive, while all other interactions are purely repulsive. We study the translocation mode that the block A threads the channel ahead of the block B since it is the most probable translocation event. Simulation results show that the translocation process is remarkably dependent on Lα and there are two maxima of translocation time. The physical mechanisms are discussed from the free energy landscape of polymer translocation. The place of the first maximum, at which there is a deepest free energy well before the block A completely enters the channel, is independent of NA. The place of the second one is at NAbx with bx the mean bond length along the channel, resulted from the matching of block A with channel α. Results show that the translocation time of diblock copolymer can be tuned by using compound channels.

1. INTRODUCTION

The translocation process is sufficiently slow under weak driving force, so it is proper to assume that the polymer is at an equilibrium state during the whole translocation process. Then the free energy of the polymer at every step of the translocation can be calculated.13−16,20−22 Theoretically, based on the free energy landscape, the translocation time τ can be obtained by the Fokker−Planck equation, and scaling relations between τ and some important parameters, such as the polymer length and the driving force, can be predicted.13,14 Furthermore, the free energy landscape is often used in interpreting qualitatively some important results from experiment, theory, and simulation. So, analyses of the free energy landscape is very useful in understanding the dynamical behaviors of polymer translocation process. It was found that polymer−channel interaction plays an important role in the translocation process.11,16,29,30,36 At repulsive or weak attractive polymer−pore interactions, there is a free energy barrier at the filling stage of the translocation, making it difficult for polymer to enter the channel. And at strong attractions, there is a free energy well at the last stage of the translocation, making it difficult for polymer to leave the channel. Whereas at moderate attractions, polymer can translocate through the channel quickly.29 The translocation dynamics is not only influenced by the strength of the attractive polymer−channel interaction but also affected by the variation of the attractive interaction along the

The translocation of polymer through nanochannels has attracted many researchers from physics, chemistry, and biology due to its universal existence in nature and its potential application in technology. Examples contain the transport of proteins through membrane channels,1−3 the translocation of RNA molecules across nuclear pores, the transfer of DNA molecules from virus to host cells and that of genes between bacteria,4 DNA separation,5,6 gene sequencing, gene therapy, controlled drug delivery, and gel electrophoresis,7 etc. Furthermore, polymer translocation is affected by many factors, such as driving force or field, polymer−solvent and polymer− channel interactions, polymer concentration, flow of fluid, pH value of solution, geometry and constitution of the channel, etc. Consequently, many experiments,5,6,8−12 theoretical studies,13−22 and simulations23−40 have been extensively performed to uncover the underlying physical mechanisms of the translocation process. It is difficult for polymer to thread spontaneously through narrow channels or narrow pores. In experiment, the translocation of polyelectrolyte, such as DNA, through channels is usually achieved under the driving of electrical field. Because of the small dielectric constant and low mobility of DNA, the ionic current through narrow channel is blocked during the translocation of DNA. Therefore, the translocation details of DNA through the channel can be obtained by recording the blockage in the channel current.8 In addition, the motion of DNA can also be detected using optical microscopes by labeling the DNA with fluorescent dye.5,6,12 © XXXX American Chemical Society

Received: June 24, 2014 Revised: September 6, 2014

A

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wall and channel are constructed by stationary beads. Inside the channel, a uniform electrical field with strength E is applied parallel to the channel for driving the charged polymer through the channel, whereas there is no external electrical field at both the cis side and the trans side. In our simulations the polymer chain ANABNB, composed of NA monomers for block A and NB monomers for block B, is mimicked by a coarse-grained off-lattice bead spring model.32 The interactions between bonded beads are described by the finitely extensible nonlinear elastic (FENE) potential:

channel. It was found that the translocation process is extremely sensitive to the detailed structure of the patterned stickiness inside the channel, and channels with well-patterned stickiness have the potential to detect the polymer sequence efficiently.37 In the recent works, we have studied the translocation of polymer through gradient channels39 and compound channels,40 respectively. For the gradient channel case, the polymer−channel interaction varies linearly along the channel. It was found that the polymer chain can translocate through the channel quickly under proper interaction gradient. Moreover, the scaling relation between the translocation time and the polymer length was also found to be dependent on the interaction gradient. For the compound channel case, the channel is composed of two parts, part α and part β, which have different lengths and different interactions with the polymer. Simulation results indicated that the translocation process is affected remarkably by the length of part α as well as the polymer−channel interaction. In this work the driven translocation of diblock copolymers (ANABNB) through compound channels under electrical field is studied by using Monte Carlo (MC) simulation. To highlight the influence of the compound channel on the polymer translocation, we use a weak electrical field. But this will lead to a large amount of simulation time. In order to save the simulation time, simulations are performed in a two-dimensional (2D) space. The channel is composed of parts α and β with lengths Lα and Lβ, respectively. The interaction between monomer A and channel α is strongly attractive, while all other interactions are purely repulsive. Under these interactions the translocation with block A entering the channel first is the most probable mode, so we only calculate the translocation time τ for this event. We find that the translocation process is remarkably affected by Lα, and it is difficult for polymer chain to thread through the αβ interface when the length of block A just matches the length of channel α.

