Abstract
Macromolecular diffusion in homogeneous fluid at length scales greater than the size of the molecule is regarded as a random process. The meansquared displacement (MSD) of molecules in this regime increases linearly with time. Here we show that nonrandom motion of DNA molecules in this regime that is undetectable by the MSD analysis can be quantified by characterizing the molecular motion relative to a latticed frame of reference. Our lattice occupancy analysis reveals unexpected submodes of motion of DNA that deviate from expected random motion in the linear, diffusive regime. We demonstrate that a subtle interplay between these submodes causes the overall diffusive motion of DNA to appear to conform to the linear regime. Our results show that apparently random motion of macromolecules could be governed by nonrandom dynamics that are detectable only by their relative motion. Our analytical approach should advance broad understanding of diffusion processes of fundamental relevance.
Introduction
Brownian motion, as famously explained by Albert Einstein in 1905, is a process during which tiny particles move randomly in a homogeneous isotropic fluid as they experience independent molecular collisions from the thermally excited fluid molecules^{1}. This erratic particle motion has fascinated scientists for most of the last two centuries. Probing biomolecular interactions^{2}, imaging of cell organelles and nanostructures in three dimensions^{3,4} and building molecular motors^{5} are among the major scientific applications of Brownian motion. The full molecularscale context of Brownian motion occurs in three basic regimes. In the first regime before any molecular collisions, the particle shows a ballisticlike motion^{6}. In the second regime, which begins when the particle interacts with fluid molecules and the resulting friction creates local vortices that act on the particle, the molecular motion is affected by hydrodynamic forces of the fluid^{7}. These ballistic and hydrodynamic regimes, therefore, deviate from the random Brownian motion. In the third regime during which the particle diffuses its own radius, statistically independent collisions dominate the motion, causing the overall motion of the particle to be random. The random Brownian motion causes the random positions of the particle, and therefore, the meansquared displacement (MSD) of molecules increases linearly with time.
Among the dynamics of particles in fluids^{6,7,8}, those of DNA are rather unique^{9,10,11}. DNA is a semiflexible polymer; its motion in a homogeneous isotropic medium and within its radius of gyration is governed by the constraints imposed by its chain connectivity and by various intramolecular hydrodynamic interactions^{10,11}. These dynamics, which complicate the hydrodynamic regime of DNA molecules, often cross over and partially affect their linear diffusive regime^{12,13}. Consequently, at diffusion distances close to the radius of gyration, the motion of DNA molecules is nonlinear and subdiffusive. At longer diffusion distances and at long time scales, DNA molecules diffuse, following the expected behaviour of a polymer as a whole, and MSD is linear with time^{12,13}. Although slow conformational fluctuations have been observed within the time scale of the diffusive motion of DNA^{14}, the physical origin of these fluctuations and whether they affect the diffusive motion of DNA have not been determined. This fundamental understanding has thus far been hampered by a lack of both theory and analytical tools that give access to the diffusive regime of macromolecules.
Here we report the development of new theoretical framework and analytical tool that can capture the motion of macromolecules in their diffusive regime by characterizing the motion relative to a latticed frame of reference. Our new method—lattice occupancy analysis—reveals unexpected submodes of motion of DNA molecules that deviate from the expected random motion in the linear, diffusive regime.
Results
Theoretical framework of lattice occupancy analysis
Brownian motion is typically viewed in terms of the absolute positions of single molecules, which are the hallmark of MSD analysis^{6,7,9,10,15,16,17}. In this study, we consider their linear, diffusive regime not in terms of MSD’s absolute measurement but in terms of a relative measurement. In this measurement, we study the motion of single molecules with respect to a virtual latticed frame of reference with which we analyse how often the molecule steps into new lattice sites in a diffusion space during its motion (Fig. 1a, Supplementary Data 1). The experimental probability of lattice occupancy (P_{t}) at time t is given by
where n is the number of steps and <k_{t}> is the average number of visits to new lattice sites. According to onedimensional (1D) random diffusion theory, the probability distribution (p) of finding the particle at different lattice sites, q, is described by^{18,19}
where n, l and σ respectively denote the number of step(s), the step size and the s.d. of p. For simplicity, we set l=m, where m is the side length of the lattice. The particle executes n steps in time t=nτ, where τ is a unit of time. Since the diffusive spreading^{20} of the particle (that is, spreading of the diffusing particles in the space away from their initial position) is a measure of the rate at which the particle spreads out in space during time t^{19}, it is expressed as the increase in σ as the square root of t increases (Fig. 1b)^{20}. The rate of diffusive spreading in the lattice’s 1D space (that is, the number of visits to new lattice sites) is equivalent to the probability of occurrence of visits to new lattice sites at time t (P_{t}). P_{t} is expressed as
P_{t} obtained from a simulated 1D random diffusion trajectory agrees well with equation (3) (Supplementary Fig. 1). In twodimensional (2D) random diffusion, the new position of the molecule depends on the step size, the direction of the motion and where the last step ends. Successive steps could therefore occur in the same lattice site, and the rate of the power law decay (that is, the rate of visiting new lattice sites in 2D) could thus accordingly decreases. Thus, P_{t} in 2D space (P_{t}^{2D}) can be given by
where β is the scaling exponent of the powerlaw decay. P_{t}^{2D} obtained from a simulated 2D random diffusion trajectory agrees well with equation (4) (Fig. 1c). Our lattice occupancy analysis of simulated directed and confined 2D motions showed that the values of β were dependent on the diffusion mode (Supplementary Fig. 2).
