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Feb 17, 2016 - We show that the dynamic nonequilibrium segmental concentration profile of a single nanochannel confined DNA molecule can be described ...
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Nonequilibrium Dynamics of Nanochannel Confined DNA Ahmed Khorshid,*,† Susan Amin,† Zhiyue Zhang,† Takahiro Sakaue,‡ and Walter W. Reisner*,† †

Physics Department, McGill University, Montreal, QC H3A 2T8, Canada Department of Physics, Kyushu University, Fukuoka 819-0395, Japan

‡

S Supporting Information *

ABSTRACT: We show that the dynamic nonequilibrium segmental concentration profile of a single nanochannel confined DNA molecule can be described via a partial differential evolution equation based on nonlinear diffusion, using an approach analogous to that used in the description of many-molecule systems such as polymer solutions. This equation can describe the segmental concentration profile of a single polymer along the nanochannel as a function of time for chain behavior ranging from states of high compression to equilibrium. In particular, to demonstrate the generality of our approach, we show that our model can describe two distinct types of experimental behavior generated via a sliding bead assay, symmetric relaxation resulting from free expansion of the polymer after compression, and the evolution of DNA concentration “shock waves” as a molecule is driven from equilibrium to a compressed state.

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also resembles the “tension fronts” that have been postulated in the theoretical description of polymer translocation through a nanopore,5,6 but in contrast to the pore system we are able to directly observe the shock wave dynamics via imaging of the evolving fluorescence profile along the nanochannel. The bead is then retracted and the molecule relaxes symmetrically with a time-dependent nonuniform concentration profile that gradually evolves to a uniform equilibrium profile (Figures 1c,d and 2b). While simple evolution equations, based on the Flory free energy,7 have been proposed to model relaxation dynamics of the scalar confined DNA extension,8 the processes we observe (transient compression and symmetric relaxation) are highly nonequilibrium phenomena that give rise to dynamically evolving nonuniform concentration profiles that could not be described via the classic Flory free energy picture. We show that the dynamic concentration profile c(x,t) we observe can be described via an evolution equation model based on a classic nonlinear diffusion equation. The nonlinear diffusion equation, a one-dimensional PDE, is supplemented by two equations, imposed at the chain edges, that determine the rate at which the chain boundaries spread outward. Physically, the evolution equation arises from cooperative diffusion9,10 that drives contour outward to the chain boundaries; the chain boundaries then dynamically evolve due to the bulk osmotic pressure drop at the chain edges. The model can be solved using a commercially available one-dimensional PDE package. Knowledge of c(x,t) is critical for modeling a wide range of transient and steady-state nonequilibrium phenomena in

INTRODUCTION A vast range of phenomena in nonequilibrium soft matter physics, including spreading droplets,1 solute diffusion, gel dynamics,2,3 and flow in porous media,4 can be described by single or coupled systems of partial differential equations (PDE’s) for the coarse-grained, time-dependent, and spatially varying velocity distributions, solute concentrations, or interfacial profiles. These systems are often characterized by irreversible time evolution from an initial nonuniform, nonequilibrium state to a final uniform equilibrium state or, conversely, upon application of an external driving force, by evolution from equilibrium toward a nonuniform steady state. In particular, application of external forcing at a rate exceeding that of internal relaxation will often lead to the development and propagation of wave fronts or “shock waves”. For example, when an ideal gas is compressed faster than the speed of sound propagation in the gas, a concentration front will build up near the piston. Using a sliding bead assay, we demonstrate that analogous nonequilibrium phenomena arise in the context of single nanoconfined polymer chains. Moreover, we show that these single-chain phenomena can be described using coarse-grained evolution equation models analogous to those used in the context of multimolecule systems. In our sliding-bead assay, an optically trapped bead gasket is translated down the nanochannel at a constant sliding speed V toward a nanochannel extended molecule. When the bead gasket impinges upon the molecule (Figures 1a,b and 2a), the molecule concentration locally increases near the bead, creating a transient “shock wave” that grows and propagates down the molecule (analogous to the shock wave formed upon sudden compression of an ideal gas). This compression shock wave © XXXX American Chemical Society

Received: October 13, 2015 Revised: February 3, 2016

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DOI: 10.1021/acs.macromol.5b02240 Macromolecules XXXX, XXX, XXX−XXX

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may have considerable appeal as it requires less specialized knowledge to implement. Moreover, simulation and PDE methodologies could be combined for cross-validation and aid in comparing theory to experimental data, as it is difficult to efficiently change simulation conditions over a large parameter space. For the same reason, combining simulations and the PDE approach could help guide best engineering solutions for nanofluidic device layouts (for example, to enhance throughput into nanochannel arrays for high-throughput mapping19,20 or to prevent device clogging).

