Journal of Mathematical Biology manuscript No.(will be inserted by the editor)

Helical swimming can provide robust upwardstransport for gravitactic single-cell algae; amechanistic model.

R. N. Bearon

Received: date / Accepted: date

Abstract In still fluid, many phytoplankton swim in helical paths with anaverage upwards motion. A new mechanistic model for gravitactic algae sub-ject to an intrinsic torque is developed here, based on Heterosigma akashiwa,which results in upwards helical trajectories in still fluid. The resultant up-wards swimming speed is calculated as a function of the gravitactic and in-trinsic torques. Helical swimmers have a reduced upwards speed in still fluidcompared to cells which swim straight upwards. However a novel result is ob-tained when the effect of fluid shear is considered. For intermediate values ofshear and intrinsic torque, a new stable equilibrium solution for swimmingdirection is obtained for helical swimmers. This results in positive upwardstransport in vertical shear flow, in contrast to the stable equilibrium solutionfor straight swimmers which results in downwards transport in vertical shearflow. Furthermore, for strong intrinsic torque, when there is no longer a sta-ble orientation equilibrium, we show that the average downwards transportof helical swimmers in vertical shear flow is greatly suppressed compared tostraight swimmers. We hypothesise that helical swimming provides robustnessfor upwards transport in the presence of fluid shearing motions.

Keywords algae · helical swimming · gravitaxis · fluid shear

Mathematics Subject Classification (2000) 92B05 · 76Z10 · 92C05

R. N. BearonDept. Mathematical Sciences, University of Liverpool,Liverpool, L69 7ZL, UKTel.: +44 (0)151 794 4022Fax: +44 (0)151 794 4061E-mail: rbearon@liv.ac.uk

2 R. N. Bearon

1 Introduction

Swimming micro-organisms are ubiquitous in fluid environments. Phytoplank-ton in particular are important in a wide variety of natural phenomena. Thesesingle-celled organisms, many of which are motile, are integral players in theoceanic ecosystem. For example, massive phytoplankton blooms can have ma-jor economic and environmental impact (Horner et al, 1997). In the contextof climate change, the oceans are among the largest sinks of carbon dioxide(Raven and Falkowski, 1999), which is transformed by phytoplankton intoorganic carbon, providing food for a wide range of organisms; from marinebacteria which recycle carbon from lysed phytoplankton, to blue whales whichsieve the phytoplankton from the water.

Many phytoplankton are gravitactic, that is they swim on average upwardsin still fluid which can be beneficial for reaching regions of optimal light.For some species this is due to being bottom-heavy: the centre of gravity forthese cells is offset from the centre of buoyancy, and the combination of theeffects of gravity with the buoyancy force gives rise to a gravitational torquewhich serves to reorient the cell allowing it to swim upwards. However in shearflow, these organisms are reorientated from the vertical because of viscoustorques (Pedley and Kessler, 1992). This can result in cells being focussedinto down-welling regions (Kessler, 1985) or trapped in regions of high shear(Durham et al, 2009). In simple vertical shear flow, organisms tend to bereorientated towards regions of downwards flow resulting in downwards nettransport, which may be disadvantageous for cells seeking upper well-lit regionsof the water column. In simple horizontal shear flow, cells which are able tomaintain upwards swimming will undergo significant shear dispersion as theyare advected by the horizontal flow, whereas cells which lose their ability tomaintain upwards swimming in regions of high shear may get trapped in suchregions and undergo reduced lateral dispersion.

It has been observed that many phytoplankton do not swim in a straightline, rather they undergo helical trajectories (Crenshaw, 1996; Gurarie et al,2011; Boakes et al, 2011). Several explanations have been proposed for theubiquity of helical movement among micro-organisms. Over a century ago,Jennings (1901) postulated that the helical trajectory allows an otherwiseasymmetric organism to move along a nearly straight trajectory. More re-cently, Crenshaw (1996) demonstrated how organisms can modify their helicalmotions in order to undergo phototaxis and chemotaxis, through a process hetermed helical klinotaxis. Here we propose a mechanistic model based on thealgal species Heterosigma akashiwa which can explain upwards helical trajec-tories. Furthermore, we show that organisms undergoing helical motion in thepresence of vertical shear flow avoid swimming into regions of downwards flowand retain upwards net transport. We thus hypothesise that helical swimmingcan provide robustness for upwards transport in the presence of vertical shearflow.

