SUPPLEMENTARY INFORMATION

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1 SUPPLEMENTARY INFORMATION doi: /nature12682 Supplementary Information 1 Theoretical models of flagellum growth on the outside of a bacterial cell A classic problem in polymer physics concerns the diffusion of an unfolded, thermally fluctuating polymer chain constrained in a channel, called reptation motion 46, where reptation time is a strong function of channel length. Reptation motion could be considered a mechanism by which flagellar structural subunits, which have been unfolded and translocated across the cytosolic membrane by the well-studied export machinery at the base of the flagellum 45, might diffuse through the narrow (c.2nm diameter) central channel in the external flagellum 47. The diffusion constant for individual protein reptation motion is given by: D T = k T d 2 B 2 2 3N ζ a (equation 6.40 of ref. 46 ) where k B T is the Boltzmann temperature ( J) and ζ is the friction constant for the motion of one subunit. For the estimates below we shall need to know the value of ζ, which can be obtained approximately from ζ = 6πηa with η being the viscosity of the suspending fluid. Assuming water (η =0.7 mpa.s) and a=0.3 nm the approximate length of an amino acid residue, we have ζ (in SI units) and then taking a single flagellar subunit N 400, we estimate: D m 2 /s (for comparison, the diffusion constant of an unrestricted folded protein is usually about m 2 /s, a thousand times greater). The time to diffuse a distance R is given by the diffusion law: R 2 D T t. For an ideal smooth channel length R=1 µm, we estimate t 10 seconds, but for a channel length R=10 µm, t 17 minutes. Consequently, the rate of flagellum growth would show a very strong dependence on flagellum length. 1

2 RESEARCH SUPPLEMENTARY INFORMATION However, Berg and colleagues find that the rate of flagellum growth (1 subunit crystallized every ~2s), and therefore the rate of subunit transit through the central channel, is independent of the length of the growing flagellum 4. This finding rules out individual reptation motion of subunits through the flagellar channel as the primary mechanism of filament growth. This is because if the rate of insertion of new subunits into the channel is one per 2s, and the time of individual subunit diffusion is much longer than that (e.g. 10s), then the subunits would gradually build up their density in the channel and soon enter into the crowded regime where the resistance force is high due to the multiple interactions between subunits in close contact, and the channel becomes effectively clogged. Alternatively, it has been suggested that subunits might move through the flagellum channel by a Brownian ratchet mechanism 48, however, within the field there is a certain confusion of terminology when considering what is a Brownian ratchet. One description is of a Brownian ratchet motor producing a continuous linear motion, originally defined by Prost and co-workers 18. This mechanism powers much of the intracellular active transport of cells. However it cannot be the mechanism that powers transit of subunits through the channel in the external flagellum as the channel is too narrow to accommodate motor proteins (e.g. analogous to eukaryotic dynein), and there is no continuous supply of ATP to power movement along the entire length of the channel. A passive mechanism 4,48 that might also be called a Brownian ratchet, involves the random diffusion of subunits (Brownian motion along the channel axis) being rectified by blocking the channel at the cell proximal end. In this model, the flagellar membrane export machinery, energised by ATP hydrolysis and the proton motive force 1,8,9, translocates subunits across the membrane into the external channel at a constant rate and, given sufficient time, the subunits accumulate at such a density along the length of the channel that they are ejected at the distal end. 2

3 SUPPLEMENTARY INFORMATION RESEARCH This mechanism could work if subunits are translocated across the membrane at a low rate, producing a low density of unfolded subunits in the channel with a lot of space between subunits (essentially, one subunit in the channel at any time), as the rate-controlling process of subunit transit through the channel would be the rate of insertion. However, when subunits are densely packed along the channel, in the crowded regime, the mechanical work required for insertion of a new subunit is a product of the unfolded subunit size and the resistance force that is generated by the increasingly long file of closely packed subunits in front. There are many studies that allow one to estimate the resistance force of a chain travelling along a narrow channel 49,50,51 but the main point is that the total force would be the sum from all subunits, i.e. resistance would be proportional to the channel length. This means that the export machinery would have to continuously increase its power output to maintain a constant rate of subunit translocation across the membrane into the channel. It is difficult to imagine how such an unprecedented biological pump might function. This is the main objection to the rectified Brownian motion mechanism in the crowded regime. An additional factor arguing against this model is that the flagellar subunits are unfolded prior to their entry into the channel, but unfolded thermally fluctuating polymer chains cannot be pushed 16, they can only be pulled: the thermally fluctuating polypeptide chain does not transmit compressive force, even confined in a channel of 2nm diameter. Stern and Berg have studied this theoretical model of ratcheted diffusion 17, in which flagellin subunits move through the channel folded as extended α-helices, essentially rods of 1.2nm diameter and 74nm length 4. This hypothetical flagellin conformation is unprecedented and might be based on an assumption that confinement in the channel forces this unnatural fold. However, tight confinement in nanotubes can instead destabilize helical structure, even in the case polypeptides that are predicted to fold naturally as α-helices 52. As flagellin contains only ~40% 3

