Length dependence of viscoelasticity of entangled-DNA solution with and without external stress

We observed the diffusive motion of a micron-sized bead in an entangled-DNA solution to investigate the effect of the viscoelasticity on the bead motion. In the absence of external stress (passive microrheology), subdiffusion appears in the timescale of 0.1–10 s, and the normal diffusion recovers in longer timescales. We evaluated the apparent viscosity and elasticity, which yields a simple relaxation time for the viscoelastic medium. We found that the absence of DNA-length dependence for the time-dependent diffusion is explained by the simple relaxation of the viscoelastic media rather than the reptation dynamics, including the disentanglement. On the other hand, in the presence of a small external stress in active microrheology, the bead motion showed clear length dependence owing to the viscoelasticity. These results suggest that the viscoelasticity of the entangled DNA is highly sensitive to the external stress, even in the linear response regime.We observed the diffusive motion of a micron-sized bead in an entangled-DNA solution to investigate the effect of the viscoelasticity on the bead motion. In the absence of external stress (passive microrheology), subdiffusion appears in the timescale of 0.1–10 s, and the normal diffusion recovers in longer timescales. We evaluated the apparent viscosity and elasticity, which yields a simple relaxation time for the viscoelastic medium. We found that the absence of DNA-length dependence for the time-dependent diffusion is explained by the simple relaxation of the viscoelastic media rather than the reptation dynamics, including the disentanglement. On the other hand, in the presence of a small external stress in active microrheology, the bead motion showed clear length dependence owing to the viscoelasticity. These results suggest that the viscoelasticity of the entangled DNA is highly sensitive to the external stress, even in the linear response regime.

In the case of DNA having <10 kilobase pairs, a micron-sized bead shows normal diffusion, where the mean-squared displacement (MSD) is proportional to time t; however, its topology (linear or circular) and length affect the viscosity. 27 On the other hand, for longer DNA, timescale-dependent diffusion occurs; 28 subdiffusion (MSD ∝ t α ; α < 1) appears in timescales ranging from 10 −1 s to 10 1 s, and normal diffusion recovers at long timescales of >10 1 s. The subdiffusion is due to elasticity caused by entangled DNA, and the recovery to normal diffusion is explained by the relaxation of the reptation dynamics of entangled DNA. 28 Similar subdiffusion has been observed in dense F-actin; the exponent α depends on the ratio of the radius of the probe particle to the mesh size of the polymer network, owing to hopping between pores in the network. 16 Recently, a hopping mechanism for the diffusion of a particle in a polymer network was theoretically proposed, showing timescale-dependent diffusion; 29,30 the normal diffusion recovers in a timescale long enough to rearrange the entangled network.
If the timescale-dependent diffusion observed for dense DNA is due to the reptation dynamics of the entangled polymers or the hopping dynamics, the diffusive behavior of the probe particle depends on the DNA length and the particle size. To verify the DNA-length dependence and the probe-size dependence of the diffusion, we observed the diffusive motion of a probe particle in dense DNA, using DNA of two different length and particles of two different sizes. We observed timescale-dependent diffusion: normal diffusion in short timescales (< 10 −2 s), subdiffusion in medium timescales (10 −1 − 10 1 s), and recovery to normal diffusion in long timescales (> 10 1 s). However, no DNA-length dependence was observed. To explain the timescale-dependent diffusion, we evaluated the apparent viscosity and elasticity, which yields a simple relaxation time for the viscoelastic medium. The timescale-dependent diffusion is explained by the simple relaxation of the viscoelastic media rather than the reptation dynamics, including the disentanglement. In contrast, in active microrheology, clear DNA-length dependence was observed. The difference observed in between the passive and active microrhology suggest that the viscoelasticity of the entangled DNA is highly sensitive to the external stress, even in the linear response regime.

