File Name: brownian motion fluctuations dynamics and applications .zip
It is one of the four groundbreaking papers Einstein published in , in Annalen der Physik , in his miracle year. In , botanist Robert Brown used a microscope to look at dust grains floating in water.
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Our lattice occupancy analysis reveals unexpected sub-modes of motion of DNA that deviate from expected random motion in the linear, diffusive regime.
We demonstrate that a subtle interplay between these sub-modes 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 non-random dynamics that are detectable only by their relative motion.
Our analytical approach should advance broad understanding of diffusion processes of fundamental relevance. Brownian motion, as famously explained by Albert Einstein in , 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 molecular-scale context of Brownian motion occurs in three basic regimes. In the first regime before any molecular collisions, the particle shows a ballistic-like 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 mean-squared 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 , 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 , These dynamics, which complicate the hydrodynamic regime of DNA molecules, often cross over and partially affect their linear diffusive regime 12 , Consequently, at diffusion distances close to the radius of gyration, the motion of DNA molecules is non-linear 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 , 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 sub-modes of motion of DNA molecules that deviate from the expected random motion in the linear, diffusive regime.
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 , 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.
The experimental probability of lattice occupancy P t at time t is given by. The probability of occurrence of visits to new lattice sites at time t P t decreases as t increases. The s. The red line shows the fitting to equation 4.
According to one-dimensional 1D random diffusion theory, the probability distribution p of finding the particle at different lattice sites, q , is described by 18 , P t is expressed as.
P t obtained from a simulated 1D random diffusion trajectory agrees well with equation 3 Supplementary Fig. In two-dimensional 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.
P t 2D obtained from a simulated 2D random diffusion trajectory agrees well with equation 4 Fig. Based on this theoretical framework of molecular motion, we characterized the spatiotemporal pattern of the diffusive motion of DNA linear ColE 1 DNA ref.
Standard MSD analysis of the diffusion trajectories Fig. This regime arises due to the crossover of the hydrodynamic regime of DNA 10 shown in red in Fig.
On the other hand, the MSD of spherical polymer nanospheres exhibited a linear increase with time at all scales Supplementary Fig. This indicated the pure random walk of these nanospheres; we therefore used them as a control throughout the study Supplementary Fig.
The temporal profile of P 25 was then obtained by sliding the time window along the trajectory Fig. The time-dependent 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. The blue circles show the 2D positions of the molecule determined by the tracking.
Next, we analysed the diffusive dynamics of DNA based on the time-dependent 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 On the other hand, non-random motion occurring in the linear, diffusive regime causes time-scale-dependent 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 To provide statistically robust HE estimates, we joined 98 single-molecule tracks end-to-end and generated a long probability time series Fig.
Any systematic errors that could arise from the end-to-end connections were evaluated by calculating the HE of shuffled replicates by randomizing the order of the connections between the original trajectories Supplementary Fig. 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 results clearly demonstrate that the non-random motion of DNA in the linear, diffusion regime, which is not captured by MSD analysis, is revealed by lattice occupancy analysis. The same colour coding as in e is used.
These results indicate that the sub-diffusive behaviour of DNA observed in the hydrodynamic non-random regime Fig. Next, we investigated the origin of the hidden non-random 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. 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 , 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 non-random motion of DNA and the apparent deviations directed-like and confined-like modes from the random motion.
Indeed, lattice occupancy analyses of simulated trajectories displaying directed-like and confined-like 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. As a first step in distinguishing the apparent non-random diffusion caused by statistical variations intrinsic to Brownian motion and actual non-random 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 non-random diffusion and the temporal P 25 profile actual non-random diffusion. Comparison of the two temporal profiles identified the time instances at which the correlation between the P 25 and MSD profiles was positive directed-like motion with low lattice occupancy high P 25 mode d-LO sub-mode and confined-like motion with high lattice occupancy low P 25 mode c-HO sub-mode Fig.
