Hook length of the bacterial flagellum is controlled to nanometer-scale for optimal motility performance

1Humboldt-Universität zu Berlin, Institute for Biology – Bacterial Physiology, Berlin, Germany 9 2Junior Research Group Infection Biology of Salmonella, Helmholtz Centre for Infection Research, 10 Inhoffenstraße 7, 38124 Braunschweig, Germany 11 3School of Physics and Astronomy, The University of Edinburgh, Peter Guthrie Tait Road, Edinburgh, EH9 12 3FD, UK 13 4Department of Medicine/MED3, Chair of Pharmacology, University of Fribourg, Chemin du Musée 5, CH14 1700 Fribourg, Switzerland 15 5Microbiology and Molecular Genetics, Michigan State University, 567 Wilson Road, East Lansing, MI 16 48824, USA 17 6Braunschweig Integrated Centre of Systems Biology (BRICS), Rebenring 56, D-38106 Braunschweig, 18 Germany 19 7Department of Medicine/MED3, Microbiologie, University of Fribourg, Chemin du Musée 5, CH-1700 20 Fribourg, Switzerland 21 8Central Facility for Microscopy, Helmholtz Centre for Infection Research, Inhoffenstraße 7, 38124 22 Braunschweig, Germany 23 9Systems Immunology, Helmholtz Centre for Infection Research, Inhoffenstraße 7, 38124 Braunschweig, 24 Germany 25 26


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Most bacteria swim in liquid environments by rotating one or several flagella. Each 35 flagellum consists of a long external filament that is connected to a membrane-embedded 36 basal-body by a short, curved structure: the hook. The length of the hook is controlled on 37 a nanometer-scale by a sophisticated molecular ruler mechanism and it functions as a 38 flexible universal joint allowing transmission of motor torque to the filament. However, 39 why its length is stringently controlled has remained elusive. We engineered and studied 40 a diverse set of hook length variants of Salmonella enterica. Measurements of plate-41 assay motility, single-cell swimming speed and directional persistence in quasi 2D and 42 population-averaged swimming speed and body angular velocity in 3D revealed that the 43 motility performance is optimal around the wild type hook length. We conclude that too 44 short hooks may be too stiff to function as a junction and too long hooks may buckle and 45 create instability in the flagellar bundle. Accordingly, peritrichously flagellated bacteria 46 move most efficiently as the distance travelled per body rotation is maximal and body 47 The bacterial flagellum is composed of three main structural parts: (i) a membrane-81 embedded basal body; (ii) a several micrometer long external filament; and (iii) the hook, 82 a linking structure that connects the basal body and the rigid filament (5). The basal body 83 complex functions as a motor to rotate the flagellum and includes a proton motive force 84 (pmf) dependent flagellum-specific protein export machine (6, 7). Assembly of the 85 flagellum initiates with formation of the export machinery within the cytoplasmic 86 membrane, followed by formation of a rod structure that traverses the periplasmic space. 87 Assembly of the hook starts upon completion of the P-and L-rings, which form a pore in 88 the outer membrane and polymerize around the distal rod (8,9). The extracellular, flexible 89 hook structure functions as a universal joint and allows the conversion of the torque 90 generated by the cytoplasmic motor into rotational motion of the flagellar filament 91 irrespective of the cell body orientation (10). The self-assembly of the rod, hook and 92 filament structures are controlled by different regulatory mechanisms. The length of the 93 flagellar rod of ~25 nm is determined by the width of the periplasmic space (11), whereas 94 the growth rate of flagellar filaments decreases with length and is controlled through pmf-95 dependent injection and diffusive movements of filament subunits inside the flagellar 96 secretion channel (12). The length of the hook structure is controlled to ~55 nm through 97 a molecular ruler mechanism (Fig. 1A) (13,14). Upon termination of hook growth, the 98 export apparatus switches secretion specificity from early ('rod/hook'-type) to late 99 ('filament'-type) substrate secretion (15). This mechanism ensures that a functional hook-100 basal-body