The conformation and dynamics of circular polymers is a subject of considerable theoretical and experimental interest. DNA is an important example because it occurs naturally in different topological states, including linear, relaxed circular, and supercoiled circular forms. A fundamental question is how the diffusion coefficients of isolated polymers scale with molecular length and how they vary for different topologies. Here, diffusion coefficients D for relaxed circular, supercoiled, and linear DNA molecules of length L ranging from Ϸ6 to 290 kbp were measured by tracking the Brownian motion of single molecules. A topology-independent scaling law D ϳ L ؊ was observed with L ‫؍‬ 0.571 ؎ 0.014, C ‫؍‬ 0.589 ؎ 0.018, and S ‫؍‬ 0.571 ؎ 0.057 for linear, relaxed circular, and supercoiled DNA, respectively, in good agreement with the scaling exponent of Х 0.588 predicted by renormalization group theory for polymers with significant excluded volume interactions. Our findings thus provide evidence in support of several theories that predict an effective diameter of DNA much greater than the Debye screening length. In addition, the measured ratio DCircular͞ DLinear ‫؍‬ 1.32 ؎ 0.014 was closer to the value of 1.45 predicted by using renormalization group theory than the value of 1.18 predicted by classical Kirkwood hydrodynamic theory and agreed well with a value of 1.31 predicted when incorporating a recently proposed expression for the radius of gyration of circular polymers into the Zimm model. circular ͉ polymer ͉ polyelectrolyte ͉ hydrodynamics ͉ excluded volume W hile many eukaryotic genomes are linear, prokaryotic genomes and most cloned DNA constructs are circular (1). Indeed, a commonly stated motivation for theoretical calculations on circular polymers is that they may be applicable to understanding the behavior of DNA. However, four of the five previously reported studies on the diffusion of circular polymers have used synthetic polymers, and only two of these, both using synthetic polymers, examined the dependence of the diffusion coefficient on molecular length. The dependence of D on length for relaxed circular DNA has never been measured. Here, we examine linear, relaxed circular, and supercoiled DNA molecules covering a wide range of lengths (Ϸ6 to 290 kbp).For long linear polymers in a good solvent, where excluded volume effects are appreciable, polymer physics theory (2) predicts D ϳ 1͞R G ϳ L Ϫ with Х 0.588, where R G is radius of gyration. The same scaling exponent has been calculated for both dynamic (D ϳ L Ϫ ) and static (R G ϳ L ) quantities, with nearly identical results determined by using a wide range of methods. Static scaling has been examined by using Monte Carlo simulations (3), bead-rod simulations (4), and cylindrical selfavoiding polygon models (5). Renormalization group theory methods have been used in both static (3, 6) and dynamic (7) calculations, and bond-fluctuation simulations (8) were used to measure and compare both scaling relationships. The predicted scaling exponent is close ...

Molecular motors drive genome packaging into preformed procapsids in many dsDNA viruses. Here, we present optical tweezers measurements of single DNA molecule packaging in bacteriophage λ. DNA-gpA-gpNu1 complexes were assembled with recombinant gpA and gpNu1 proteins and tethered to microspheres, and procapsids were attached to separate microspheres. DNA binding and initiation of packaging were observed within a few seconds of bringing these microspheres into proximity in the presence of ATP. The motor was observed to generate greater than 50 picoNewtons (pN) of force, in the same range as observed with bacteriophage ϕ29, suggesting that high force generation is a common property of viral packaging motors. However, at low capsid filling the packaging rate averaged ~600 bp/s, which is 3.5-fold higher than ϕ29, and the motor processivity was also 3-fold higher, with less than one slip per genome length translocated. The packaging rate slowed significantly with increasing capsid filling, indicating a buildup of internal force reaching 14 pN at 86% packaging, in good agreement with the force driving DNA ejection measured in osmotic pressure experiments and calculated theoretically. Taken together, these experiments show that the internal force that builds during packaging is largely available to drive subsequent DNA ejection. In addition, we observed an 80 bp/s dip in the average packaging rate at 30% packaging, suggesting that procapsid expansion occurs at this point following the buildup of an average of 4 pN of internal force. In experiments with a DNA construct longer than the wild-type genome, a sudden acceleration in packaging rate was observed above 90% packaging in many cases, and greater than 100% of the genome length was translocated, suggesting that internal force can rupture the immature procapsid.