Helical swimming can provide robust upwards transport 3

2 Model

Our model organism is based on Heterosigma akashiwa. This organism hastwo flagella; a leading flagellum which sends waves along its length and pullsthe organism through the fluid, and a secondary flagellum which emerges fromthe side of the cell whose purpose is less clear (Hara and Chihara, 1987). Thiscell undergoes helical motions (Gurarie et al, 2011) and can switch betweenhelical and straight swimming (Bearon et al, 2004). As depicted in figure 1, themodel organism we consider has a leading flagellum that provides a propulsiveforce which generates a swimming velocity vp, where p is a unit vector in thedirection of cell swimming. The second flagellum provides an intrinsic torque inthe direction n, also a unit vector, which results in the cell undergoing helicalmotions. The model organism is bottom-heavy so that there is an additionalreorientating torque due to gravity, and we take the center of mass to bedisplaced from the center of buoyancy in direction −p.

To calculate the cell position, x, we suppose that in addition to the cellswimming, the cell is advected by the local fluid velocity, V, and thus satisfiesthe following equation of motion:

dx

dt= V + vp, (1)

where t is time.As derived in appendix A, due to the combined effects of gravity, intrinsic

torque and viscous torque, the cell rotates with angular velocity, Ω, given by

Ω = Gp ∧ k +Rn +1

2ω, (2)

where ∧ denotes the vector cross product,G and R are parameters representingthe strength of the gravitactic and intrinsic rotational torque respectively, kis a unit vector in the vertical direction and ω = ∇∧V is the vorticity of thefluid flow. The vectors p and n which are embedded in the cell rotate at theangular velocity of the cell according to the following equations of motion:

dp

dt= Ω ∧ p, (3)

dn

dt= Ω ∧ n. (4)

For simplicity, in this paper we focus on fluid velocity V constrained to liein the x− z plane so that the vorticity of the fluid ω = ∇ ∧V = ωj, where jis a unit vector in the y direction. Furthermore it is convenient to define theunit vectors p and n in terms of spherical co-ordinates, e.g.

p = (sin θp cosφp, sin θp sinφp, cos θp), (5)

so that θp is the angle the vector p makes with the vertical. The upwardsswimming speed is then given by

vup = v cos θp. (6)

4 R. N. Bearon

As shown in appendix A we can obtain scalar differential equations for theangles defining the cell orientation:

dθpdt

= −G sin θp +R sin θn sin(φn − φp) +ω

2cosφp, (7)

sin θpdφpdt

= −R (sin θn cos θp cos(φn − φp)− cos θn sin θp)

−ω2

cos θp sinφp, (8)

dθndt

= −G sin θp cos(φp − φn) +ω

2cosφn, (9)

sin θndφndt

= cos θn

(G sin θp sin(φn − φp)− ω

2sinφn

). (10)

We define γ as the angle between the vectors p and n, that is

p.n = sin θp sin θn cos(φp − φn) + cos θp cos θn = cos γ. (11)

As the vectors p and n are assumed to be embedded within the cell and ro-tate at the angular velocity of the cell, the quantity γ is a constant. Notethat when γ = 0, the intrinsic torque acts in the same direction as the swim-ming direction. The swimming trajectories are therefore determined only bythe equations for θp and φp, and are identical to non-helical swimmers, i.e.trajectories obtained by setting R = 0.

Equations (7-10) were solved numerically in Matlab R2009a using theode45 solver. All parameters were assumed positive throughout. For the cal-culation of mean upwards displacement in the presence of fluid motion, wesimulated 100 trajectories with random initial orientations. Specifically wechose a random orientation p uniformly from the unit sphere, and then chosea random vector, q, perpendicular to p and obtained n by rotating p by anangle γ about the axis q.

3 Results

3.1 Still fluid

In still fluid, ω = 0, as shown in appendix B, there exists a periodic solutionto the system of equations (7-10) which represents helical trajectories withswimming orientation given by

cos θp = cos θep, φp(t) = Rcos γ

cos θept+ φp(0), (12)

where 0 < θep < π/2, and x = (cos θep)2 satisfies the following quadratic equa-tion

x2 + ((R/G)2 − 1)x− (R/G)2 cos2 γ = 0. (13)

Helical swimming can provide robust upwards transport 5

Fig. 1 Diagram of model organism and co-ordinate system. The cell orientationis described by two unit vectors, p and n, representing the direction of the two flagella. Thevector p is represented in spherical polar co-ordinates by the angles (θp, φp). The angle γ isthe (constant) angle between the vectors p and n. The cell swims in the direction p and issituated in vertical shear flow with velocity V represented by the vertical arrows. The cell isreorientated due to three torques indicated by the curved arrows: gravity, of magnitude G,due to being bottom heavy as indicated by the shaded ellipse; viscosity, in that it is rotateddue to the vorticity of the flow, of magnitude ω, and an intrinsic torque of magnitude R inthe direction of n generated by the second flagellum.