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PARTIAL DIFFERENTIAL EVOLUTION EQUATION Let c(x,t) represent the local ensemble-averaged segment concentration (Kuhn segments per unit volume) as a function of position x along the nanochannel (see Figure 3a). The molecule has a total contour length L or number of segments N = L/ak, where ak is the Kuhn length equal to twice the persistence length P. Chain concentration is locally conserved, leading to

Figure 1. Sliding-bead assay. (a) The optically trapped bead gasket is used to compress a single nanochannel-extended DNA molecule. (b) Two-dimensional schematic showing compression process. Initially (t1) the molecule is in its equilibrium state. When the molecule contacts the bead (t2), the segmental concentration c(x) grows at the edge near the bead, creating a concentration wave that propagates down the molecule. Eventually, the concentration wave reaches the far edge of the molecule, and the profile adopts a steady-state ramp form (t3). (c) The bead gasket is retracted to induce a symmetric relaxation process with both molecule ends free. (d) Two-dimensional schematic showing relaxation process. Initially, the molecule is in a ramped steady-state profile resulting from the compression process (t1). This ramped profile begins to relax (t2) and eventually reaches the flat equilibrium profile (t3).

∂J ∂c =0 + ∂x ∂t

(1)

with J representing a 1-D segmental current. In the presence of a uniform flow V the current is given by J = −Dc(c(x , t ))

∂c + cV ∂x

(2)

The quantity Dc is a cooperative diffusion constant; critically, it is a function of of c(x,t),22 quantifying how osmotic pressure gradients, balanced by local friction, give rise to segmental current within the molecule (the dependence of Dc on c is shown explicitly in eq 2; we do not indicate the dependence elsewhere in the text to make the notation less cumbersome). Combining eq 1 and eq 2 leads to

nanochannel physics.7 These phenomena include compression and stretching observed in crossing from regions of low to high confinement, dynamics of molecules spanning channel junctions,11 compression against defects,8 entropic recoil,13,14 dynamics in funneling channels,15 and dynamic compression.12 Finally, understanding the detailed concentration dynamics is essential technologically as locally increased chain concentration leads to higher self-exclusion free energy that can modify free energy barriers, promoting chain escape, affecting performance of devices such as entropic traps arrays.16,17 While powerful polymer simulation methodologies exist that can access c(x,t),18,21 our coarse-grained evolution equation approach, based on the solution of a one-dimensional PDE,

⎞ ∂c ∂ ⎛⎜ ∂c − − cV ⎟ = 0 Dc ⎠ ∂t ∂x ⎝ ∂x

(3)

Equation 3 is sufficient to describe chain dynamics when the chain is highly compressed. In particular, when subject to a uniform compressional flow V, the chain will eventually reach a steady-state profile with J = 0 leading to the relation Dc(∂c/∂x) = cV. This equation can be solved to determine the steady-state ramp profile. In particular, the dependence of Dc on c directly determines the functional form of the ramp. Khorshid et al.12

Figure 2. Experimental setup and chip design. (a) Montage of a single T4 DNA being compressed using the nanodozer assay inside a 300 × 300 nm channel. (b) The bead gasket is rapidly retracted to allow the molecule to relax symmetrically from both ends. Note that the stage translation used to retract the bead shifts the entire device in the camera field of view and creates an apparent motion of the molecule. Both (a) and (b) have a horizontal scale bar of 1 s and vertical scale bar of 10 μm. (c) A photo of the nanofluidic device next to a quarter for scale. (d) The nanochannel array connected to the microchannels imaged at 20× with a scale bar of 50 μm. (e) An SEM image of the nanochannels with a scale bar of 2 μm. B

