Single-molecule trajectory

The matter wave function is formed as a coherent superposition of states or a state ensemble, a wave packet. As the phase relationships change the wave packet moves, and spreads, not necessarily in only one direction the localized launch configuration disperses or propagates with the wave packet. The initially localized wave packets evolve like single-molecule trajectories. 

Excitations of molecules with femtosecond laser pulses lead to excited-state matter wave packets coherently, launching them with such well-defined spatial resolution and coherence in nuclear motions that they evolve like single-molecule trajectories. Both electronically excited and vibrationally excited ground-state species may be studied. The structural change versus time profile of a reaction turns out to be compatible with classical modes of thinking. 

In a classical Bohr orbit, the electron makes a complete journey in 0.15 fs. In reactions, the chemical transformation involves the separation of nuclei at velocities much slower than that of the electron. For a velocity 105 cm/s and a distance change of 10 8 cm (1 A), the time scale is 100 fs. This is a key concept in the ability of femtochemistry to expose the elementary motions as they actually occur. The classical picture has been verified by quantum calculations. Furthermore, as the deBroglie wavelength is on the atomic scale, we can speak of the coherent motion of a single-molecule trajectory and not of an ensemble-averaged phenomenon. Unlike kinetics, studies of dynamics require such coherence, a concept we have been involved with for some time. 

Fig. 24.2. Single-molecule recording of T4 lysozyme conformational motions and enzymatic reaction turnovers of hydrolysis of an E. coli B cell wall in real time, (a) This panel shows a pair of trajectories from a fluorescence donor tetramethyl-rhodamine blue) and acceptor Texas Red (red) pair in a single-T4 lysozyme in the presence of E. coli cells of 2.5mg/mL at pH 7.2 buffer. Anticorrelated fluctuation features are evident. (b) The correlation functions (C (t)) of donor ( A/a (0) Aid (f)), blue), acceptor ((A/a (0) A/a (t)), red), and donor-acceptor cross-correlation function ((A/d (0) A/d (t)), black), deduced from the single-molecule trajectories in (a). They are fitted with the same decay rate constant of 180 40s. A long decay component of 10 2s is also evident in each autocorrelation function. The first data point (not shown) of each correlation function contains the contribution from the measurement noise and fluctuations faster than the time resolution. The correlation functions are normalized, and the (A/a (0) A/a (t)) is presented with a shift on the y axis to enhance the view, (c) A pair of fluorescence trajectories from a donor (blue) and acceptor (red) pair in a T4 lysozyme protein without substrates present. The acceptor was photo-bleached at about 8.5 s. (d) The correlation functions (C(t)) of donor ((A/d (0) A/d (t)), blue), acceptor ((A/a (0) A/a (t)), red) derived from the trajectories in (c). The autocorrelation function only shows a spike at t = 0 and drops to zero at t > 0, which indicates that only uncorrelated measurement noise and fluctuation faster than the time resolution recorded (Adapted with permission from [12]. Copyright 2003 American Chemical Society)...

Figure 15. Ensemble average of the MSD as a function of time calculated from 531 single-molecule trajectories. From the linear fit of MSD against time the average 2D diffusion constant of D = 1.42 + 0.23 x 10 8cm2s I is determined for the POPE molecules in the POPC membrane at room temperature. (Adopted from [66].)...

Manley, S., Gillette, J. M., Patterson, G. H., Shroff, H., Hess, H. E, Betzig, E., and Lippincott-Schwartz, J. 2008. High-density mapping of single-molecule trajectories with photoactivated localization microscopy, Nat Methods 5,155-157. 

Figure 7.33 Single molecule trajectories showing balance between molar mass and long period. Note the presence of non-crystallizable chain branches in the right-hand case.

In a recently published single-molecule study (Habuchi etal, 2010), fluorophore-labeled cyclic poly(tetrahydrofuran) (3.8kg/mol) was mixed with unlabeled linear poly(THF) (3kg/mol) and examined at a semidilute concentration in toluene. Single-molecule trajectories were measured and diffusion coefficient distributions were constructed. Analysis of the distributions yielded two diffusion coefficients for cyclic poly(THF), 1.1 and 4.9 X 10 m /s, which were attributed to threaded and unthreaded rings, respectively. Comparison with the study discussed above using PFG NMR (Nam et al, 2009) reveals intriguing similarities low molecular weight cyclic poly ethers, diffusion coefficients of 10 m /s, ratio of threaded to unthreaded diffusion coefficients from 3 (NMR) to 4.5... 

