Study of Quantitative and Dynamics Polymer Particle Tracking

The theory of polymer dynamics is one of the most elegant and cogent subjects in macromolecular science (Doi and Edwards, 1986). The fundamental difference between the physics and chemistry of a small molecule and that of a macromolecule is that the latter can adopt a vast number of approximately equally probable conformations. Any individual polymer molecule within a population constantly fluctuates among these conformations; the populations must be characterized and analyzed in terms of a distribution. In the past, almost all experimental measurements on polymers, either thermodynamic or hydrodynamic, yielded averaged behavior of the whole population distribution (Flory, 1969; Doi and Edwards, 1986). These measurements have validated the statistical theory of polymer conformations and have provided much important information about a wide range of both synthetic and biological polymers. They do not directly reveal, however, the conformational dynamics of individual polymer molecules. For biological polymers like DNA the rates of conformational fluctuations can be of great significance in biochemical processes such as the binding of Lac repressor protein to its operon for gene expression (Finzi and Gelles, 1995; Shore et al., 1981; Horowitz and Wang, 1984). With the technological advancement in micromanipula- tion and video-enhanced optical microscopy, one now can study single DNA molecules in aqueous solution (Smith et al., 1992). Large spontaneous dynamic fluctuations of indi-vidual macromolecules can be observed by optical micros-copy. In a typical experiment, one end of the polymer molecule is anchored to a substratum. A microsphere bound to the other end as an optical marker is tracked using video-enhanced optical microscopy with a nanometer precision (Gelles et al., 1988). This approach has recently been applied to measure the kinetics of DNA looping induced by Lac repressor protein. The spatial range of the spontaneous conformation fluctuations of the tethered DNA molecule indicated whether its effective length had been shortened by the formation of an interior loop (Finzi and Gelles, 1995). Although there was no attempt to study the intrinsic properties of the DNA molecule itself in these experiments, this method of tethered particle motion (TPM) (Yin et al., 1994) can also yield much information about the equilibrium and dynamics of conformational fluctuations of the tethering DNA molecule and of other flexible polymer tethers. The method belongs to a class of optical methods based on particle tracking (Gelles et al., 1988; Qian et al., 1991; Mason et al., 1997; Gittes et al., 1997). In this paper we propose the use of single-particle tracking (SPT) to study polymer conformational dynamics at the level of the indi-vidual molecule. This novel approach provides direct ob-servations of conformational fluctuations of individual mac- romolecules under relatively nonperturbing conditions, complementary to recent studies of single DNA molecules under extensional force (Smith et al., 1992, 1996; Perkins et al., 1994, 1995; Cluzel et al., 1996; Strick et al., 1996). The fundamental difference between traditional macro-scopic studies and the recent single-molecule measurements is that the thermal fluctuations are an intrinsic and essential contribution rather than an undesirable source of noise (Elson and Webb, 1975; Elson and Qian, 1994; Shapiro and Qian, 1997, 1998), reminiscent of the pioneering studies on single channel proteins in membranes that have revolutionized physiology (Neherand Stevens, 1977). In this paper we develop a quantitative basis for interpreting measurements of the stochastic behavior of single macromolecular tethers (e.g., DNA) based on the theory of polymer dynamics (Rouse, 1953; Zimm, 1956; Doi and Edwards, 1986). We anticipate that the quantitative analysis of fluctuating polymers will become a powerful tool for studying the relationship between structures and dynamical properties of flexible polymers as well as biological macromolecules. Measurements of the transient relaxation of a linear ther- modynamic system after a perturbation yield information identical to that obtained from measurements of spontane-ous fluctuations (Keizer, 1987). For a nonlinear system, e.g., a random coil polymer at large extension, the stretching experiments are dominated by the effects of the nonlinear- ity. The fluctuation measurements, however, probe the linear elasticity of the system. A fluctuation measurement requires many measurements over time (i.e., the measurements of many fluctuations) to achieve statistical accuracy. The time resolution of the kinetics is not determined, however, by the duration of the measurement. A longer measurement time yields a higher accuracy, by averaging over both random measurement errors and variations due to the stochastic character of the spontaneous