Hydrodynamic interactions are key in thrust-generation of microswimmers with hairy agella

The important role of ﬂagellates in aquatic microbial food webs is mediated by their ﬂagella that enable them to swim and generate a feeding current. The ﬂagellum in most predatory ﬂagellates is equipped with rigid hairs that reverse the direction of thrust compared to the thrust due to a smooth ﬂagellum. Conventionally, such reversal has been attributed to drag anisotropy of individual hairs, neglecting their hydrodynamic interactions. Here, we show that hydrodynamic interactions are key to thrust-generation and reversal in hairy ﬂagellates, making their hydrodynamics fundamentally diﬀerent from the slender-body theory governing microswimmers with smooth ﬂagella. Using computational ﬂuid dynamics and model analysis, we demonstrate that long and not too closely spaced hairs and strongly curved ﬂagellar waveforms are optimal for thrust-generation. Our results form a theoretical basis for understanding the diverse ﬂagellar architectures and feeding modes found in predatory ﬂagellates.

Flagella and cilia are ubiquitous in both prokaryotic and eukaryotic organisms. They accomplish different tasks such as locomotion in bacteria, unicellular protists and spermatozoa 1;2 , feeding and pumping in unicellular and multi-cellular organisms [3][4][5][6] , and other transport functions in dense ciliary arrays in plankton and mammalians [7][8][9] . The flagellum is often a smooth, slender organelle and the drag-based propulsion of such 'naked' 1 flagella is extensively studied and well understood [10][11][12] . In many unicellular protists, however, the flagellum is equipped with either a vane 13 , small and thin, non-tubular hairs 14;15 , or thick and rigid, tubular hairs 16 . Little is known about the hydrodynamics of such flagella despite their great significance.
Predatory flagellates are important in the aquatic microbial food webs and in the marine carbon cycle 3;19-21 , and their survival relies on the feeding flow generated by the flagellum that in most species is equipped with tubular hairs 22;23 . In hairy flagellates with flagellar wave motion in a two-dimensional plane, the tubular hairs are oriented in the beat plane and perpendicular to the flagellum (Fig. 1a). The hairs cause the thrust to reverse compared with the thrust of a smooth flagellum and to point in the opposite direction of the propagating wave (Fig. 1b). Thus, swimming cells are pulled rather than pushed by the flagellum, and in tethered cells, the flagellum generates a feeding current toward the cell 17;24;25 . The mechanism of thrust reversal has been ascribed to the drag anisotropy of individual hairs: the presence of hairs on the flagellum effectively increases the tangential drag and decreases the normal drag relative to the flagellum, hence reversing the thrust direction 26;27 .
In these analyses using resistive force theory, the effect of the induced flow generated by the flagellum and the neighboring hairs is neglected, implicitly assuming that hairs are hydrodynamically independent of one another. However, the distance between neighboring hairs is typically much smaller than the length of the hairs, and disregarding hydrodynamic interactions appears unjustified [26][27][28] . In this study we use computational fluid dynamics (CFD) to examine hairy flagella, and we demonstrate how the hitherto unexplored interactions dominate the hydrodynamics of thrust-generation and reversal in predatory flagellates.

