Hydrodynamics of bacteriophage migration along bacterial flagella
Summary (3 min read)
- Hydrodynamics of bacteriophage migration along bacterial flagella Panayiota Katsambaa and Eric Lauga† Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, United Kingdom (Dated: November 2, 2018) Bacteriophage viruses, one of the most abundant entities in their planet, lack the ability to move independently.
- Indeed, phages have been killing bacteria for way longer than humanity has been fighting against bacterial infections, with as many as 1029 infections of bacterial cells by oceanic phages taking place every day [11, 12].
- Once in contact with a rotating flagellar filament, it is anticipated that the phage fibres will wrap along the short-pitch grooves.
- The authors in Ref.  examined flagellar filaments with different polymorphic forms, since the different arrangements of the flagellin subunits give rise to grooves with different pitch and chirality , as shown in Fig. 3B.
- The hydrodynamic torque actuating the translocation is provided by the parts sticking out in the bulk.
- As their first model, the authors consider the flagellar filament as a straight, smooth rod aligned with the z-axis and of radius Rfl.
- The phage has a capsid head of size 2ah, a tail of length Lt and fibres that wrap around the flagellar filament.
- The helical shape of the fibres has helix angle α, as shown in Fig.
- Assuming the phage to move rigidly and working in the laboratory frame, every point r on the phage moves with velocity Uez + ωpez ∧ r.
- The purpose of their calculation is to compute the two unknown quantities, U and ωp, in terms of ωfl by enforcing the overall force and torque balance on the phage along the z-axis.
II.2. Forces and moments
- In order to calculate the forces and torques acting on the tail and fibres the authors use the resistive-force theory of viscous hydrodynamics (RFT in short) [30, 31].
- This drag anisotropy is at the heart of the propulsion physics for microorganisms such as bacteria and spermatozoa .
- The symbols ζ⊥,t, ζ‖,t are the drag coefficients for motion perpendicular and parallel to the local tangent, with ζ‖,t ≡ ρtζ⊥,t and the velocity of the tail relative to the fluid is ureltail(s) = ωp (ez ∧ rt) + Uez. (6) For the fibres, the authors use the version of RFT modified to capture the motion of slender rods near a surface.
- These results are valid in the limit in which the distance d between the fibre and the surface of the flagellar filaments is much smaller than the radius of the flagellar filament (d Rfl), such that the surface of the smooth flagellar filament is locally planar.
- The authors thus proceed by considering the two limiting geometries of long- and short-tailed phages.
II.4.1. Long-tailed phages
- The authors use below the χ-phage as a typical long-tailed phage, whose detailed dimensions are reported in Ref. .
- Å between parallel sides (that is 2ah ≈ 650− 675 Å).
- From this the authors see that they can safely assume that Rfl, ah Lt, Lfib.
- The authors variables are thus divided into the short lengthscales of ah, Rfl and the long lengthscales of Lfib, Lt. With these approximations the authors obtain the translocation speed as Ulong ≈− hωflRfl(1− ρfib) sinα cosα Glong, (32) Glong = ζ⊥,fibLfib [ 1 3ζ⊥,tLt(1− t 2 z) + [ ζ⊥,tRfltx + 6πµah(1− t2z) ]] 1 3ζ⊥,tLt(1− t2z) [ ζ⊥,tLt [1− (1− ρt)t2z] + ζ⊥,fibLfib(sin2 α+ ρfib cos2 α) ] (33) ≈ ζ⊥,fibLfib[ ζ⊥,tLt [1− (1− ρt)t2z] + ζ⊥,fibLfib(sin2 α+ ρfib cos2 α) ] , (34) with a relative error of O (ah/Lt, ah/Lfib, Rfl/Lt, Rfl/Lfib).
- Details of the approximation are given in the Supplementary Material (see ).
