Translocation through a narrow pore under a puling force
aa r X i v : . [ phy s i c s . c o m p - ph ] J a n Translocation through a narrow pore under a pullingforce
Mohammadreza Niknam Hamidabad, Rouhollah Haji Abdolvahab ∗ Physics Department, Iran University of Science and Technology (IUST), 16846-13114,Tehran, Iran.
Abstract
We employ a three-dimensional molecular dynamics to simulate transloca-tion of a polymer through a nanopore driven by an external force. Thetranslocation is investigated for different four pore diameters and two differ-ent external forces. In order to see the polymer and pore interaction effectson translocation time, we studied 9 different interaction energies. Moreover,to better understand the simulation results we investigate polymer center ofmass, shape factor and the monomer spatial distribution through the translo-cation process.Our results unveil that while increasing the polymer-pore interaction en-ergy slows down the translocation, expanding the pore diameter, makes thetranslocation faster. The shape analysis of the results reveals that the poly-mer shape is very sensitive to the interaction energy. In great interactions,the monomers come close to the pore from both sides. As a result, the ∗ Corresponding author
Email address: [email protected] (Rouhollah Haji Abdolvahab)
Preprint submitted to arxiv.org January 23, 2019 ranslocation becomes fast at first and slows down at last.
Keywords:
Translocation time, Attractive pore, Pore size, Mean waitingtime, Polymer shape
1. Introduction
Biopolymers translocation through nanopores is a critical and ubiquitousprocess in both biology and biotechnology. This leads to extensive and com-prehensive studies over the past few decades. Undoubtedly, the study of thetranslocation of a polymer through nanopores can be considered as one of themost active fields of research in the whole soft matter physics [1–8]. It shouldbe noted that the importance of this process, polymer translocation (PT),is not limited to understanding its physical and biological dimensions, butalso the essential technological applications, including DNA sequencing [9–13], controlled drug delivery [14, 15], gene therapy [14, 16–18] and biologicalsensors [9, 19].Moreover, the passage of biopolymers such as DNA and RNA throughnuclear pore complexes [20–23], virus RNA injection into the host cell [24, 25]and passing proteins through the cell organelle membrane channels [3] aresome other biologic processes which have doubled the importance of thisissue.In the process, the biopolymer must overcome the entropy barrier [1, 26–28]. Hence, the methods of PT through the nanopores include the use ofexternal force which is one of the most common methods used both in the2aboratory and computational simulations [5, 6, 29–32]. However, in vivo
PTdriven by assisted proteins called chaperone is proposed [4, 33–37].In the following simulation, we have used the polymer-mediated passageof polymer through nanopores driven by the external force. In this typeof translocation, several parameters, such as the length and radius of thenanopore, the applied external force, and the friction coefficient of both Cisand Trans environments, are investigated [2, 38–41]. In the meantime, oneof the cases that are rarely investigated is the interaction energy (potentialdepth) between the nanopore wall and the polymer passing through it andits effect on the time of PT.In this paper, we used a coarse-grained molecular dynamics method tosimulate the translocation of the polymer through the nanopore in three di-mensions. The simulation includes nine different interactions, three nanoporediameters, and two different external forces.
2. Theory and simulation details
In the following 3D simulations, the polymer is modeled by a mass andspring chain in such a way that adjacent monomers have the nonlinear po-tential of FENE: U F ENE = − kR ln (1 − r R ) . (1)Here, r is the distance between two adjacent monomers, k, and R are thespring constant and the maximum permissible spacing for adjacent monomers.3e employ the Lenard-Jones potential, equation 2, to model the nanoporesuch that the cutoff radius of the nanopore interactions with the polymer isdifferent from the other interactions. U LJ = ( ǫ (cid:20)(cid:18) σr (cid:19) − (cid:18) σr (cid:19) (cid:21) + ǫ r ≤ r cut sigma is the diameter of each monomer, ǫ is the potential depthof the Lenard-Jones and r cut calls for the potential cutoff radius.We do the simulation employing the Langevin dynamics method. In thismethod, the following statement can be written for each monomer: m ¨ r = F Ci + F Fi + F Ri (3)where m is the monomer mass. Moreover, the F Ci , F Fi , and F Ri are theconservative, frictional, and the random forces applied on the i ′ s monomer,respectively. The frictional forces are connected to the monomer’s speed bythe following equation: F Fi = − ξV i (4)in which ξ is the frictional coefficient. One also can write for the conser-vative forces: F Ci = −∇ ( U LJ + U F ENE ) + F external (5)4here the last term is the external force, excreted on the polymer throughthe nanopore and is defined as: F external = F ˆ x (6)in which, the direction of the force corresponds to the nanopore-axis andtowards the Trans side. The initial configuration of the system is such that the first monomer is atthe end of a nanopore of length 6 σ and with different diameters of 3 , σ .We then place the remaining monomers close to their equilibrium positionrelative to each other, and in the front of the nanopore. It should be notedthat the pore-axis is parallel to the x-axis. After placing the monomers, wegive them the opportunity to achieve their equilibrium as a whole polymer.In this equilibration process, we fix a few monomers in the nanopore andallow the rest of the monomers, the polymer tail, to move freely until wereach the equilibrium. Afterward, the process of translocation begins. Thispart of the simulation lasts from about 20%, for the slowest, up to about40%, for the fastest translocation, of each PT time through the nanopore.Here, we translocated the polymer for at least 1,500 times to reach a rathergood time distribution.In order to find the equilibrium point, we calculate the radius of gyrationof the polymer through the time. The equilibration process continues until5he changes in the radius of gyration becomes as small as 2 σ .We calculate the time scale of the simulation using the t LJ which is [34]: t LJ = ( mσ ǫ ) (7)We pick the forces from two different regions of strong and medium asthe external force in the pore. The relation determining this regions for theaverage force is [29]: k B TσN ν ≤ F ≤ k B Tσ (8)in which ν is the Flory exponent, and N stands for the total number ofmonomers. The magnitude of the strong and medium forces we employ inthe simulation are 2 ǫ /σ and 1 ǫ /σ , respectively ( ǫ is defined below.). Simulation parameters:
The other simulation parameters include thecutoff radius for interactions of the nanopore and the polymer which is 2 . σ and in other interactions, between monomers and monomers and wall, are2 σ . For the energy we use ǫ which is ǫ = 1 . k B T , except the interactionbetween polymer and the nanopore which changed and are multiplies of ǫ .Moreover, the friction coefficient is ξ = 0 . m/t LJ and for the FENE potential,the spring constant is k = 30 ǫ /σ and the cutoff radius R = 1 . σ [34].6 .1 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 ε/ε T r a n s l o c a t i o n t i m e External Force = 1.0 f = 1, r = 1.5σf = 1, r = 2.0σf = 1, r = 2.5σf = 1, r = 3.0σ
Figure 1: Translocation time versus energy for four different diameter in external force f = 1
3. Results and analysis
Translocation time of the polymer versus polymer and pore interactionenergy is plotted in the figure 1 and figure 2. The interaction energy ischanged from ǫ = 0 . ǫ = 8. The external force is changed from f = 1in figure 1 to f = 2 in figure 2. As it appears from both figures, increasingthe pore diameter will decrease the translocation time. Moreover, whileincreasing the interaction energy, generally will increase the translocationtime, this increase is very small in the low interaction energies ( ǫ = 0 . , .1 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 ε/ε T r a n s l o c a t i o n t i m e External Force = 2.0 f = 2, r = 1.5σf = 2, r = 2.0σf = 2, r = 2.5σf = 2, r = 3.0σ
Figure 2: Translocation time versus energy for four different pore radii in external force f = 2
10 20 30 40 50 s W a i t i n g t i m e f = 1, r = 1. 5σ ǫ = 1ǫ = 3ǫ = 6ǫ = 8 0 10 20 30 40 50 s W a i t i n g t i m e f = 2, r = 1. 5σ ǫ = 1ǫ = 3ǫ = 6ǫ = 8 Figure 3: Mean waiting time of the polymer versus monomer number from a nanopore ofradius r = 1 . σ
10 20 30 40 50 s W a i t i n g t i m e f = 1, r = 2. 5σ ǫ = 1ǫ = 3ǫ = 6ǫ = 8 0 10 20 30 40 50 s W a i t i n g t i m e f = 2, r = 2. 5σ ǫ = 1ǫ = 3ǫ = 6ǫ = 8 Figure 4: Mean waiting time of the polymer versus monomer number from a nanopore ofradius r = 2 . σ
10 20 30 40 50 s W a i t i n g t i m e f = 1, r = 3. 0σ ǫ = 1ǫ = 3ǫ = 6ǫ = 8 0 10 20 30 40 50 s W a i t i n g t i m e f = 2, r = 3. 0σ ǫ = 1ǫ = 3ǫ = 6ǫ = 8 Figure 5: Mean waiting time of the polymer versus monomer number from a nanopore ofradius r = 3 . σ
