A mathematical analysis for the Brownian dynamics of a DNA tether

Statistical physics and mesoscopic modeling to interpret tethered particle motion experiments

Tethered particle motion experiments are versatile single-molecule techniques enabling one to address in vitro the molecular properties of DNA and its interactions with various partners involved in genetic regulations. These techniques provide raw data such as the tracked particle amplitude of movement, from which relevant information about DNA conformations or states must be recovered. Solving this inverse problem appeals to specific theoretical tools that have been designed in the two last decades, together with the data pre-processing procedures that ought to be implemented to avoid biases inherent to these experimental techniques. These statistical tools and models are reviewed in this paper.

Translation factor mRNA granules direct protein synthetic capacity to regions of polarized growth

mRNA localization serves key functions in localized protein production, making it critical that the translation machinery itself is present at these locations. Here we show that translation factor mRNAs are localized to distinct granules within yeast cells. In contrast to many messenger RNP granules, such as processing bodies and stress granules, which contain translationally repressed mRNAs, these granules harbor translated mRNAs under active growth conditions. The granules require Pab1p for their integrity and are inherited by developing daughter cells in a She2p/She3p-dependent manner. These results point to a model where roughly half the mRNA for certain translation factors is specifically directed in granules or translation factories toward the tip of the developing daughter cell, where protein synthesis is most heavily required, which has particular implications for filamentous forms of growth. Such a feedforward mechanism would ensure adequate provision of the translation machinery where it is to be needed most over the coming growth cycle.

Statistics and Related Topics in Single-Molecule Biophysics

Since the universal acceptance of atoms and molecules as the fundamental constituents of matter in the early twentieth century, molecular physics, chemistry and molecular biology have all experienced major theoretical breakthroughs. To be able to actually "see" biological macromolecules, one at a time in action, one has to wait until the 1970s. Since then the field of single-molecule biophysics has witnessed extensive growth both in experiments and theory. A distinct feature of single-molecule biophysics is that the motions and interactions of molecules and the transformation of molecular species are necessarily described in the language of stochastic processes, whether one investigates equilibrium or nonequilibrium living behavior. For laboratory measurements following a biological process, if it is sampled over time on individual participating molecules, then the analysis of experimental data naturally calls for the inference of stochastic processes. The theoretical and experimental developments of single-molecule biophysics thus present interesting questions and unique opportunity for applied statisticians and probabilists. In this article, we review some important statistical developments in connection to single-molecule biophysics, emphasizing the application of stochastic-process theory and the statistical questions arising from modeling and analyzing experimental data.

Single-molecule approaches to probe the structure, kinetics, and thermodynamics of nucleoprotein complexes that regulate transcription

Single-molecule experimentation has contributed significantly to our understanding of the mechanics of nucleoprotein complexes that regulate epigenetic switches. In this minireview, we will discuss the application of the tethered-particle motion technique, magnetic tweezers, and atomic force microscopy to (i) directly visualize and thermodynamically characterize DNA loops induced by the lac, gal, and lambda repressors and (ii) understand the mechanistic role of DNA-supercoiling and DNA-bending cofactors in both prokaryotic and eukaryotic systems.

Quantitative analysis of DNA-looping kinetics from tethered particle motion experiments

In this chapter we show the application of a maximum-likelihood-based method to the reconstruction of DNA-looping single-molecule time traces from tethered particle motion experiments. The method does not require time filtering of the data and improves the time resolution by an order of magnitude with respect to the threshold-crossing approach. Moreover, it is not based on presumed kinetic models, overcoming the limitations of other approaches proposed previously, and allowing its applications to mechanisms with complex kinetic schemes. Numerical simulations have been used to test the performances of this analysis over a wide range of time scales. We have then applied this method to determine the looping kinetics of a well-known DNA-looping protein, the lambda-repressor.

Studying RecBCD helicase translocation along Chi-DNA using tethered particle motion with a stretching force

Escherichia coli RecBCD helicase unwinds blunt-end duplex DNA to repair damaged DNA molecules in the homologous recombination pathway. Previous single-molecule experiments showed that RecBCD recognizes an 8 nt DNA sequence, chi, and lowers its unwinding rate afterward under saturating ATP condition. We have developed a single-molecule force-tethered particle motion (FTPM) method, which is modified from the conventional TPM method, and applied it to study RecBCD motion in detail. In the FTPM experiment, a stretching force is applied to the DNA-bead complex that suppresses the bead's Brownian motion, resulting in an improved spatial resolution at long DNA substrates. Based on the equipartition theorem, the mean-square displacement of the bead's Brownian motion measured by FTPM correlates linearly to DNA extension length with a predicted slope, circumventing the difficulties of conventional TPM experiments, such as nonlinearity and low resolution of long DNA substrates. The FTPM method offers the best resolution in the presence of only a small stretching force (0.20 pN). We used the FTPM method to investigate RecBCD helicase motion along 4.1 kb long chi-containing duplex DNA molecules, and observed that the translocation rate of RecBCD changes after the chi sequence under limited ATP concentrations. This suggests that chi recognition by RecBCD does not require saturating ATP conditions, contrary to what was previously reported.