complex is present, on top of which the long flagellar filament made of several 101 tens of thousands flagellin molecules can assemble (16). Control of hook length is lost in 102 mutant strains defective of the fliK gene (17) and polymerization of hook subunits 103 continues beyond the physiological length of 55 nm. This results in a wide, uncontrolled 104 distribution of hook lengths, called polyhooks, and coincides with a failure of the export apparatus to switch secretion specificity from early to late substrate secretion mode. 106 Thus, the FliK protein is responsible for both the measurement of hook length and the 107 transmission of the hook growth termination signal to the export apparatus, after which 108 the switch in substrate specificity occurs. Evidence has accumulated demonstrating that 109 FliK functions as a molecular ruler. FliK takes intermittent length measurements 110 throughout the assembly of the hook structure and terminates further hook growth when 111 the hook has reached a length of ~55 nm or longer (14,18,19). speed ω, causing the cell body to rotate in the opposite direction, and propel the cell 117 forward at speed v (run phase). Additionally, the body has a secondary motion 118 superimposed on the rotating axis called body wobbling, characterized by an angular 119 velocity Ω, due to the flagella bundle pushing the body off-axis (20,21). When one or 120 more of the motors turn clockwise (CW), their associated flagella separate from the 121 bundle, causing a random change in the cell orientation (tumble). After all the motors have 122 reverted to CCW rotation, the flagellar bundle reforms and the bacterium swims in a new 123 direction. In homogenous environment (absence of gradient), the run-and-tumble 124 behavior results in an effective diffusive motion over large distances. Biasing the tumbling 125 rate allows chemotaxis, i.e. seeking attractants or escaping repellents in a chemo-126 gradient (22)(23)(24)(25). For flagellated bacteria swimming in a Newtonian liquid, e.g. buffer, v 127 is proportional to Ω and ω, which are determined by the geometry of both body and 128 flagellar bundle (26). 129 The relationship between hook length and motility is poorly studied. Loss of hook length 130 control diminishes motility (15); however, the mechanisms have not yet been examined. 131 The physiological benefit of tight hook length control remains unclear and was highlighted 132 recently as one of the big questions in bacterial hydrodynamics (27). We hypothesized 133 that the hook in its function as a universal joint might play a crucial role in the bundle state 134 and/or its stability. Thus, the need for proper hook length control would have evolved to optimize motility in diverse environments. Here, we analyze the impact of variations in 136 hook length on the motility performance in semi-solid agar plate as well as liquid medium 137 in quasi 2D and 3D by single-cell tracking, differential dynamic microscopy (DDM) and 138 dark-field flicker microscopy (DFM). Our results reveal an optimal motility performance for 139 a wild type hook length of ~55 nm in respect to swimming speed, directional persistence 140 and propulsive efficiency. We conclude that the molecular ruler mechanism evolved to 141 control an optimal length of the hook structure for maximized motility performance via a 142 more stable bundle formation. Thus, a mechanism to control flagellar hook length 143 optimizes the cell's motility performance by minimizing random directional change during 144 the run phase. 145

FliK molecular ruler deletion and insertion variants are secreted and functional 149
