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Polymer ejection from strong spherical confinement

We examine the ejection of an initially strongly confined flexible polymer from a spherical capsid through a nanoscale pore. We use molecular dynamics for unprecedentedly high initial monomer densities. We show that the time for an individual monomer to eject grows exponentially with the number of ejected monomers. By measurements of the force at the pore we show this dependence to be a consequence of the excess free energy of the polymer due to confinement growing exponentially with the number of monomers initially inside the capsid. This growth relates closely to the divergence of mixing energy in the Flory-Huggins theory at large concentration. We show that the pressure inside the capsid driving the ejection dominates the process that is characterized by the ejection time growing linearly with the lengths of different polymers. Waiting time profiles would indicate that the superlinear dependence obtained for polymers amenable to computer simulations results from a finite-size effect due to the final retraction of polymers' tails from capsids.

The ejection dynamics through a nanoscale pore of a flexible polymer that is initially strongly confined inside a spherical capsid is examined. By extensive simulations using the stochastic rotation dynamics method we show that the time for an individual monomer to eject grows exponentially with the number of ejected monomers under constant initial monomer density. This dependence is a consequence of the excess free energy of the polymer due to confinement growing exponentially with the initial monomer number inside the capsid, which we address to strong monomer-monomer interactions. Consequently, for sufficiently strong initial confinement and long polymers ejection times for polymers of different lengths depend linearly on the length. At polymer lengths amenable to computer simulations the dependence is superlinear due to the finite-size effect related to the retraction of polymer tails at final stages of ejection. Processes involving macromolecules in confinements are studied intensely due to their significance in biology and potential technological and medical applications. An important class of such processes is the capsid ejection where a polymer is initially in a compact conformation inside a capsid and then ejects outside through a pore a few nanometers wide. By far the most important biological process of this class is the viral packaging in and ejection from bacteriophages [1][2][3][4][5][6][7][8][9].
In the bacteriophages found in vivo the viral DNAs are packed to almost crystalline densities [3], realization of which is beyond most molecular-dynamics-based (MD) computer simulations. Computational investigations of some specific characteristics using close-to-realistic model polymers packed to high densities are typically done using some form of probabilistic Metropolis sampling, see e.g. [10]. While the studies of the effect and dominance of the different interactions are definitely of high importance, it is disturbing that not even the generic model of a flexible chain ejecting a spherical capsid is completely understood.
In an early work ejection of a flexible polymer starting from a random conformation inside a capsid was investigated by Monte Carlo (MC) simulations [1]. Assuming the excess energy due to the confinement to be ∆F/k B T ∼ N 0 /R 1/ν 0 a scaling prediction for the ejection time of the form τ ∼ N 0 R 1/ν 0 = N 0 (N 0 /ρ 0 ) 1/3ν was obtained, where N 0 is the degree of polymerization, R 0 the capsid radius, ρ 0 the initial monomer density in the capsid, and ν the Flory exponent. This scaling was confirmed by MC simulations. In a later work [7] the excess energy due to spherical confinement was taken to follow the scaling 0 is the initial monomer volume fraction. This scaling law, first introduced by Grosberg and Khokhlov [11], was shown to be correct for the spherically confined polymer at φ 0 where the blob scaling in the semidilute conditions is valid [12]. This led to the scaling relation τ ∼ N 1+ν , again confirmed by MC simulations. * Corresponding author: riku.linna@aalto.fi A unified framework for polymer decompression processes was presented by Sakaue and Yoshinaga [8]. Assuming uniform polymer conformation the excess confinement energy inside a spherical capsid was derived in the same form as in [11], The resultant osmotic driving force ∼ k B T /ξ(t) is exerted on the monomer residing at the pore. The overall dissipation takes place close to the pore, within the range of the correlation length ξ(t), where there exists a velocity gradient of segments of the order ∼ aṄ (t)/ξ(t). Accordingly, the dissipation term was evaluated as TṠ(t) = η[Ṅ (t)/ξ(t)] 2 ξ(t) 3 . The excess confinement energy is dissipated at the rate ∆Ḟ (t) = −TṠ(t), from which the time evolution was obtained as where the exponent β = (1 − 3ν)/[2(1 − ν)] and the time constant is the monomer scale time constant, where η is the solution viscosity. For the osmotically driven part the scaling τ ∼ τ 0 φ −(ν+2)/(3ν) 0 N (2+ν)/(3ν) 0 was obtained. [8] In the present paper the ejection dynamics is explored at larger volume fractions than in the research cited above. Again, we obtain scaling of τ with N 0 . However, the waiting times, i.e. the time each monomer, labeled s, has to wait at the pore opening on average to be ejected, are seen not to scale with s. Since the ejection time is just the integral of the waiting time over all the monomers, τ = N0 0 t w (s)ds, where the bead s at the pore is taken as the reaction coordinate for the process, polymer ejection is not inherently a scale-invariant process. This is in stark contrast with polymer translocation, where the major part of the waiting time profile scales with s.