Thrust in model flagellates with different densities of hairs
We focus on the generic situation in which the flagellum beats in a two-dimensional plane and is equipped with identical and equidistantly spaced hairs that are perpendicular to the flagellum and positioned in the beat plane ( Fig. 1c). In most CFD simulations, we use the waveform: where φ(s, t) is the angle of the tangent of the flagellum with respect to the flagellar axis, s the arc length from the point of attachment on the spherical cell body, A φ the amplitude, λ φ the wavelength, f the beat frequency, and δ the amplitude modulation factor 29 . Parameters used in Eq. 1 and henceforth are listed in Table S1. The predatory flagellate Pteridomonas danica is a representative model organism 3 , and we use its morphological and kinematic parameters (Methods). Our CFD simulations of P. danica predict that the flagellum pulls the    (1) are chosen based on the beat analysis of Pteridomonas danica, i.e., f = 50 Hz, λ φ = 10 µm, A φ = 2π/5, and δ = 2 µm, and the length of the flagellum is L f = 12.5 µm. The length of the hairs is reduced to l = 0.9 µm to avoid physical interference during the beat. a, At low density of hairs (N = 1 µm −1 ), the hydrodynamic interactions are weak. b, At high density of hairs (N = 7 µm −1 ), as for P. danica, the hydrodynamic interactions are dominant. c, The time-averaged thrust is approximately the same as the time-averaged thrust due to a flexible, plane sheet with the same beat kinematics when the density of hairs is high. To explore the hydrodynamic interactions we simulate the flow for different densities of hairs, N , i.e., the number of hairs per unit length of the flagellum (Fig. 2a,b). The time-averaged thrust is negative for the smooth flagellum, and the presence of hairs results in thrust-reversal when N > 1 µm −1 (Fig. 2c). The thrust increases with the density of hairs, and it saturates when N > 6 µm −1 , consistent with the observed density of hairs in P. danica. The hydrodynamic interactions between the hairs are negligible when they are far apart, and the thrust therefore increases roughly proportional to the number of hairs when the density is low 26 . However, at higher densities, the hydrodynamic interactions become predominant, and densely spaced hairs generate approximately the same thrust as a flexible, plane sheet (Fig. 2c).

Hair arrays: hydrodynamic interactions and drag anisotropy
At such complete dominance of hydrodynamic interactions, the question is whether the previously suggested anisotropy in the drag of individual hairs can cause thrust reversal 26;27 . To answer this question, we first consider a segment along the straight part of the hairy flagellum for which resistive force theory predicts that hairs give rise to thrust generation and reversal 26;27 . We neglect the flagellum itself and assume that hairs along the segment have the same lateral motion. For simplicity, we consider parallel hairs of length l, equidistantly positioned in an array of side length l and moving lengthwise or sideways with speed U cf. Fig. 3a,b. The magnitudes of the normal and the tangential drag on the array approach the asymptotic value of an equivalent square sheet when increasing the density of hairs, thus neutralizing the drag anisotropy of an isolated hair ( Fig. 3c). Although the above results are obtained for a specific length of hairs, similar reduction of drag anisotropy is expected for any length of hairs, as long as the hairs are slender, since the induced far field flow primarily depends on the length of the hairs and only weakly on their diameter.
To shed light on the hydrodynamic interactions and their effect on drag anisotropy, we explore the interactions between two parallel hairs (Fig. 3a). The low Reynolds number forces on the second hair F where C t and C n are the drag coefficients for a single hair in length and sideways motion, respectively (equations 5-11.52 and 5-11.54 in reference 32 ). As an approximation of the induced velocity past the second hair, consider the velocity in the parallel and the normal direction due to a point force positioned at the center of the first hair and with the same magnitude as the drag on an individual hair 11 : Hence, the drag ratio in such a system of two hairs can be approximated: The hydrodynamic interactions between two hairs are stronger when they move sideways than when they move lengthwise. b, By adding more hairs and increasing their density, N , the flows become similar for the two directions of motion. c, The drag on the array in sideways, F n , and lengthwise, F t , motion approach the asymptotic value of an equivalent square sheet (top), and the drag anisotropy of the system vanishes (bottom).
The theoretical prediction for a single hair is F n /F t = 1.6 in agreement with the CFD results (blue symbols), and equation (6) predicts F The approximation is in good agreement with our CFD simulations (Fig. 3c), and the analysis shows that the hydrodynamic interactions reduce the strength of the drag anisotropy because U