The flagellar hook length is determined by secretion of a molecular ruler protein, FliK. We 150 engineered FliK mutants varying from 310 to 785 amino acids (aa) length using insertions 151 of α-helical parts of a homologous molecular ruler of the related virulence-associated type 152 III secretion system of Yersinia, YscP, or deletions of the central, α-helical domain of FliK 153 ( Fig. 1B) (14,18,28). All FliK mutants were secreted and retained the ability to induce 154 the switch in secretion specificity to late substrate secretion as evidenced by comparable 155 levels of external flagellin FliC and induction of a Class 3 gene reporter (Fig. 1C, 156 Supplementary Fig. S1). The flagellation levels of the FliK mutants were not significantly 157 different with an average of 3.8 ± 1.5 flagella per cell (averaged across all FliK mutants) 158 compared to 3.6 ± 1.3 flagella per cell for the wild type (wt) (Fig. 1D, Supplementary Fig.  159 S2). We next purified hook basal body (HBB) complexes of the FliK length variation 160 mutants and determined the length of the hook structures by transmission electron 161 microscopy. FliK mutants ranging from 363 to 785 residues lengths retained a tight control 162 of hook length from 42 nm to 135 nm with a linear increase in hook length of 0.2 nm per 163 inserted amino acid (Fig. 1E, Supplementary Fig. S3). An increase of 0.2 nm per inserted 164 amino acid is consistent with an alpha-helical conformation of the ruler domain of FliK, as 165 suggested before (18). The two shortest FliK mutants, FliK310 and FliK335, displayed 166 partially uncontrolled hook lengths, indicating that a minimal FliK length is needed for 167 effective hook length control (Fig. 1E, Supplementary Fig. S3). 168 169 Hook length mutants display a pronounced motility defect in semi-solid medium 170 We next analyzed the set of hook length variation mutants for their motility phenotype in 171 semi-solid agar plates (Fig. 1F+G, Supplemental Fig. S4). The motility halo size was 172 substantially decreased for mutants with shorter or longer hooks (FliK ≤ 363 aa and FliK 173 ≥ 520 aa) and peaked at hook lengths around the wt length of 55 nm (FliK = 405 aa). The 174 observed motility defect for long and short hook mutants was independent of the motility 175 buffer and agar concentration, while prolonged incubation highlighted the motility 176 differences ( Supplementary Fig. S4+S5). These observations suggest that the hook 177 length of the wt is in some way optimal (Fig. 1F+G). We note, however, that the motility 178 phenotype in semi-solid agar plates is a complex combination of bacterial growth, motility 179 and chemotactic behavior (29, 30). We found that bacterial growth rate was not impaired 180 in the FliK mutants ( Supplementary Fig. S5). Further, the presence of motility halos for all 181 hook length mutants indicated functional chemotaxis. However, in order to decouple 182 chemotaxis from motility and to reveal the effect of hook length on the swimming behavior, 183 we next performed single-cell tracking experiments. 184 185 Single-cell tracking reveals that hook length mutants have lower swimming speed 186 and shorter directional persistence 187 To identify which behavioral parameter explains the phenotype of hook length mutants in 188 semi-solid medium, we characterized single-cell behavior in a quasi 2D environment as 189 described before (31). In the absence of chemotactic signals, we observed the effect of 190 hook length variation on the basal run-and-tumble behavior. We found that cells with 191 longer hook lengths (>75 nm; FliK > 520 aa) have reduced average swimming speed and 192 tumble more frequently (Fig. 2A+B, Supplementary Movie S1). Consequently, the 193 directional persistence and the effective diffusion coefficient (32) of the mutants are reduced (Fig. 2C+D). Therefore, longer hooks affect the cell's ability to explore their 195 environment. We propose that these observations are largely responsible for the motility 196 phenotypes that we observed in semi-solid agar plates. 197 We hypothesized that variations in the hook length increase instability during formation Accordingly, we assessed the motility behavior of smooth-swimming hook length mutants 208 using 2D single-cell tracking. In the ΔcheY genetic background (che − ), hook length 209 mutants still have reduced swimming speeds and smaller diffusion coefficients when 210 compared to the wt hook length (Fig. 2E+H). Cells with hooks longer than 75 nm have a 211 higher probability of changing direction suddenly; events, which we will refer to as 212 'pseudo-tumbles' (Fig. 2F). In non-chemotactic, smooth-swimming ΔcheY mutants, these 213 'pseudo-tumbles' cannot be caused by reversions of the flagellar motor. Finally, che − hook 214 length mutants displayed a reduced directional persistence (Fig. 2G), because of a 215 general increase in the probability of large changes in swimming direction 216 ( Supplementary Fig. S6). These observations suggest that the poorer swimming 217 performance in hook-length mutants is likely to be primarily due to decreased stability of 218 the flagellar bundle. 