The aim of this paper is to show that the dynamics of the ejection process that comprises two parts, the osmotically driven and the diffusive part, as presented within the unified framework [8], is already for fairly moderate initial monomer densities completely determined by the form of the excess energy due to confinement. In other words the scaling expected from the diffusive part does not show in the ejection time.
In what follows, we first briefly describe the computational stochastic rotation dynamics (SRD) method and the capsid and polymer models. In SRD hydrodynamics in the solvent can be switched on and off. For clarity, we present results for polymers that are immersed in Brownian heat bath and report our results on the effect of hydrodynamics in a forthcoming paper [13]. Finally, we summarize our conclusions on the generic polymer ejection process.
We use a hybrid computational method where the timeintegration of the polymer is performed by MD implemented by the velocity Verlet (vV) algorithm [14,15]. The polymer is immersed in a solvent that, in turn, is modeled by SRD [16,17]. The spherical capsid is modeled as a shell with rigid walls imposing slip and no-slip boundary conditions for polymers and solvent, respectively. The pore is modeled as a cylindrical hole in the shell. The simulation geometry and snapshots of an ejecting polymer are depicted in Fig. 1. The capsid geometry was created using constructive solid geometry technique [18]. The pore radius R p is 0.4 for the polymer and 0.8 for the solvent. This geometry allows a smooth flow for the solvent bounded by no-slip conditions. The thickness of the capsid wall is 3.
The polymer is modeled as a chain of point-like beads with mass m b . Adjacent beads are connected via the FENE poten- is the distance between adjacent beads and K and r max are potential parameters describing the strength and maximum distance limit of adjacent beads. A Lennard-Jones potential acts between all beads: U LJ = 4.8ǫ for r ij ≤ 6 √ 2σ and U LJ = 0 for r ij > 6 √ 2σ. ǫ and σ are potential parameters and r ij is the distance between beads i and j. The potential parameters are chosen as σ = 1.0, ǫ = 1.0, K = 30/σ 2 , and r max = 1.5σ in reduced units [19].
We use initial monomer density ρ 0 = N 0 /( 4 3 πR 3 0 ) instead of volume fraction φ 0 . Accordingly, the number of monomers N 0 corresponding to ρ 0 = 1 is by the factor 4 3 π larger than N 0 corresponding to φ 0 = 1. On the other hand, φ 0 was used for hard spheres, the use of which is not possible in high-density MD simulations. A value of φ 0 constitutes a slightly lower compression in a system using soft potentials than in one using hard-sphere potentials. This effect is more than compensated by our using ρ 0 instead of φ 0 , so effectively the systems simulated here start from more compressed states than those dealt with in [1,7,8]. In the initial conformations four beads are inside the pore so the total length of polymers is N 0 + 4.
SRD solvent consists of point-like particles whose dynamics can be divided into two steps. In the streaming step the solvent particle positions are moved ballistically, r i (t+∆t) = r i (t) + v i (t)∆t, where t is the simulation time, ∆t is the SRD time step, r i is the position, and v i is the velocity of the solvent particle i. The interactions between particles are taken into account in the collision step. The simulation space is divided into a grid of cubic cells of edge length 1.0. There are on average 5 solvent particles per unit cube. To mimic collisions the random parts of velocities of both polymer and solvent particles within each cell are rotated according to the equation (Images created using VMD [21] and POV-Ray [22]) is the center-of-mass velocity of the particles in the cell, and Ω is a rotation by angle α = 3π/4 around an axis that is chosen randomly at each step. In the present case of Brownian heat bath velocities are randomly exchanged between all particles after the collision step. The solvent is kept at the constant temperature of k B T = 1.0 by scaling the random parts of particle velocities, v i (t) − v cm (t), such that the equipartition theorem holds at all times. In order to maintain Galilean invariance, the grid is shifted randomly at each time step [20]. The vV time step δt = 0.0002 and the SRD time step ∆t = 0.5. A relatively small δt in vV is required to prevent numerical errors accumulating inside the capsid when the monomer density is high. MD and SRD steps are performed in turns such that after ∆t/δt = 2500 vV steps a single SRD step is performed. An initial polymer conformation is created by injecting a polymer into the capsid by force at the pore that is ramped up until the desired ρ 0 is reached. The polymer is then thermalized by scaling the bead velocities. During the ejection the polymer maintains its temperature via the SRD solvent collision step. Fig. 2 (a) shows the ejection time τ vs number of monomers N 0 initially inside the capsid for different initial densities ρ 0 . As reported earlier [1][2][3][4][6][7][8][9], scaling τ ∼ N β 0 is obtained. β = 1.362 ± 0.05, 1.325 ± 0.04, 1.293 ± 0.04, 1.299 ± 0.03,    The lower data points (no tail) depict the force measured without beads outside the capsid. (e) Waiting times tw for all simulated values of N0. Data here is averaged over 50 runs, which results in poorer statistics for the date presented in (c). (f) Beads in the capsid N (t) during the ejection for the same N0 as in (e) with fits to Eqs. (2) and (4). and 1.300 ± 0.04 for ρ 0 = 0.5, 0.75, 1, 1.25, and 1.5, respectively. The exponent β decreases slightly with increasing initial monomer density for ρ 0 ≤ 1 in accordance with our previous findings [9]. This is in accord with what we have stated in the case of driven translocation when the driving bias force increases [23,24]. The deterministic driving force (pressure) increases with ρ 0 . Most of the effective friction is exerted on the ejecting polymer in the vicinity of the pore through which the pressure pushes it. Hence, with increasing ρ 0 the contribution from driving becomes more dominant over diffusion. As the driving is frictional the dependence can only approach linear, i.e. τ ∼ N β 0 , β → 1. For extremely dense conformations ρ 0 ≥ 1, β increases with ρ 0 due to the increasing contribution from the friction between monomers in the capsid. For polymers of N 0 ≥ 50 τ ∝ 1/ρ 0 is obtained very precisely, which is in accordance with the above reasoning. A surface fit to the whole data for different N 0 and ρ 0 gives τ ∼ ρ α 0 N β , where α = −0.980 ± 0.039, and β = 1.316 ± 0.019. Fig. 2 (b) shows τ as a function of capsid radius R 0 . Comparing the obtained scaling τ ∼ R ζ 0 , where ζ ∈ [2.71, 3.23], with the scaling in Fig. 3 (a) of [8], where ζ decreases from 5 toward 2 when R 0 /a decreases from 10 2 toward 1, suggests that the ejection takes place in the very strongly confined regime. We also measured the radii of gyration R g for the polymers' initial conformations inside the capsids. Spherical scaling R g ∼ N θ 0 , where θ ≈ 0.33, was obtained for all ρ 0 also confirming strong confinement.
Although ejection times τ scale with N 0 , as seen from the endpoints of the cumulative waiting times, t(s) as such do not scale with s, but grow exponentially with s, see Figs. 3 (a) and (b). Accordingly, waiting times of the ejecting monomers t w = t(s) − t(s − 1) follow the exponential form, t w ∼ exp(As), as seen in Fig. 3 (c). This is in accord with the waiting time profile found in [9] for the ejection through a symmetric pore.
As for large enough ρ 0 the waiting time at any time is expected to be largely dominated by the instantaneous pressure inside the capsid driving the ejecting polymer the dependence t w ∼ exp(As) is at odds with the excess energy due to confinement based on the blob scaling under semidilute conditions, Eq. (1). To rule out the possibility that t w (s) deviated from what is expected from the semidilute assumption due to the polymer segment outside the capsid being driven out of equilibrium, we measured R g of this segment at different stages. This proved to be only slightly smaller than the measured corresponding equilibrium R g scaling as R g ∼ s 0.6 , so monomers do not crowd close to the pore and so change t w (s). Fig. 3 (d) shows the force that has to be exerted on the monomer at the pore to keep it in position at different s. Fluctuations at small ρ are large and f -s dependence can be determined only for ρ 0.2. This is well above φ = 0.1 where ∆F was seen to deviate from the form predicted by the semidilute blob picture [25]. Force due to confinement was measured also via the change in the momenta of the particles hitting the capsid walls. This way we obtain the force, or pressure, exerted on the whole capsid surface. The force measured this way and at the pore have the same dependence on ρ. We also measured force at the pore for different N for a model with only the polymer segment inside the capsid present and the rest of the polymer removed. f for this modified model has the exact same dependence on s as for the full model, only the force magnitude is less by 2 (see Fig. 3 (d)). Hence, the ejected part of the polymer significantly contributes to the ejection throughout the process but does not essentially change the generic features of dynamics.
The measured force has very precisely the exponential dependency f ∼ exp[BN (t)] anticipated from t w . In Fig. 3 (c) we plot 1/f for the force measured for ρ 0 = 1 and N 0 = 50. t w ∝ 1/f is seen to hold very precisely for all s/N 0 0.9, that is, throughout the ejection until the final retraction of the monomers remaining in the capsid. The exponential growth arises due to excluded volume interactions: inside the capsid monomers interact individually rather than as ensembles of blobs and the higher order interaction terms become important. This observation is in contrast with the claimed screening of the excluded volume interactions at high concentrations giving rise to a different scaling than given by the semidilute blob picture, Eq. (1) [25].