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dxi ∂c = −Dc dt ∂x

+ c(xi , t )V (5)

x = xi

Equation 5 expresses the balance between the flux created via movement of the chain edges and the in-flow of segments created by concentration gradients at the chain edge and the flow velocity V. In order to solve the coupled system eqs 3−5, we must determine the dependence of Π on c. The osmotic pressure can be determined from the Flory theory, now well established in the classic and extended de Gennes regime. Flory theory gives rise to a force−compression relation that can be expressed entirely in terms of r/r0:7,23,24 f ⎛ r ⎞2 ⎤ A⎡ r = ⎢− + ⎜ 0 ⎟ ⎥ ⎝r⎠ ⎦ kBT rb ⎣ r0

(6)

with A a numerical constant (= 2.81 according to ref 23) and rb the lateral scale of a single blob. The extended de Gennes theory would suggest that rb is related to channel dimensions via ⎛ a 2 ⎞1/3 rb = ⎜ k ⎟ D2/3 ⎝ w ⎠

Figure 3. Theoretical model. (a) Our model predicts the time evolution of the local segment density c(x,t) along the nanochannel. The model output (blue curve) is convolved with a Gaussian to create the experimental profile (red). The quantities X1 and X2 represent the molecule edge positions, and R represents the extension in units of the equilibrium extension r0. (b) Theoretical output for a relaxation process with both molecule ends free starting from an initial ramped concentration profile. The molecule evolves toward a flat profile. (Inset) The extension as a function of time for the profiles shown (black line). A best fit to the uniform expansion model is also shown (red dashed line). Note that the uniform expansion model is not exact but gives a very good description of the extension relaxation behavior.

(7)

where w is the effective width of the chain. As eq 6 can be expressed entirely as a function of the r/r0 ratio, we argue that it would apply locally to a segment of molecule with extension δr (as long as δr is larger than rb). Note that the quantity δr/δL is a local “fractional extension”. Let δr0 be the equilibrium extension of the section of chain of contour length δL, then δr/ δr0 = (δr/δL)(δr0/δL) = c0/c. Moreover, the chain osmotic pressure Π = f/D2 so that ⎡ ⎛ c ⎞2 ⎤ c Π = Π 0⎢ − 0 + ⎜ ⎟ ⎥ ⎢⎣ c ⎝ c0 ⎠ ⎥⎦

find that the ramp shape is linear, leading to a Dc that is linearly proportional to c and which we parametrize via Dc = D0(c/c0). Yet, eq 3 is not sufficient to describe the dynamics of a chain approaching equilibrium. As a chain approaches equilibrium (in the limit of large t), the concentration profile of the chain asymptotes to a flat profile with a constant equilibrium concentration c0 and finite extension r0, set by the balance of entropic elasticity with self-exclusion. Equation 3, in contrast, predicts a chain concentration that unphysically decays to zero as t gets large. In order to correctly describe the approach to equilibrium, we supplement eq 3 with two equations governing the chain edge dynamics. We assume, first, that the chain concentration at the molecule edges falls off sharply over a length scale on order of the channel width D (a small fraction of the chain extension in our experiments). The position of the chain edge leftward and rightward edge is then determined from

(8)

with Π0 ≡ kBTA/rbD2. This form for the osmotic pressure can alternatively be derived from the free energy density for a nanoconfined chain and a general thermodynamic result relating energy density to osmotic pressure (see Supporting Information). Note that Π(c/c0) = 0 when c = c0. Thus, we see that boundary conditions eqs 4 and 5, with the osmotic pressure determined by eq 8, will halt the chain expansion when the chain reaches equilibrium (i.e., when c = c0). For convenience we choose to nondimensionalize c, r, t, and x as follows: C≡

c , c0

R≡

r , r0

T≡

t , τ

X≡

x r0

(9)

If the time scale τ ≡ r0 /D0, we find that eq 3 becomes 2

dx ξi i = ∓ΠD2 + ξiV (4) dt The quantity Π is the osmotic pressure at the chain edges, itself a function of the edge concentrations c(xi,t) (see Figure 3a). In our notation, the left edge corresponds to i = 1 and the right edge corresponds to i = 2 (+ sign corresponds to rightward edge that moves toward positive x; − sign corresponds to leftward edge that moves toward negative x). The quantity ξi is a friction factor associated with edge motion. Second, we must have a flux-balance condition imposed at the chain edges:

⎞ ∂C ∂ ⎛⎜ ∂C − − CV0⎟ = 0 C ⎠ ∂T ∂X ⎝ ∂X

(10)