10 20 30 40 50 s C u m u l a t i v e w a i t i n g t i m e f = 1, r = 1.5σ, ε = 1f = 1, r = 1.5σ, ε = 8f = 1, r = 2.5σ, ε = 1f = 1, r = 2.5σ, ε = 8f = 2, r = 1.5σ, ε = 1f = 2, r = 1.5σ, ε = 8f = 2, r = 2.5σ, ε = 1f = 2, r = 2.5σ, ε = 8 Radius of Nanopore = 2.5 σ
Radius of Nanopore = 1.5 σ
Figure 6: Cumulative waiting times versus monomers number s velocity. The mean waiting time of each monomer for different pore radiiof 1 . σ , 2 . σ and 3 . σ is plotted against the monomer number, s, in figure3, 4 and figure 5, respectively. The maximum of the translocation time isrelated to the middle monomers due to the entropic barrier of the cis andtrans monomers. Thus the mean waiting plots are nearly bell-shape. Thebehavior of the final monomers in the interaction energy of ǫ = 8 andnanopore of radius r = 1 . σ is interesting. As it appears in figure 3, the finalmonomers waiting times for ǫ = 8 and for both external forces of f = 1and f = 2 are ascending. Because of the large interaction energy, the finalmonomers do not want to come out. 12umulative waiting time versus monomer number, s, is shown in figure6. It compares, different interaction energies of ǫ = 1 ,
8, different externalforces of f = 1 ,
2, and different pore radii of r = 1 . σ, . σ . Insets are thezoom of the plots at first monomers. As the top inset shows for r = 1 . σ the polymer with ǫ = 8 is faster than the interaction energy of ǫ = 1 at 6first monomers for both forces of f = 1 ,
2. In the wider pore where r = 2 . σ the intersection of plots becomes on s = 13 (see the low inset of the figure6). It means that the high interaction pulls the polymer through the poreand makes it faster at first, but slows its translocation through the pore inthe middle stages. This effect becomes more important as the pore radiusbecomes larger. We expect that by increasing the radius of the pore till thepoint where it is still smaller than the gyration radius of the polymer, andalso the interaction of the nanopore with the polymer is large enough, thismonomer number will rise.To justify such behaviors in the polymer translocation, we need to lookat other parameters such as the center of mass (COM) of the polymer duringthe passage, the overall shape of the polymer (shape factor) and the spatialdistribution of monomers through the translocation process, etc.The figures 7 show the X components of the COM of the polymer versusmonomer’s number, s, respectively. The pore center coordinate is (40, 38, 40).It is important to mention here that the polymer is initially in equilibrium.To discuss the translocation in more detail, we focus on X COM which is thepore direction in figure 7. As it shows, in the first stage of the translocation13
10 20 30 40 50 s X C M S End of NanoporeBeginning of Nanopore f = 1 r = 1. 5σ, ǫ = 1r = 1. 5σ, ǫ = 8r = 2. 5σ, ǫ = 1r = 2. 5σ, ǫ = 8r = 3. 0σ, ǫ = 1r = 3. 0σ, ǫ = 8 s X C M S End of NanoporeBeginning of Nanopore f = 2 r = 1. 5σ, ǫ = 1r = 1. 5σ, ǫ = 8r = 2. 5σ, ǫ = 1r = 2. 5σ, ǫ = 8r = 3. 0σ, ǫ = 1r = 3. 0σ, ǫ = 8
Figure 7: x component of the location of the center of mass (COM) of the polymer versusmonomer number, s.
10 20 30 40 50 s α f = 1, r = 1. 5σ ǫ=1ǫ=3ǫ=6ǫ=8 s α f = 1, r = 2. 5σ ǫ=1ǫ=3ǫ=6ǫ=8 s α f = 1, r = 3. 0σ ǫ=1ǫ=3ǫ=6ǫ=8 s α f = 2, r = 1. 5σ ǫ=1ǫ=3ǫ=6ǫ=8 s α f = 2, r = 2. 5σ ǫ=1ǫ=3ǫ=6ǫ=8 s α f = 2, r = 3. 0σ ǫ=1ǫ=3ǫ=6ǫ=8 Figure 8: α versus monomer number s. the polymers with high interaction energy of ǫ = 8 have greater X COM fromthe polymers with low interaction energy of ǫ = 1 which means they reachto equilibrium nearest to the pore as the interaction supports. They are alsonearest to the pore in the last stage of the translocation with the same reason.To see the polymer’s behavior in more detail, we study the polymer shapeusing the parameters α and shape factor. α compares the distribution ofthe monomers in pore axis (x) and from the translocation axis (in yz plane), α = ∆ x/ (2 r ). ∆ x is the maximum of the polymer distance from the porein the trans side in the x-direction and r is the maximum distance of thepolymer from the pore axis (x) in the trans side, r = p y max + z max [42].As the figures 8, 9 and 10 show the distribution of monomers in wider pore15
10 20 30 40 50 s α r = 1. 5σ, ǫ = 1 f=1f=2 s α r = 1. 5σ, ǫ = 8 f=1f=2 s α r = 3. 0σ, ǫ = 1 f=1f=2 s α r = 3. 0σ, ǫ = 8 f=1f=2 Figure 9: α versus monomer number s. s α f = 1, (cid:4) = 1 r=1.5ǫr=2.5ǫr=3.0ǫ s α f = 2, (cid:4) = 1 r=1.5ǫr=2.5ǫr=3.0ǫ s α f = 1, (cid:4) = 8 r=1.5ǫr=2.5ǫr=3.0ǫ s α f = 2, (cid:4) = 8 r=1.5ǫr=2.5ǫr=3.0ǫ Figure 10: α versus monomer number s.
