Geometrical Constraints on the Tangling of Bacterial Flagellar Filaments

Abstract

Many species of bacteria swim through viscous environments by rotating multiple helical flagella. The filaments gather behind the cell body and form a close helical bundle, which propels the cell forward during a "run". The filaments inside the bundle cannot be continuously actuated, nor can they easily unbundle, if they are tangled around one another. The fact that bacteria can passively form coherent bundles, i.e. bundles which do not contain tangled pairs of filaments, may appear surprising given that flagella are actuated by uncoordinated motors. In this article, we establish the theoretical conditions under which a pair of rigid helical filaments can form a tangled bundle, and we compare these constraints with experimental data collected from the literature. Our results suggest that bacterial flagella are too straight and too far apart to form tangled bundles based on their intrinsic, undeformed geometry alone. This makes the formation of coherent bundles more robust against the passive nature of the bundling process, where the position of individual filaments cannot be controlled.There is arguably no complex geometrical structure more central to biology than the helix. Two helical strands of nucleic acid spiral around each other to form the blueprint of life 1 , the arteries and vein that make up the human umbilical cord are twisted in a helical pattern 2 , and our human hearts beat with a spiralling contraction of the myocardium 3 . Many large eukaryotic cells, particularly in plants, set up helical flows to enhance chemical transport within the cell 4 and some viruses also take this shape 5 . Prokaryotes too have learnt how to assemble blocks of proteins into a helical structure and put it to good use 6-9 . Indeed, many species of bacteria are able to propel themselves through the viscous medium they inhabit by exploiting an apparatus known as flagellum, whereby a relatively rigid helical filament is actuated by a specialised constant-torque rotary motor 10-12 . In the case of multi-flagellated bacteria, the filaments are swept behind the cell body and rotate together in a coherent bundle, which leads to an interval of straight swimming called a "run" 13 . To change swimming direction, at least one motor must switch its sense of rotation, upon which the associated flagellum will leave the bundle and generate an imbalance of forces. The subsequent reorientation of the cell is called a "tumble" 14 .This run-and-tumble behaviour lies at the heart of bacterial motility. It is known that the initial stage of bundling is enabled by an elastohydrodynamic instability of the hook 15 , and from there onwards both the counter-rotation of the cell-body and hydrodynamic interactions between the filaments contribute to the synchronization and formation of the bundle [16][17][18][19][20][21][22] . Computational studies have investigated the effect of the number of flagella on the motility of the cell 23 , and the role played by polymorphic transformations 24 and mismatched motor torques 25 in the bundling and unbundling...

Bacteria-inspired robotic propulsion from bundling of soft helical filaments at low Reynolds number

“…To fully understand the physics behind bacterial locomotion, the phenomenon such as synchronization and tangling of the bacterial flagella are being investigated. [14][15][16] Besides its complexity in physics, the multi-flagellated mechanism is vital from both robotic and biological perspectives due to the following features: (1) directional stability, 2 (2) redundancy of actuation, 17 (3) chemical secretion using flagella, 18,19 (4) improved efficiency in the swarm and propagation. 20,21 Compared to the multi-flagellated mechanism, locomotion used by bacteria with a single flagellum lacks interaction with one or more flagella, exploiting a similar yet different mechanism.…”

An integral-based spectral method for inextensible slender fibers in Stokes flow

“…Interactions of long, thin, inextensible filaments with a viscous fluid abound in biology, engineering, physics, and medicine. In biology, the swimming mechanisms of flagellated organisms have been of interest for decades, with an initial cluster of studies on how force and torque balances lead to swimming [14,9,12,42], and a more recent focus on flagellar bundling and propulsion [46,52,54]. In physics and engineering, suspensions of high-aspect-ratio fibers have been observed to display non-Newtonian, viscoelastic behavior both experimentally [10] and computationally [51,76].…”

Direct vs indirect hydrodynamic interactions during bundle formation of bacterial flagella

“…Moreover, because the flagellar filaments and especially the hook are flexible, and the interaction between these slender elastic structures, the cell body and the fluid lead to non-linearities that are difficult to compute [27]. Despite these challenges, the dynamics of bundling and unbundling have been the subject of numerous theoretical [28][29][30][31], numerical [32][33][34][35][36][37][38][39][40][41][42][43] and experimental [6,13,16,[44][45][46][47][48][49] studies. Early studies examined the dynamics of filaments without taking into account the cell body [28, 29, 32-35, 39, 45], or ignored elastic effects [43].…”

“…With advances in computational power and the sophistication of numerical methods [50][51][52], more recent work has been able to accurately simulate the swimming dynamics of multi-flagellated motile bacteria [36,38,40]. Some studies also focus on a particular aspect of the problem, such as flagellar polymorphism [41] or the geometrical constraints of entanglement [31].…”