Periodic forces trigger knot untying during translocation of knotted proteins

Abstract

Proteins need to be unfolded when translocated through the pores in mitochondrial and other cellular membranes. Knotted proteins, however, might get stuck during this process, jamming the pore, since the diameter of the pore is smaller than the size of maximally tightened knot. The jamming probability dramatically increases as the magnitude of the driving force exceeds a critical value, Fc. In this numerical study, we show that for deep knots Fc lies below the force range over which molecular import motors operate, which suggest that in these cases the knots will tighten and block the pores. Next, we show how such topological traps might be prevented by using a pulling protocol of a repetitive, on-off character. Such a repetitive pulling is biologically relevant, since the mitochondrial import motor, like other molecular motors transforms chemical energy into directed motions via nucleotide-hydrolysis-mediated conformational changes, which are cyclic in character.

Figure 1. The simulation setup: the protein (here 1j85) and the pore (blue).

Figure 2. The movement of knotted core during the translocation of the protein 1ns5 pulled into the pore by the N terminus with F = 2.2ε/Å (a) and with the force F = 1.4ε/Å by the C terminus (b).

The colors mark the two ends of the knot as they move along the chain. In (a) the knot gets tightened and blocks the pore, whereas in (b) - it slides off the chain. The insets of (a) show the conformations of the protein backbone at the beginning of the translocation and at jamming. The final position of the knot is between the aminoacids 119 and 134. Only a portion of a longer trajectory (up to trajectory is shown in panel (a), however no further changes in the knot position were observed beyond .

Figure 3. Pulling the knotted rope through a model pore.

Trefoil (upper left) and figure-of-eight knot (lower left) on the climbing rope is tugged sharply into the spool opening, which results in tightening of the knot (right panels).

Figure 4. Jamming probability as a function of force for the protein 2k0a pulled by the N terminus.

The data points in this and other figures have been obtained based on the average over 100 simulations runs. The red line represents the fit to the two-pathway model (1) with , and . The inset shows schematically the kinetic partitioning between the translocation and knot tightening . Error bars mark 68% Wilson confidence intervals.

Figure 5. The movement of knotted core during the translocation of the protein 1ns5 pulled into the pore by the N terminus with an on-off cycle of 4500τ when the force is switched between F  = 4ε/Å in on-state and F = 0 in off-state.

The colors (red and blue) mark the two ends of the knot as they move along the chain. The circles on the time axis represent the moments of switching the force on (white) and off (red). The conformations below correspond to the time moments marked by the dashed lines.

Figure 6. Mean translocation time as a function of the force period for a knotted protein (1ns5) pulled by the N terminus.

The amplitude of the force is F₀ = 3ε/Å. The curve represents Eq. (2), with and , estimated as described in the text. The error bars mark the standard deviation from the mean.

Figure 7. The distribution of knot loosening times for the protein 1ns5 after the force is relaxed.

Initially the knot is fully tightened with the knotted core spanning 15 aminoacids (corresponding to the plateaus in Fig. 5) and the final state corresponds to the knotted core spanning ~30 aminoacids (corresponding to the minima of the red curve in Fig. 5). The blue curve is a fit to the inverse Gaussian distribution with and .

Figure 8. Mean translocation time as a function of the magnitude of the pulling force for repetitive protocol for the protein 2k0a pulled by the N terminus (top) and 1ns5 pulled by the C terminus (bottom).

The period of the force in both cases is equal to . The red curve in the top panel represents the formula , as given by Eq. (4) . The values of the parameters are , , , whereas is given by Eq. (1) with and . In the bottom panel, the red curve is given by , since the forces involved are much larger than F_c for this protein (cf. Table 1). The remaining parameters are , , and . The error bars mark the standard deviation from the mean.