Figure Cl.5.8. Spectral jumping of a single molecule of terrylene in polyethylene at 1.5 K. The upper trace displays fluorescence excitation spectra of tire same single molecule taken over two different 20 s time intervals, showing tire same molecule absorbing at two distinctly different frequencies. The lower panel plots tire peak frequency in tire fluorescence excitation spectmm as a function of time over a 40 min trajectory. The molecule undergoes discrete jumps among four (briefly five) different resonant frequencies during tliis time period. Arrows represent scans during which tire molecule had jumped entirely outside tire 10 GHz scan window. Adapted from...

Figure Cl.5.15. Molecular orientational trajectories of five single molecules. Each step in tire trajectory is separated by 300 ms and is obtained from tire fit to tire dipole emission pattern such as is shown in figure Cl.5.14. The radial component is displayed as sin 0 and tire angular variable as (ji. The lighter dots around tire average orientation represent 1 standard deviation. Reprinted witli pennission from Bartko and Dickson 11481. Copyright 1999 American Chemical Society.

Weston, K D., Dyck, M., Tinnefeld, P., Muller, C., Herten, D. P. and Sauer, M. (2002) Measuring the number of independent emitters in single-molecule fluorescence images and trajectories using coincident photons. Anal. Chem., 74, 5342-5347. 

Direct observation of molecular diffusion is the most powerful approach to evaluate the bilayer fluidity and molecular diffusivity. Recent advances in optics and CCD devices enable us to detect and track the diffusive motion of a single molecule with an optical microscope. Usually, a fluorescent dye, gold nanoparticle, or fluorescent microsphere is used to label the target molecule in order to visualize it in the microscope [31-33]. By tracking the diffusive motion of the labeled-molecule in an artificial lipid bilayer, random Brownian motion was clearly observed (Figure 13.3) [31]. As already mentioned, the artificial lipid bilayer can be treated as a two-dimensional fluid. Thus, an analysis for a two-dimensional random walk can be applied. Each trajectory observed on the microscope is then numerically analyzed by a simple relationship between the displacement, r, and time interval, T,... 

Systems of chemical interest typically contain particles in molar quantity. Mathematical modelling of all interactions in such a system is clearly impossible. Even in a system of non-interacting particles, such as an ideal gas, it would be equally impossible to keep track of all individual trajectories. It is consequently not a trivial matter to extend the mechanical description (either classical or non-classical) of single molecules to macrosystems. It would be required in the first place to define the initial state of each particle in terms of an equation... 

Figure 18-15 Tracks of two molecules of 20 pM rhodamine 6G in silica gel observed by fluorescence integrated over 0.20-s periods at 0.78-s intervals. Some points are not connected, because the molecule disappeared above or below the focal plane in the 0.45-ixnrvthick film and was not observed in a particular observation interval. In the nine periods when molecule A was in one location, it might have been adsorbed to a particle of silica. An individual molecule emits thousands of photons in 0.2 s as the molecule cycles between ground and excited states. Only a fraction of these photons reaches the detector, which generates a burst of —10-50 electrons. [From K. s. McCain, D. C. Hanley, andJ. M. Harris. "Single-Molecule Fluorescence Trajectories tor Investigating Molecular Transport in Thin Silica Sol-Gel Films," Anal.

State, orientation, wave packet coherence Uncertainty principle and coherence Single-molecule, not ensemble, trajectory Dynamics, not kinetics Complex systems, robustness of phenomena... 

McKinney, S. A., Joo, C., and Ha, T. (2006). Analysis of single-molecule FRET trajectories using hidden Markov modeling. Biophys. J. 91, 1941—1951. 

In all of these computations, there is a dense manifold of excited states present [83], Thus the computations are sensitive to dynamic electron correlation and the details of the reaction coordinates involved. In the cytosine-guanine base pair simulations, trajectory calculations proved to be necessary to determine the extent of the conical intersection that is actually accessible. Subsequent improvements in the level of theory used for the static calculation of single molecules will be possible, but these should be balanced against a more realistic treatment of vibrational kinetic energy and environmental effects (solvent/protein). 

Two-dimensional trajectories of a typical gas molecule and a typical aerosol particle can be compared in Fig. 9.2. The molecule shows sharp changes in direction, each change occurring when it strikes another molecule. As discussed in Chap. 3, the average distance between hits is defined as the mean free path of the molecule. For the particle, a hit by a single molecule does not appreciably affect its motion. Therefore, its path is not characterized by sharp changes in direction, but by smooth curves representing the combined effect of hits by many molecules. 

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