fluctuations. Similarly, averaging over many transients in a relaxation experiment improves the accuracy. Finally, a measurement in which a single polymer is extended by external force selectively probes the modes with fast relaxation and less thermal fluctuations, whereas an equilibrium fluctuation measure-ment probes the modes equally with fast and slow relax-ations (equipartition). To apply the theory of polymer dynamics to SPT, one has to take explicit account of the microsphere (optical marker). This requires a reformulation of the established theory (Doi and Edwards, 1986) and leads to a different mathematical problem with a transcendental eigen equation. To clearly illustrate the theoretical development, we present analyses for both free-draining polymer (Rouse, 1953) and polymer with hydrodynamic interaction (Zimm, 1956). The former is conceptually simple but applicable only to short polymer molecules, whereas the latter is more appropriate for long polymers. The paper is organized as follows. In the next section we describe a simple quantitative method for analyzing conformational fluctuations of flexible macromole- cules such as DNA. The third section presents an analysis of the dependence of the observed polymer dynamics on the presence of the tethered particle and the length of the tether. The fourth section provides an example of the application of the analytical method to experimental measurements. The last section discusses the significance of this approach to the study of biochemical processes and polymer dynamics.

In SPT measurements the position of the microsphere, tethered by a flexible polymer molecule to the substratum, fluctuates over time. The microsphere position can be sim-ply defined as the center of its optical image. Its distance to the site of anchorage to the substratum defines the end-to- end distance (Flory, 1969; Doi and Edwards, 1986) of the tether. A histogram and a time correlation function (Elson and Webb, 1975; Qian et al., 1991) of the microsphere position respectively characterize statistically the equilibrium and the dynamic properties of the tether (Fig. 1). Our first task is to interpret the measured fluctuations of the position of the tethered particle in terms of the structural and dynamic properties of the polymer molecule. The simplest analysis is based on the phenomenological dumbbell model for polymer dynamics (Bird et al., 1987), according to which the particle is assumed to be tethered to the substratum by a simple linear spring with effective spring constant kdb. This spring constant is determined both by the flexibility and by the length of the polymer tether, as discussed below. The viscous resistance to the motion of the micro- sphere defines an overall relaxation time constantThe time constant characterizes the rate of motion of the har-monically bound particle in its viscous environment. The equilibrium distribution of the positions of the mi- crosphere should be independent of both its frictional coef-ficient and size if we neglect its excluded volume effect with both the substratum and the tether itself. Therefore, under these assumptions the SPT measurements should faithfully yield the distribution of polymer end-to-end dis- tancesThe dumbbell model characterizes the dynamics FIGURE 1 A schematic illustration of the analysis of the fluctuating position of the optical marker, in terms of a distribution or histogram and an autocorrelation function of the position measurements. The position fluctuation data are in arbitrary length units. The distribution gives the mean and variance of the fluctuating position. The autocorrelation function yields a correlation timeand its amplitude is the same as the variance. For a sequence of data the autocorrelation function is obtained as where m usually is much smaller than N. of the polymer tether in terms of a (Langevin) equation of motion (Bird et al., 1987) for the end-to-end vector r in a viscous solvent: where we neglect the inertial term and fdb(t) is a fluctuating force due to the thermal motions of the solvent molecules. The subscript db specifies the dumbbell model. In effect, the stochastic differential equation (Eq. 1) describes the diffusion of the microsphere in the harmonic potential to which it is bound and so is mathematically equivalent to a partial differential equation in the form of a diffusion equation for the probability distribution P(r, t) = P(x, t)P(y, t)P(z, t) (Wax, 1954, Van Kampen, 1992): where D is a diffusion coefficient; and ( • ) denotes an ensemble average. The equilibrium solution of Eq. 2 is a Gaussian distribution (Peq(r)) for the end-to-end distance with variance Thus far the relationship between the stochastic differential equation (Eq. 1) and the partial differential equation (Eq. 2) is strictly mathematical (Van Kampen, 1992). We can further determine the relationship between and D from thermal physics. At thermal