219 To further analyze the dependence of 'pseudo-tumble' events on flagellar hook length 220 and to estimate a tumbling rate, we calculated the directional autocorrelation function and  The observed trajectories of non-chemotactic, hook length mutants with a 'pseudo-226 tumble' phenotype were fitted to a simple run-and-tumble migration model 227 (Supplementary Text 2, Supplementary Fig. S8), taking into account the curvature of the 228 tracks due to hydrodynamic interactions of bacteria swimming in quasi 2D close to the 229 surface. The fit of the run-and-tumble model to the trajectories of the non-chemotactic, 230 hook length mutants suggest that the directional persistence decreases with increasing 231 hook length (Fig. 2G, Supplementary Fig. S7). We concluded that the pseudo tumble 232 events of the non-chemotactic mutants arise from a mechanical instability of the hook, 233 similar to the tumbling mechanism of V. alginolyticus (34). 234 235

Non-chemotactic long hook mutants display a pseudo-tumble behavior 236
The conclusion we have reached leads to an interesting prediction of the behavior of hook 237 mutants in agar gels. Chemotactic (che + ) bacteria can navigate through an agar gel matrix 238 due to their run-and-tumble motility behavior ( Fig. 3A) (29). However, non-chemotactic, 239 smooth-swimming (∆cheY) cells are trapped because tumbling is required to efficiently 240 escape the agar gel matrix (Fig. 3B). However, if the increased flagellar bundle instability 241 in hook mutants conferred on them a 'pseudo-tumble' phenotype, then we may expect 242 che − long hook mutants to regain the ability to migrate through an agar gel matrix. A 243 similar pseudotaxis behavior has previously been observed for non-chemotactic E. coli 244 mutants, where point mutations in the switch complex proteins FliG and FliM allowed 245 random motor rotational switching in the absence of chemotactic stimuli (29), as well as 246 in a non-chemotactic A. tumefaciens strain, where suppressor mutations in the hook, the 247 fliK gene and the motor force generators were isolated that allowed the che − cells to 248 navigate through semi-solid agar (35). 249 We analyzed the motility behavior of non-chemotactic, smooth-swimming hook length 250 mutants to test our prediction. The motility halo size of che + hook length mutants peaked 251 around the wt hook length (Fig. 1F, Fig. 3A). In contrast, che − long hook mutants 252 performed substantially better than che − mutants with shorter and wt hook length (Fig.  253 3B), in accordance with our suggestion that long hook lengths confer a pseudo-tumble phenotype. We also noticed a big difference in the speed of spreading, che − mutants 255 needed to be incubated ~5× longer to reach comparable halo sizes. This indicates that 256 pseudo-tumbling events occur less frequently than tumbling, i.e. due to motor reversal. 257 There is an interesting difference in appearance, che − colonies form disks rather than 258 typical chemotactic rings confirming the pseudo-tumble events are not related to 259 chemotaxis and thus not associated with motor reversal. 260 261 Motility parameters are optimal for wt hook length in 3D liquid environment 262 All the experiments described so far pertain to bacteria swimming close to a hard surface 263 or in a gel matrix. In a final set of experiments, we measured swimming speed v and body 264 angular velocity Ω using high-throughput DDM and DFM in a 3D liquid environment (26, 265 36, 37). DDM and DFM experiments were performed with bacteria grown in TB medium 266 at 30 °C, which did not affect the overall motility behavior ( Supplementary Fig. S9, 267 Supplementary Text 3). We recorded over 10 4 cells swimming in a 3D liquid medium (i.e., 268 in bulk) for each of the hook length mutants. We found that the population-averaged These experimental findings are independent of the tumbling