4 RESEARCH SUPPLEMENTARY INFORMATION α-helix 10 the cost in free energy of folding incorrectly as a 495-residue helix would be extremely high. We cannot find a precedent for such a forced misfolding, even for polypeptides confined in cylindrical nanotubes. Furthermore it is known, for example, that nascent polypeptides constrained in the ribosome exit tunnel do not adopt unfavourable secondary structures 53,54. Rather than folding as a rigid α-helix inside the flagellum channel, it seems more likely that flagellin would remain predominantly in an unfolded random coil configuration, as other proteins confined in nanochannels do. Two other criteria not considered in Stern and Berg s theoretical model 17 are subunit interactions with the wall of the proteinaceous channel and interactions between subunits in transit. Modeling these factors is not straightforward, making it extremely difficult to estimate realistic diffusion coefficients for polymers moving through channels. Nevertheless, it is established 55,56,57 that polymer interactions with the wall of a confining channel reduce the value of the diffusion coefficient according to the expression: D D 2 0 = R R V ( x1 ) / kbt V ( x2 ) / kbt dx1e dx2e 0 0 R where D 0 is the diffusion constant in the ideal smooth channel of length R, and V(x) is the potential of interaction with the channel wall. The product of the two integrals in the denominator is always greater than unity and, therefore, for any potential V(x) on the channel wall the diffusion is much hindered - the more so for stronger-interacting interfaces such as protein on protein, as would be the case for flagellin constrained in the narrow channel. As a result, the diffusion constants used by Stern and Berg are likely to be optimistic overestimates for flagellin transit through the channel. When considering interactions between subunits transiting through a channel, it is difficult to 4

5 SUPPLEMENTARY INFORMATION RESEARCH estimate the point at which resistance caused by frequent collisions between neighbouring subunits becomes the dominant factor (see above for an explanation of the pushed subunit column). As crossover to this crowded regime is estimated to occur when the diffusion time for individual subunits is of the same order of magnitude as the inverse rate of subunit delivery into the channel, we estimate that the proposed theoretical model 17 of flagellin transit crosses the threshold into the crowded regime long before mature filament length is reached. Supplementary Information 2 An entropic chain mechanism to pull subunits into and through the channel We propose a new physical mechanism of directed transport of unfolded subunits to the assembly tip of the growing flagellum, based on the entropic contractile force of the subunit polymer chain and the breaking of directional symmetry through chain crystallization at the distal assembly tip. This mechanism provides a pulling force on the subunit at the cell proximal end, instead of a pushing (compression) force. We show that this pulling force automatically adjusts, irrespective of the length of the growing flagellum, and therefore is consistent with our data showing that subunits link in a chain in the channel (Fig. 2) and the observation of constant rate flagellum growth 4. There are three key elements in this transport mechanism: 2a. Strong anchoring of the subunit polypeptide chain at the distal assembly site through crystallization The binding energy for an assembled hook subunit is estimated 25 to be ΔE A = 487 kcal/mol. In order to advance our analysis, we convert this binding energy to physical energy units, using: 1 kcal = 4184 Joules and 1mol = , giving the energy ΔE A = Joules per hook subunit. It is interesting to compare this binding energy with the characteristic thermal energy, which is a 5