Diffusive motion of DNA in a latticed 2D space
Based on this theoretical framework of molecular motion, we characterized the spatiotemporal pattern of the diffusive motion of DNA (linear ColE_{1} DNA (ref. 21)) using singlemolecule fluorescence microscopy (Fig. 2a,b and Supplementary Video 1, see Supplementary Note 1). Standard MSD analysis of the diffusion trajectories (Fig. 2b) showed subdiffusive dynamics of DNA at length scales close to its radius of gyration (R_{g}=0.186 μm (50 ms))^{21} (Fig. 2c). This regime arises due to the crossover of the hydrodynamic regime of DNA^{10} (shown in red in Fig. 2c). The MSDtimelag (Δt) profile at length scales larger than the radius of gyration of DNA, that is, at a time scale longer than 50 ms, reflected the linear, diffusive regime of its Brownian motion (shown in blue in Fig. 2c and Supplementary Fig. 3a). On the other hand, the MSD of spherical polymer nanospheres exhibited a linear increase with time at all scales (Supplementary Fig. 3b). This indicated the pure random walk of these nanospheres; we therefore used them as a control throughout the study (Supplementary Fig. 3b and Supplementary Video 2). To characterize the diffusive motion of the DNA and nanospheres in the linear, diffusive regime with respect to the latticed frame, we calculated a temporal profile of P_{25} (the probability of new visits in a time lag of 25Δt, Fig. 1c). P_{25} at each time point was determined by applying a 50Δt time window. The temporal profile of P_{25} was then obtained by sliding the time window along the trajectory (Fig. 2b,d). The timedependent P_{25} values exhibited fluctuations between a high lattice occupancy mode (low P_{25} value or a few visits to new lattice sites) and a low lattice occupancy mode (high P_{25} value or more visits to new lattice sites). We collectively refer to the modes that result from the lattice occupancy analysis as relative modes (Fig. 2d).
Analysis of the hidden nonrandom diffusive dynamics of DNA
Next, we analysed the diffusive dynamics of DNA based on the timedependent P_{25} profile. Because Brownian motion is fractal in nature, its temporal fluctuations are random at all scales. Any fluctuations, including P_{25}, are hence invariant regardless of the time scale used to probe the motion^{22}. On the other hand, nonrandom motion occurring in the linear, diffusive regime causes timescaledependent fluctuations. Such fluctuations can be captured by using detrended fluctuation analysis (DFA) and by calculation of the Hurst exponent (HE) (Supplementary Data 2, see Methods for the details)^{23}. The timescaledependent fluctuations due to the nonrandom motion results in larger HE (HE>0.5) compared with that obtained from random motion (HE=0.5).
To provide statistically robust HE estimates, we joined 98 singlemolecule tracks endtoend and generated a long probability time series (Fig. 3a, Supplementary Fig. 4). Any systematic errors that could arise from the endtoend connections were evaluated by calculating the HE of 100 shuffled replicates by randomizing the order of the connections between the original trajectories (Supplementary Fig. 4a). We then compared the HE of these experimental replicates with those of simulated replicates to identify any deviations from random behaviour, if any, and also to identify the physical origin of these deviations. The simulated trajectories were generated by randomizing the order of both the step sizes (S) and the step directions (angles (A)) of the original experimental replicates (denoted as S_{r}A_{r} simulated replicates), by randomizing the angles while maintaining the order of the step sizes (S_{i}A_{r}) or by randomizing the step sizes while maintaining the order of the angles (S_{r}A_{i}). The analyses uncovered a dramatic positive shift of the calculated HE profiles of experimental replicates of DNA (Fig. 3c blue lines, Fig. 3e blue line, Fig. 3g blue line), but not of the nanospheres (Fig. 3b,d,f, Supplementary Fig. 5), the simulated replicates of S_{r}A_{r} (Fig. 3e green lines, Fig. 3g green line), and the simulated replicates of S_{r}A_{i} (Fig. 3e black lines, Fig. 3g black line). The HE profiles of the experimental replicates of DNA (Fig. 3c blue lines) exhibit this positive shift compared with those of the simulated replicates of S_{r}A_{r} (Fig. 3c green lines) even at 10Δt at which the MSD10Δt profile exhibits linear behaviour (Supplementary Fig. 3a). The results clearly demonstrate that the nonrandom motion of DNA in the linear, diffusion regime, which is not captured by MSD analysis, is revealed by lattice occupancy analysis.
Interestingly, the HE profile of the simulated replicates of S_{i}A_{r} (Fig. 3e red lines, Fig. 3g red line) partially overlaps that of the experimental replicates, indicating that the HE shift is unique to DNA motion and it is partially related to the order of the step sizes but not to the order of the angles. Furthermore, MSD analyses of the S_{r}A_{i} replicates (Fig. 3h black line) produced a subdiffusive profile, whereas MSD analyses of the S_{r}A_{r} (Fig. 3h green line) and the S_{i}A_{r} (Fig. 3h red line) replicates produced linear profiles. These results indicate that the subdiffusive behaviour of DNA observed in the hydrodynamic nonrandom regime (Fig. 2c) results from the order of the directions of the steps of the DNA walk.
Identification of the diffusion modes of DNA
Next, we investigated the origin of the hidden nonrandom motion of DNA molecules in the linear, diffusive regime as revealed by lattice occupancy analysis. According to the random walk theory, statistical variations in step sizes and step directions yield trajectories that resemble by chance those of directed and confined modes of diffusion (Supplementary Fig. 6a–f)^{24,25}. The MSDΔt profiles obtained from the trajectories with directedlike and confinedlike modes of motion thus respectively exhibited concave and convex curves as described by equations (10) and (11) (see Methods; Supplementary Fig. 6g,h). These apparent deviations from random motion arise from the limited length of the experimental trajectories. Because these apparent deviations are viewed as parts of the random fractal nature of the diffusive regime of Brownian motion, they are persistent at all time scales^{22,24,25}. Thus, the temporal fluctuations of the P_{25} value occurring at the time scale of the linear, diffusive regime are accounted for by both the nonrandom motion of DNA and the apparent deviations (directedlike and confinedlike modes) from the random motion. Indeed, lattice occupancy analyses of simulated trajectories displaying directedlike and confinedlike modes of diffusion respectively yield a low lattice occupancy mode (high P_{25} value) and a high lattice occupancy mode (low P_{25} value) (Supplementary Fig. 2).