UFENE = −

⎡ ⎛ b − b0 ⎞2 ⎤ kF (bmax − b0)2 ln⎢1 − ⎜ ⎟⎥ ⎢⎣ 2 ⎝ bmax − b0 ⎠ ⎥⎦

(1)

with the spring constant kF = 40, the average bond length b0 = 0.7, the maximum bond length bmax = 1, and the minimum bond length bmin = 0.4.32 Here b is the bond length. The interactions between nonbonded monomers with distance r in polymer are described by the Morse potential: ⎧ ε{exp[− 2α (r − r )] − 2 r ≤ rcut M min ⎪ UM(r ) = ⎨ exp[− αM(r − rmin)]} − Ucut ; ⎪ r > rcut ⎩ 0;

(2)

where αM = 24, rmin = 0.8, rcut = 1, ε = 1,32 and Ucut is a special value that ensures UM(rcut) = 0. All the monomer−channel interactions and the monomer−wall interactions are modeled by the Morse potential by eq 2 with rcut = 1 and ε = εAα for the interaction between monomer A and channel α and rcut = 0.8 and ε = 1 for the others. Therefore, the interaction between monomer A and channel α is attractive, whereas all other interactions are purely repulsive. Besides, each monomer in polymer carries one effective charge q = 1. Therefore, every monomer inside the channel experiences a uniform electrical force f = qE = E, by which the polymer is driven to pass through the channel. The dynamics of the polymer chain is achieved by random motion of polymer monomers with MC method and the standard Metropolis algorithm. For each trial move, a monomer is randomly selected and is attempted to move from its position (x0, y0) to a new site (x, y) with increments Δx and Δy chosen randomly from the intervals (−0.5, 0.5).32 If the new position does not violate the bond length constraints, the trial move will be accepted with a probability min[1, exp(−ΔU/kBT)], where ΔU represents the energy change due to the trial move. The time unit is one Monte Carlo step (MCS) during which N moves are tried. Here N = NA + NB represents the length of the copolymer chain. For a copolymer ANABNB, it was found that the block with strong attractive interaction has a high probability to enter the channel first.33 As we consider a strong attraction between block A and the channel α, we therefore only consider the translocation of block A entering the channel first in this work. Therefore, the first monomer A is placed at the entrance of the channel while the remaining monomers are at the cis side at the beginning of the simulation. The polymer chain is then allowed to undergo a long time of Brownian motion to reach an equilibrium state, but under the constraint that the first monomer is fixed. Once the polymer chain reaches the equilibrium state, we release the first monomer and set this moment as time t = 0. Under the random thermal fluctuation and the driving of the electrical force inside the channel, several monomers may move into the channel by back and forth

2. SIMULATION MODEL AND METHOD A sketch of the 2D model system is presented in Figure 1. The whole space is separated by a wall with thickness L. The cis side at the left and the trans side at the right are connected by a channel of width W in the center of the wall. The channel is composed of part α with length Lα and part β with length Lβ. The two parts have different interactions with the polymer. The

Figure 1. A sketch of the copolymer model and the compound channel used in the simulation. The copolymer is composed of block A and block B. The compound channel with width W is composed of part α with length Lα and part β with length Lβ. Along the channel (the x direction), an external electrical field with strength E is applied inside the channel. B

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motions. But this step does not necessarily lead to a successful translocation. The monomers have entered into the channel may be withdrawn from the channel under the influence of the entropy barrier. This contributes an attempted transition. Once the head segment moves back into the cis side again, we begin a new translocation process until the successful translocation occurs eventually. In this article the time interval for the successful translocation is defined as the translocation time τ and the translocation probability ptrans is defined as 1/(Natt + 1) with Natt the attempted translocation number in one simulation. The simulation results shown in this work are averaged over 2000 independent samples. In this work kBT and bmax are set as the units of energy and length, respectively, where kB is the Boltzmann constant and T is the temperature. Therefore, the electrical field strength E is in the units of kBT/qbmax. In most of our simulations, we fix N = 64, W = 2, E = 0.2, εAα = 3, and L = 40 but vary Lα and NA. The choice of εAα = 3 ensures that the attraction between monomer A and channel α is strong. The strong attraction and driving force can compensate for the loss of entropy; therefore, the free energy of polymer decreases with the translocation of monomers A. We mainly study the influence of channel composition on the dynamics of copolymer translocation and uncover the underlying physical mechanism by using the free energy landscape.

