loop for tensions above the stability curve; although the straight conformation is stable, loop growth eventually leads to an energy decrease and the subsequent growth of a plectoneme.
In this analysis, we neglect thermal fluctuations. The non-equilibrium energy displayed in Fig. 6.3 suggests that loop formation requires an energetic nudge in order to overcome a barrier for sufficient end tension (conditions above the stability diagram in Fig. 6.1). A molecular strand is subject to thermal fluctuations with characteristic energy kBT, thus energies comparable to kBT are attainable by a fluctuating strand. For the conditions in Fig. 6.3, thermal energy can easily overcome the boundary to loop formation, thus an AFM or tweezer experiment conducted under these conditions is predicted to exhibit transient loop formations leading to plectonemic supercoils, even under stable conditions.
In the following section, we numerically solve the equations of motion found in Sec. 6.2 in order to show the progression from straight, twisted chain through the instability and the subsequent nonlinear dynamics. In particular, we compare and contrast the predictions of this section to assess the validity of the pseudo-equilibrium approximation.
the post-buckling, nonlinear dynamics using a discrete representation of the polymer-chain model. We solve the discrete analogues of Eqs. 6.2 and 6.3 for an initially straight polymer chain, with a small random transverse perturbation, with twist B = 150 and tension F = 4700 including a contact term in the energy (Eq. 6.1) such that the chain cannot cross itself.
In Fig. 6.4, we show snapshots from the relaxation dynamics at several points in time, which is nondimensionalized by the time-scale of instability growth τinst = Γ⊥L4/(Aσ1).
The first snapshot, occurring at t/τinst = 7.30, shows the conformation just beyond the initial instability, which looks similar to conformation D in Fig. 6.1. Further growth of the instability shows the localization of the helical undulations into loops near the ends of the chain. Since the chain is of fixed length, the chain ends must retract during the transverse growth, which is responsible for the helical localization near the chain ends.
The loop formation leads to the growth of plectonemes, which twirl out into the quiescent fluid as the ends retract. From the fourth snapshot (t/τinst = 29.27) to the final snapshot (t/τinst = 747.34), the two plectonemes grow as they are fed by the connecting chain segment, thus the chain is reptating within each plectoneme as they grow. We note the plectonemes trail off the ends due to the viscous drag as the chain ends are pulled towards each other. The final snapshot in Fig. 6.4 shows the plectonemes in the final conformation eventually cease to grow as they encompass the entire chain. Further relaxation shows the two plectoneme straighten perpendicular to the tension direction; however, this process is extremely slow in comparison to the dynamics shown here.
We find an interesting phenomenon occurs if we decrease the tension further than the quench in Fig. 6.4; the retraction of the chain ends occur faster than the loop collapses on itself. This has different ramifications depending on the method that the chain ends are clamped. If the ends are clamped between two bars, the chain will loop around the bars
Figure 6.4: Snapshots from the relaxation dynamics for B =
150 and F = 4700. The snapshots are for t/τinst =
7.30,11.60,18.43,29.27,46.50,73.87,117.35,186.42,296.15,470.45,747.34, respectively.
at each end and climb them as they are pulled towards each other. If the chain ends are fixed in space, as in an optical tweezers experiment, the chain is looped around the ends several times, and then the chain is extended due to the elimination of the twist during this process.
The dynamics presented in Fig. 6.4 suggest the chain ends play the role of nucleation sites for plectoneme growth; however, in the case of DNA, several other factors may seed the growth of supercoils. For example, sequence-specific elasticity has a dramatic effect on the coiling of DNA around nucleosome particles [21]; therefore, we expect sequence- dependent weak spots to serve as nucleation sights for supercoiling as well. Also, if the chain conformation is not initially straight, as is certainly the case in DNA, the formation of plectonemes is more likely to occur at locations of high curvature, since they form the plectoneme end and also serve as a tension gradient to feed the growing plectoneme. In this analysis, we neglect the role of thermal fluctuations; however, their effect would change the dynamic behavior. For example, the stable region in the stability diagram (Fig. 6.1) results in a stable straight chain; however, thermal fluctuations would allow the formation of the loop structures shown in Fig. 6.2, leading to plectonemic supercoiling.
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Chapter 7
DNA Packaging in Bacteriophage:
Is Twist Important?
We study the packaging of DNA into a bacteriophage capsid using computer simulation, specifically focusing on the potential impact of twist on the final packaged conformation. We perform two dynamic simulations of packaging a polymer chain into a spherical confinement:
one where the chain end is rotated as it is fed, and one where the chain is fed without end rotation. The final packaged conformation exhibits distinct differences in these two cases:
the packaged conformation from feeding with rotation exhibits a spool-like character that is consistent with experimental and previous theoretical work, whereas feeding without rotation results in a folded conformation inconsistent with a spool conformation. The chain segment density shows a layered structure, which is more pronounced for packaging with rotation. However in both cases, the conformation is marked by frequent jumps of the polymer chain from layer to layer, potentially influencing the ability to disentangle during subsequent ejection. Ejection simulations with and without Brownian forces show that Brownian forces are necessary to achieve complete ejection of the polymer chain in the absence of external forces.