6 RESEARCH SUPPLEMENTARY INFORMATION measure of thermal vibrations: k B T at T=37 C is approximately Joules, which means that ΔE A is approximately equal to 800 k B T for a hook subunit. This means the binding of subunits to the crystallized flagellum is very strong and cannot be broken spontaneously by thermal motion. To estimate the external force required to break such binding, we assume that a displacement of approximately one amino acid length (a 0.3nm) against this binding potential is required. The definition of a force needed to achieve this displacement is F A = ΔE A /a. This gives the critical force F A = [J/m] = 11nN for a hook subunit. This is a very high force, compared with typical forces generated by molecular mechanisms in a cell 58, and indicates strong anchoring of flagellar subunits in the crystallized structure. The chain in the channel is likely to contain a mix of subunits that link head-to-tail e.g. hook and hook-cap (Fig. 1b and Extended Data Fig. 4), or flagellin, filament cap and hook-filament junction proteins 45. On reaching the tip of the flagellum, the excess proteins in the chain will fold as they exit the channel but will not incorporate into the structure, being discarded instead into the extracellular space 26. Similar export without assembly is observed for effectors secreted through related virulence needles, which are discussed further in Supplementary Information 3. Our calculation suggests that even in the absence of assembly, subunit folding at the tip alone provides sufficient anchoring for the chain mechanism to function. For example, the folding energies for hook 25 and flagellin 59 subunits are estimated to be ΔEA =263 kcal/mol and ΔE A = 283 kcal/mol, respectively. Repeating the calculation, we obtain ΔEA = Joules (equal to 428 k B T) per hook subunit, and ΔE A = Joules (equal to 461 k B T) per flagellin subunit. To break this anchoring would require forces F A = ΔE A /a = 6nN for hook and F A = 6.7nN for flagellin, which are still very high. 6

7 SUPPLEMENTARY INFORMATION RESEARCH 2b. Linking of subunits through a parallel coiled-coil formed by the extreme C-terminus of the unfolded subunit chain in the channel and the extreme N-terminus of the next subunit docked at the export gate We present experimental evidence (Fig. 1e and Fig. 2a) showing that each hook subunit terminus contributes ~14-25 residues and each flagellin subunit contributes ~21-32 residues to each coiled-coil link between subunits. The binding energy for a coiled-coil is obtained as 1.2 kcal/mol per residue 60, or equivalently, Joules (approximately 2k B T), giving the total binding energy ΔE L = Joules (or approximately k B T) for the range of ~14-25 residues in hook subunits and ΔE L = Joules or approximately k B T for the range of ~21-32 residues in flagellin subunits. Again, assuming that a displacement of one residue length, a, is required to break the bond, the critical forces to disrupt the parallel coiled-coils that link subunits is: F L = ΔE L /a = pN for hook and F L = ΔE L /a = pN for flagellin. These are high forces, but at least an order of magnitude lower than F A required to break subunit anchoring at the distal end. That is, it is easier to break the chain of linked subunits than to release the subunit anchoring at the flagellum tip, which we shall regard as permanent for the subsequent analysis. 2c. Generation of an entropic force (F) at the end of the flexible subunit chain pulls linked subunits from the export machinery into the channel and towards the distal assembly tip This entropic force increases as the anchored N-terminus of the subunit crystallizes at the flagellum tip and fewer unfolded residues remain in the channel, eventually pulling the next linked subunit off the membrane export machinery into the channel (Fig. 3b and Extended Data Fig. 7). 7

8 RESEARCH SUPPLEMENTARY INFORMATION The physical problem of a polymer chain confined in a tube and anchored at one end has been studied extensively 50. However, the problem we need to solve is easier than the classic question of the confined chain. It has been shown that when the ends of a flexible polymer with N monomers of size a are separated by a distance R, in the regime when R exceeds Na(a/d) 2/3 the effect of confinement in the tube of diameter d becomes irrelevant and an ordinary entropic force of an overstretched chain comes into effect 27. That is, if a chain of N monomers is stretched by R > Na(a/d) 2/3, then the transverse size of this highly stretched chain is less than d, so that the confinement in the channel does not have any additional effect. In our case a/d = The overstretched regime starts when the length of the flagellum channel R exceeds Na(0.15) 2/3 = 0.28Na (here Na is the arc length of the free polymer chain in the tube). The best current model by Blundell 23 gives the expression for the free energy A(R) of a stretched chain that is valid in all regimes (both weak stretching and overstretching): [ ] + 2N π k B T 1 A = π 2 2N k BT 1 ( R /Na )2 ( ) 2 1 R /Na A differential of this free energy with respect to the stretching R gives the returning force: F = A R = π 2 k B T N 2 a In the strong stretching regime, when (R/Na) approaches 1, the second term dominates and the simpler approximate expression for pulling force at the free end of the highly stretched chain takes the form: ( R / Na ) + 4k BT π a ( R / Na) ( ) 2 # $ 1 R / Na 2 % & F 1.27 k B T a ( R / Na) ( ) 2 # $ 1 R / Na 2 % & 8