10 20 30 40 50 s δ f = 1, r = 1. 5σ ǫ=1ǫ=3ǫ=6ǫ=8 s δ f = 1, r = 2. 5σ ǫ=1ǫ=3ǫ=6ǫ=8 s δ f = 1, r = 3. 0σ ǫ=1ǫ=3ǫ=6ǫ=8 s δ f = 2, r = 1. 5σ ǫ=1ǫ=3ǫ=6ǫ=8 s δ f = 2, r = 2. 5σ ǫ=1ǫ=3ǫ=6ǫ=8 s δ f = 2, r = 3. 0σ ǫ=1ǫ=3ǫ=6ǫ=8 Figure 11: shape factor δ versus monomer number s of r = 2 . σ is thinner than the pore with radius r = 1 . σ . Moreover, it showsthat in accordance to the previous discussion, the narrowest of distributionof the monomers is in the case of high interaction energy of ǫ = 8 and in r = 2 . σ .The shape factor δ versus monomer number have been shown in figures11, 12 and 13. This parameter compares the gyration radius and the hydro-dynamic radius [28]. The upper limit of the shape factor δ is for a rod andequals δ max = 4 . δ = 0 .
77 [28].They show that increasing the interaction energy will decrease the shapefactor variation. Moreover, increasing the external force δ will increase, and17
10 20 30 40 50 s δ r = 1. 5σ, ǫ = 1 f=1f=2 s δ r = 1. 5σ, ǫ = 8 f=1f=2 s δ r = 3. 0σ, ǫ = 1 f=1f=2 s δ r = 3. 0σ, ǫ = 8 f=1f=2 Figure 12: shape factor δ versus monomer number s s δ f = 1, (cid:4) = 1 r=1.5ǫr=2.5ǫr=3.0ǫ s δ f = 2, (cid:4) = 1 r=1.5ǫr=2.5ǫr=3.0ǫ s δ f = 1, (cid:4) = 8 r=1.5ǫr=2.5ǫr=3.0ǫ s δ f = 2, (cid:4) = 8 r=1.5ǫr=2.5ǫr=3.0ǫ Figure 13: shape factor δ versus monomer number s δ will increase by increasing the interactionenergy. It means that the polymer with lower interaction energies is morecompact with respect to those with higher ǫ .
4. Conclusions
We use a 3D molecular dynamics to simulate the polymer translocationthrough a narrow pore driven by an external force. Simulation results showthat increasing the polymer-pore interaction energy slows down the translo-cation. Moreover, increasing the pore diameter makes the translocation fasterwhich is in complete accordance with previous results [43, 44].The detailed analysis of the polymer shape shows that the polymer wantsto be more near the pore in high energies at both first and last part of thetranslocation process with respect to the polymers with lower interaction en-ergies. This cause the translocation of the high interaction polymers becomesfaster at first and slower at last. Moreover, our detailed shape analysis revealsthat the polymers with lower energy and in wider pores are more rod shapethrough the translocation. Also, while the polymer shape is not sensible tothe external force (at least in the forces of f = 1 and f = 2), its shape isvery sensitive to the interaction energy between the polymer and nanopore.Waiting time analysis shows that the middle monomers take more timethan others. In high interaction energy of ǫ = 8 and the small pore radiusof r = 1 . σ , the last monomer’s waiting times versus monomer number are19scending. Due to the high interaction and accumulation of the monomersat the trans side, the polymer does not want to come out of the pore.In summary, changing the pore diameter and polymer-pore interactionwill cause the translocation time, polymer shape through the translocation,accumulation of the monomer at first and last stage of the translocation andwaiting time of each monomer to variate widely.
5. Acknowledgments
The Molecular Dynamics simulations were performed with the ESPResSopackage [45–47]. Simulation plotting’s been done by using Matplotlib [48].
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