Using the measured dependence f = C exp[BN (t)], we can calculate τ as a function of ρ 0 in the framework presented in [8]. If the monomers are packed inside the capsid by force applied at the pore, then the excess energy that results from packing N monomers is ∆F ≈ where f i is the force required to move the bead i at the inner pore opening into the capsid and the bead i + 1 in its place, ∆l i is the distance the bead i + 1 needs to be moved. In the case of f i being measured for individual beads ∆l i = a ≈ 1, ∀i. In the limit ∆l i → 0 and the minimum necessary force being applied continuously on the polymer at the inner pore opening ∆F = Approximating the correlation length to be constant ξ(t) = ξ and using the initial condition N (t = 0) = N 0 , the solution is given in the form Fig. 3 (f) shows the fit of Eq. (4) to the measured N (t). A fit of N (t) given by Eq. (2) for the largest N 0 = 283 is given for reference. Eq. (4) is seen to describe N (t) with excellent precision as it should given that t w ∝ 1/f almost throughout the ejection.
To demonstrate that ∆F mainly determines the scaling τ ∼ N β 0 for the driven part we apply the criterion from the semidilute picture for the termination of osmotic driving. Let τ 2 be the time when the number of monomers in the capsid has decreased to the number corresponding the relaxed chain conformation of average radius R 0 : N (t = τ 2 ) = g 0 = (R 0 /a) 1/ν . For constant φ 0 and different N 0 and R 0 , g 0 = (N 0 /φ 0 ) 1/(3ν) . This gives τ 2 is seen to grow linearly with N 0 . We used g 0 as a criterion for a relaxed chain to demonstrate that even with its scaling form the linear relation τ ∼ N 0 is obtained. However, for the volumes that give the initial densities ρ 0 = 0.5 . . . 1.5 there would be only a few monomers per blob. The times τ 2 given by N (t = τ 2 ) = g 0 are marked in Fig. 3 (b). τ 2 are at points where the short remaining tail inside the capsid rapidly retracts through the pore due to the large entropy difference on both sides of the capsid. Clearly, ρ 0 are so large that τ 2 are not relevant. Instead, we can use the times τ * 2 when the measured force exerted by the polymer segment inside the capsid on a monomer at the pore is zero as a criterion for the termination of osmotic driving. Also τ * 2 are marked in Fig 3 (b). We obtain N (t = τ * 2 ) ∼ N 0 . Again, linear relation τ * 2 ∼ N 0 is obtained. From Figs. 3 (c) and (e) it is seen that t w (s/N 0 ) for different N 0 deviate from the common form only at the final stage of ejection when the remaining part of the polymer retracts from the capsid. Shorter polymers start retracting at a relatively earlier point in the process. Retraction speed increases identically for all polymers. This results in τ ∼ N β 0 , where β > 1. For very strong initial confinement this is a finite-size effect. After τ * 2 the force due to entropic imbalance between polymer segments outside and inside the capsid exerted at the pore causes the tension to propagate from the pore along the polymer segment in the capsid. The tension propagating similarly, the tensed segment ∆s will grow identically for all N 0 . Hence, ∆s/N 0 will be larger for smaller N 0 resulting in retraction starting at smaller s/N 0 for shorter polymers. Hence, for asymptotically long polymers linear dependence τ ∼ N 0 is obtained. For a weaker initial confinement the exponentially decaying force driving the diffusive polymer segment can give superlinear dependence.
In conclusion, we have investigated in detail the ejection of flexible polymers from spherical capsids through a nanoscale pore. Polymers were simulated by molecular dynamics and the surrounding heat bath by stochastic rotation dynamics. The ejection dynamics and the pertaining excess energy due to confinement ∆F were analyzed via measured waiting time profiles and forces exerted on polymers at the pore. We found that the waiting times t w , that is, the times it takes for individual monomers to exit the capsid, grow exponentially with the number of ejected monomers. The force f measured at the pore, in turn, increases exponentially with the number of monomers in the capsid N . This we address to follow from ∆F growing exponentially with N due to the higherorder terms in monomer-monomer interactions for the sufficiently high monomer densities. We also found that t w ∼ 1/f holds well for the simulated densities. This means that the form of ∆F completely determines the ejection dynamics for strongly confined polymers. We showed that the dependence ∆F ∼ exp(AN/N 0 ) can only give linear dependence for the ejection time on the polymer length τ ∼ N 0 . For strong initial confinement the apparent scaling τ ∼ N β 0 , where β > 1 results from a finite-size effect that is likely related to tension propagation. We have here shown that already for densities far lower than those found in bacteriophages ejection dynamics is fully determined by the form of the excess energy due to confinement that results from strong monomer-monomer interactions. We have also shown that even for densities that are moderate compared to those in bacteriophages no assistance for example by flow is necessary for the ejection to complete.