This section presents a follow-up project to the previous one on GS-repeats and is thus largely analogous to the previous section. The difference is the following: Instead of GS-repeats with different chain lengths, here we investigate a different amino acid sequence with a single chain lengthN=52. It is given by
Atto655-CTNNGGGFNLKTTNSNTSNNFLSSSGFTNGGATGNSKTQQASSSGFLSTGA and, other than the GS-repeats, is highly heterogeneous. The quencher tryptophan is not placed at the other chain end, but at different positionswithinthe chain, which are marked in red.
These marked amino acids are replaced by the quencher tryptophan one at a time such that a total of four internal-to-end contact rates are measured. In contrast to the previous section, the quencher position (let us call its) now plays the role of chain length, and thus a power law is fitted to the functionk+(s). Since there is only a single chain length, no tail effect is measured or simulated. For a given parameter set, all internal-to-end contact rates can be calculated from the same simulated trajectories – the four relevant inter-bead distances are simply kept track of simultaneously. The PET-FCS experiments were performed by Man Zhou (and published in his doctoral thesis [130]), and the 2fFCS experiments by Arindam Ghosh.
Experiments The experimental results are summarized in table 5.8.
s 13 23 33 43
RH/ Å 17.6 17.9 19.2 19.2 k+·10−6s 5.3 3.2 2.1 1.4 k−·10−6s 4.3 4.1 4.8 4.7
Table 5.8: The results of PET-FCS and 2fFCS measurements performed on FG-proteins, with the quencher tryptophan being placed at different sitesswithin the chain. Error bars are shown in figures 5.16 and 5.18.
Hydrodynamic radius The model predicts (see figure 5.16) that the hydrodynamic radius of a closed chain is smaller when the quencher is positioned further away from the fluorophore.
Intuitively, this makes sense – a chain’s size is reduced more strongly when its ends are pulled together than when one end is pulled towards a center-position. Furthermore, the model predicts that the hydrodynamic radius of an open chain is not significantly altered as the quencher position changes. This also makes sense as the respective ensembles of open conformations are to a great extend identical.
Meanwhile, the experiments show that the proteins spend more time in the open state the greater the dye-quencher distance is. This makes the model’s weighted mean of open and closed states shift more towards the open state’s hydrodynamic radius assincreases. This effect counteracts the drop of hydrodynamic radii in the closed state and hence, the model’s predicted hydrodynamic radius depends only weakly on the quencher position, which holds true for all candidate parameter sets. The experimental data, on the other hand, show an increase as a function of quencher position. This increase is weak, however, and only slightly exceeds the experimental error bars.
0 10 20 30 40 50
0 5 10 15 20 25
quencher position s R
H/ Å
MC closed state experiment
MC weighted state MC open state
Figure 5.16: Experimental (red) and model (black) hydrodynamic radius values as functions of quencher position. MC results for RH were calculated using equation 4.23 (page 96), considering the sub-ensembles of open (green) and closed states (blue) separately. The red curve was calculated by weighting open and closed states according to their experimentally measured ratiok−/k+=popen/pclose. Shown are model curves forσ=3 Å (persistence length 6.81 Å) and the best matching parameter a(σ=3 Å)=2.90 Å. Model curves for the other parameter sets (σ,a) look similar and are not shown.
Candidate parameter sets Turning to the candidate parameter sets which best fit the exper-imental hydrodynamic radii (see figure 5.17), we observe a stronger dependence ofaonσthan for the previously studied GS-repeats. This may be due to the fact that the FG-protein is longer and thus a change in persistence length entails a steeper growth of the effectively occupied volume by the chain, which must be compensated by lowering the bead radiusafurther.
0 1 2 3 4 5 0
1 2 3 4 5 6 7
bending stiffness σ / Å
bead radius a / Å
A exp(-σ/B) +C
candidate parameter sets
Figure 5.17: Parameter sets for which Brownian dynamics simulations were run. For four discrete values ofσ(σ=1, . . . , 4 Å), values ofawere determined that lead to hydrodynamic radius values that match best the experimental values, see figure 5.16. This procedure yields tuples (σ,a) (shown as black dots) which can be fitted with the heuristic function (red line) a=Aexp(−σ/B)+Cwhere A=5.35 Å,B=2.93 Å andC=0.96 Å.
Contact rates The simulated contact rates as a function of quencher position are again well-fitted by power laws (see figure 5.18), and again the corresponding power law exponent changes systematically as a different candidate parameter set is chosen. However, the simulated rates’
values do not reproduce the experimental values regardless of the chosen parameter set – they exceed the experimental ones by a factor of 2 to 5, which may be considered as acceptable considering the systematic uncertainty of simulated rates due to the chosen contact radius RC=5.4 Å. However, this factor of 2 to 5 does not fit together with the one found for GS-repeats (a factor of≈1.17) and even more importantly, the experimental power law exponent differs significantly from all simulated ones.
Conclusion The model failed in reproducing the functional form of the measured internal-to-end contact rates in FG-proteins. Possible causes are mainly the inhomogeneity of the amino acid sequence as well as the possible presence of internal friction. This will be discussed in the next chapter (see page 145).
1.6 1.8 2.0 2.2 6.0
6.5 7.0 7.5
log
10( quencher position / Å ) log
10( rate / s
-1)
experiment parameters 4 parameters 3 parameters 2 parameters 1
1 2 3 4
-2.0 -1.5 -1.0 -0.5
parameter set
power law exponent ϵ
experimentsimulation
Figure 5.18: Top: The simulated contact rates for all chain lengths and candidate parameter sets, together with the experimentally measured contact rates (black dots). We show the data points together with respective power law fits (lines). Their slopes (power law exponents) change as the parameter set is changed and the experimental curve is notably flatter than all simulated ones. Furthermore, the measured rates are smaller than the simulated ones roughly by a factor of 2–5. Bottom: For each candidate parameter set, simulated internal-to-end contact rates as a function of the quencher position were fitted with a power law. The resulting power law exponents are shown here as a function of the candidate parameter set (σ,a). As can be seen, exponent values increase monotonically, converging towards a value of∼ −1.6. The power law exponent for the experimental data is shown in red (−1.07±0.09).
The shaded area is its fit error.