behavior, as run-and-tumble 286 strains displayed similar hook length dependency compared to the smooth-swimmers 287 ( Fig. 4, Supplementary Fig. S12). This was expected because the motility parameters are 288 measured over a length-scale corresponding to the run length (<15 μm, distance between 289 two tumble events). Differences are found in absolute values of R between run-and-290 tumble and smooth-swimmers at a given hook length due to tumbling events 291 ( Supplementary Fig. S10). However, the relative path straightness In summary, our results provide substantial evidence that hook length control in 296 peritrichously flagellated bacteria has been evolutionary selected to optimize the stability 297 of the flagellar bundle and thus maximize swimming performance. We found that 298 swimming speed and directional persistence of the trajectories of swimming bacteria are 299 tightly related to the length of the hook structure. Recently, small polymorphic changes in 300 the flagellar bundle have been observed without filaments leaving the bundle. These 301 polymorphic changes create deflection in the swimming trajectory and have been 302 associated speculatively with motor reversal (33). Our present work suggests these 303 events might be associated with the run phase rather than motor reversal as they also 304 occur in smooth swimming mutants. Thus, the role of hook length control may be to 305 minimize such polymorphic changes by maximizing stability of the flagellar bundle during 306 the run phase. We thus propose that the hook length control mechanism evolved to match 307 the requirements for effective formation of the flagellar bundle. Stiffness is an inverse 308 function of the hook length, e.g. stiffness increases with decreasing hook length. Too 309 short hooks are too stiff to function as a universal joint whereas too long hooks may buckle 310 and create too much instability, thus necessitating the need to control for an optimal length 311 of the hook structure. Further, the observed higher processivity of hooks with wt length appears to result in more efficient propulsion i.e. the cells are able to achieve higher 313 swimming speeds for the same energy expenditure. Our results are also consistent with 314 previous observations, where the wt hook of E. coli was modified to be twice more rigid 315 by binding of streptavidin and the flagellar bundle was unable to form (38). Hook length 316 control may also play a major role in other bacteria. Son and co-workers demonstrated 317 for the polar flagellated Vibrio alginolyticus that a buckling instability of the hook enables 318 the cell to 'flick' and re-orient (34). They suggested buckling occurs when the viscous 319 loads, i.e. force and torque exerted on the hook by the cell body and flagellum, exceed 320 the hook's buckling critical threshold. The associated critical torque and force are 321 inversely proportional to the hook length and its square, respectively. Thus, we expect 322 that changing the hook length of polar flagellated bacteria could dramatically alter the 323 instability region and enable more "flicking" events, which in turn would affect motility and 324 chemotactic behavior. 325 The function of the bacterial flagellum relies on the self-assembly of major components 326 with defined or self-limiting lengths (11, 12). Incomplete assembly or small deviations from 327 the blueprint result in a non-functional motility organelle with devastating consequences 328 for the organism. Thus, regulatory mechanisms evolved to determine the correct 329 dimensions of flagellar sub-assemblies and to ensure robust assembly of a functional 330 motility organelle. Our results demonstrate the selective advantage of a mechanism that 331 controls the length of the hook structure. In particular, our results suggest that the 332 molecular ruler mechanism evolved to determine the optimal length of the flagellar hook.  Hook-basal-body purification. Purification of hook-basal body complexes without C-365 ring was performed as described (43) with slight modifications. Briefly, bacteria were 366 grown in 500 ml culture until OD600 1 -1.5. Cells were harvested (8,000g, 4 °C, 10 min) 367 and resuspended in 30 ml ice-cold sucrose solution (0.5 M sucrose, 0.1 M Tris-HCl, pH 368 8). 