II. MATERIALS AND METHODS
Toinvestigate the DNA-length dependence of the diffusion, we used λ-phage and T4 GT7 DNA (Nippon Gene). The contour length of λ-DNA (48.5 kbp) and T4-DNA (166 kbp) are 16.5 µm and 56.4 µm, respectively; the length is obtained, using 0.34 nm/bp for a double stranded DNA. Both cohesive single-stranded ends of the λ-phage DNA had been degraded by a Klenow fragment. The concentration of the commercial DNA solution was condensed to approximately 1 mg/mL via evaporation, and the final concentration was determined according to the optical density at 260 nm. To investigate the dependence of the viscoelasticity on the particle size, polystyrene beads (carboxylate microspheres, Polysciences, Inc.) 0.5 and 3 µm in diameter were used. The DNA and beads were diluted in a Tris-EDTA buffer (40 mM Tris-HCl, 4 mM EDTA, pH 7.4) and placed into a handmade chamber comprising a 0.13-mm silicon spacer sandwiched by two coverslips. We varied the DNA concentration c from 0 to 1.0 mg/mL. The mean-squared endto-end distance R 0 can be estimated using the scaling exponent 3/5 for a real chain and the persistence length of 50 nm for DNA. Assuming that R 2 G = R 2 0 /6, the overlap concentration is c * = m/(4/3πR 3 G ), where m is the mass of the single DNA. The estimated values of R G and c * are listed in Table I. We performed more than five measurements using different beads at each DNA concentration. To minimize the wall effects, first, the bead was trapped by optical tweezers and gently placed 10 µm (for the 3-µm bead) or 5 µm (for the 0.5-µm-bead) from the bottom surface. After waiting for >120 s to relax the storage elasticity, the bead was released from the optical trap, and then its motion was observed. The temperature was maintained at 25.0 ± 0.5 • C during the experiments.
The bead images were captured using an objective lens (100× oil, N.A. 1.35, Olympus) and a charge-coupled device camera (ICL-B0620M, Imperx) and recorded for 1,000 s with 2-ms time resolution. The images were binarized, and the center of mass (x, y) was determined. To minimize the effect of the stage drift for long-time measurements, we monitored the adsorption of the bead on the surface, and the position of the target bead was calibrated according to that of the adsorbing bead.
To investigate the effect of the viscoelasticity of the DNA solution on the diffusion of the bead, the time series x was divided into n intervals of t int , and the variance σ 2 was obtained for each time interval, as follows: 22 where n represents the mean value averaged over n. Here, (x(t) − x(t )) 2 represents the MSD at t int = t − t . Thus, when the diffusion is normal (MSD = 2Dt int ) with a diffusion constant D, The relationship between σ 2 and the MSD is analogous to that between the gyration radius and the end-to-end distance of a polymer. Figure 1a shows σ 2 with respect to t int for the 3-µm bead in a λ-DNA solution. In the presence of DNA, timescale-dependent diffusion appears. In short timescales (t int < 10 −2 s), the diffusion is almost normal ( σ 2 ∝ t 1 int ). Subdiffusion ( σ 2 ∝ t α int ; α < 1) appears in the medium timescales (10 −1 −10 1 s), and the normal diffusion recovers in long timescales (t int > 10 1 s). The timescale-dependent feature becomes distinct as the DNA concentration increases. For a longer time interval (t int > 10 2 s), we cannot ignore the flow effect that results in α 2. Thus, Fig. 1a plots data points for which the flow  effect can be neglected; the σ 2 values shown in the figure are more than 10 times larger than the interpolation values, assuming that α = 2 at t int = 900 s. To elucidate the timescale-dependent feature, the exponent α is plotted with respect to t int in Fig. 1b; α was calculated using ∆ log σ 2 /∆ log t int between two successive data points, and the average values for the neighboring five data points are plotted in Fig. 1b. α is minimized in the range of 0.1 − 1 s and decreases as the DNA concentration increases. Timescale-dependent diffusion was observed in a previous study, 28 where the subdiffusion was explained by the crossover of the elastic storage G and viscous loss G . This crossover has recently been observed in active microrheology. 31 The subdiffusion in the medium timescales is caused by the elasticity of the entangled DNA.

