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Journal ArticleDOI

Hydrodynamics of bacteriophage migration along bacterial flagella

04 Jan 2019-Vol. 4, Iss: 1, pp 013101
TL;DR: In this paper, the authors confirm a 40-year old hypothesis and show that infection can be induced by hydrodynamic forces due to rotation of flagellar filaments used by bacteria for propulsion.
Abstract: Bacteriophage viruses infect and replicate within bacteria. Some phages ride along the flagellar filaments used by bacteria for propulsion. Here we confirm a 40-year old hypothesis and show that infection can be induced by hydrodynamic forces due to rotation of flagellar filaments.

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.

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Hydrodynamics of bacteriophage migration along bacterial flagella
Panayiota Katsamba
a
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 our planet, lack the ability to move
independently. Instead, they crowd fluid environments in anticipation of a random encounter with
a bacterium. Once they ‘land’ on the cell body of their victim, they are able to eject their genetic
material inside the host cell. Many phage species, however, first attach to the flagellar filaments
of bacteria. Being immotile, these so-called flagellotropic phages still manage to reach the cell
body for infection, and the process by which they move up the flagellar filament has intrigued the
scientific community for decades. In 1973, Berg and Anderson (Nature, 245, 380-382) proposed
the nut-and-bolt mechanism in which, similarly to a rotated nut that is able to move along a bolt,
the phage wraps itself around a flagellar filament possessing helical grooves (due to the helical
rows of flagellin molecules) and exploits the rotation of the flagellar filament in order to passively
travel along it. One of the main evidence for this mechanism is the fact that mutants of bacterial
species such as Escherichia coli and Salmonella typhimurium that possess straight flagellar filaments
with a preserved helical groove structure can still be infected by their relative phages. Using two
distinct approaches to address the short-range interactions between phages and flagellar filaments,
we provide here a first-principle theoretical model for the nut-and-bolt mechanism applicable to
mutants possessing straight flagellar filaments. Our model is fully analytical, is able to predict the
speed of translocation of a bacteriophage along a flagellar filament as a function of the geometry
of both phage and bacterium, the rotation rate of the flagellar filament, and the handedness of the
helical grooves, and is consistent with past experimental observations.
a
Current address: School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK
e.lauga@damtp.cam.ac.uk

2
Head
Collar
Tail
Long Tail
Fibres
Base Plate
DNA
Protein
2D 3D
FIG. 1. Bacteriophages: (A) A typical morphology of a bacteriophage, such as the Enterobacteria T4 phage (Adenosine,
Wikimedia Commons); (B) Electron micrograph of bacteriophages attached to a bacterial cell, (Dr. G. Beards, Wikimedia
Commons); (C-E) Phages can come in various shapes: (C) Myoviridae; (D) Podoviridae; (E) Siphoviridae [8]. Panels C-E:
Reprinted by permission from Suttle CA, “Viruses in the sea”, Nature, 437 (356), 356-361, Copyright 2005 Springer Nature.
I. INTRODUCTION
As big as a fraction of a micrometre, bacteriophages (in short phages), are ‘bacteria-eating’ viruses (illustrated in
Fig. 1) that infect bacteria and replicate within them [1]. With their number estimated to be of over 10
31
on the
planet, phages are more abundant than every other organism on Earth combined [2–8].
Phages have been used extensively in genetic studies [1, 4, 9], and their future use in medicine is potentially of even
greater impact. The global rise in antibiotic resistance, as reported by the increasing number of multidrug-resistant
bacterial infections [10], poses one of the greatest threats to human health of our times, and phages could offer the
key to resolution. Indeed, phages have been killing bacteria for way longer than humanity has been fighting against
bacterial infections, with as many as 10
29
infections of bacterial cells by oceanic phages taking place every day [11, 12].
Phage therapy is an alternative to antibiotics that has been used for almost a century and offers promising solutions
to tackle antibiotic-resistant bacterial infections [13]. Furthermore, the unceasing phage-bacteria war taking place in
enormous numbers offers the scientific community great opportunities to learn. For example, the ability of phages
to update their infection mechanisms in response to bacterial resistance could offer us valuable insight into updating
antibiotics treatment against multi drug-resistant pathogenic bacteria [14]. In addition, the high selectivity of the
attachment of a phage to the receptors on the bacterial cell surface and the species it infects could help identify possible
target points of particular pathogenic bacteria for drugs to attack [15]. In general, extensive studies of bacteriophage
infection strategies could not only reveal vulnerable points of bacteria, but may help uncover remarkable biophysical
phenomena taking place at these small scales.
Infection mechanisms can vary across the spectrum of phage species [15, 16]. Lacking the ability to move indepen-
dently, phages simply crowd fluid environments and rely on a random encounter with a bacterium in order to land
on its surface and accomplish infection using remarkable nanometre size machinery. Typically, the receptor-binding
proteins located on the long tail fibres recognise and bind to the receptors of the host cell via a two-stage process
called phage adsorption [15]. The first stage is reversible, and is followed by irreversible attachment onto the cell
surface. Subsequently, the genetic material is ejected from their capsid-shaped head, through their tail, which is a
hollow tube, into the bacterium [17, 18].
While all phages need to find themselves on the surface of the cell body for infection to take place, there is a
class of phages, called flagellotropic phages, that first attach to the flagellar filaments of bacteria. Examples include
the χ-phage infecting Escherichia coli (E. coli) and Salmonella typhimurium (Salmonella), the phage PBS1 infecting
Bacillus subtilis (B. subtilis) and the recently discovered phage vB VpaS OWB (for short OWB) infecting Vibrio
parahaemolyticus (V. parahaemolyticus) [19], illustrated in Fig. 2.
Given the fact that phages are themselves incapable of moving independently and that the distance they would have
to traverse along the flagellar filament is large compared to their size, they must find an active means of progressing
along the flagellar filament. In Ref. [22], electron microscopy images of the flagellotropic χ-phage, shown in Figs. 2C
and D, were provided to show that the mechanism by which χ-phage infects E. coli consists of travelling along the
outside of the flagellar filament until it reaches the base of the flagellar filament where it ejects its DNA.
A possible mechanism driving the translocation of χ-phage along the flagellar filament was first proposed in Berg
and Anderson’s seminal paper as the ‘nut-and-bolt’ mechanism [23]. Their paper is best known for establishing that
bacteria swim by rotating their flagellar filaments. One of the supporting arguments was the proposed mechanism