The quantity V0 is a dimensionless sliding speed defined via V0 ≡ V/Vc with Vc = r0/τ a characteristic velocity scale. Equation 10 has the form of a classic nonlinear diffusion equation that appears in many contexts in soft-matter physics, including capillary spreading along chemically patterned stripes1 and hydrodynamics in porous media.4 Lastly, we normalize ξi using the total friction factor of the chain 6πηr0,22 introducing a C

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Figure 4. Transient compression. (a) Ensemble averaged profiles (blue) for transient compression compared with transient-ramp model (green dashed curve) and best-fitting PDE-model output (black curve). The profile at t < 0 corresponds to the initial condition used to generate the PDEmodel output: this frame was taken one frame preceding the process starting point. The sliding speed used in the fit is fixed and set equal to the experimental value V = 5 μm/s. The experimental frame rate is 0.05 s. The best fits are obtained with a α = 0.25. Eight single molecule compression events were used to generate the experimental profiles: the error bars on the concentration values represent a one-sigma error on the mean for the set of events used. The molecules used had an equilibrium extension r0 = 18.7 ± 1 μm. (b) The experimental extension (red) determined via the transient ramp fitting function compared to PDE-model output for the profiles shown in (a). The theoretical extension was generated using parameters identical to that of (a). The error bars were determined as one-sigma confidence intervals from the nonlinear fit. (c) The maximum concentration from experimental profiles (red) compared to PDE model output for the profiles shown in (a). Again, the theoretical concentration values were generated using parameters identical to that of (a). The error bars were determined from the one-sigma error on the maximum experimental concentration value.

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dimensionless edge-friction βi ≡ ξi/6πηr0. The boundary conditions eqs 4 and 5 then become dX i 1⎡ 1⎤ = ∓ ⎢C 2 − ⎥ + V0 ⎣ dT αi C⎦

(11)

and ∂C ∂X

=± X = Xi

1⎡ 2 1⎤ ⎢C − ⎦⎥ ⎣ αi C

EXPERIMENTAL METHODS

Fluorescently stained DNA molecules were driven into fused-silica nanochannels and then dynamically compressed via the nanodozer setup (see Figure 1). The nanochannels have a dimension of 300 × 300 nm corresponding the extended de Gennes confinement regime.7 The bead gasket was then retracted and the relaxation process observed (see Supporting Information for information on fabrication, buffer conditions, and nanodozer operation). Figure 2 shows a kymograph representation of a transient compression (Figure 2a) and relaxation event (Figure 2b). The single-molecule data are then rescaled and normalized (see Supporting Information) in order to create ensemble-averaged results. The extension r0 and intensity level I0 of the equilibrium profiles are obtained from analysis of concentration profiles following relaxation using a classic flat-profile type fitting function (see Supporting Information). The dimensionless quantity C(x) = c(x)/c0 = I(x)/I0 where I(x) represents the experimental molecule intensity measured along the nanochannel.

(12)

The parameter αi = βi(6πηD0/Π0D2) is a dimensionless parameter that controls the rate at which the chain edges evolve. In the simplest implemention of our approach, αi is assumed constant and treated as an empirical fitting parameter. Note that τ is related to Dc and is physically distinct from the equilibrium relaxation time τr observed from monitoring chain extensional fluctuations:25 τ is the time scale for a concentration fluctuation to diffuse across the entire molecule extent. Finally, note that our use of nondimensionalized variables will X automatically ensure normalization of C(X): ∫ X21C(X) dX = 1. Equations 10−12 can be efficiently solved with the commercial PDE package flexPDE. Figure 3b shows an example relaxation process demonstrating that our model can produce concentration profiles that evolve from an initial condition to the expected flat equilibrium result. Our prediction for R(t) (Figure 3b, inset) is consistent with earlier theoretical models that assumed a uniform expansion hypothesis and constant total friction factor8 (see Supporting Information and inset of Figure 3b).