As a first step in distinguishing the apparent nonrandom diffusion caused by statistical variations intrinsic to Brownian motion and actual nonrandom diffusion of DNA, we compared the temporal profiles retrieved from the relevant analytical tool in each case. Specifically, we compared the temporal behaviour characterized by MSD analysis (apparent nonrandom diffusion) and the temporal P_{25} profile (actual nonrandom diffusion). We first normalized the temporal profile between 0 and 1 and split the trajectory into high and low lattice occupancy modes at the mean (μ_{nP}) (Fig. 4a, Supplementary Data 3). The characterization of the temporal behaviour by MSD analysis was conducted by first calculating the MSDΔt plots at each time point by applying a sliding window with a 50Δt time width (see Methods for details). The directed and confinedlike motions were quantified by drift velocity (ν, equation (10)) and the length of the confined area (L, equation (11)) and normalized to μ_{nP} and 1 and μ_{nP} and 0, respectively (Fig. 4b, Supplementary Data 4). We refer to these directedlike and confinedlike modes that result from MSD’s absolute measurement as absolute modes. We then superimposed the normalized P_{25} and MSD (that is, ν and L) temporal profiles for analysing the modes of diffusion qualitatively (Fig. 4c). Comparison of the two temporal profiles identified the time instances at which the correlation between the P_{25} and MSD profiles was positive (directedlike motion with low lattice occupancy (high P_{25}) mode (dLO submode) and confinedlike motion with high lattice occupancy (low P_{25}) mode (cHO submode)) (Fig. 4c). However, the temporal profiles also revealed that they are not always positively correlated with each other (Fig. 4c). Specifically, the temporal profiles showed that the confinedlike motion was sometime correlated with the low lattice occupancy mode (cLO submode) and that the directedlike mode was sometime correlated with the high lattice occupancy mode (dHO submode) (Fig. 4c).
Characterization of the nonrandom diffusion modes of DNA
To determine if any of these four submodes (dLO, cHO, cLO and dHO) causes the nonrandom motion of DNA, we devised a fourstep analytical approach (Fig. 5 and Supplementary Note 2). First, we split the normalized P_{25} and MSD temporal profiles (Fig. 4a,b) into two modesets at μ_{nP} (step A in Fig. 5, Supplementary Note 3) and calculated the amplitudes of the local variations (ALV) between the two profiles (step B in Fig. 5, Fig. 6a) using the dynamic time warping (DTW) algorithm (Supplementary Data 5, see Methods for details) to quantify the deviations between the two time profiles. The DTW algorithm provided a negative ALV (−ALV zone) when the amplitude in the P_{25} profile was larger than the amplitude in the MSD profile. On the other hand, the DTW algorithm provided a positive ALV (+ALV zone) when the amplitude in the P_{25} profile was smaller than the amplitude in the MSD profile. Thus, extremely negative ALV values in the first and second modesets reflect the cLO and dHO submodes, respectively, and extremely positive ALV values in the first and second modesets reflect the dHO and cLO modes, respectively (Fig. 6a). We then used the ALV values above or below the thresholds to distinguish between different submodes that exist in the experimental replicates (step C in Fig. 5 and Fig. 6b,c). We used the thresholds defined by the mean (ALV_{μG}) and the s.d. of the ALV profiles (σ_{ALV}, between 2.2σ_{ALV} (1.4 % of the data) and 0.1σ_{ALV} (46% of the data)), which were calculated from the 100 experimental replicates. We next extracted the diffusion subtrajectories corresponding to these submodes. Using these subtrajectories, we calculated MSDΔt plots and stepsize distributions to analyse the diffusion modes (step D in Fig. 5). Finally, we compared these trajectories with their respective trajectories obtained from the nanospheres and the simulated S_{r}A_{r} replicates so that we could discern whether or not the experimental submodes exhibited nonrandom behaviours.
Figure 6a–c shows that the deviations between the two profiles are easily identified by calculating ALV using the DTW algorithm. We first examined the stepsize distributions of the extracted diffusion subtrajectories whose time regions are defined by setting the ALV threshold to ±2.2σ_{ALV} (Fig. 7a, Supplementary Data 6). While the distributions do not display any deviations from the 2D random diffusion theory^{26}, the distributions obtained from the –ALV zones (Fig. 7a top) exhibit either larger (first modeset) or smaller (second modeset) mean step sizes compared with those obtained from the +ALV zones (Fig. 7a bottom). The mean step sizes in the –ALV zones clearly show the dependency on the threshold level, demonstrating that the negative peaks in the ALV plots are responsible for the larger and smaller step sizes in the first and the second modeset, respectively (Fig. 7b). We did not observe this threshold dependency in the negative peaks detected in the ALV plots obtained from the nanospheres (Fig. 7c) and the simulated S_{r}A_{r} replicates (Fig. 7b) (that is, those peaks are assigned to the apparent nonrandom diffusion caused by the statistical variations intrinsic to Brownian motion). These results further demonstrate that the subtrajectories corresponding to the ALV peaks in the negative zones display nonrandom diffusion modes. Since the negative ALV peaks correspond to the larger amplitudes of the P_{25} profile compared with those of the MSD profile (that is, in our analytical approach, the diffusive motion is mainly characterized by the relative diffusion modes—cLO and dHO submodes for the first and second modesets, respectively), the results also demonstrate that lattice occupancy analysis can capture nonrandom diffusion modes. The step sizes obtained from the peaks detected in the +ALV zones, which mainly reflect the absolute diffusion modes (that is, directed or confinedlike motion), do not show any deviations from the average step size (Fig. 7a–c), demonstrating that these submodes captured by MSD analysis do not exhibit nonrandom behaviour.