9 SUPPLEMENTARY INFORMATION RESEARCH Note that this force depends only on the non-dimensional ratio (R/Na), which measures the current stretching as a fraction of the total contour length of the free chain (Na). In our case the characteristic pre-factor in this expression for the force, which gives a measure of its value is: (1.27 k B T/a) 18 pn. This entropic force ultimately may reach the covalent-bond breaking force as the ratio R/Na 1 and the factor in the denominator of the force expression gets close to zero. Obviously, this will not happen. The increase of force occurs because the unfolded monomers crystallize at the distal tip, reducing the value of N, the number of residues remaining free in the channel, thus increasing the ratio R/Na (Fig. 3b and Extended data Fig. 7). As the force increases, it will first reach a specific critical value - which we name F M - when the next unfolded subunit protein is pulled off the membrane export machinery into the channel. Once this occurs, the number of free monomers in the tube increases in a step-wise manner as a whole new unfolded subunit, linked to the previous one, emerges in the channel. As a consequence, the entropic force drops and the slack of the chain will continue to gradually crystallize at the anchored end. Then the process repeats: N gradually reduces, the pulling force F(R/Na) increases until the critical value F M is reached again and the subunits are thus transported to the distal assembly site in a chain sequence (Fig. 3b). This is a comparable sequential process to that observed for unfolding of titin immunoglobulin under applied force 61. We now need to make an estimate of the binding energy of subunit to the export machinery, ΔE M, which in turn will provide the estimate for the required minimum pulling force, F M. The ITC experiment reported in Extended Data Fig. 2a, shows the heat released as subunits bind to the membrane machinery export gate in vitro. Extrapolating the data to the limit of molar ratio 0, that is, when an individual subunit protein is guaranteed to bind to a large selection of export gate sites in solution, we obtain the binding energy, ΔE M = 0.9 kcal/mol. Repeating the calculation 9

10 RESEARCH SUPPLEMENTARY INFORMATION above to convert this into mechanical energy units we obtain ΔE M = Joules. As above, to overcome this barrier by displacing over a single amino acid gives the estimate of the critical force F M = ΔE M /a = ~30pN. This is a small force, much lower than both F L and F A, which means that the integrity of the chain of linked subunits anchored at the distal end would not be compromised at any point. The magnitude of F M is comparable to typical forces exerted in single protein stretching experiments 61. Note, however, that the force calculations provide only the lower estimates of the actual binding force. Several factors, such as macromolecular crowding and transient interactions of subunits with other components of the export machinery would contribute to strengthening of the binding interactions in vivo 62. Supplementary Information 3 Adjustment of entropic force of the subunit chain with length of the channel to support constant rate flagellum growth For the chain mechanism to work, the critical pulling force F M must be lower than F L, which would break the parallel coiled-coil links between consecutive subunits, which in turn is much smaller than the anchoring force F A. Let us now test the proposed transport model by applying the values of parameters known for flagellum growth. The cell proximal part of the flagellum channel is divided into two parts: rod (in which we include the length of channel that passes through the MS ring) and hook. This is followed by the flagellum filament, which can be very long. Extended Data Table 2 shows the established data on the length of flagellar substructures from electron microscopy experiments 10,63,64,65 and the average number of residues in rod, hook or filament subunits. The table also gives the number of subunit residues that are free to fluctuate in the unfolded state in the channel, as opposed to the N- and C- terminal residues that contribute 10