3
FIG. 2. Flagellotropic phages: (A) Attachment of phage OWB to V. parahaemolyticus [19]. Red arrows indicate phage
particles. (B) Phage PBS1 adsorbed to the flagellar filament of a B. Subtilis bacterium with its tail fibres wrapped around the
flagellar filament in a helical shape with a pitch of 35 nm [20]. The phage hexagonal head capsid measures 120 nm from edge
to edge [21]. Reprinted (amended) by permission from American Society for Microbiology from Raimondo LM, Lundh NP,
Martinez RJ, “Primary Adsorption Site of Phage PBS1: the Flagellum of Bacillus”, J. Virol., 1968, 2 (3), 256-264, Copyright
1968, American Society for Microbiology. (C) χ-phage of E. coli [22]. The head measures 65 to 67.5 nm between the parallel
sides of the hexagon [22]; (D) χ-phage at different times between attachment on the flagellar filament of E. coli and reaching
the base of the filament [22]. Arrows point to the bases of the flagella. Panels C-D: Reprinted (amended) by permission from
American Society for Microbiology from Schade SZ, Adler J, Ris H, “How Bacteriophage χ Attacks Motile Bacteria”, J. Virol.,
1967, 1 (3), 599-609, Copyright 1967, American Society for Microbiology.
where the phage plays the role of the nut and the bolt is the flagellar filament, with the grooves between the helical
rows of flagellin molecules making up the flagellar filament serving as the threads [23] (Figs. 3A and B). A phage
would then wrap around the flagellar filament and the rotation of the latter would result in the translocation of the
phage along it.
A mutant of Salmonella that has straight flagellar filaments, but possesses the same helical screw-like surface due
to the arrangement of the flagellin molecules [24] is non-motile due to the lack of chiral shape yet fully sensitive to
χ-phage [26], i.e. the phages manage to get transported to the base of the flagellar filament. This is consistent with
the nut-and-bolt mechanism and was used as evidence that the flagellar filament is rotating [23].
More evidence in support of the nut-and-bolt mechanism were provided 26 years after its inception in a work studying
strains of Salmonella mutants with straight flagellar filaments whose motors alternate from rotating clockwise (CW)
and counter-clockwise (CCW) [27]. The directionality of rotation is crucial to the mechanism as CCW rotation
will only pull the phage toward the cell body if the phage slides along a right-handed groove. In order to test the
directionality, the authors used a chemotaxis signalling protein that interacts with the flagellar motor, decreasing the
CCW bias. They found that strains with a large CCW bias are sensitive to χ-phage infection, whereas those with
small CCW bias are resistant, in agreement with the proposed nut-and-bolt mechanism.
Details of the packing of the flagellin molecules that give rise to the grooves can be found in Ref. [25] and examples
are shown in Figs. 3A and B. It is important to note that the packing of flagellin molecules produces two overlapping
sets of helical grooves, a long-pitch and a short-pitch set of grooves which are of opposite chirality [24]. Once in
contact with a rotating flagellar filament, it is anticipated that the phage fibres will wrap along the short-pitch
grooves. Indeed, the findings of Ref. [27] show that the directionality of phage translocation correlates with the
chirality of the short-pitch grooves.
The flagellar filaments of bacteria can take one of the twelve distinct polymorphic shapes as illustrated in Figs. 3B
and C. 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 L-type straight flagellar filament (f0) that was used by Ref. [27] has both left-handed long-pitch grooves and right-
handed short-pitch grooves [24]. Given that the short-pitch grooves are relevant to the wrapping of the fibres, the
findings of Ref. [27] that bacteria with their flagellar filament in the f0 polymorphic state (i.e. with right-handed short-
pitch grooves) and with a large CCW bias are sensitive to χ-phage infection, are in agreement with the nut-and-bolt