III. RESULTS AND DISCUSSION
If the timescale-dependent diffusion shown in Fig. 1 is governed by the reptation dynamics, including the disentanglement of DNA, 28-31 DNA-length dependence appears in the diffusion. However, we found no clear difference in the σ 2 − t int plot between λ-and T4-DNA. In order to check the DNA-length dependence, we calculated the σ 2 ratio and the α ratio of T4-DNA to λ-DNA for the 3-µm bead as shown in Fig. 1c and 1d, respectively. For both σ 2 and α, although the ratio slightly increases with increasing t int , the majority of the data is included in 0.9 − 1.3, excepting 0.6 mg/mL of the σ 2 ratio. To precisely examine the DNA-length dependence, we evaluated the time interval t * int at which the exponent α is minimized; t * int was obtained via fitting with a quadratic function for the α − log t int plot in the range of 0.04 s ≤ t int ≤ 10 s. Figure 2 shows t * int with respect to the DNA concentration for two different DNA lengths and two different bead sizes. Although the T4-DNA is more than three times longer than the λ-DNA, no clear difference appears between their data points; the large deviation for 0.2 mg/mL arises from the uncertainty in the fit with the quadratic function. Moreover, little difference appears between the two different bead sizes. If the timescale t * int is determined by the Rouse relaxation time (τ R ) of a single DNA molecule or the disentanglement time (τ d ) of DNA, t * int depends on the DNA length; τ R ∝ L 2 and τ d ∝ L 3 . 32 Figure 2 indicates that t * int is determined only by the concentration, regardless of the length. Although the exponent α shows little bead-size dependence, σ 2 depends on the bead size, as shown in Fig. 3a. The values of σ 2 for the 0.5-µm bead are vertically shifted from those for the 3-µm bead over almost the entire range of t int . In the case of normal diffusion, the diffusion constant is expressed as follows; D = k B T/6πηr, where k B is the Boltzmann constant, T is the absolute temperature, η is the viscosity, and r is the radius of the particle. According to Eq. (4), the ratio of σ 2 between the two different-diameter beads is the ratio of the radii between the two beads. Figure 3b shows the σ 2 ratio of the 0.5-µm bead to the 3-µm bead for λ-DNA. In the absence of DNA, the ratio is approximately 5. The value is slightly smaller than 6, which might be caused by the deviation of the bead size. Although the values are scattered for t int > 1 s, the values of the ratio are in the range of 5-10, and no distinct concentration dependence appears; a similar trend was observed for FIG. 2. t * int with respect to the DNA concentration for λ-and T4-DNA and beads 0.5 and 3 µm in diameter. t * int was obtained via fitting with a quadratic function for the α − log t int plot in the range of 0.04 s ≤ t int ≤ 10 s before averaging the neighboring five data points shown in Fig. 1b. T4-DNA, as shown in Fig. S1. These results indicate that the vertical shift in Fig. 3a arises from the viscous drag, depending on the size of the bead. According to our experimental results, we evaluate the apparent spring constant and viscosity of the entangled DNA. In our experiments, there is no external force, and the motion of the bead is in equilibrium. The absence of length dependence of α indicates that the elasticity appearing in the medium timescales is caused by the equilibrium structure of the entangled DNA with a mesh size of ξ: Here, the length dependence disappears when the scaling exponent of the polymer is 3/5. We evaluated the apparent spring constant of the entangled DNA using the law of equi-partition: where σ 2 * is the σ 2 value at t int = t * int . The evaluated apparent spring constant k * with respect to the mesh size ξ is shown in Fig. 4a. k * is proportional to ξ −2 , which is consistent with the fact that the spring constant of entangled DNA with mesh size ξ can be evaluated as k DNA = k B T ξ 2 ; 27,33 the elastic modulus is given by µ DNA = k DNA ξ = k B T ξ 3 . The vertical shift of σ 2 between the two different-sized beads is caused by the difference of the viscous drag force, as previously mentioned. Therefore, we conclude that the elasticity due to the equilibrium mesh structure results in the subdiffusion in the medium timescales.