Oligomerization of the HIV-1 protein Rev on the Rev Response Element (RRE) regulates nuclear export of genomic viral RNA and partially spliced viral mRNAs encoding for structural proteins. Single-molecule fluorescence spectroscopy has been used to dissect the multistep assembly pathway of this essential ribonucleoprotein, revealing dynamic intermediates and the mechanism of assembly. Assembly is initiated by binding of Rev to a high-affinity site in stem-loop IIB of the RRE and proceeds rapidly by addition of single Rev monomers, facilitated by cooperative Rev-Rev interactions on the RRE. Dwell-time analysis of fluorescence trajectories recorded during individual Rev-RRE assembly reactions has revealed the microscopic rate constants for several of the Rev monomer binding and dissociation steps. The high-affinity binding of multiple Rev monomers to the RRE is achieved on a much faster timescale than reported in previous bulk kinetic studies of Rev-RRE association, indicating that oligomerization is an early step in complex assembly.ribonucleoprotein assembly ͉ single-molecule fluorescence spectroscopy ͉ viral RNA trafficking R ev, a key regulatory protein of HIV-1, activates nuclear export of unspliced and partially spliced viral mRNAs, encoding genomic RNA and the structural proteins Gag, Pol, and Env, respectively (reviewed in ref. 1). Rev binds to a highly conserved region of the viral mRNA known as the Rev Response Element (RRE). The RRE contains a single high-affinity binding site for Rev, although as many as 8 Rev molecules can bind to a single RNA (2-4). In fact, binding of a single Rev molecule to the RRE is incapable of activating mRNA export, indicating that oligomerization of Rev on the RRE is required for Rev function (5, 6). Because Rev-mediated RNA export is essential for viral replication, the Rev-RRE complex is a potential therapeutic target for treatment of HIV/AIDS, although effective inhibitors of Rev function have yet to be developed.Despite the important role of the Rev-RRE ribonucleoprotein (RNP) in the HIV-1 life cycle, the mechanism of oligomeric complex assembly is not well understood. Previous studies have suggested that a single Rev monomer initially binds to the high-affinity site, after which additional monomers assemble on adjacent regions of the RRE (3, 6, 7). Alternatively, preformed oligomers of Rev might bind directly to the RRE (8, 9), because Rev is known to self-associate in the absence of RNA, with an association constant of 1 ϫ 10 6 M Ϫ1 (10). To date, it has not been possible to distinguish these assembly models, because all previous studies have been performed under ensemble-averaged conditions at relatively high protein concentrations. Here, we use single-molecule fluorescence spectroscopy to observe individual steps of RNP assembly. Our results clearly demonstrate that multiple Rev monomers bind sequentially to the RRE and they establish a detailed kinetic framework for the early steps of Rev-RRE assembly. Moreover, the method described here could be used to visualize th...

When long polymers such as DNA are in a highly concentrated state they may become entangled, leading to restricted self-diffusion. Here, we investigate the effect of molecular topology on diffusion in concentrated DNA solutions and find surprisingly large effects, even with molecules of modest length and concentration. We measured the diffusion coefficients of linear and relaxed circular molecules by tracking the Brownian motion of single molecules with fluorescence microscopy. Four possible cases were compared: linear molecules surrounded by linear molecules, circular molecules surrounded by linear molecules, linear molecules surrounded by circles, and circles surrounded by circles. In measurements with 45-kbp DNA at 1 mg/ml, we found that circles diffused Ϸ100 times slower when surrounded by linear molecules than when surrounded by circles. In contrast, linear and circular molecules diffused at nearly the same rate when surrounded by circles, and circles diffused Ϸ10 times slower than linears when surrounded by linears. Thus, diffusion in entangled DNA solutions strongly depends on topology of both the diffusing molecule and the surrounding molecules. This effect also strongly depends on DNA concentration and length. The differences largely disappeared when the concentration was lowered to 0.1 mg/ml or when the DNA length was lowered to 6 kb. Present theories cannot fully explain these effects.polymers ͉ reptation A mong polymers DNA is rather unique in that it is naturally found in a number of different topological forms, including linear, supercoiled circular, relaxed circular, knotted circular, and branched. DNA solutions handled in vitro in molecular biology research are often relatively concentrated (Ϸ1-10 mg/ ml, for example, after lysis of bacterial cells during DNA isolation, or when DNA is redissolved after ethanol precipitation). According to classical theories and experiments in polymer physics, long flexible molecules form random coils that overlap and become entangled as the concentration of solutions is increased (1, 2). In the field of polymer physics and rheology there is considerable fundamental interest in understanding the effect of molecular topology on entangled polymer dynamics (3, 4). In gel electrophoresis it is well known that molecular topology strongly affects the mobility of DNA. However, with few exceptions, most theories and experiments on diffusion in concentrated polymer solutions have examined only linear molecules. Here, we investigate the effect of molecular topology on diffusion of entangled DNA. Although relaxed circular molecules differ from linear molecules only by the presence of one additional pair of phosphodiester bonds (linking head to tail) and diffuse at nearly the same rate in dilute solution (5), we observe large differences in their diffusion rates with molecules of modest size and concentration. ResultsA 45-kbp fosmid DNA construct (pCCFOS1-45) was prepared as described (6). It was treated with topoisomerase I to prepare the relaxed circular form and with ApaI ...

Self-diffusion coefficients (D) of DNA molecules of varying length and concentration were measured by tracking the Brownian motion of individual fluorescently labeled tracer molecules. Four possible cases were examined: linear tracer molecules surrounded by linear molecules (L-L), circular tracers surrounded by linears (C-L), linear tracers surrounded by circles (L-C), and circles surrounded by circles (C-C). With 6 and 11 kilobasepair (kbp) DNA D was largely insensitive to topology and varied consistent with Rouse scaling (D ∼ L -1 C -0.5 ). In contrast, with 25 and 45 kbp DNA topology had a strong influence. At 1 mg/mL we foundIn the L-L, L-C, and C-C cases a crossover from scaling consistent with the Rouse model to scaling consistent with the reptation model (D ∼ L -2 C -1.75 ) was observed at ∼6 times the molecular overlap concentration. In contrast, D C-L decreased much more steeply with concentration, indicating that a process much slower than reptation governs that case.