However, the temporal profiles also revealed that they are not always positively correlated with each other Fig. Specifically, the temporal profiles showed that the confined-like motion was sometime correlated with the low lattice occupancy mode c-LO sub-mode and that the directed-like mode was sometime correlated with the high lattice occupancy mode d-HO sub-mode Fig. The green shadings highlight the d-LO and c-HO sub-modes. The red and blue shadings highlight the c-LO and d-HO sub-modes, respectively.
We then used the ALV values above or below the thresholds to distinguish between different sub-modes that exist in the experimental replicates step C in Fig. We next extracted the diffusion sub-trajectories corresponding to these sub-modes. 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 sub-modes exhibited non-random behaviours.
The green circles indicate that the amplitudes of the directed-like mode are higher than those of the relative mode. The red circles indicate that the amplitudes of the low occupancy mode are higher than those of the absolute mode. The green squares indicate that the amplitudes of the confined-like mode are lower than those of the relative mode. The blue circles indicate that the amplitudes of the high occupancy mode are lower than those of the absolute mode.
Figure 6a—c shows that the deviations between the two profiles are easily identified by calculating ALV using the DTW algorithm. While the distributions do not display any deviations from the 2D random diffusion theory 26 , the distributions obtained from the —ALV zones Fig. 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 mode-set, respectively Fig.
We did not observe this threshold dependency in the negative peaks detected in the ALV plots obtained from the nanospheres Fig.
These results further demonstrate that the sub-trajectories corresponding to the ALV peaks in the negative zones display non-random 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—c-LO and d-HO sub-modes for the first and second mode-sets, respectively , the results also demonstrate that lattice occupancy analysis can capture non-random diffusion modes.
The mean step-sizes were determined by fitting the step-size distributions to equation 5. Error bars in b , c correspond to the s. This result together with the ALV threshold-dependent step sizes Fig. Insets in a , b show examples of single-molecule sub-trajectories obtained from the —ALV zone of the first mode-set and the —ALV zone of the second mode-set, respectively.
The green shaded area in a , c , d highlights part of the MSD profiles whose time scale shows sub-diffusive behaviour. The yellow shaded area in c , d highlight part of the MSD profile whose time scale shows linear diffusive behaviour. This part of the profile shows either confined-like c or directed-like motion d compared with the theoretical profile. To investigate the effect of the temporal order of the step sizes on the non-random motion, we replaced the larger steps of the c-LO sub-mode Fig.
These findings further demonstrate that the non-random temporal order of the step sizes causes the non-random motion of the DNA and is consistent with c-LO and d-HO sub-modes.
We then examined whether or not the relative c-LO and d-HO sub-modes exist in other regimes of molecular motion.
Brownian motion , also called Brownian movement , any of various physical phenomena in which some quantity is constantly undergoing small, random fluctuations. It was named for the Scottish botanist Robert Brown , the first to study such fluctuations If a number of particles subject to Brownian motion are present in a given medium and there is no preferred direction for the random oscillations, then over a period of time the particles will tend to be spread evenly throughout the medium. The physical process in which a substance tends to spread steadily from regions of high concentration to regions of lower concentration is called diffusion. Diffusion can therefore be considered a macroscopic manifestation of Brownian motion on the microscopic level.
Understanding the fluctuations by which phenomenological evolution equations with thermodynamic structure can be enhanced is the key to a general framework of nonequilibrium statistical mechanics. These fluctuations provide an idealized representation of microscopic details. We consider fluctuation-enhanced equations associated with Markov processes and elaborate the general recipes for evaluating dynamic material properties, which characterize force-flux constitutive laws, by statistical mechanics. Markov processes with continuous trajectories are conveniently characterized by stochastic differential equations and lead to Green—Kubo-type formulas for dynamic material properties. Markov processes with discontinuous jumps include transitions over energy barriers with the rates calculated by Kramers. We describe a unified approach to Markovian fluctuations and demonstrate how the appropriate type of fluctuations continuous versus discontinuous is reflected in the mathematical structure of the phenomenological equations.
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