Thrust-generation with sparse and dense hairs
While hydrodynamic interactions neutralize the fundamental anisotropy underlying drag-based propulsion, it brings about another potential mechanism of thrust-generation. Consider the previous array of parallel hairs in the side-wise motion. Upon saturation of the thrust at N ≈ 6 µm −1 , further increasing the density of hairs does not result in more force (Fig. 3c). However, extending the row of hairs, i.e. increasing the sideways length of the array will result in an increased force (Fig. S3). This implies that for a given number of hairs, a net force is produced in a periodic and non-reciprocal fashion, where the sideways length of the array is increased during a "power" stroke and decreased during a "return" stroke. We propose that this mechanism is exploited by the hairy flagellates where the curvature of the flagellum plays a direct and key role by periodically bringing about an anisotropy in the arc length covered by several hairs at either sides of the flagellar crest. This anisotropy in the arc length (analogous to the sideways length of the array) at the two sides, moving in the opposite direction due to the action of the flagellum, results in a net force (thrust).
The proposed mechanism suggests that in a flagellum with dense hairs, the location of thrust generation is shifted from the straight parts to the crests of the passing wave. To illustrate the mechanism, we consider a one quarter wavelength long segment of the flagellum with sparse and dense hairs, respectively. As a simple model that allows us to compare with known expressions from resistive force theory 26;27 , we use the flagellar kinematics: where d is the displacement of the material points on the flagellum in the lateral x-direction, y the centerline axis, A the amplitude, and λ the wavelength. In the sparse system, the thrust correlates with the motion of individual hairs, and accordingly, only hairs at the straight part (with a dominant transversal motion) generate thrust, but not the hairs at the crest (with a dominant rotational motion) cf. Fig. 4a,b. In the dense system, on the other hand, the role switches and most of the thrust is generated at the crest (Fig. 4c,d). The resulting T /4 phase shift (T = 1/f being the period of the flagellar wave) of the generated thrust due to the hairs from the sparse to the dense system is a signature of the change from drag-based thrust to hydrodynamic interaction based thrust (Fig. 4e). For comparison, consider the time-averaged drag-based thrust generated by a single hair suggested by resistive force theory 23;26;27 :   (8), whereas the thrust in the dense system decreases strongly with increasing wavelength (decreasing curvature), and it is significantly overestimated by RFT.
where C t and C n are the drag coefficients used in equations (2) and (3). The thrust expression is proportional to the difference between C n and C t , emphasising the significance of drag anisotropy in drag-based thrust 12 . As expected from equations (2) and (3), the resistive force theory model overestimates the generated thrust by the hairs, slightly in the sparse system but significantly in the dense system (Fig. 4f). Importantly, the thrust has a weak dependence on the wavelength for the sparse system, but in contrast in the dense system, it increases significantly by decreasing the wavelength highlighting effect of the curvature of the flagellum (Fig. 4f).

Thrust mechanism and analytical model
Here we present an analytical model that allows us to shed light on the mechanism and key parameter of thrustgeneration by flagella with dense hairs (N > 6 µm −1 ). We assume that hairs are relatively long (2l/D f ≫ 1, where D f is the diameter of the flagellum), and several hairs, together, effectively function as a sheet cf. Fig. 5a,b.
For simplicity of the analytical model, we assume that the amplitude modulation factor, δ, the radius of the flagellum, and the radius of the hairs only affect the thrust weakly. The remaining important parameters are µ, f , λ φ , A φ , L f , and l. From these dimensional parameters, we can construct two dimensionless parameters.
In total, we have the three dimensionless parameters l/λ φ , A φ , and L f /λ φ , and in our minimal model we can express the thrust:F where Φ is a dimensionless function. From dimensional analysis it is clear thatF y is proportional to µ and f .
To determine Φ and develop a mechanistic model, we consider the sinusoidal beat pattern: Considering that the thrust is predominantly generated at the crests of the flagellar wave (Fig. 4c, d), we assume that the basic thrust-generating unit spans one quarter of the wavelength and is centered at the crest ( Fig. 5c). Each unit consists of two parts on either side of the flagellum where the hairs come close and fan out, respectively. For such unit, the part where the hairs come close moves upwards and creates a negative thrust, whereas the part where the hairs fan out moves downwards and creates a positive thrust. The net thrust, F y, unit , is the difference between the two contributions, and we make the low Reynolds number estimate: where U unit is the characteristic speed of the two parts, θ (r c −l/2) the characteristic length of the part where the hairs come close (corresponding to the arc length in the middle of the part), and θ (r c + l/2) the characteristic length of the part where the hairs fan out (Fig. 5c). Note that θ = √ 2A φ , if the unit spans one quarter of the wavelength. To estimate U unit we use the magnitude of the rotation rate of the hairs at the crests, Ω unit = 2πf A φ , since ∂φ/∂t = 2πf A φ cos (2πf t − 2πs/λ φ ). Therefore, we obtain that U unit = Ω unit l/2 = πf A φ l. By combining the expressions, we find the estimate:F where α is a dimensionless shape factor. There are 2L f /λ φ thrust-generating units along the full length of the flagellum, and the total thrust generated by the hairy flagellum becomes: where κ f = 2πA φ /λ φ is the magnitude of the curvature at the crest. The model gives a reasonable estimate and a consistent functional behaviour of the generated thrust, as compared with the CFD results for the three dimensionless parameters l/λ φ , A φ , and L f /λ φ (Fig. 5d-f). In the CFD simulations there is also influence of the flagellum itself as well as the cell body (Fig. S5), which are not accounted for in our model. Note that a flagellum with short hairs function like a slender body, in which drag-based thrust is dominant giving rise to a negative thrust (Fig. 5d). However, hydrodynamic interaction-based thrust dominates for longer hairs, as in hairy flagellates with tubular mastigonemes, consistent with the model presented here (Eq. 13).