Example trajectories are shown in figure 2(A-C) demonstrating that this so-lution is stable and globally attracting. When comparing the helical trajec-tories, we chose to rescale velocities on the swimming speed, v, and time onthe gravitactic reorientation time, 1/G. After a transient period, the cells es-tablish upwards helical motion which is determined by two non-dimensionalparameters: R/G and γ. Increasing the ratio of intrinsic rotation to gravitacticreorientation, R/G, results in tighter helices and a reduction in upswimmingspeed. Increasing the angle γ also causes a reduction in upswimming (figure2D). In the absence of helical motion, that is when either R = 0 or γ = 0,the time taken to establish upwards helical motion only depends on 1/G,the gravitational reorientation time. For helical swimmers, taking time to benon-dimensionalised on 1/G, increasing R/G or γ increases the time taken toestablish upwards helical motion (figure 2E).

6 R. N. Bearon

Fig. 2 Trajectories in still fluid. Example helical trajectories and calculation of meanupswimming speed in still fluid, for parameters see text. (a-c) Trajectories of duration 50in non-dimensional units starting from the origin (indicated by black dot) with a randominitial orientation: (a) R/G = 1, γ = π/4; (b) R/G = 1, γ = 7π/16; (c)R/G = 20, γ = π/4.To aid visualisation, only final 5 non-dimensional time units of trajectory are shown in(c). (d) Upswimming speed as a function of R/G for angle γ = [0, π/4, 3π/8, 7π/16], asgiven by equation (13). For large R/G the upswimming speed cos θp asymptotes to cos γ,see equation 14 (e) Mean time to attain equilibrium as a function of R/G for angle γ =[0, π/4, 3π/8, 7π/16]. The mean is calculated from 100 trajectories starting with randominitial orientation at the origin. Equilibrium is defined as being attained when cos θp remainswithin 10−2 of its value at time 50.

3.1.1 Experimental estimates for parameters

Experimental data on single-cell algae can be analyzed to estimate parame-ters used in this model. Specifically, from Bearon and Grunbaum (2008) weestimate the inverse gravitactic reorientation timescale, G, for Heterosigmaakashiwo to be 0.1s−1, which is in agreement with values given by Pedley andKessler (1992) for species of Chlamydomonas. The angular velocity of Het-

Helical swimming can provide robust upwards transport 7

erosigma akashiwo undergoing helical trajectories is estimated as 2s−1 (Gu-rarie et al, 2011). The intrinsic torque thus acts much more rapidly than thegravitactic reorientation, and we may approximate G/R to be a small param-eter of the problem. At leading order in G/R, the stable periodic solution hasthe unit vector n pointing directly upwards, and the swimming direction protating around the vertical at (dimensional) angular velocity R. Specifically,the solution is given by:

cos θp = cos γ, φp = Rt+ φp(0). (14)

We thus obtain an estimate of R/G ≈ 20. Furthermore, Gurarie et al (2011)found that the mean tangential angle, θt = π/2 − θep, varies significantly be-tween strains from 0.376 to 0.876 radians (21.5 deg and 50.0 deg). This corre-sponds to cos γ varying from 0.367 to 0.876.

3.2 Vertical shear flow

We now consider transport in the presence of vertical shear flow, V = −ωxk.On comparing figure 3a with figure 2a, we see that helical motions are main-tained under the influence of weak shear, but the net transport is modifieddue to advection by the flow. Note that, for these parameter values, px, thehorizontal component of swimming orientation, undergoes periodic oscillationsaround a positive mean value. When combined with advection by the flow, thisnet swimming motion in the positive x direction generates a negative verticaldisplacement after a sufficient time. For stronger shear, from figure 3b, wesee that helical motions are suppressed, and the cell orientation attains a newequilibrium. Increasing the shear further results in a loss of this equilibriumsolution and the cell orientation undergoes more complex dynamics, figure 3c.However, for these parameter values, we note that the mean value of px is stillpositive and thus cells undergo large negative vertical displacements.

As found previously (Pedley and Kessler, 1992), in the absence of an in-trinsic torque, provided ω < 2G, cells attain a stable equilibrium orientationwith swimming orientation given by

pe =

(ω

2G, 0,

√1− ω2

4G2

). (15)

Solving equation (1), for vertical shear flow, V = −ωxk, the position of a cellstarting from the origin with equilibrium orientation is given by

x = vpext, z = −ω2vpext

2 + vpezt. (16)

The dominant effect of gyrotactic reorientation is to cause the cell to swimin the positive horizontal direction towards downwards moving fluid. The cellis thus advected downwards by the flow resulting, after sufficient time, indownwards vertical displacement proportional to the square of time.

