Contact rate

While the hydrodynamic radius can be quickly calculated using MC sampling (seconds to minutes), the contact rate requires a full Langevin simulation (hours to days). For the sake of well-converged contact rate values, we thus only studied short model chains of length N=10 to demonstrate the influence of_σand aon the end-to-end contact rate k₊. Additionally, we demonstrate the so-calledtail effectby comparing the end-to-end contact rate with the contact rate between the chain’s end and a bead placed at different sites withinthe chain. We set the critical contact radius toR_C=5.4 / 3.8≈1.42 because this is the same nondimensional radius that we later use for interpreting PET-FCS measurements.

The functional behavior ofk₊(N) is the treated in the next section which constitutes the main project of this thesis.

Contact rate as a function of bead size We must be careful when comparing contact rates from simulations with different values of afor the following reason. The contact rate is the inverse of the mean first passage time〈τ〉and as such, when reconverted into physical units, is proportional to the time scale t₀∝a, which may differ between simulations. Without hy-drodynamic interactions, for instance, the bead radius a does not influence the numerics at all, and thus k₊∝a⁻¹. To take this into account, we do not study k₊ but the dimensionless quantityk₊·t₀ as a function ofa. In the absence of hydrodynamic interactions, the resulting curves would be constant. The influence of hydrodynamic interactions on the contact rate can thus be observed in the form of deviations from a horizontal line – a decreasing function means that hydrodynamic interactions lower the contact rate, and a growing function means that they enhance it. Thereby, we disentangle the effects that a larger bead (i) diffuses more slowly on its own, but (ii) is hydrodynamically coupled to other beads more strongly which dampens their relative motion.

Figure 5.4 shows the functional form of k₊(a) (normalized to the first data point for visual purposes) for_σ∈{0, 1} and N=10. We observe that the curves are decreasing for both tested chain stiffnesses. Thus, the hydrodynamic coupling between the beads reduces the contact rate – their relative movement is dampened because they fluctuate more cooperatively. Furthermore, the curve of the more flexible chain drops faster. Mathematically, that is because the limit of the Rotne-Prager matrices becoming identity matrices is reached faster for smaller inter-bead distances. Physically, the closer the beads’ centers are, the more the beads overlap. Thus their relative movement is dampened faster in flexible chains as the beads grow.

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

hydrodynamic bead radius a normalized k

· t

σ=0 σ=1

Figure 5.4: The simulated nondimensional contact ratesk₊·t₀as a function of the hydro-dynamic bead radiusa, shown for two different bending stiffness values_σ. The two curves were normalized such that both start at 1. They both decrease, meaning that hydrodynamic coupling decreases the contact rate. The stiffer chain shows a slower decrease than the more flexible one.

Contact rate as a function of bending stiffness Figure 5.5 shows the functional form of k₊(_σ) (normalized to the first data point for visual purposes) fora∈{0, 1} andN=10. We observe that bending stiffness decreases the simulated contact rate, which is intuitive – the energetic barrier which must be overcome for the chain ends to meet limits the rate of successful loop formation. Moreover, the curve for chains with smaller bead size drops faster. That is because hydrodynamic interactions decrease the contact rate, and the bending stiffness in turn reduces hydrodynamic interactions because the inter-bead distances grow the stiffer the chain is. As expected, the contact rate seems to vanish in the limit_σ→ ∞.

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

bending stiffness σ normalized k

· t

a=0 a=1

Figure 5.5: The simulated contact ratesk₊as a function of the bending stiffness_σ, shown for two different bead sizes. The chains consist of 10 beads. The curves were normalized such that they start at 0 in order to be visually comparable. The curve of the chain consisting of smaller beads drops faster.

Contact rate as a function of quencher position In figure 5.6, we show the simulated contact rate of two chains consisting ofN=10 andN=20 beads respectively, both of which have parametersa=σ=1. The contact rate was evaluated for looping events taking place between the first bead (fluorophore) and another bead (quencher) at positions withinthe chain, with the rest of the chain being present behind it. For growings, entropic effects can be expected to lowerk₊ as the vector connecting fluorophore and quencher grows, but “searches” for a small volume of fixed size∝R³_C. Fors≈l_p≈2, the (bending) energetic barrier which must be overcome for loop formation limitsk₊but sinceR_C≈1.42 is finite, we do not expect to observe this effect here.

We indeed observe that the contact rate decreases as the fluorophore-quenchersdistance in-creases. This decrease flattens out towards the ends of the respective chains: The growingsis being balanced by the fact that the end of the chain can diffuse more freely than the middle of the chain. This is because for the chain end’s diffusive dynamics, fewer beads have to be moved through the fluid. This reduces hydrodynamic friction and thus speeds up the dynamics. This

“free-end effect” is not present in theN=20 chain at s=10 – it has additional beads behind the quencher, which is called the tail. The so-calledtail effectis thus that it slows down the dynamics between fluorophore and quencher, reducing the contact rate. It cannot be inferred from the chain’s conformational probability distributions. For instance, the random walk described by MC sampling samples the Boltzmann distribution but does not include any tail effect – the variations of bond angles in the middle of the chain occur completely independently of those at the chain’s ends. In the Langevin simulation, however, correct physical dynamics are implemented:

the hydrodynamics obey the Stokes equation which alone implies the tail effect. This is further investigated in the next section and discussed in chapter 7.

Note that the tail effect hasnotbeen explicitly built into the model but emerges naturally!

tail effect

5 10 15 20

-2.5 -2.0 -1.5 -1.0 -0.5 0.0

quencher position s log

( k

· t

)

N=20 N=10

Figure 5.6: The simulated contact rate as a function of the distancesbetween fluorophore and quencher. The fluorophore is the first bead of the chain, and the quencher’s positions is altered while the total length of the chain remains fixed. We compare the curvesk₊(s) for two chains of different length but otherwise identical parametersa=σ=1. The curves initially coincide, but towards their respective ends, they flatten out instead of dropping further. Hence, ats=10, the curve forN=20 lies below the curve forN=10 by around 21 %.

This is called the “tail effect”.

5.2 End-to-end contact rates and hydrodynamic radii of