Perspective
The optimal design of a flagellum with respect to thrust production is suggested by equation (13): The flagellum should be long, curvy, and equipped with long hairs. Indeed, hairy flagella have noticeable shorter wavelengths than naked flagella 17 . However, for a planar beat, as in our model organism, hairs longer than the radius of curvature at the crests would physically interfere with each other. One solution is to orient hairs slightly off the beat plane (Fig. S2), but this implies marginally less thrust than if hairs were in the beat plane. This solution is used by flagellates where hairs are oriented at a small angle relative to the beat plane defined by the central pair of axonemal microtubules 33;34 . A second solution may be bundling of hairs (Fig. S6a) 33 , where interference is avoided if hairs become aligned rather than intertwined. Thirdly, the beat of the flagellum itself may not be perfectly planar 33 , and many hairy flagellates in fact have complex three-dimensional beat patterns 3;35 , providing room for longer hairs without interference. Yet another way to increase hydrodynamic interaction based thrust is through terminal branching of the hairs, as found in many species (Fig. S6b) 33 , which may dramatically increase the thrust (Fig. S6d). Finally, the fibrous hairs between the tubular hairs in some species make the hairs further resemble a sheet (Fig. S6b) 33 . The diverse flagella designs discussed above can thus all be understood in the context of the mechanism for thrust production suggested here, and they may provide novel ideas for design of artificial swimmers and flow generation in microfluidic devices.
The hydrodynamics of flagella is most often studied in the context of propulsion 7;11;12 . However, for predatory flagellates, the main consumers of bacteria in the ocean, efficient foraging is likely a much more important component of their fitness than propulsion per se. The presence of hairs on the flagellum is key to foraging in flagellates. Not only does the reversal of the flow increase the capture efficiency of prey arriving in the feeding current 23 , the hairs also increase the thrust generation by a factor of 5-10 ( Fig. 5d). At low Reynolds number, viscosity impedes predator-prey contact, but the hairy flagellum yields the necessary force and feeding current structure to secure the success and key role of flagellates in the microbial food webs.

Computational fluid dynamics
We where σ is the stress tensor, n the unit normal vector on the surface S. The generated thrust is the y-component

Additional Information
Beat pattern of Pteridomonas danica Details of the analysis and reconstruction of the beat in Pteridomonas danica are given in Fig. S1, and parameter used in the CFD and analytical models are listed in table S1.
sideway length of the array of hairs µm F y, unit thrust generation of the basic thrust unit pN θ angle between the straight edges of the basic thrust unit rad U unit characteristic speed of the unit µm s −1 α shape factor of the unit -Ω unit rotation rate of the hairs at the crests rad s −1 r c radius of the curvature of the flagellum µm F y, sheet thrust generation of the sheet model pN D cell diameter of the cell µm Table S1: List of symbols used in the CFD and analytical models.