compound channel of length L = 40. Figure 3a shows the dependence of τ on Lα for different NAs. The dependence of τ

3. RESULTS AND DISCUSSION 3.1. Translocation Probability. At first, the translocation probability ptrans and the translocation time τ were calculated. Figure 2 shows the dependence of ptrans on Lα for different NAs.

Figure 3. (a) Dependence of the translocation time τ on Lα for different NAs. (b) Dependence of τp and Lαp on NA at moderate NA regions; solid straight line shows the relation Lαp ≈ 0.5NA. Simulation parameters are N = 64, E = 0.2, and εAα = 3.

on Lα is rather complicated, and the behavior is dependent on NA. For large NAs, there is a small peak, which is named as satellite peak, at small Lα. Both the value of the satellite peak τs and its place Lαs are roughly independent of NA. For moderate NAs, a second peak, which is named as principal peak, appears. But the value of the principal peak τp and its place Lαp are dependent on NA. Figure 3b shows the dependence of τp and Lαp on NA. We find that τp decreases quickly with the increase in NA, whereas Lαp increases linearly with NA with a relation Lαp ≈ 0.5NA. As τp ≫ τs at small NA, the satellite peak at Lαs is covered gradually by the principal peak at Lαp as shown in Figure 3a for NA = 22 and 20 as examples. In addition, both peaks are affected by the driving force and the polymer− channel interaction. For the satellite peak, τs increases with the decrease in E or increase in εAα, and Lαs increases with the decrease in E or εAα (results not shown), which are the same as the results of the translocation of homogeneous polymer through compound channels.40 For the principal peak, shown in Figure 4 for NA = 35 as an example, we find that the place Lαp is independent of E and εAα, and the peak value τp increases with εAα but decreases with E. However, the principal peak vanishes gradually with the increase in E or decrease in εAα. Our simulation results show that the channel composition plays an important role in the translocation of copolymers, and the translocation process can be tuned by the compound channel. One can also see from Figure 3a that the translocation time τ is extremely sensitive to NA at small and moderate NA regions when Lα is moderate, which means that compound channel may be very useful for copolymer sequencing. 3.3. Theoretical Analysis. The polymer inside the channel is not straight. The average bond length along the channel bx is about 0.5, which is found to be almost a constant as it is only slightly dependent on the polymer−channel interaction and the

Figure 2. Translocation probability ptrans as a function of Lα for different NAs, where the chain length N = 64. Other parameters are E = 0.2 and εAα = 3.

Clearly, two different regimes are found. We find that ptrans is roughly zero when Lα < 4, whereas it jumps quickly to a saturation value when Lα > 4. Our results show that the length of the attractive channel is an important factor affecting the translocation probability. The value of ptrans increases with the increase in εAα and E; i.e., the translocation probability increases with the attraction strength and the driving force.30 This is in agreement with already known facts that the attraction of the channel and the driving force are two important factors which can facilitate the polymer to enter channel. Moreover, we also find that the point separating the two different regimes shown in Figure 2 shifts to small Lα with the increase in εAα and E (results not shown). 3.2. Translocation Time. We have studied the translocation time τ of a polymer with length N = 64 through a C

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translocation is controlled by the filling process which is more complicated than the escaping process. We then discuss the filling process. The dynamics of polymer in the filling process can be understood qualitatively from the view of the free energy landscape. Assuming that the polymer chain is in the equilibrium state during the translocation (the quasi-static approximation), the free energy (in units of kBT) of the filling process can be expressed as for the case NAbx ≤ Lα ⎧ E ⎪− x 2 − ⎪ 2bx ⎪ ⎪ E 2 ⎪− 2b x − x ⎪ ⎪ F(x) = ⎨ E 2 ⎪− x − ⎪ 2bx s ⎪ + 0x ⎪ bx ⎪ E ⎪− x 2 + ⎪ 2b ⎩ x

Figure 4. Dependence of the translocation time τ on Lα for (a) different εAαs at E = 0.2 and (b) different Es at εAα = 3; here N = 64 and NA = 35. The insets of (a) and (b) present the dependence of τp on εAα and E, respectively.