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EXPERIMENTAL RESULTS When the sliding bead contacts the DNA, concentration will gradually build up at the bead edge, creating a concentration shock wave. Figure 4a shows a time series of ensemble-averaged concentration profiles undergoing transient compression with V = 5 μm/s. Eventually the shock wave reaches the far edge of the molecule, and the profile will evolve toward the steady-state ramp profile. We choose to parametrize the shock wave data via a “transient ramp” concentration model (see Supporting Information). This model consists of a flat portion connected to a linear ramp convolved with a point spread function (PSF) to simulate the finite instrumental resolution (Figure 3a). We assume the PSF has an approximately Gaussian form (see Supporting Information). The total profile extension, ramp slope, and the junction of the flat portion and ramp are D

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Figure 5. Relaxation process. (a) Ensemble averaged profiles (blue) for symmetric relaxation compared with parabolic fits (green-dashed) and bestfitting PDE-model output (black curve). Eight single molecule compression events were used to generate the experimental profiles: the error bars on the concentration values represent a one-sigma error on the mean for the set of events used. The molecules used had an equilibrium extension r0 = 18.7 ± 1 μm. The profile at t < 0 corresponds to the initial condition used to generate the PDE-model output: profile was taken one frame preceding the process starting point. A retraction speed of 20 μm/s and a frame rate of 0.05 s were used to take the data (only selected frames are shown). The best fit α = 0.24. The profile evolves toward a flat profile shape at large times. (b) The extension obtained from the parabolic model (red) compared to PDE-model output (black) for the profiles shown in (a). The blue dashed curve shows PDE model output for instantaneous retraction: note that the finite-retraction speed leads to a lower extension for up to around 0.7 s. (c) The maximum concentration of the experimental profile (red) plotted versus theory (black). The dashed curve is a fit to a t1/3 scaling, demonstrating the approximate self-similarity of the relaxation process. For both (b) and (c) the error bars on experimental values are one-sigma confidence intervals taken from the nonlinear fit of the parabolic model to the experimental profiles.

the parabolic fits for the relaxation process are shown in Figure 5b,c.

adjustable parameters to be determined on a frame-by-frame basis via least-squares fitting to experimental data (Figure 4a). The transient-ramp model is not a perfect approximation to the shock wave; in particular, it assumes that the junction of the flat portion and the ramp are too abrupt, but it provides an acceptable model-independent parametrization of the profile shape. Figure 4b shows the values of the DNA extension r determined from the transient-ramp fitting model. The maximum concentration Cmax is shown in Figure 4c. The maximum concentration Cmax is obtained by finding the maximum experimental concentration value in the position window X = 0.9−1 at the molecule edge. Upon bead retraction the ramped concentration profiles relax. Figure 5a shows the concentration profiles during the relaxation process averaged over eight DNA molecules. The concentration profiles are strongly nonuniform with an approximately parabolic form, flattening out only for very long times when the molecule conformation approaches equilibrium. We fit each profile to a parabolic concentration model (parabola convolved with Gaussian PSF; see Supporting Information). This parabolic fitting function is characterized by an extension r, amplitude Cmax, and curvature: each of these three parameters is determined on a frame-by-frame basis via a nonlinear least-squares fit. The flattening of the profiles as equilibrium is approached can be described by fitting with a vanishing curvature. The values of r and Cmax determined from

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MODEL COMPARISON WITH EXPERIMENT Our evolution equation model has two adjustable parameters: the global time scale τ, fixed by D0 and r0, and the parameter αi that determines the rate of evolution of the chain edges. In order to compare model output with experiment these two parameters must be determined. First, the constant D0 is fixed by the shock-wave ramp slope in steady state. In our rescaled variables, Dc = D0(c/c0) predicts a steady-state ramp of the form12 Csteady = Ci + V0X

(13)

The constant Ci is simply the concentration at the profile edge. Note that according to eq 13, the steady-state ramp slope in rescaled variables is equal to the dimensionless sliding speed V0, from which we can extract D0 via D0 = Vr0/V0. The quantity V0 is obtained from the concentration profiles during transient compression using the fitted ramp slope obtained from the transient-ramp concentration model (Figure 4a). In practice, we average the ramp slope obtained for all frames for which t > 2.5 s (corresponding to the time range where the shock wave has reached the far molecule edge and the ramp is welldeveloped). We find that D0 = 7.1 ± 0.5 μm2/s, leading to a τ = 49 ± 4 s. E