equilibrium, the mean square end-to-end distance, i.e., the variance of the distribution of end-to-end distances, is where kB is Boltzmann's constant and T is the temperature in Kelvins (Wax, 1954; Van Kampen, 1992). Hence we have D = which is the well-known Einstein formula, a special case of the general fluctuation-dissipation relation for equilibrium thermal fluctuations (Keizer, 1987; Klapper and Qian, 1998). Therefore, the equilibrium distribution for the end-to-end distance r is a simple Boltzmann distribution: and For the microsphere diffusing in the harmonic potential well, we can determine analytically from Eq. 2 not only the equilibrium distribution Peq(r) but also the temporal behav- ior from the fundamental solution (Green's function) which yields the probability that, if the microsphere is at position r' at some reference time, it is at position r after an elapsed time interval t. The measured time-dependent fluctuations in r are most readily characterized statistically, in terms of the time correlation function (Elson and Webb, 1975; Qian et al., 1991): where the conformational relaxation time of the tethering polymer molecule. At any moment r is likely to deviate from its equilibrium value. The time required to relax to the equilibrium value is characterized bywhich depends on the elasticity (kdb) of the polymer segment and the overall viscous resistance to motionof the polymer and the microsphere through the solvent. From the time record of the measured coordinates of the microsphere, the correlation function in Eq. 4 can be computed to yield the relaxation time(see more details in Fig. 1). For a weakly bending polymer molecule, the length over which it can substantially bend is called a persistence length Molecules for which the contour length is much greater than the persistence length can be considered as random coils. The dumbbell model can be applied to microspheres tethered either by stiff polymers with less than one persistence length or by random coils with many persistence lengths, but the energetics of the elastic restoring force are different for the two types of tethers. Contributing to the force constant kdb in Eq. 3 are both entropic forces originating from the distribution of the polymer over its random coil conformations and enthalpic forces that resist bending over a range of distances that are small compared to the polymer persistence or segment length. Hence there should be contributions to kdb that are both proportional to and independent of temperature T: For a polymer with many persistence lengths (elastic dumbbell), where L is the contour length of the polymer (Flory, 1969). Hence, kdb varies inversely with the length of the tether, L, i.e., as 1/L. If, however, the polymer is short (rigid dumbbell, less than one persistence length), then the backbone bending (the a term) dominates kdb. For a stiff tether the kdb due to transverse bending varies as 1/L3 (Barkley and Zimm, 1979). The temperature dependence of kdb (i.e., Eq. 5) provides an informative indicator of the source of the flexibility and elasticity of the tethering polymer, and a way to distinguish whether the molecule is behaving as a flexible random coil or a weakly bending beam and whether the bending is static or dynamic (Schell- man and Harvey, 1995). Helical DNA behaves as a weakly bending beam for lengths shorter than 150 bp and as a wormlike random coil for much greater lengths. Hence, in SPT experiments on DNA, it is useful to carry out measurements of conformational dynamics as a function of temperature to characterize the energetics of the elastic restoring force and to determine the nature of the conformational fluctuations being observed.

To illustrate the principles underlying dynamic SPT mea-surements, we have discussed only the simplest models for macromolecular conformations and dynamics. Even at this level, it is clear that much interesting information about the dynamics of motion of the tether can be obtained using this approach. More realistic models would consider the excluded volume and hydrodynamic interactions of the tethered particle and the polymer segments, the detailed relationship between the structure and dynamic properties of the tether, the influence of the substratum, which limits the polymer conformations, and time averaging effects on the actual measurements. These subjects will be carefully investigated in future work. Let us consider whether the time resolution of present SPT technology is sufficient to study the conformational dynamics of single DNA molecules. Analysis of dynamic light scattering measurements on DNA in terms of Rouse- Zimm theory has yielded the frictional coefficient N s/m and spring constant k3 X 10~7 N/m (Lin and Schurr, 1978). On the other hand, according to Stokes' law, the frictional coefficientfor a spherical particle with diameter d is 3rn

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