We further characterized the LO and HO modes detected in the above analyses by reconstructing corresponding MSDΔt plots (Supplementary Data 6). The MSDΔt plots reconstructed from the region of the diffusion trajectories that display the LO submode clearly showed confinedlike motion (cLO submode) (Fig. 8a,c). On the other hand, the MSDΔt plots obtained from the HO submode region exhibited directedlike motion (dHO submode) (Fig. 8b,d). The MSDΔt plots of the simulated S_{r}A_{r} replicates that were reconstructed from the regions with negative ALV peaks exhibited normal diffusion (Supplementary Fig. 7a), confirming the existence of unexpected cLO and dHO submodes that are responsible for the nonrandom motion of the DNA in homogeneous isotropic environments. On the other hand, the MSDΔt plots obtained from the positive ALV zones of the DNA trajectories were indistinguishable from the simulated S_{r}A_{r} replicates (Supplementary Fig. 7b). This result together with the ALV thresholddependent step sizes (Fig. 7) confirms that the submodes characterized by the MSD analysis exhibit random behaviour.
To investigate the effect of the temporal order of the step sizes on the nonrandom motion, we replaced the larger steps of the cLO submode (Fig. 7a red) in the experimental diffusion trajectory with the randomly ordered smaller steps obtained from other modes (Fig. 7a blue and green) (denoted c_{sh}LO) (Supplementary Fig. 8). The MSDΔt plot obtained from the simulated c_{sh}LO trajectory showed a deviation from the linear MSDΔt profile of the original experimental replicate towards confinedlike motion (Fig. 9 red), suggesting that the shorter step sizes in the time regions of the cLO submode—instead of long step sizes—caused this deviation towards confinedlike motion (Supplementary Data 7). On the other hand, the MSDΔt plot obtained from the simulated trajectory whose short steps in the dHO submode were replaced by the randomly ordered longer steps obtained from other modes (d_{lg}HO trajectory) displayed the opposite behaviour (that is, a shift towards directedlike motion, Fig. 9 blue), suggesting that the larger stepsizes in the dHO submode caused the deviation towards the directedlike motion (Supplementary Data 7). These findings further demonstrate that the nonrandom temporal order of the step sizes causes the nonrandom motion of the DNA and is consistent with cLO and dHO submodes. A characteristic time scale of the cLO submode (τ_{cLO}) was estimated to be τ_{cLO}=0.33±0.016 s by Fourier transform analysis (Supplementary Figs 9,10). Interestingly, the τ_{cLO} is in good agreement with the conformational relaxation time (τ_{R}=0.34 s) of DNA (Supplementary Fig. 11), indicating possible involvement of the conformational relaxation of DNA in nonrandom motion (see Discussion for the detail).
Motion of DNA in the crossover regime
We then examined whether or not the relative cLO and dHO submodes exist in other regimes of molecular motion. To that end, we analysed diffusion trajectories of lambda phage DNA (48,500 kbp, R_{g=}0.7 μm, Fig. 4a)^{27}. Because the radius of gyration of lambda DNA is much larger than that of ColE_{1} DNA, lambda DNA displayed a subdiffusive MSDΔt profile (Fig. 10a) in the time scale that is compatible with lattice occupancy analysis. This indicates that we capture the motion of lambda DNA in its crossover regime by lattice occupancy analysis. In contrast to ColE_{1} DNA molecules, we did not find a significant difference between the calculated HE of the experimental replicates and their S_{r}A_{r} replicates (Fig. 10b, Supplementary Fig. 12a). Furthermore, we captured neither the cLO/dHO submodes (Fig. 10c) nor the nonrandom temporal order of the step sizes that were observed in ColE_{1} DNA (Supplementary Fig. 12b). The results demonstrate that the nonrandom motion of DNA we captured using lattice occupancy analysis (that is, lattice occupancy modes) is observed characteristically in the linear, diffusive regime. The nonrandom motion of DNA in its crossover regime is better characterized by MSD analysis. Since the diffusion coefficients of ColE_{1} and lambda DNA in our experiments were close to each other, these findings also serve as an important confirmation that the threedimensional (3D) motion of the molecules does not affect lattice occupancy analysis.
Discussion
Studying the relative motion of single molecules provides a means to extract essential information on nonrandom dynamics that has remained inaccessible via conventional theories of absolute measurements (Supplementary Fig. 13). We used the motion of DNA relative to a square lattice to unlock a subtle dynamic regime in the Brownian motion of DNA and to uncover that diffusion speed of polymer molecules and the mode of motion have unexpected effects on 2D lattice occupancy (that is, the presence of unexpected cLO and dHO submodes). Therefore, our analytical approach is different from, yet complementary to, other analytical methods such as cumulative area tracking (CA tracking)^{16,17,28}. The CA tracking method analyses the diffusion constant of single molecules by relating the mean cumulative area difference to the elapsed time. CA tracking has two major advantages. First, in contrast to our lattice occupancy analysis, which uses singlemolecule localization algorithms to determine the position of the molecule, CA tracking circumvents the localization step of the molecule while employing simple tracking of a limited number of pixels (proxy pixels) that define the position of the molecule. Second, by controlling the number of proxy pixels, conformational dynamics can be simultaneously analysed by CA tracking. A major difference between our lattice occupancy algorithm and CA tracking is that the latter cannot be used to calculate 2D lattice occupancy because of the random shape of the proxy pixels. Although the effect of a change in the shape of the proxy pixels is averaged out during the calculation of the diffusion constant, a large error can be introduced in the timedependent lattice occupancy profile. Lattice occupancy analysis is therefore complementary to CA tracking because it allows us to characterize the relative motion of DNA and to correlate it with the conformational relaxation time of the molecule.