8 R. N. Bearon

Fig. 3 Effect of vertical shear flow on helical trajectories. Example trajectoriesof duration 20 non-dimensionalised on G for intrinsic torque R/G = 1, γ = π/4 for arange of shear strength. (a) ω/G = 0.5, (b) ω/G = 2, (c) ω/G = 5, for explanation ofparameters see text. Cells are initially located at the origin (indicated by black dot) withrandom orientation. (i) Trajectories and (ii) horizontal component of swimming orientationpx. Dashed lines in a(ii) and c(ii) indicate the mean value calculated over each trajectory.Dashed lines in b(ii) indicate equilibrium solution given by equation 17. To aid visualisation,only final 5 non-dimensional time units are shown in c(ii).

Helical swimming can provide robust upwards transport 9

As indicated in figure 3b, we have found that some cells which undergohelical trajectories in still fluid can attain a stable equilibrium orientationin shear flow. As derived in appendix B, the swimming orientation of thisequilibrium solution is given by

pex =ω

2G− R

G

√1− 4G2

ω2cos2 γ, (17)

pey = −2R

ωcos γ, (18)

pez =√

1− (pex)2 − (pez)2. (19)

Clearly this equilbrium solution is only valid when 2Gω cos γ ≤ 1. This can

explain differences observed in figure 3. In 3a, where 2Gω cos γ = 2

√2 the cell

orientation does not attain an equilibrium solution, whereas in 3b we havethat 2G

ω cos γ = 1/√

2 and the cell orientation does attain the equilibrium so-lution given by equation (17). In appendix B, we discuss further the region ofR/G-ω/2G-γ parameter space for which equilibria exist. In general, it is nec-essary to calculate the region numerically, but for special values of γ, certainanalytic results are possible. For example if γ = π/2, the equilibrium solutionhas the intrinsic torque acting in the x − z plane, alongside the viscous andgravitational torque. In that case, we require that R − G ≤ ω

2 ≤ R + G inorder for the equilibrium solution to exist. The lower bound represents the in-trinsic torque balancing the viscous and gravitational torques acting together,whereas the upper bound represents the intrinsic torque combining with thegravitational torque to balance the viscous torque. The loss of equilibrium so-lution at sufficiently high shear is evident in figure 3c, where the shear causesthe cell to rotate more rapidly than the periodic motion generated via theintrinsic torque.

From equation (17), for small values of R/G, the horizontal component ofswimming is reduced from the value found for a non-helical swimmer, and thusfrom equation (16), we predict a reduction in downwards vertical transport.Furthermore, provided the equilibrium is feasible and stable, if R/G is suffi-ciently large the equilibrium horizontal component of swimming may changesign, and thus we predict net upwards vertical transport.

As shown in figure 4a, for sufficiently small values of R/G, the cell swimsin the positive x direction towards downwards moving fluid and is advecteddownwards by the flow. As R/G is increased, the horizontal component ofswimming orientation changes sign, and the cell swims away from the down-wards moving fluid and is advected upwards by the flow, figure 4b. As R/G isfurther increased, this equilibrium solution is no longer feasible, and the cellundergoes more complex trajectories, see figure 5. When R/G = 20, the largestvalue considered, the cell orientation appears to undergo periodic motion.

The effect of helical swimming on the mean vertical transport is depictedin figure 6. We see that for sufficiently large γ, that is when the reorientatingtorque, n, is close to being perpendicular to the swimming direction, p, thenet vertical transport changes sign from being negative to being positive, and

10 R. N. Bearon

furthermore there is an optimal value of R/G which maximises the upwardsvertical transport, figure 6a. The optimal is achieved close to the maximal valueof R/G for which the equilibrium solution is feasible. For the large values ofR/G found in experiments (see above, section 3.1.1), as shown in figure 6b,helical motions greatly suppress the net downwards transport, but do notresult in a net upwards transport. Also in figure 6b, despite being in a regionof parameter space such that the equilibrium is not feasible, there appearsto be a critical shear value around 2G

ω cos γ = 1 at which point the verticaltransport diverges from that found for non-helical swimmers.