Brownian particles in transient polymer networks

We discuss the thermal motion of colloidal particles in transient polymer networks. For particles that are physically bound to the surrounding chains, light-scattering experiments reveal that the submillisecond dynamics changes from diffusive to Rouse-like upon crossing the network formation threshold. Particles that are not bound do not show such a transition. At longer time scales the mean-square displacement (MSD) exhibits a caging plateau and, ultimately, a slow diffusive motion. The slow diffusion at longer time scales can be related to the macroscopic viscosity of the polymer solutions. Expressions that relate the caging plateau to the macroscopic network elasticity are found to fail for the cases presented here. The typical Rouse scaling of the MSD with the square root of time, as found in experiments at short time scales, is explained by developing a bead-spring model of a large colloidal particle connected to several polymer chains. The resulting analytical expressions for the MSD of the colloidal particle are shown to be consistent with experimental findings.

Real-time observation of DNA looping dynamics of Type IIE restriction enzymes NaeI and NarI

Many restriction enzymes require binding of two copies of a recognition sequence for DNA cleavage, thereby introducing a loop in the DNA. We investigated looping dynamics of Type IIE restriction enzymes NaeI and NarI by tracking the Brownian motion of single tethered DNA molecules. DNA containing two endonuclease recognition sites spaced a few 100 bp apart connect small polystyrene beads to a glass surface. The position of a bead is tracked through video microscopy. Protein-mediated looping and unlooping is then observed as a sudden specific change in Brownian motion of the bead. With this method we are able to directly follow DNA looping kinetics of single protein–DNA complexes to obtain loop stability and loop formation times. We show that, in the absence of divalent cations, NaeI induces DNA loops of specific size. In contrast, under these conditions NarI mainly creates non-specific loops, resulting in effective DNA compaction for higher enzyme concentrations. Addition of Ca²⁺ increases the NaeI-DNA loop lifetime by two orders of magnitude and stimulates specific binding by NarI. Finally, for both enzymes we observe exponentially distributed loop formation times, indicating that looping is dominated by (re)binding the second recognition site.

An asymptotic comparative analysis of the thermodynamics of non-covalent association

There is an ambiguity in the theoretical models for computing association constants, the key observable in a laboratory, of non-covalent associations. We show that three different models give unique result asymptotically in the limit of strong associate. For weak associations, the disagreement reflects the nature of ill-defined "associated complex'' which can be defined, among various ways, either geometrically or thermodynamically depending on measurement techniques. Furthermore, even when the free energy of association is unique, the corresponding entropy and enthalpy can still be different from different types of measurements--a surprising source of entropy-enthalpy compensation. This work provides a mathematical basis for modeling non-covalent association processes in biology.

Changes in DNA flexibility after irradiation with gamma rays and neutrons studied with the perturbed angular correlation method

Neutron and gamma irradiation of buffered solutions of calf thymus DNA resulted in changes in the dynamics of the macromolecule. In the low-dose region (0.8-10 cGy of 239Pu-Be neutrons and 0.34-3 Gy of 60Co gamma rays), the flexibility of DNA decreased as indicated by slower rotation of the molecules. Neutrons appeared to be approximately 35 times more effective than 60Co gamma rays. The rotational correlation time, tau C, was measured using the perturbed angular correlation (PAC) method. Its variation appears to follow a linear-exponential behavior. An attempt is made to formulate this behavior as a function of the energy deposited on the macromolecule (radiation dose), the average threshold energy (dose) required to form new lesions, and the available population of intact DNA sites.

Relative entropy: free energy associated with equilibrium fluctuations and nonequilibrium deviations

Using a one-dimensional macromolecule in aqueous solution as an illustration, we demonstrate that the relative entropy from information theory, Sigma(k)p(k) ln(p(k)/p(*)(k)), has a natural role in the energetics of equilibrium and nonequilibrium conformational fluctuations of the single molecule. It is identified as the free energy difference associated with a fluctuating density in equilibrium, and is associated with the distribution deviate from the equilibrium in nonequilibrium relaxation. This result can be generalized to any other isothermal macromolecular system using the mathematical theories of large deviations and Markov processes, and at the same time provides the well-known mathematical results with interesting physical interpretations.