⎛ εAα − s0 ⎞ ⎟x ⎜ ⎝ bx ⎠ εAαNA +

s0 x bx

(x ≤ NAbx) (NAbx < x ≤ Lα)

⎛ L − x⎞ ⎟εAα (Lα < x ≤ Lα + NAbx) ⎜NA + α bx ⎠ ⎝

s0 x bx

(Lα + NAbx < x < L)

(3)

for the case NAbx > Lα ⎧ E ⎪− x 2 − ⎪ 2bx ⎪ ⎪ E 2 ⎪− 2b x − x ⎪ ⎪ F(x) = ⎨ E 2 ⎪− x − ⎪ 2bx s ⎪ + 0x ⎪ bx ⎪ ⎪− E x 2 + ⎪ 2b ⎩ x

driving force. The entire polymer can be inside the channel during the translocation for polymer if Nbx < L. For this case, the whole polymer translocation process can be divided into two subprocesses: (1) filling process with time τf, during which the head monomer of the polymer chain moves toward the exit until the end monomer reaches the entrance, and (2) escaping process with time τe, during which the end monomer threads through the channel and finally leaves the exit. The translocation time is τ = τf + τe. We first discuss the escaping process. For homogeneous polymers escaping from homogeneous channels, the escaping process was found to be strongly dependent on the interaction between the polymer and the channel.35 When the polymer− channel interaction is repulsive or weakly attractive, the polymer can leave the channel quickly that results in a small τe. When the polymer−channel interaction is strongly attractive and the driving force is weak, it is very difficult for polymer to leave the channel since polymer will lose high attractive energy to finish the escaping process. In this case, τe is big. For diblock copolymers translocation through compound channels in our model, the escaping process starts when the end monomer (monomer B) reaches the entrance of the channel and finishes when the end monomer leaves the exit of the channel. The loss of the attractive energy, ΔUp, is determined by the length of block A inside channel α, Lα − NBbx or Lα + NAbx − Nbx, at the beginning of the escaping process. If Lα + NAbx − Nbx ≤ 0 or there is no monomer A inside the channel α, then the attractive energy loss ΔUp = 0 and the polymer can escape the channel quickly. Whereas if Lα + NAbx − Nbx > 0, then ΔUp > 0 since part of block A is inside the channel α. It is clear that ΔUp increases with the increase in Lα and (or) NA as there will be more monomer A inside the channel α. Therefore, in the case that both Lα and NA are large, it is very difficult for polymer to leave the channel; thus τe is big. Then the whole translocation is controlled by the escaping process, leading to the increase of τ with Lα or NA, as shown in Figure 3a. While in other cases, τe is small and the whole

⎛ εAα − s0 ⎞ ⎟x ⎜ ⎝ bx ⎠

(x ≤ Lα)

s εAα Lα + 0 x bx bx

(Lα < x ≤ NAbx)

⎛ L − ⎜NA + α bx ⎝

x⎞ ⎟εAα (NAbx < x ≤ Lα + NAbx) ⎠

s0 x bx

(Lα + NAbx < x < L)

(4)

where x represents the position of the head monomer inside the channel and s0 (in units of kB) represents the decrement in entropy for a monomer inside the channel. Here the free energy of the initial state (x = 0) is set as 0. s0 is an important parameter but is difficult to be calculated directly. However, our simulation results can be well understood by adopting s0 = 2.1. Figure 5 shows the free energy of the filling process F(x) as a function of x for NA = 25 and different Lαs. For the filling process, there are three important states: state I, the first monomer of block A arriving at the αβ interface between channel α and channel β, i.e., x = Lα; state II, the first monomer of block B arriving at the entrance of the channel, i.e., x = NAbx; and state III, the first monomer of block B arriving at the αβ interface, i.e., x = Lα + NAbx. For the case Lα < NAbx, the polymer chain reaches these states, I, II, and III, orderly. While for the case Lα > NAbx, the order of the three states is II, I, and III. The places of these states are at x1 = Lα, x2 = NAbx, and x3 = x1 + x2. For both cases, there are two energy wells at states I and II, respectively, leading to the trapping behavior at the two states. The trapping behavior as well as the time is dependent on the heights of three free energy barriers, ΔF0, ΔF1, and ΔF2, as shown in Figure 5 for Lα = 3 as an example. At Lα ≪ NAbx, ΔF1 is big but ΔF0 is small, it is easy for polymer returning to the cis side by overcoming ΔF0. But it is very difficult for block A to pass through the entrance of the channel completely, leading to the small successful translocation probability ptrans in Figure 2. As the time is reset once D

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Figure 7. Dependence of the residence time tr on the length of polymer inside the channel x for NA = 25 and four different Lαs. Dashed vertical lines show the places of peaks near x = Lα. Simulation parameters are N = 64, E = 0.2, and εAα = 3.