By using lattice occupancy and MSD analyses, we found that the relative motion of DNA is dramatically affected by the distribution of its step sizes. Specifically, an increase in the step sizes during confinedlike motion pushes the molecular positions apart, and the relative motion thus exhibits low lattice occupancy (cLO submodes). Conversely, a decrease in the step sizes during directedlike motion causes the relative motion to exhibit high lattice occupancy (dHO submode). We termed the coincidence between specific distributions of step sizes and the modes of motion as steptomode matching. Validation of this matching was obtained by randomizing the step directions of the experimental trajectories to break the matching of the diffusional modes to the original step sizes (the S_{i}A_{r} simulated replicates). We found that this breaking caused the HE profiles to have partial rather than full overlap with that of the experimental replicates (Fig. 3g). Another validation was obtained by altering the stepsizes of the cLO or the dHO submodes at a time to break the matching of the step sizes to the original diffusive modes (the c_{sh}LO and the d_{lg}HO simulated trajectories). We found that manipulating the stepsizes caused welldefined deviations from the linear MSDtime profile (Fig. 9). Thus, we conclude that the combined behaviour of these nonrandom dynamics, rather than a simple stochastic process, is essential for the overall singlemolecule behaviour to conform to the linear trend of MSD. This remarkable conservation of the linear trend suggests that these dynamics are in a subtle mechanistic balance, suggesting that they could be attributed to the same physical origin. While this physical origin is still not entirely clear, the good agreement between τ_{cLO} and τ_{R} is noteworthy. This agreement partially accounts for the nonrandom dynamics and suggests that they are related to the conformational relaxation dynamics of DNA in which the relaxed conformations diffuse with shorter step sizes than do the compact conformations that diffuse with longer step sizes. Although a similar yet fundamentally different behaviour was previously elucidated as autocorrelated fluctuations in step sizes, these fluctuations are caused by changes in the radius of gyration and are attributed to the internal conformational fluctuations of DNA^{11}. These internal fluctuations relax by diffusion, and their characteristic time is defined as the time required by the molecule to diffuse a distance that equals its radius of gyration (t_{Rg}(τ_{Rg}))^{29}. The relative dynamics that we report in this study can be distinguished from these internal fluctuations because the time scale of the relative dynamics is much longer than the characteristic time scale of the internal fluctuations (τ_{cLO}≫ τ_{Rg}) (Fig. 2c, Supplementary Fig. 9b) and because of the characteristic steptomode matching that we observed in our analyses.
Further to the abovementioned rational for the mechanism of the relative dynamics, we believe these dynamics could be related—in part—to the anisotropic diffusion of DNA because of its transient relaxation. During the time when the DNA molecule is relaxed, its shape is anisotropic, which can be modelled as an elongated ellipsoid where a (length) ≫ b (width). The resulting anisotropic diffusion involves two components, D_{a}, diffusion coefficient in directions parallel to the long axis (D_{a}=k_{B}T/γ_{a}), and D_{b}, diffusion coefficient in directions perpendicular to the long axis (D_{b}=k_{B}T/γ_{b}). Because the friction coefficient γ_{a} is smaller than γ_{b}, D_{a} is greater than D_{b} and therefore the molecule is expected to show directedlike motion^{30}. The time scale of this directedlike motion is determined by τ_{θ} (the time required for the ellipsoid to diffuse 1 rad by rotational motion). At time scales longer than τ_{θ}, the rotation randomizes the motion and, eventually, results in a crossover from anisotropic diffusion to isotropic diffusion^{30}. Because the elongation of the DNA occurs transiently during the conformational relaxation process, we cannot gather conclusive evidence on whether τ_{θ} is correlated with τ_{R} and with τ_{cLO}. Because of this uncertainty and the fact that shape isotropy/anisotropy cannot similarly explain the confinedlike submode motion, we argue that anisotropic diffusion could—only partially—explain the directedlike submode motion. Taken together, we conclude that the conformational relaxation dynamics and the anisotropic diffusion partially elucidate the mechanism of the relative dynamics we describe here. Describing the full mechanism remains an open research question.
For the diffusion mode to be reliably captured using lattice occupancy analysis, the time scale of the dynamics should be slower than the frame rate of the detector and faster than the diffusion of the molecule out of the focal plane of the microscope. This limits the time scale of the dynamics that can be captured by lattice occupancy analysis. A detector with a faster frame rate^{31}, stroboscopic laser excitation^{32} and 3D singlemolecule tracking techniques^{33} may further expand the applicability of the analysis to wider time scales.
The results reported here demonstrate that studying the relative motion of single molecules provides information on the dynamics hidden in their diffusive motion. These dynamics, which we term conserved linear dynamics, were not previously observed in the motion of single molecules. Our identification of conserved linear dynamics suggest that the apparent random diffusive motion of molecules in nature could actually be governed by nonrandom dynamics. Our observations and our analytical approach provide a new method for advancing our understanding of diffusion processes that are central to studies in diverse scientific fields. For example, from studying anomalous diffusion processes in biophysics to studying dynamic disorder in polymer science, our analytical approach could uncover essential dynamics and hence could provide access to intriguing applications. To that end, understanding the relative motion of molecules in terms of the specific modes of diffusion in the relevant fields (similar to what is shown in Figs 4, 8 and 9) is essential. Such fundamental knowledge could also provide essential information on crucial diffusionlimited processes of the cell.