4
FIG. 3. Bacterial flagellar filaments and polymorphism: (A) Structure of straight flagellar filament from a mutant of
Salmonella typhimurium [24]. Reprinted with permission from O’Brien EJ, Bennett PM, “Structure of straight flagella from a
mutant Salmonella”, J. Mol. Biol., 70 (1), 145-152, Copyright 1972 Elsevier. The L-type straight flagellar filament (left) has
two types of helical grooves, left-handed long-pitch and right-handed short-pitch grooves, whereas the R-type straight filament
(right) has right-handed long-pitch and left-handed short-pitch grooves. Examples of short-pitch grooves are marked by thin
blue lines and indicated by arrows. (B) Schematic of some polymorphic states of the flagellar filament, from left to right: L-type
straight, normal (left-handed shape), curly (right-handed) and R-type straight. The top panel shows the shape of the filaments
while the bottom panel displays the arrangements of flagellin subunits. Examples of short-pitch grooves are marked by thin
blue lines and indicated by arrows [25]. Reprinted with permission from Namba K, Vonderviszt F, “Molecular architecture of
bacterial flagellum”, Q. Rev. Biophys., 1997, 30 (1), 1-65, Copyright 1997 Cambridge University Press.
mechanism.
The same study also argued that the translocation time of the phage to the cell body is less than the flagellar
filament reversal interval, a necessary condition for successful infection by the virus of wild-type bacteria whose
motors alternate between CCW and CW rotation. Their estimated translocation speeds on the order of microns per
seconds give a translocation time which is less than the CCW time interval of about a 1 s [27].
Relevant to the nut-and-bolt mechanism are also the findings of an alternative mechanism for adsorption of the
flagellotropic phages φCbK and φCb13 that interact with the flagellar filament of Caulobacter crescentus using a
filament located on the head of the phage [28], instead of the tail or tail fibres that other flagellotropic phages use,
such as χ and P BS1. This study also reports on a higher likelihood of infection with a CCW rotational bias that
is consistent with the nut-and-bolt mechanism. Notably, phages can also attach to curli fibres, which are bacterial
filaments employed in biofilms; however due to the lack of helical grooves and rotational motion of these filaments,
phages are unable to move along them [29].
In this paper, we theoretically examine the nut-and-bolt mechanism from a quantitative point of view and perform
a detailed mathematical analysis of the physical mechanics at play. We focus on the virus translocation along straight
flagellar filaments in mutants such as the mutant of Salmonella used in Ref. [23]. A flagellotropic phage can wrap
around a given flagellar filament using its tail fibres (fibres for short), its tail, or in some cases a filament emanating
from the top of its head [28] and the models we develop can address all these relevant morphologies. A schematic
diagram of the typical geometry we consider is shown in Fig. 4.
A phage floating in a fluid whose fibres suddenly collide with a flagellar filament rotating at high frequency will
undergo a short, transient period of wrapping, during which the length of the fibres that are wrapped around the
filament is increasing. In this paper we study the translocation of the phage once it has reached a steady, post-wrapping
state, and assume that it is moving rigidly with no longer any change in the relative virus-filament configuration.
In order to provide first-principle theoretical modelling of the nut-and-bolt mechanism, we build in the paper a
hierarchy of models. In §II, we start with a model of drag-induced translocation along smooth flagellar filaments that
ignores the microscopic mechanics of the grooves yet implicitly captures their effect by coupling the helical shape
of the fibres with anisotropy in motion in the local tangent plane of the flagellar filament. Having acquired insight
into the key characteristics of the mechanism, we proceed by building a refined, more detailed model of the guided