3 ml lysozyme (10 mg/ml in sucrose solution) and 3 ml 0.1 M EDTA (pH 8) were added 369 to digest the peptidoglycan layer and to prepare the spheroplasts. The suspension was 370 stirred for 30 min on ice. 3 ml 10% Triton-X and 3 ml 0.1M MgSO4 were added to lyse 371 the spheroplasts. For complete lysis of the spheroplasts the mixture was stirred overnight 372 at 4 °C. Afterwards 3 ml 0.1M EDTA (pH 11) was added. Unlysed cells and cell debris 373 was pelleted by centrifugation (17,000 g, 4 °C, 20 min) and the pH was adjusted to pH 11 374 by adding 5 N NaOH. Following centrifugation (17,000g, 4 °C, 20 min) the flagella were 375 pelleted by ultracentrifugation (100,000 g, 4 °C, 1 h). The pellet was resuspended in 35 376 ml ice-cold pH 11 buffer (10% sucrose, 0.1 M KCl, 0.1% Triton-X) and centrifuged (17,000 377 g, 4 °C, 20 min) and the flagella were pelleted by ultracentrifugation (100,000 g, 4 °C, 1 378 h). The pellet was resuspended in 35 ml ice-cold TET-solution (10 mM Tris-HCl pH 8, 5 379 mM EDTA, 0.1% Triton-X) and the mixture again subjected to ultracentrifugation (100,000 380 g, 4 °C, 1 h). For depolymerization of the filaments, the flagella were resuspended in 35 381 ml pH 2.5 buffer (50 mM glycine, 0.1% Triton-X) and left at room temperature (RT) for 30 382 min. The debris was pelleted by centrifugation (17,000 g, 4 °C, 20 min) and the HBB were 383 collected by ultracentrifugation (100,000 g, 4 °C, 1 h). The pellet was air-dried and 384 resuspended in 200 µl TE solution (10 mM Tris-HCl pH 8, 5 mM EDTA). The samples 385 were stored at 4 °C and imaged the following day by negative staining electron 386 microscopy. 387 388 Fluorescent microscopy. FliC-phase locked bacteria were grown until mid-log phase in 389 LB or TB medium and immobilized on pre-coated 0.1% poly-L-lysine coverslips in a flow 390 cell as described (14). Cells were fixed with formaldehyde (2% final) and glutaraldehyde 391 (0.2% final). Flagella were stained using polyclonal FliC antibody (anti-rabbit) and anti-392 rabbit coupled to Alexa Fluor 488. The membrane was stained using N-3-393 Triethylammoniumpropyl-4-6-4-Diethylamino Phenyl Hexatrienyl Pyridinium Dibromide (FM 4-64) and the DNA using 4′,6-diamidino-2-phenylindole (DAPI). Images were taken 395 using a Zeiss AxioObserver microscope at 100x magnification. 396 397 Electron microscopy. Hook samples were stored at 4 °C to straighten the hooks and 398 facilitate length measurements. The samples were negatively stained by 2% aqueous 399 uranyl acetate on a carbon film. Samples were imaged using a Zeiss TEM 910 at an 400 acceleration voltage of 80 kV with calibrated magnifications. Images were recorded using 401 a Slow-Scan CCD camera (ProScan, 1024x1024, Scheuring, Germany) with ITEM-402 Software (Olympus Soft Imaging Solutions, Münster, Germany) software. Hook length 403 was measured using ImageJ. 404 405 Luciferase assay. The luciferase assay was performed as described before (19) placed onto a microscope, and movies were recorded. Importantly, these five steps are 442 performed in parallel for a range of hook mutants, which allows precise measurement of 443 potential small difference in motility. Details are given below. with e.g. L1 < L2. Theoretically, swimmers with a straight path yield R = 1. Deviations 481 from straight-path, due to e.g. rotational diffusion or tumbling events, will increase this 482 ratio above 1.
From DFM analysis of the dark-field movies, the body angular velocity Ω averaged over 484 ~10 4 cells in a few seconds was extracted. Details can be found elsewhere (26). Briefly, 485 the power spectrum of the flickering image of individual cells was Fourier transformed 486 and the lowest frequency peak in the average power spectrum was identified as Ω / 2 π. 487 Acknowledgments: 488 We thank Nadine Körner for expert technical assistance, Kelly T. Hughes for generous 489 donation of strains and Howard Berg for discussions.                             Uncontrolled hook length mutants are shown in red. All parameters were obtained by 691 normalizing the time-dependent parameters to the wt (FliK405) and by averaging over the 692 time-window 10 min<t<60 min.