Methods
Materials
Supercoiled ColE_{1} (6.6 kbp) DNA was obtained from Nippon Gene (Toyama, Japan) whereas the lambda phage DNA was obtained from New England Biolabs (Hitchin, UK). The restriction enzyme SmaI and the digestion buffer were obtained from New England Biolabs and were used to prepare the linear form of the ColE_{1} DNA. The DNA molecules were covalently labelled with Cy5 using a Label IT Cy 5 labelling kit obtained from Mirus Bio (Madison, WI, USA).
Suncoast yellow fluorescent polymer nanospheres (excitation/emission maxima 540/600 nm) of nominal size (0.19 μm; 2.653 × 10^{12} nanospheres ml^{−1}) were purchased from Bangs Laboratories, Inc. (Fishers, IN, USA). The nanospheres were diluted with 70% glycerol in 10 mM TRIS buffer (pH 8) to yield a concentration of 1.5 × 10^{6} nanospheres ml^{−1}.
Preparation of the linear form of the supercoiled DNA
To prepare the linear form of ColE_{1}, 8 μg of the supercoiled form were mixed with 50 units of SmaI in 50 μl of the digestion buffer (50 mM potassium acetate, 20 mM Trisacetate, 10 mM magnesium acetate, 1 mM DTT, pH=7.9). The reaction mixture was incubated at 25 °C for 8 h before removing the enzyme and the buffer components using isopropanol precipitation.
Isopropanol precipitation
The SmaI enzyme and the buffer components were removed after enzymatic digestion by the standard isopropanol precipitation method. Twentyfive microlitres of sodium acetate solution (3 M) and 40 μl of isopropanol were added to the digested DNA solution followed by ultracentrifugation at 15,000 r.p.m. for 20 min at 4 °C. The supernatant was carefully removed and the DNA pellets were washed three times with 70 % ethanol. The washing step was repeated three times before the DNA pellets were dried in air.
Labelling the DNA with Cy5 dye
The ColE_{1} DNA was covalently labelled with Cy5 dye according to the protocol accompanying the labelling kit. The DNA pellets obtained after isopropanol precipitation were dissolved in 37 μl water. Then, 5 μl of the labelling buffer and 8 μl of the labelling reagent were added to the DNA solution. The mixture was incubated at 37 °C for 2 h. The DNA was purified from the labelling reagents using isopropanol precipitation as described above. The labelled DNA pellets were dissolved in Tris EDTA (TE) buffer (10 mM Tris, 1 nM EDTA, pH=8) to yield a concentration of 0.2 μg ml^{−1}. These labelling procedures should yield labelling efficiency of approximately one label every 10–30 base pairs according to the manufacturer’s specifications.
Preparation of the DNA imaging buffer
To record singlemolecule trajectories of appropriate lengths (more than 100 frames), the diffusion constant (D) was slowed from D=1.3 μm^{2} s^{−1} (in TE buffer)^{17} to D=0.17 μm^{2} s^{−1} by adding glycerol to the imaging buffer. One hundred microlitres of glycerol was mixed with 78 μl of TE buffer and then degassed for 1 h. Then, 13 μl of an antioxidant cocktail (6 μl of 0.1 μM PCA, 6 μl of 1 μM PCD and 1 μl of 1 nM Trolox)^{34} was added directly before the imaging experiment. Then, 5 μl of the 0.2 μg ml^{−1} DNA solution was mixed with 5 μl of the imaging buffer to yield a final glycerol concentration of 25%. The solution was then sandwiched between a clean coverslip and a glass slide and sealed by a doublesided adhesive (0.12 mM, GraceBiolabs, Bend, OR, USA). The labelled lambda DNA was dissolved in TE buffer at a concentration of 0.1 μg ml^{−1}. The calculated D from the MSD plot was 0.29 μm^{2} s^{−1}.
Singlemolecule fluorescence imaging measurements
The singlemolecule fluorescence imaging experiments were conducted on a custombuilt epifluorescence microscopy setup^{17}. The setup is based on an inverted microscope (IX71, Olympus, Tokyo, Japan) illuminated with a CW 100 mW 532nm laser (Samba, Cobolt, Solna, Sweden) and a CW 60 mW 640nm laser (MLD, Cobolt). The 532nm and the 640nm lasers passed through FF01530/11 and LD01640/8 excitation filters, respectively (Semrock, Lake Forest, IL, USA). The lasers were introduced into the microscope through two 5 × beam expanders (Thorlab, NJ, USA) and then through a focusing lens (f=300 mm). The 532 and 640nm laser lines were reflected to a UAPON 100XO TIRF NA 1.49 oil immersion objective lens by Di01R532 and FF660Di02 dichroic mirrors (Semrock), respectively. By means of an acoustooptical tunable filter (AOTF; AA Optoelectronics), the output of the excitation lasers was synchronized to an iXon Ultra EMCCD camera (Andor Technology, Belfast, Ireland) to illuminate the sample only during image acquisition and thus to minimize photobleaching. After illuminating the sample with either the 532 or 640nm laser lines, the fluorescence from single molecules was collected by the same objective lens and then passed through FF01580/60 and LP02664RU emission filters (Semrock), respectively, before being introduced into the camera. All single molecule fluorescence images were recorded at a 0.16μm pixel size and at 156 Hz with a 6.4 ms exposure time.