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Macromolecules The edge parameters αi are obtained from direct leastsquares fitting of the PDE model output to our dynamic data. Note that in order to compare with experiment, the theoretical profiles, like the fitting models, are convolved with a Gaussian point spread function (PSF). For the shock wave data, the PDE model is solved using an initial flat-type equilibrium profile and a fixed V0 determined by the ramp profile slope. The PDE model is able to capture the entire compression process, from shock wave formation to the subtle details of the evolution toward steady state (Figure 4a). We find the best fit α = 0.25+0.05 −0.07 with the error bars determined from the error on Dc. In order to show the quality of agreement across the entire process time course, we compare measurements of the dynamic extension and maximum concentration during compression (see Figure 4b,c) against theory generated using the same value of α used to produce Figure 4a. Lastly, we include in the fit a time offset T0 that reflects uncertainty in the exact time at which the bead contacts the DNA and the shock wave begins (the offset is small, typically within 1−2 frames of the bestguess experimental starting time). For the relaxation data, the PDE model is solved with an initial concentration profile C(x,t=0) determined by the fit of a linear-ramp concentration model (see Supporting Information) to the frame directly preceding t = 0. A complexity of our assay for producing symmetric relaxation is that the bead is withdrawn at a finite retraction speed Vr. To incorporate this effect, we argue that the velocity of the edge-point x2 is fixed at the retraction speed until the free velocity of the molecule edge (e.g., the speed determined by eq 11) falls below Vr. At this point the bead is moving away faster than the molecule is expanding, and the molecule will no longer directly contact the bead. The retraction speed Vr is fixed and set equal to its experimental value (20 μm/s). We also assume that the edge parameters αi of both free edges are equal. While this assumption may not be true for small times, it should hold for intermediate times when the profile shape is approximately symmetric. The resulting profile fits are shown in Figure 5a; corresponding theoretical r and Cmax values are shown in Figure 5b,c. We find good agreement for the intermediate time range 0.2−2 s with a best fit α = 0.24 ± 0.05. This value compares well with the value of α obtained from compression. The theoretical profiles during the relaxation process are well-described by the parabolic model. This is expected, as eq 3 has a well-known similarity solution with a parabolic similarity profile (Barenblatt solution).4 The relaxation process, however, is not exactly self-similar. While Cmax ∼ t−1/3 (Figure 5c) as expected from the Barenblatt solution, we do not find that r ∼ t1/3 to good approximation (see Figure S2a in Supporting Information) as the edge boundary condition breaks similarity unless C ≫ 1 (in particular, only the quadratic term in eq 12 is consistent with the similarity transformation). We also show in Figure 5b the predicted result if the bead is removed instantaneously after compression. The finite retraction speed leads to a slightly lower value of r than would be observed upon infinitely rapid retraction. Note that predicted Cmax values for instantaneous retraction coincide with the prediction for a finite speed (see Figure S2b in Supporting Information). This suggests that the agreement of the Cmax results with the r ∼ 1/ t1/3 law is a fundamental feature of our model and does not arise from the effect of a finite retraction speed. The model does underestimate the relaxation time for very long times; it also fails for very short times (first two frames), overestimating the maximum concentration. The long-time disagreement may

arise due to subtleties in the osmotic pressure’s dependence on c or variation in Dc and edge friction close to equilibrium that are not included in our model.

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DISCUSSION We have used a purely phenomenological description of Dc based on the linearity of the steady-state ramp profiles for a compressed chain. It is however possible to relate the cooperative diffusion constant to the osmotic pressure via2,22 Dc ≡

cD2 ∂Π ξ ∂c

(14)

with the parameter ξ is a friction factor per unit length. We can introduce a dimensionless friction factor per unit length Ξ by normalizing ξ to the total chain friction factor: Ξ ≡ ξ/6πη. Recalling that C ≡ c/c0 and using eqs 8 and 14, we find Dc =

2Π 0D2 C 2 ⎛ 1 ⎞ ⎜1 + ⎟ 6πη Ξ ⎝ 2C 3 ⎠

(15)

Equation 15 differs in two respects from the linear form assumed throughout this study (Dc = D0C). First, at low C there is an extra factor of 1/C3. This factor is small for most of the relaxation process where C > 1. Including it in the fit yields negligible changes in the fit quality and fairly small changes in the best-fit values of β (5% change for compression data, 20% for relaxation). Second, there is an additional factor of C/Ξ. In order to make eq 15 consistent with the ansatz Dc = D0C, we require Ξ = Ξ0C. Thus, for high C the phenomenological relation Dc ∼ C leads directly to a friction factor per unit length that is also proportional to chain concentration. We can determine the proportionality constant Ξ0 via the following argument. The relation Ξ = Ξ0C yields a total chain friction factor ζ = ∫ r0ξ dx independent of chain extension and equal to (6πηr0)Ξ0. If we set Ξ0 = 1, we obtain the correct equilibrium total friction factor of the chain. Finally, this specification of Ξ implies that D0 = 2Π0D2/6πη, so that αi = 2βi and our best estimate βi = 0.12 ± 0.03. The edge-friction ξi reflects the total friction contribution due to portions of the molecule that participate in the motion of a particular edge. Crudely speaking, the value of the edge friction is related to the extent of the chain re that participates in the edge motion: βi = re/r0. We might expect, at very early times, that the edge friction should be determined by the friction factor of a single hydrodynamically coupled region, so that βi = D/r0 = 0.016. At longer times, as a flow profile is established and contour flows in from interior regions of the chain, we might expect a greater extent of the chain to participate in determining the edge friction leading to a larger value of βi. Our result suggests that the scale of the edge region lies intermediate between the blob scale and the total chain extent.