5
FIG. 4. Schematic model of the translocation of a flagellotropic phage along the straight flagellar filament of
a mutant bacterium. The phage, illustrated in dark green, is wrapped around the straight flagellar filament (light blue
cylinder) using its fibres, with its tail and head protruding in the bulk fluid.
translocation of phages along grooved flagellar filaments by incorporating the microscopic mechanics of the grooves in
§III. This is done by including a restoring force that acts to keep the fibres in the centre of the grooves, thereby guiding
their motion, as well as a resistive force acting against the sliding motion. In both models, we proceed by considering
the geometry, and the forces and torques acting on the different parts of the phage. We use the resistive-force theory
of viscous hydrodynamics in order to model the tail and tail fibres which are both slender [30, 31]. The portion of
the phage wrapping around the flagellar filament is typically the fibres. They experience a hydrodynamic drag from
the motion in the proximity of the rotating flagellar filament along which they slide in the smooth flagellum model,
or a combination of a guiding and resistive forces in the grooved flagellum model. Parts sticking out in the bulk away
from the flagellar filament experience a hydrodynamic drag due to their motion in an otherwise stagnant fluid.
We build in our paper a general mathematical formulation relevant to a broad phage morphology. In our typical
geometry of phages wrapping around flagellar filaments using their fibres, two limits arise for long-tailed and short-
tailed phages. Long-tailed phages have their tail and head sticking out in the bulk, away from the flagellar filament,
whereas for short-tailed phages only the head is exposed to the bulk fluid. The hydrodynamic torque actuating the
translocation is provided by the parts sticking out in the bulk.
We compare the results from the two models addressing the two geometrical limits and find these to be consistent
with each other and with the predictions and experimental observations of Refs. [23, 27]. In particular, we predict
quantitatively the speed of phage translocation along the flagellar filament they are attached to, and its critical
dependence on the interplay between the chirality of the wrapping and the direction of rotation of the filament, as
well as the geometrical parameters. Most importantly we show that our models capture the correct directionality of
translocation, i.e. that CCW rotation will only pull the phage toward the cell body if the phage slides along a right-
handed groove, and predict speeds of translocation on the order of µms
1
, which are crucial for successful infection
in the case of bacteria with alternating CCW and CW rotations.
II. DRAG-INDUCED TRANSLOCATION ALONG SMOOTH FLAGELLAR FILAMENTS
II.1. Geometry
As our first model, we consider the flagellar filament as a straight, smooth rod aligned with the z-axis and of radius
R
fl
. The phage has a capsid head of size 2a
h
, a tail of length L
t
and fibres that wrap around the flagellar filament.
We 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. The helical shape of the fibres
has helix angle α, as shown in Fig. 5. With the assumption that the gap between the fibres and the flagellar filament
is negligible compared to the radius R
fl
of the flagellar filament, the centreline r
fib
(s) of the fibres, parametrised by

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Journal ArticleDOI
TL;DR: Using Escherichia coli biofilms and the lytic phage T7 as models, it is discovered that an amyloid fibre network of CsgA (curli polymer) protectsBiofilms against phage attack via two separate mechanisms.
Abstract: In nature, bacteria primarily live in surface-attached, multicellular communities, termed biofilms 1–6 . In medical settings, biofilms cause devastating damage during chronic and acute infections; indeed, bacteria are often viewed as agents of human disease 7 . However, bacteria themselves suffer from diseases, most notably in the form of viral pathogens termed bacteriophages 8–12 , which are the most abundant replicating entities on Earth. Phage–biofilm encounters are undoubtedly common in the environment, but the mechanisms that determine the outcome of these encounters are unknown. Using Escherichia coli biofilms and the lytic phage T7 as models, we discovered that an amyloid fibre network of CsgA (curli polymer) protects biofilms against phage attack via two separate mechanisms. First, collective cell protection results from inhibition of phage transport into the biofilm, which we demonstrate in vivo and in vitro. Second, CsgA fibres protect cells individually by coating their surface and binding phage particles, thereby preventing their attachment to the cell exterior. These insights into biofilm–phage interactions have broad-ranging implications for the design of phage applications in biotechnology, phage therapy and the evolutionary dynamics of phages with their bacterial hosts. At late stages of biofilm development, Escherichia coli cells express the curli polymer CsgA. CsgA assembles into a fibre network that protects biofilms from attack by lytic phages.

212 citations

Journal ArticleDOI
TL;DR: Observations support the idea that a primary role of host populations in phage ecology and evolution is to serve as vectors for genetic exchange among phage and host communities.

180 citations

Frequently Asked Questions (17)
Q1. What have the authors contributed in "Hydrodynamics of bacteriophage migration along bacterial flagella" ?

Using two distinct approaches to address the short-range interactions between phages and flagellar filaments, the authors provide here a first-principle theoretical model for the nut-and-bolt mechanism applicable to mutants possessing straight flagellar filaments. 

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. 

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. 

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. 

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 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. 

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. 

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. 

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 + 

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). 

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]. 

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. 

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. 

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 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. 

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. 

there will be an additional hydrodynamic drag on the phage due to the rotation and translation of the flagellar filament.