Singlemolecule localization and tracking
Analysis of the singlemolecule images of the diffusion trajectories was performed using a versatile tracking algorithm (see Supplementary Note 1 for details). This algorithm splits away the poorly localized data points from the trajectory and thus it is applicable to studying the motion of macromolecules, such as the motion of chromosomes as well as the motion of small particles. A description of how we determined the position is in Supplementary Note 1. Ninetyeight singlemolecule movies of 304±214 frames each (mean±s.d.) (min=101 frames, max=1,097 frames) were imported into the Matlab software and then the 2D spatial positions were exported as text files.
Simulation of a singlemolecule random walk
The simulated trajectories were constructed using a routine written in Matlab starting at (x,y)=(0,0). The random step sizes were generated using a distribution function (R) expected from the normal diffusion theory of a Brownian particle:^{26}
where r and Δt denote the step size and the time lag, respectively. The step directions (angles between successive displacements) were generated based on random angles between 0 and 360°.
Probability of square lattice occupancy
We performed lattice occupancy analyses of the diffusion trajectories using a routine written in Matlab (Supplementary Data 1, Supplementary Note 2). The spatial positions obtained from the tracking algorithm were mapped onto a square lattice of sidelength (m), which equalled the pixel size of the camera (m=0.16 μm). The experimental probability was fitted to equation (4) to calculate the P_{25} value.
Detrended fluctuation analysis
For the DFA to be statistically robust, we joined the singlemolecule trajectories endtoend. The combined trajectories of both the DNA and the nanospheres have an approximate length of N=30,000 Δt. The DFA^{23} was performed using a routine written in Matlab (Supplementary Note 2). We generated integrated time series of P_{25} (P_{25} (k) (k=1,2,…, N)), Z(i) by subtracting the mean P_{25} value (〈P_{25}〉)5 and integrating the time series:
The profile Z (i) of length N was then divided into nonoverlapping segments (s) of equal size (l). The local trend in each segment (Z_{l}(i)) was calculated by subtracting the linear fit of the data:
where f_{s} (i) is the linear fit value in the sth segment. The root mean square variation (RMSV) for the segment size of l (RMSV(l)) was calculated by
We next calculated the power law exponent (the HE) that quantifies how fast the RMSV grows as the segment width increases^{23,35}. The HE were calculated by using logarithmically spaced segment sizes (l=50Δt − 5,000Δt). The width of the shortest detrending segment (50Δt=0.32 s) was set to be sufficiently longer than the time required by the DNA molecule to diffuse its radius of gyration (τ_{Rg}=0.05 s, Fig. 2c) so that the analysis would target the linear regime of Brownian motion (diffusive Brownian motion). The width of the longest detrending segment was set to 5,000Δt (32 s) because it approximately represents 1/6 of the full length of the experimental replicate (30,000Δt) and hence the data from six segments can be averaged to give a statistically valid RMSV value.
HE of the P_{25} temporal profiles recorded at different time lags (iΔt) (i=1,2,…, 10) (Fig. 3c) were obtained by calculating the RMSV at each iΔt with segment sizes of l=50Δt – (5,000/i)Δt. Cumulative HE (Fig. 3e) were obtained by subtracting the HE of a pure random walk (HE=0.5) from the calculated HE values at each time lag followed by integrating those subtracted values. Note that, in the case of a random walk, such as the walk of nanospheres, the calculated HE is larger than 0.5 due to the local broadening of the probability time series, which is caused by the width of the sliding window.
Meansquared displacement analysis
MSD was calculated by using the following expression:
where x_{i+n} and y_{i+n} describe the spatial positions after time interval Δt, given by the frame number, n, after starting at positions x_{i} and y_{i}. D is the diffusion constant. The theoretical MSDs at 1Δt (MSD1Δt) and at 10Δt (MSD10Δt) were calculated from the theoretical diffusion constant (D) by using equation (9). The theoretical D was calculated by using the experimental MSD value at 1Δt and 10Δt.
To generate a temporal profile of the absolute diffusion modes, we used the experimental replicates and the S_{r}A_{r} replicates to calculate the MSDΔt profile of a sliding window of width 50Δt. The MSDΔt profiles (time lags between 1Δt and 25Δt) were fitted to the normal (equation (9)), directed^{36} (equation (10)), and confined^{37} (equation (11)) diffusion models.
where v is the drift velocity,
where L is the side length of the confined area and σ_{xy} is the positional accuracy in x and y dimensions. The directed and confinedlike motions were quantified by v and L, respectively. Diffusion modes whose MSDtime profile fitted to a linear trend or those that showed extreme irregularities that could not be fitted by using equations (9)–(11), were considered with no specific diffusion mode. We set a sidelength limit of L=700 nm (equation (11)) to avoid mistakenly classifying the MSDΔt plots. This limit, which approximately equalled 4R_{g}, was empirically driven from the MSDtime profile and represented the maximum sidelength that could describe confinedlike motion of DNA within the 50Δt window. When the fitting operation of the MSDtime profile to equation (S6) retrieved L larger than 700 nm, we considered the chisquare of the linear (equation (9)) and the directed (equation (10)) motion profiles.