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CONCLUSION In conclusion, we have demonstrated that the nonequilibrium dynamics of nanochannel confined DNA can be reduced to solution of a nonlinear partial differential evolution equation supplemented by two equations governing the rate at which the molecule edges expand outward. In particular, we demonstrate that this approach can describe the symmetric relaxation and transient compression processes generated via our sliding bead assay. We are hopeful that our approach can be extended to describe the dynamic chain concentration profile in different F

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ACKNOWLEDGMENTS This work was funded through the Natural Sciences and Engineering Research Council of Canada (NSERC) discovery grants program (RGPIN 386212), Fonds de Recherche Nature et Technologies de Quebec (FRQNT) team grant (PR180418) and KAKENHI (No.26103525,“Fluctuation and Structure”) (No.24340100, Grant-in-Aid for Scientific Research (B)) from MEXT, Japan and JSPS Core-to-Core Program (Nonequilibrium Dynamics of Soft Matter and Information). In addition, we thank McGill’s Nanotools facility and the Institut Nationale de Recherche Scientifique (INRS) for providing fabrication tools and McGill’s Facility for Electron Microscopy (FEMR) for providing a scanning electron microscope.

transient as well as steady-state regimes subject to a wide range of boundary conditions and imposed forcing, for example, behavior of chains crossing defects, mobility of chains in channels of varying diameter, and even problems involving multiple interacting polymers in channels.31 Varying channel dimensions, for example, as well as applied thermophoretic and dielectrophoretic forces, could be incorporated via imposed spatially varying potentials and varying free energy densities. While we are excited that the complex dynamics of nanconfined DNA can be described by such a simple phenomenology, it is important to list several caveats. The particular form of Dc(c) suggested here is preliminary, phenomenological in nature, and likely requires further refinement. A number of complex confinement regimes may exist for c > c0 that could yield a much more subtle dependence of Dc on c.26,27 The disagreement we observe at large times in our relaxation data could be direct evidence of such regimes, for example, arising from subtleties in hydrodynamic effects28,29 in the calculation of chain friction, or emergence of new confinement regimes as the correlation length falls below the channel width.27 The channel width dependence of Dc is also unclear: Dc may have different qualitative forms in the transition and Odijk confinement regimes.7 Another question is the origin of edge friction and an explanation of the intermediate value we obtain. More detailed theories might explicitly include the time scales of stress propagation from the molecule edge and how these scales couple into the local hydrodynamics. We also neglect the effect of thermal fluctuations, which may alter the profile shape, particularly near equilibrium (another possible cause of the large-time disagreement of our theory with relaxation data) and entanglement.30 A very promising theoretical approach is to combine our coarse-grained PDE model with simulations to obtain detailed estimates of the phenomenological inputs (in particular, dependence of cooperative diffusion on chain concentration and channel diameter). The PDE model then serves as an intermediate f ramework that can take simulation determined inputs and efficiently deduce the corresponding implications for a range of dynamic experiments under varying conditions.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.5b02240. (1) Alternative argument for the derivation of osmotic pressure, (2) a discussion of the relation between τ and τr and why our model yields results consistent with the uniform approximation model, (3) additional discussion of the approximate self-similarity of the relaxation process, including a supplemental version of Figure 5, and (4) additional information regarding the experimental methods and data analysis procedures (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail [email protected] (A.K.). *E-mail [email protected] (W.W.R.). Notes

The authors declare no competing financial interest. G

DOI: 10.1021/acs.macromol.5b02240 Macromolecules XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.macromol.5b02240 Macromolecules XXXX, XXX, XXX−XXX