Observations from modeling of single-cell behavior 700
The angular velocity ψ is related to the tangential velocity v and the radius s of the 701 curvature of the cell trajectory as = ⁄ , making this parameter proportional to the cell 702 velocity and inversely proportional to the radius of curvature. We observed that the 703 angular velocity initially increases with hook length. This corresponds to a decrease in the 704 curvature radius of the trajectories, similar as observed before for sperm cells swimming 705 near surfaces (47). The angular velocity ψ decreases for longer trajectories, however, as 706 a consequence of decreased run speed v. We also note that we were unable to reproduce 707 the experimental observables of the short-hook (FliK335) che − mutant by a single 708 parameter couple (ψ,λ) in the context of our run-and-tumble model, which suggested that 709 the movement of those mutants is not diffusive (Supplementary Fig. S7). We suggest that (2) ⃗( ) = (− sin , cos ) 726 From Eq. 2, we see that the initial velocity of the bacterium is 728 (3) ⃗(0) = (0, ν). 729 730 Next, we assume that after some time τ V has passed, the bacterium will tumble and start 731 moving along a different circle of the same radius and with the same angular velocity, i.e. 732 the velocity will be changed by a phase shift ϕ X ∈ [0,2π). We assume that tumbles can 733 make the bacterium reorient towards any possible direction, and that the reorientations 734 after each tumble are independent of one another (Markov property) i.e. ϕ X , i ∈ ℕ are 735 stochastic variables distributed as 736

738
We also assume that the intertumbling times τ X , i ∈ ℕ are completely independent of each 739 other, i.e. the intertumbling times are stochastic variables distributed as 740 where the parameter λ > 0 of the distribution is known as the tumbling rate. 743

744
The angular autocorrelation, defined as 745 where Θ(x) is the Heaviside function, due to the Markov property and statistical properties 752 of the phase shifts ϕ X , and for the purposes of calculating the angular correlation this 753 expression can be rewritten as 754 (7) v j⃗(t) = ν(− sin(νt + ξϕ) , cos(νt + ξϕ)), 755 756 where ϕ =: ϕ V + ∑ Θ(t − τ X )ϕ X X:_ is an effective phase shift, distributed following Eq. 4 757 and 758 The stochastic variable ξ has a value of ξ = 0 if the bacterium has not tumbled for the first 762 time, and ξ = 1 if it has tumbled at least once. From Eqs. 5 and 8, it is evident that the 763 probability of ξ = 1 is just the cumulative distribution function of τ X , that is 764 The velocity autocorrelation function is defined as the expected value of the angular 773 autocorrelation, 〈g(t)〉. Therefore, in this model, the velocity autocorrelation fucntion is 774 given by 775

Effects of growth medium and temperature on single-cell behavior 796
For experimental reasons, single cell tracking in quasi 2D was performed with cells grown 797 in LB medium at 37 °C, whereas bacteria for the DDM and DFM experiments were grown 798 in TB medium at 30 °C. While the overall motility behavior was remarkably similar 799 between the experimental conditions (Fig. 2, Fig. 4, Supplementary Fig. S9), we observed 800 a substantial decrease in swimming speed for long hook mutants grown in LB at 37 °C. 801 These differences may be attributed to increased flagella numbers per cell for bacteria 802 grown in LB at 37 °C, which presumably exacerbate the increased bundle instability of 803 the long hook mutants (Supplementary Fig. S9).

Supplementary Tables
805 Supplementary Table S1    immunostaining. The flagellar filament was detected using anti-FliC immunostaining 823 (green), membranes were stained using FM-64 (red) and DNA was stained using DAPI 824 (blue). Scale bar = 5 µm. containing 2% yeast extract. Quantification of the swimming motility assay (left panel) and 835 representative soft agar plate (right panel). FliK length in amino acids is indicated; (-) 836 indicates the non-flagellated control strain ΔfliHIJ. Motility halo size was measured using 837 ImageJ and normalized to the wt (dots represent single data points; bars represent the 838 mean; red dots represent the uncontrolled hook length mutants). (B) Time course motility 839 assay of the hook length mutants in 0.3% swimming motility agar plates. Motility halo size 840 was determined every 30 minutes and the average motility halo area of 5 biological 841 replicates is shown. Uncontrolled hook length mutants are shown in red. Measurement of swimming speed v, body rotational speed Ω and processivity P = v / Ω 893 relative to the wild type for hook length mutants grown in in LB at 37 °C and TB at 30 °C.

894
(C) Measurement of swimming speed v, body rotational speed Ω and processivity P = v / 895 Ω relative to the wild type for chemotactic deficient (∆cheY) hook length mutants grown 896 in LB at 37 °C and TB at 30 °C.