II.4.2. Short-tailed phages
- Phages with very short tails that use their fibres to wrap around flagellar filaments are equivalent geometrically to phages that use their entire tail for wrapping since in both cases there is a filamentous part of the phage wrapped around the flagellar filament and the head is sticking out in the bulk close to the surface of the filament.
- In order to avoid any confusion, the authors will carry out the calculations of this section using the geometry of short-tailed phages, and assume that (i) the tail is negligible and (ii) the fibres are wrapping around the flagellar filament.
- Firstly, and most importantly, both results for the translocation speeds in Eqs. 32 and 35 have the common factor −hRflωfl(1 − ρfib) sinα cosα which is multiplying the positive dimensionless expressions Glong and Gshort respectively.
- Secondly, the factor (1− ρfib) reveals that translocation requires anisotropy in the friction between the fibres and the surface of the flagellar filament (i.e. ρfib 6= 1).
- The authors observe that terms involving Lfib appear in both the numerator and denominator of Eq. 34.
- As a more refined physical model, the authors now include in this section the mechanics arising from the microscopic details of the grooved surface of the flagellar filament due to the packing of the flagellin molecules and modify the previous calculation in order to account for the motion of the phage fibres sliding along the helical grooves.
III.2. Forces and moments
- The details of the interactions between the phage fibres and the grooves are expected to be complicated as they depend on the parts of the flagellin molecules that make up the groove surface and interact with the proteins that the fibres consist of.
- These interactions could originate from a number of short range intermolecular forces, for example electrostatic repulsion or Van der Waals forces.
- The authors model here the resultant of the interaction forces acting on the fibre sliding along the grooves as consisting of two parts, a drag and a restoring force, as shown in the inset of Fig.
- Finally, from Eq. 60, the translocation velocity along the z-axis is calculated as U = V cosα.
- III.4. Two limits: long vs short-tailed phages.
III.4.1. Long-tailed phages
- Under the approximations relevant for long-tailed phages such as χ-phage described in §II.4, i.e. Rfl, ah Lt, Lfib, the translocation velocity along the z-axis gets simplified to Ulong = −hRflωfl sinα cosαGlong, (62) Glong = L2t [ 1 3ζ⊥,tLt + 6πµah ] (1− t2z) L2t [ 1 3ζ⊥,tLt + 6πµah ] (1− t2z) sin2 α+ µ̃R2flLfib · (63) with the details of the approximation given in the Supplementary Material (see ).
III.4.2. Short-tail phages
- In the case of short-tail phages, the authors assume that the tail is negligible and that the fibres are wrapping around the flagellar filament.
- Interpretation and discussion of the results Similarly to §II.4.3, the authors interpret and compare the results in Eqs. 62 and 64.
- Here again, the crucial factor −hRflωfl sinα cosα appears in both equations multiplying a positive, non-dimensional expression, and the authors obtain the correct directionality and speed of translocation in agreement with Ref. .
- The presence of the term µ̃R2flLfib in the denominator implies that the sliding drag from the fibre decreases the translocation speed, and longer fibres give a decreased speed.
- The authors now illustrate the dependence of the translocation speed on the geometrical parameters of the phage, namely Lt and Lfib, according to their model of translocation along grooved flagellar filaments.
- The authors carried out a first-principle theoretical study of the nut-and-bolt mechanism of phage translocation along the straight flagellar filaments of bacteria.
- The main theoretical predictions from their two models, Eqs. 32, 35, 62 and 64, give the phage translocation speed, U , in terms of the phage and groove geometries and the rotation rate of the flagellar filament, in the two relevant limits of long- and short-tailed phages.
- The common prefactor in the formulae for the translocation speed along the filament, U ∼ −hωflRfl sinα cosα, appears in the expressions from both models.
- The authors conjecture that the second model with its explicit inclusion of the grooves should be closer to the real-life situation.
- The authors hope that the modelling developed in this paper will motivate not only further theoretical studies along those lines but also more experimental work clarifying the processes involved in the wrapping and motion of the fibre in the grooves.
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