Dynamic time warping analysis
The DTW analysis was performed using a routine written in Matlab (Supplementary Note 2). The DTW algorithm^{20} was used to generate a timedependent motion profile illustrating the similarities and differences between the P_{25} and the MSD time series within a sliding window of width 50Δt. The pairwise Euclidean distance (d) from each data point (i=1,2,3,…, W) of the normalized MSD time series (Y) to all the points (i=1,2,3,…, W) of the normalized P_{25} time series (X) was calculated by
A matrix, M, of size (W, W) was constructed such that the value that the DTW algorithm recovered at position M (i_{w}, j_{w}) of the matrix M was the one with the minimum cumulated distance:
where w is an integer between 1 and W and it represents the corresponding position of i and j in matrix M. Dynamic programming was initialized such that the cumulated distance M (i_{w}, j_{w}) was recursively calculated based on both the minimum value from the previous cumulated distances [M (i_{w}−1, j_{w}−1), M (i_{w}−1, j_{w}), M (i_{w}, j_{w}−1)] and the distance recovered from the pair d (i_{w}, j_{w}):
The output of the DTW algorithm of each window, W, was a single value accumulated at M (W, W) suggesting how close both the MSD and the P_{25} time series are. When the normalized profile of the MSD time series is higher in magnitude than that of the P_{25} time series, the amplitude of local variation (ALV) is a positive value (equation (15)). Conversely, when the profile of the P_{25} time series is higher in magnitude, the ALV value is a negative value (equation (15)). The ALV value approaches zero as the two motion profiles become similar:
Conformational dynamics of DNA
The conformational relaxation time of ColE_{1} DNA (τ_{R}) was estimated by the cumulativearea tracking method^{17}. We analysed singlemolecule movies with signaltobackground ratios greater than 3 for at least 500 consecutive frames. The singlemolecule images were converted to binary images by using the tracking software. In the binary images, the pixels that are set to 1 identify the local maxima of the original images and hence they define the area occupied by the DNA molecule (A_{f}). The binary images were used to calculate the time dependentfluctuations of A_{f}. The area fluctuations of approximately 35 movies (500 frames each) were autocorrelated and then the autocorrelation plots were averaged into a single autocorrelation plot. To account for the fluctuations that could arise due to brief partial escape of the molecule from the field of view (defocusing fluctuations), we analysed the polymer nanospheres using the same analytical method. The characteristic time of the defocusing fluctuations of the nanospheres (τ_{n}) was estimated by fitting the averaged autocorrelation plot to a singleexponential decay using the following formula:
Because the depth of the field of view was the same for the DNA and the nanospheres, the characteristic time of the defocusing fluctuations of DNA (τ_{d}) could be approximated by calculating the time required by the DNA to have an MSD value equal to that of the nanospheres using the equation
where D^{DNA} and D^{Nsph} are the diffusion constants of the DNA and the nanospheres, respectively. τ_{R} was estimated by fitting the averaged autocorrelation plot to a doubleexponential decay using the following formula:
Data availability
The data that support the findings of this study are available from the corresponding authors on request.
Additional information
How to cite this article: Serag, M. F. & Habuchi, S. Conserved linear dynamics of singlemolecule brownian motion. Nat. Commun. 8, 15675 doi: 10.1038/ncomms15675 (2017).
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Acknowledgements
The research reported in this publication was supported by funding from King Abdullah University of Science and Technology (KAUST) and the KAUST Office of Sponsored Research (OSR) under Award No. CRF20152646CRG4. We would like to thank Matthijs van Waveren, Antonio M. Arena and Alain Clo of KAUST IT Research Computing and Amine El Helou of MathWorks Ltd for their precious help in speeding up the MATLAB analysis and for providing the KAUST high performance computing (HPC) Addon for the direct submission of the MATLAB script to the KAUST Noor computer clusters. We thank Virginia Unkefer and Lina Mynar for editing the manuscript.
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M.F.S developed the theory, conceived and reached the scientific target, and devised the analytical approach. M.F.S. and S.H. wrote the manuscript. S.H. supervised the study.
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Supplementary information
Supplementary Information
Supplementary Figures, Supplementary Notes, Supplementary Methods and Supplementary References (PDF 2622 kb)
Supplementary Movie 1
Time lapse movie of a single ColE1 DNA molecule diffusing in the imaging buffer. The frame rate is 6.4 ms. The blue circles show the spatial 2D positions of the molecule obtained by single molecule tracking. (MOV 2605 kb)
Supplementary Movie 2
Time lapse movie of a single nanosphere diffusing in the imaging buffer. The frame rate is 6.4 ms. The blue circles show the spatial 2D positions of the particle obtained by single molecule tracking. (MOV 5795 kb)
Supplementary Data 1
Stepbystep description of lattice occupancy analysis. Equations 1 and 3 are derived in this scheme. (TIFF 9235 kb)
Supplementary Data 2
Stepbystep description of detrended fluctuation analysis. The calculations of the Hurst exponents (HE) from the temporal profiles of P25 using equations 6, 7, and 8 are described in detail. (TIFF 3244 kb)
Supplementary Data 3
Normalization of P25 temporal profile. (TIFF 782 kb)
Supplementary Data 4
Normalization of temporal MSD profile. (TIFF 2350 kb)
Supplementary Data 5
Stepbystep description of dynamic time warping analysis. The calculations of the amplitude of local variations (ALV) from the temporal profiles of P25 and temporal MSD profiles using equations 12, 14, and 15 are described in detail. (TIFF 3548 kb)
Supplementary Data 6
Stepbystep description of the analysis of the submodes. The procedures of determining the stepsizes and MSD?t profiles of each submode are described in detail. (TIFF 1749 kb)
Supplementary Data 7
Stepbystep description of the manipulation of the temporal order of the stepsizes. The procedures of calculating the MSD?t profiles of the manipulated singlemolecule trajectories, in which the stepsizes of cLO or dHO submodes in the original trajectories are replaced by those of other submodes, are described in detail. (TIFF 2456 kb)
Supplementary Software
Software for lattice occupancy analysis. The analysis can be done by running the file Main.m. (ZIP 6 kb)
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Serag, M., Habuchi, S. Conserved linear dynamics of singlemolecule Brownian motion. Nat Commun 8, 15675 (2017). https://doi.org/10.1038/ncomms15675
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