Hydrodynamics of bacteriophage migration along bacterial flagella
Summary (3 min read)
Introduction
- 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. [27] examined flagellar filaments with different polymorphic forms, since the different arrangements of the flagellin subunits give rise to grooves with different pitch and chirality [25], as shown in Fig. 3B.
- The hydrodynamic torque actuating the translocation is provided by the parts sticking out in the bulk.
II.1. Geometry
- 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 [31].
- 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. [22].
- From this the authors see that they can safely assume that Rfl, ah Lt, Lfib.
- Details of the approximation are given in the Supplementary Material (see [35]).
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.
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.
- Long vs short-tailed phages, also known as III.4. Two limits.
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. [27].
- 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.
IV. CONCLUSION
- 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.
Did you find this useful? Give us your feedback
Citations
24 citations
22 citations
13 citations
9 citations
References
362 citations
282 citations
243 citations
212 citations
180 citations
Related Papers (5)
Frequently Asked Questions (17)
Q2. What future works have the authors mentioned in the paper "Hydrodynamics of bacteriophage migration along bacterial flagella" ?
This opens up the possibility of a competition between the nut-and-bolt translocation effect and the possibly opposing drag due to translation, which will vary with the helical angle of the flagellar filament. Future studies could address the transient period of wrapping, where the length of the fibres wrapped around the filament is increasing and the ‘ grip ’ is possibly becoming tighter. 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.
Q3. What is the effect of the term involving Lfib on the fibres?
The presence of the term involving Lfib in the denominator of Eq. 63 leads to a decrease of U with Lfib, and is physically due to an increase of the viscous drag on the fibres as Lfib increases.
Q4. What is the drag anisotropy of the filament?
the very drag anisotropy that allows the rotation of helical flagellar filaments to propel bacteria in the bulk will also enable the rotation of helical fibres around a smooth filament to lead to translocation along the axis of the filament.
Q5. What is the common prefactor in the formulae for the speed of the phage?
The common prefactor in the formulae for the translocation speed along the filament, U ∼ −hωflRfl sinα cosα, appears in the expressions from both models.
Q6. How do the authors capture the effect of the grooves?
The authors implicitly capture the effect of the grooves (i) by imposing that the fibres that emanate from the bottom of the tail of the phage are wrapped around the flagellar filament in a helical shape and (ii) via the anisotropy in the drag arising from the relative motion between the fibres and the rotating flagellar filament.
Q7. What is the effect of the helical wrapping of the fibres on the phage?
The assumption of a helical wrapping of the fibres coupled with this anisotropy simulates the guiding effect of the grooves in this first model by resisting motion perpendicular to the local tangent of the grooves and promoting motion parallel to it.
Q8. what is the restoring force of the fibre?
A simple modelling approach consists of viewing each side13of the groove as repelling the fibre, with the resultant of these forces providing a restoring force hkδbfib(s) per unit length, arising from a potential well 12kδ2 where δ is the distance from the centre of the well, and bfib is the local binormal vector to the fibre centreline,bfib =[ h cosα sin ( sRfl/ sinα) ,− cosα cos ( sRfl/ sinα) , h sinα ] , (44)that lies in the local tangent plane of the surface of the flagellar filament and is perpendicular to the tangent vector tfib of the fibre centreline.
Q9. What is the z-component of the torque on the phage tail?
The drag force and torque due to the motion of the head in the otherwise stagnant fluid are given byFhead = −6πµahurelhead, (13) ez ·Mhead = ez · [ −6πµah ( rh ∧ urelhead )] − 8πµa3hωp, (14)8 with urelh given byurelh = ωp (ez ∧ rh) + Uez. (15)Taking th = ttail, the centre of the head will be located at position rh = rb +
Q10. What is the helical shape of the flagellar filament?
5. With the assumption that the gap between the fibres and the flagellar filament is negligible compared to the radius Rfl of the flagellar filament, the centreline rfib(s) of the fibres, parametrised by6the contour length position s, is described mathematically asrfib(s) = ( Rfl cos ( sRfl/ sinα) , hRfl sin ( sRfl/ sinα) , s cosα ) , −LLfib < s < LRfib, (1)with total contour length Lfib = L L fib + L R fib, where the authors allow for fibres extending to both sides of the base of the tail to have lengths LLfib (left side) and L R fib (right side).
Q11. What is the significance of the factor hRflfl sin co?
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. [27].
Q12. What are the main theoretical observations and predictions of Refs?
These mathematical results capture the basic qualitative experimental observations and predictions of Refs. [23, 27] for the speed and directionality of translocation which are both crucial for successful infection.
Q13. What is the directionality of the phage?
The factor −hωfl gives a directionality for U in agreement with the qualitative prediction of Ref. [27] that CCW rotation will only pull the phage toward the cell body if the phage slides along a right-handed groove.
Q14. What is the z-component of the drag anisotropy for the fibre?
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.
Q15. What is the effect of the term Lt in Eq. 40?
The long-tailed approximation from Eq. 63, through the terms with Lt in both numerator and denominator, is able to capture the increasing behaviour of U with Lt. Physically, this trend is caused by the propulsive terms in ez ·Mtail in Eq. 40 (proportional to L3t ) that increase as Lt increases.
Q16. What is the difference between the two models?
17The important point where the two models deviate from each other is their opposite predictions for the translocation speed as a function of the phage tail length and the phage fibre length.
Q17. What is the effect of the translation on the phage?
there will be an additional hydrodynamic drag on the phage due to the rotation and translation of the flagellar filament.