Models and simulations of polymers at various scales

Rotational blobs
We have studied the properties of a linear polymer performing a stationary rotational motion around a long, impenetrable rod - using both blob theory and
computer simulations. The polymer was forced into rotation by attaching one of its ends to the rod, and imposing a fixed torque on this end. We then studied the response of the polymer as a whole, using torsional blobs. Within such a blob, the polymer obeys the statistics of a free chain, but on larger length scales, the effect of the external torque is observed. Torsional blobs are analogous to the tensile blobs used when describing a polymer that is subjected to a constant force, but are adapted to the constant torque dynamics. Blob theory predicts three different regimes.

• For small torque, we are in the equilibrium regime. The polymer behaves as a free polymer, i.e. there is only one blob.
• For intermediate torque, a trumpet regime appears. The effect of the torque becomes apparent in the configuration of the polymer. Segments of the polymer that are close to the attached end are, one average, closer to the rod than segments further away. This means there are multiple blobs, and their size grows towards the free end.
• For large torque, the polymer is in the stem-trumpet regime. The part of the polymer close to the attached end is now tightly wrapped around the rod.  Further away from this end, the polymer loosens up, and this part assumes a trumpet shape.

These three regimes were recovered through extensive computer simulations.

Using these torsional blobs, also the angular velocity of the configuration in the steady state can be predicted, in function of the applied torque. For small and large torque (i.e. the equilibrium and stem-trumpet regime), the angular velocity is expected to scale linearly with the torque. For intermediate torques, we observe a thinning regime: a small change in the applied torque changes the resulting angular velocity appreciably. This particular behavior was also observed in the simulations.

This work introduced the concept of torsional blobs,  useful in modeling the dynamics of a polymer under constant torque. We showed that this blob picture successfully captured the dynamics of a polymer rotating around a rod. This could be relevant for describing the physics surrounding gene transcription, where the emerging messenger RNA rotates around  the stiff DNA chain. Our theory could be extended to include a constant force on top of the constant torque. Furthermore the presented predictions could be put to the test experimentally, by using an optical torque wrench to force the rotation, and fluorescent beads to visualize the dynamics of the polymer. Especially our predictions of the elongation of the molecule along the rod in function of the angular velocity should be easily testable, and hint towards the applicability of our theory.

Transition path times
Macromolecules with two coexisting conformational states tend to rarely switch between those states. They remain in the same configuration for a very long time, but when a transition happens, it happens very fast. Conformational transitions between two states of a macromolecule are often modeled as transitions between two wells in a one-dimensional energy landscape. We studied the very short duration of these transition paths, for the one-dimensional case as well as for complex molecules.

Firstly, one-dimensional barrier crossing was studied. This problem was already studied in the overdamped regime, i.e. where friction dominates over inertia. However, owing to the short time scales at which these processes take place, inertia might play an important role in the transition dynamics. To this end, we developed a formalism with which the transition path time distributions of barrier crossing were calculated in the underdamped limit, i.e. without neglecting inertial effects. Specifically, we presented a formula for the transition path time distribution of parabolic barrier crossing, which is exact in the high barrier limit. Our results were backed up by numerical simulations of simple, one-dimensional processes.

Secondly, we looked at conformational transitions of complex molecules. Under the right conditions, two states of such a molecule coexist: a compact, folded
state, and an open, unfolded state. Transitions between these states are often modeled as transitions of a reaction coordinate over a one-dimensional barrier,
thus projecting the whole dynamics onto a single degree of freedom. However, recent experiments revealed some inconsistencies when applying this reaction
coordinate picture to measured transition path times. We employed a simple simulation to mimic the dynamics of complex molecules, but were not
able to find any inconsistencies upon mapping this dynamics onto one reaction coordinate. Also oxDNA, a more detailed computer model, was used to simulate folding/unfolding transitions of DNA hairpins. Different reaction coordinates were monitored, and upon comparing them we observed that the length of transition path times depends on the choice of reaction coordinate. In summary, we calculated the duration of transition paths in the underdamped
regime for one-dimensional crossing over a parabolic barrier. Furthermore we studied transitions of complex molecules, and how to map them onto a
one-dimensional reaction coordinate. We take away that one should carefully choose an appropriate reaction coordinate, otherwise important parts of the
dynamics might not be captured.

Teardrop shapes
The flexibility of DNA on longer length scales is excellently captured by the worm-like chain model. On short length scales, however, DNA appears to be
much more flexible than a worm-like chain. This anomalous behavior has been debated for many years. We study a simple construct, which should elucidate the bending mechanisms of short pieces of DNA, namely DNA teardrops. These teardrops are formed by forcing the two ends of the molecule closely together, and then keeping their distance fixed. Upon doing so, a worm-like chain adopts a shape that closely resembles a teardrop. However, through computer simulations with oxDNA, we observed that DNA molecules do not always retain this shape. If the stress is too high, melting bubbles form, in which base pairs open, and the geometrical structure of DNA is locally disrupted. At the location of these disruptions, DNA becomes much more flexible. These melting bubbles then act as a kind of hinge, and anomalous (i.e. not described by the worm-like chain) structures are formed.

In conclusion: one possible mechanism explaining the anomalous behavior is the formation of melting bubbles. More extensive simulations and comparison to
experiments should elucidate whether this mechanism indeed (partly) explains the experimentally observed enhanced flexibility at short length scales.