Next, we evaluate the viscosity of the DNA solution using the value of σ 2 at the longest time interval. In long timescales (t int > 10 s), normal diffusion recovers, as shown in Fig. 1a. We assumed Eq. (4) and evaluated the viscosity η DNA for each bead diameter at each DNA concentration, as shown in Fig. 4b. η DNA is proportional to ξ −β , and β is between 3 and 4. We can evaluate the relaxation time τ DNA for the viscoelastic media using the elastic modulus µ DNA and the viscosity η DNA : Figure 5a shows the mesh-size dependence of τ DNA obtained using Eq. (7). τ DNA tends to decrease with the increase of ξ, and there is no DNA-length dependence or bead-size dependence. These features quantitatively agree with the mesh-size dependence of t * int , as shown in Fig. 5b. Thus, we conclude that t * int corresponds to the simple relaxation time for the viscoelastic media with µ DNA and η DNA .
The anomalous diffusion scaled by the relaxation time for viscoelastic media has been shown in a recent theoretical work 34 using a two-fluid model. [35][36][37] In the case of dense DNA, the recovery to normal diffusion in long timescales is caused by the simple relaxation of the mesh structure rather than the reptation dynamics, including the disentanglement of DNA, which is governed by only the mesh size and is independent of the DNA length. The timescale-dependent diffusion observed in a dense DNA solution is explained as follows. In short timescales (< 10 −2 s), as the displacement of the bead is small, normal diffusion occurs because there is almost no deformation of the mesh structure. In medium timescales (10 −1 − 10 1 s), the displacement of the bead causes the deformation of the mesh structure, and the elasticity of the mesh structure suppress the motion of the bead. In timescales long enough (> 10 1 s) for the relaxation of the deformation, the normal diffusion recovers. In our experiments, the length scale of the fluctuation of the probe particle is limited to a mesh size that is significantly smaller than the DNA length. Our results are consistent with the length scale-dependent rheology observed for entangled F-actin. 17 Compared with our results for passive microrheology, the G and G obtained in active microrheology depend on the DNA length. 31 Furthermore, the recently proposed novel microrheology technique reveals nonlinear features, such as shear thinning governed by stress-dependent entanglements. 33 These experimental facts imply that the viscoelastic properties of entangled polymers are highly sensitive to the degree of external stress. To confirm this sensitivity, we investigated the viscoelastic response using a previously reported microrheology technique. 33 Figure 6 shows the experimental procedure. First, a 3-µm-diameter bead was optically trapped and moved in the x-direction at a constant speed. When the bead displacement reached approximately 25 µm, the optical trap was shut off, and the bead was released from the trap. In the case of a simple FIG. 6. Experimental move-and-release procedure for the bead in the DNA solution. A 3-µm-diameter bead was trapped by optical tweezers (a) and moved at a constant speed (b). The bead was released from the optical trap when the bead displacement reached approximately 25 µm (c). After the release from the optical trap, the backward motion of the bead was observed (d).  Fig. 7; (t, x) = (0, 0) was identified using the maximum value of x, and each data point is the average value for 10-12 trials using different beads. The relaxation processes differ between the two different DNA lengths at the same speed. Additionally, the backward displacement depends on the DNA length and the moving speed. Here, we emphasize that the strain rate estimated byγ = 3v/ √ 2r is 0.24 and 4.8 s −1 for v = 0.17 and 3.4 µm/s, respectively, which is small enough for the linear response regime, in which the apparent viscosity is independent of the strain rate. 33 Although further investigation is needed to elucidate the length-dependent relaxation, the DNA-length dependence of the bead motion with (Fig. 7) and without (Fig. 4) external stress suggests that the viscoelastic properties of the entangled DNA are highly sensitive to the external stress.

IV. CONCLUSIONS
We observed the diffusive motion of a micron-sized bead in an entangled-DNA solution in the absence of external stress (passive microrheology). To explain the timescale-dependent diffusion, we evaluated the apparent viscosity and elasticity, which yields a simple relaxation time for the viscoelastic medium. The timescale-dependent diffusion exhibited no DNA-length dependence, which is explained by the simple relaxation of the viscoelastic media rather than the reptation dynamics, including the disentanglement. On the other hand, DNA-length dependence was observed in the presence of a small external stress (active microrheology). These results suggest that the viscoelasticity of the entangled DNA is highly sensitive to the external stress, even in the linear response regime. Figure S1 of the supplementary material shows the σ 2 ratio of the 0.5-µm bead to the 3-µm bead with respect to t int for T4-DNA.