11 SUPPLEMENTARY INFORMATION RESEARCH to the coiled-coil link; for clarity, we estimate this number for rod and hook subunits (14-25 residues) as a value of 18 residues, and for flagellin (21-32 residues) as a value of 27 residues. This means that in subunit chains ~36 residues per rod/hook subunit and ~54 residues per flagellin subunits are not involved in generating the entropic force. The final row of Extended Data Table 2 gives the estimate value of the entropic pulling force at the export machinery, if all of a single subunit was free in the channel. These values show that for the rod substructure, just one rod subunit extended along the channel is sufficient to generate a force F M as it gradually crystallizes at the tip. In contrast, for the hook substructure, just one hook subunit would create far too great a force, i.e. there will be several linked hook subunits in the channel. In fact, using the expression for the force, F(R/Na), it is evident that there will be two hook subunits in the rod/hook channel of length R = 90nm. Of course, for the potentially long filament, the number of linked flagellin subunits will be high. Again, using F(R/Na), we estimate that there would be ~150 linked flagellin subunits over the length R = 20µm to provide F = F M at the export machinery. The proposed chain mechanism of subunit transport is essentially equilibrium: the pulling force F(R/Na) has to be the same along the whole length of the linked unfolded subunit chain (by Newton s third law). Therefore, in this model the rate of subunit delivery to the flagellum tip appears to be determined by the rate of crystallization into the structure. The motion of unfolded subunits through the channel under this force is very different from the reptation diffusion discussed earlier. However, we can use the effective friction (that leads to the reptation diffusion coefficient estimate D T ) and find the minimum velocity the chain of subunits can acquire under the pulling force. For the long flagellar filament, to achieve the maximum force F M each individual flagellin subunit is stretched to Δx 90 nm. The friction of just one such subunit against the channel is 11

12 RESEARCH SUPPLEMENTARY INFORMATION equal to k B T/D T and if we assume there are a maximum of 150 subunits in the linked chain over the longest filament length, the overall friction constant would be γ N s/m. Since the pulling force F always self-adjusts to 30 pn, the minimum velocity of motion is equal to (F/γ) 3.6 µm/s, which is equivalent to the rate of c.40 subunits of length Δx per second. If the filament is shorter, the friction constant is lower and the speed can be higher. Obviously this simply means that friction is not the rate-limiting factor in the progression of the chain of subunits along the channel the chain will move essentially freely following the rate of crystallization at the tip. Could the chain mechanism operate in assembly of needles that deliver virulence effectors? These cell surface structures utilize export machineries that are closely related to that used by flagella, and the similarities and seeming differences in the assembly of subunits into these functionally distinct cell surface structures has been discussed 1. As with flagella assembly, unfolded structural subunits are delivered across the bacterial membrane before they transit through a narrow central channel and incorporate at the structure tip 3,66,67. Furthermore, structural subunits have helical termini that could conceivably link as parallel coiled-coils. Nevertheless, it is not known if needle growth rate is constant, and experimental investigation will be needed to establish whether the chain mechanism operates during transit of structural components to the needle tip, or indeed of the virulence effectors, which are released and not incorporated into the structure. 12

13 SUPPLEMENTARY INFORMATION RESEARCH Supplementary References 46. Doi, M. & Edwards, S. F. The Theory of Polymer Dynamics (Oxford Unvi. Press, 1986) 47. Tanner, D. E., Ma, W., Chen, Z. & Schulten, K. Theoretical and computational investigation of flagellin translocation and bacterial flagellum growth. Biophys. J. 100, (2011) 48. Minamino, T., Imada, K. & Namba, K. Mechanisms of type III protein export for bacterial flagellar assembly. Mol. Biosyst. 4, (2008) 49. Prinsen, P., Fang, L.T., Yoffe, A.M., Knobler, C.M. & Gelbart, W.M. The force acting on a polymer partially confined in a tube. J. Phys. Chem. B, 113, (2009) 50. Muthukumar, M. Polymers under Confinement. Adv. Chem. Phys. 149, (2012) 51. Piguet, F. & Foster, D. P. Translocation of short and long polymers through an interacting pore. J. Chem. Phys. 138, (2013) 52. O'Brien, E. P., Stan, G., Thirumalai, D. & Brooks, B. R. Factors Governing Helix Formation in Peptides Confined to Carbon Nanotubes. Nano Lett. 8, (2008) 53. Bhushan, S. et al. α-helical nascent polypeptide chains visualized within distinct regions of the ribosomal exit tunnel. Nature Struct. Mol. Biol. 17, (2010) 54. Wilson, D.N. & Beckmann, R. The ribosomal tunnel as a functional environment for nascent polypeptide folding and translational stalling. Curr. Opin. Struct. Biol. 21, (2011) 55. Lubensky, D. K. & Nelson, D. R. Driven polymer translocation through a narrow pore. Biophys. J. 70, (1999) 56. Lifson, S. & Jackson, J. L. On the self-diffusion of ions in a polyelectrolyte solution. J. Chem. Phys. 36, (1962) 13

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