Figure 5. Free energy F(x) as a function of x for NA = 25 at different Lαs from 3 to 18 with increment 1. At small Lα, each curve is consisted by four parts jointed at three states. Parameters are E = 0.2 and εAα = 3. The blue line is F(x) for Lα = 4, and the green one is F(x) for Lα = 12.

polymer inside the channel x = ninbx for NA = 25 and different Lαs. We can see that, for Lα ≠ NAbx, there are two local maxima locating near x = Lα and x = NAbx, respectively, whereas for Lα = NAbx (here Lα= 13), only one single maximum is found at x = Lα. These simulation results of tr are in good agreement with the free energy landscapes shown in Figure 5.

the polymer goes back into the cis side in the simulations, the trapped time at state I is thus determined by the free energy well Fwell,1 (= min(ΔF0, ΔF1)). The maximum of Fwell,1 is at Lα ≈ 4, which results in the satellite peak of τ and the jump of ptrans, which are in agreement with our results, as shown in Figures 2 and 3a. With the increase in Lα, ΔF1 decreases and the two free energy wells get close; then the filling process is dominated by value Fwell,2 (= min(ΔF0, ΔF2)). Fwell,2 reaches maximum at Lα = NAbx at which state I meets state II, leading to the principal peak of τ as shown in Figure 3a. Therefore, we have Lαs = 4 and Lαp = 0.5NA as shown in Figure 3. The result shows that polymer threads forward slowly when the channel α is just full filled by the whole block A, i.e., at NAbx = Lα. In the simulations, we have calculated the distance Dih of the head monomer away from the αβ interface once the whole block A enters into the channel, that is, Dih = |NAbx − Lα|. Figure 6 shows the dependence of τ and Dih on Lα for NA = 25.

4. CONCLUSIONS We have investigated the translocation of diblock copolymer (ANABNB) through compound channels under external electrical field by using the dynamical Monte Carlo method. The compound channel is composed of part α near the cis side with length Lα and part β near the trans side with length Lβ. In the model system, the interaction between monomer A and channel α is strongly attractive, while all other interactions are purely repulsive. The translocation process of copolymer is remarkably affected by the length Lα, which is consistent with the free energy landscape of polymer translocation. There are two maxima in the translocation time τ for the translocation of a polymer through channels with different Lαs. The place of the first maximum is at Lαs, which is independent of NA because there is a deepest free energy well at Lαs. And the place of the second one is at Lαp = NAbx with bx the mean bond length along the channel. The free energy well at Lαp results from the matching of block A with channel α. For both cases, the copolymer is trapped. Results show that the translocation time of diblock copolymer can be tuned by using compound channels. Finally, although our work was carried out in the 2D system, we would predict that the simulation results are qualitatively similar to that of the 3D system since the forms of free energy landscapes governing the translocation dynamics are similar for both 2D and 3D systems. The different parameter is only the entropy parameter s0 which is the entropy of one monomer in solution.

Figure 6. Dependence of the translocation time τ and the distance Dih of head monomer away from the αβ interface on Lα for copolymer with NA = 25. Simulation parameters are N = 64, E = 0.2, and εAα = 3.

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AUTHOR INFORMATION

Corresponding Author

We find that the minimum of Dih and the principal peak of τ just locate at the same place; i.e., we have maximum τ at NAbx = Lα. The results clearly show that it is difficult for copolymer to fill the compound channel when the length of the block A (NAbx) just matches the length of channel α (Lα). 3.4. Residence Time. To describe the dynamical detail of the filling process, we have calculated the residence time tr(nin) which is defined as the duration with nin monomers inside the channel. Figure 7 shows the dependence of tr on the length of

*E-mail [email protected]; Fax +8657187951328; Phone +8613989818207 (M.-B.L.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China under Grants 21174132 and 11374255 E

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(40) Wang, C.; Chen, Y. C.; Sun, L. Z.; Luo, M. B. J. Chem. Phys. 2013, 138, 044903.

and by Zhejiang Provincial Natural Science Foundation of China under Grant LQ14A040006.

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