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 ≈6 to 290 kbp were measured by tracking the Brownian motion of single molecules. A topology-independent scaling law D ∼ L ^−ν was observed with ν _L = 0.571 ± 0.014, ν _C = 0.589 ± 0.018, and ν _S = 0.571 ± 0.057 for linear, relaxed circular, and supercoiled DNA, respectively, in good agreement with the scaling exponent of ν ≅ 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 D _Circular / D _Linear = 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.

Terminase enzyme complexes, which facilitate ATP-driven DNA packaging in phages and in many eukaryotic viruses, constitute a wide and potentially diverse family of molecular motors about which little dynamic or mechanistic information is available. Here we report optical tweezers measurements of single DNA molecule packaging dynamics in phage T4, a large, tailed Escherichia coli virus that is an important model system in molecular biology. We show that a complex is formed between the empty prohead and the large terminase protein (gp17) that can capture and begin packaging a target DNA molecule within a few seconds, thus demonstrating a distinct viral assembly pathway. The motor generates forces >60 pN, similar to those measured with phage φ29, suggesting that high force generation is a common property of viral DNA packaging motors. However, the DNA translocation rate for T4 was strikingly higher than that for φ29, averaging ≈700 bp/s and ranging up to ≈2,000 bp/s, consistent with packaging by phage T4 of an enormous, 171-kb genome in 300 s ⁻¹ . The motor velocity decreased with applied load but averaged 320 bp/s at 45 pN, indicating very high power generation. Interestingly, the motor also exhibited large dynamic changes in velocity, suggesting that it can assume multiple active conformational states gearing different translocation rates. This capability, in addition to the reversible pausing and slipping capabilities that were observed, may allow phage T4 to coordinate DNA packaging with other ongoing processes, including viral DNA transcription, recombination, and repair.

In many viruses, DNA is confined at such high density that its bending rigidity and electrostatic self-repulsion present a strong energy barrier in viral assembly. Therefore, a powerful molecular motor is needed to package the DNA into the viral capsid. Here, we investigate the role of electrostatic repulsion on single DNA packaging dynamics in bacteriophage φ29 via optical tweezers measurements. We show that ionic screening strongly affects the packing forces, confirming the importance of electrostatic repulsion. Separately, we find that ions affect the motor function. We separate these effects through constant force measurements and velocity versus load measurements at both low and high capsid filling. Regarding motor function, we find that eliminating free Mg ²⁺ blocks initiation of packaging. In contrast, Na ⁺ is not required, but it increases the motor velocity by up to 50% at low load. Regarding internal resistance, we find that the internal force was lowest when Mg ²⁺ was the dominant ion or with the addition of 1 mM Co ³⁺ . Forces resisting DNA confinement were up to ≈80% higher with Na ⁺ as the dominant counterion, and only ≈90% of the genome length could be packaged in this condition. The observed trend of the packing forces is in accord with that predicted by DNA charge-screening theory. However, the forces are up to six times higher than predicted by models that assume coaxial spooling of the DNA and interaction potentials derived from DNA condensation experiments. The forces are also severalfold higher than ejection forces measured with bacteriophage λ.

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 ≈100 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 ≈10 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.

It is well known that a jostled string tends to become knotted; yet the factors governing the “spontaneous” formation of various knots are unclear. We performed experiments in which a string was tumbled inside a box and found that complex knots often form within seconds. We used mathematical knot theory to analyze the knots. Above a critical string length, the probability P of knotting at first increased sharply with length but then saturated below 100%. This behavior differs from that of mathematical self-avoiding random walks, where P has been proven to approach 100%. Finite agitation time and jamming of the string due to its stiffness result in lower probability, but P approaches 100% with long, flexible strings. We analyzed the knots by calculating their Jones polynomials via computer analysis of digital photos of the string. Remarkably, almost all were identified as prime knots: 120 different types, having minimum crossing numbers up to 11, were observed in 3,415 trials. All prime knots with up to seven crossings were observed. The relative probability of forming a knot decreased exponentially with minimum crossing number and Möbius energy, mathematical measures of knot complexity. Based on the observation that long, stiff strings tend to form a coiled structure when confined, we propose a simple model to describe the knot formation based on random “braid moves” of the string end. Our model can qualitatively account for the observed distribution of knots and dependence on agitation time and string length.