3.3 Horizontal shear flow

We now consider transport in the presence of horizontal shear flow, V =ωzi, where the dominant transport is typically in the positive x directiondue to the combination of upwards net swimming and horizontal advection.This flow field has the same vorticity as the vertical shear flow considered inthe previous section and so the dynamics of cell orientation are identical; forexample in figure 7 we see the cell orientation undergoing periodic motion atlow shear, attaining a stable equilibrium at moderate shear, and undergoingmore complex dynamics at high shear. At low shear, cells are able to swimupwards on average, figure 7a,b, and therefore are advected by the flow in thepositive x direction. At stronger shear, figure 7c, the mean vertical componentof swimming direction is approximately zero, and so horizontal advection bythe flow is suppressed. In particular, on comparing figure 7a and 7c, we seethat despite a 10-fold increase in the shear, the horizontal displacement isapproximately unchanged due to the suppression of vertical swimming.

The effect of increasing the strength of the intrinsic torque on cell trajecto-ries is depicted in figure 8. Note that the cell orientation undergoes the samedynamics as for vertical shear flow. As shown in figure 8a,b, for sufficientlysmall values of R/G, the cells swim upwards on average and are advected inthe positive x-direction by the flow. As R/G is increased, there is a bifurcationin the dynamics, the equilibrium orientation is no longer feasible and stable,and the cell no longer swims upwards on average. As a consequence, horizontaladvection by the fluid is much reduced.

4 Discussion

We have proposed a simple mechanistic model for gravitactic organisms expe-riencing an intrinsic reorientating torque which yields stable upwards helicaltrajectories in still fluid. Helical swimmers have a reduced upwards speed instill fluid compared to cells which swim straight upwards, so why are helicaltrajectories ubiquitous amongst single-celled organisms? We have proposed anovel hypothesis for why this might be of benefit to the many species whichlive in dynamic, turbulent environments. We have shown that in simple verti-cal shear flow, whereas non-helical swimmers tend to swim towards downwards

Helical swimming can provide robust upwards transport 11

Fig. 4 How torque strength affects trajectories; weak torque Example trajectorieswith shear ω/G = 1 with γ = 7π/16 for increasing strength intrinsic torque. (a) R/G = 0.1,(b) R/G = 1, for explanation of parameters see text. Cells are initially located at theorigin (indicated by black dot) with random orientation. (i) Trajectories of duration 20 non-dimensionalised on G and (ii) horizontal component of swimming orientation px for duration50. Dashed lines in (ii) indicate equilibrium solution given by equation (17).

12 R. N. Bearon

Fig. 5 How torque strength affects trajectories; strong torque As figure 4. (a)R/G = 2, (b) R/G = 20. To aid visualisation, only final 2 non-dimensional time units areshown in b(ii).

Helical swimming can provide robust upwards transport 13

Fig. 6 Mean upwards transport The vertical displacement of trajectory of duration 20divided by the duration due to combined effects of swimming and advection by shear forangle γ = [0, π/4, 3π/8, 7π/16]: (a) as a function of R/G for fixed shear ω/G = 1; (b), asa function of shear, ω/G, for fixed R/G = 20. The vertical lines in (a) indicate an upperbound on R/G for which the equilibrium solution is feasible, as calculated numerically fromequations (39-40). The equilibrium solution is not feasible for any parameters shown in (b),but a change in behaviour is noted near 2G

ωcos γ = 1 indicated by the vertical lines. Note

the trajectories of non-helical swimmers (R = 0) are identical to trajectories obtained withγ = 0 for arbitrary R. The mean is calculated from 100 trajectories starting with randominitial orientation at the origin.

14 R. N. Bearon

Fig. 7 Effect of horizontal shear flow on helical trajectories. As figure 3 but forhorizontal shear flow V = ωzi with shear (a) ω/G = 0.5, (b) ω/G = 2, (c) ω/G = 5.(i) Trajectories and (ii) vertical component of swimming orientation pz . Dashed lines ina(ii) and c(ii) indicate the mean value calculated over each trajectory. Dashed lines in b(ii)indicate equilibrium solution given by equation 19.

Helical swimming can provide robust upwards transport 15

Fig. 8 How torque strength affects trajectories in horizontal shear flow. Celltrajectories as described in figures 4,5 but for horizontal shear flow V = ωzi for increasingstrength intrinsic torque. (a) R/G = 0.1, (b) R/G = 1, (c) R/G = 2, (d) R/G = 20.

16 R. N. Bearon

flow and experience net downwards transport, helical swimmers can swim intoupwards moving fluid and experience net upwards transport. To understandhow this occurs, it is necessary to understand the competing torques actingon helical swimmers. In particular, we have shown that the intrinsic torquewhich generates helical motions in cells swimming in still fluid can alterna-tively act against the viscous torque experienced in shear flow. For certainparameters values, this can result in the helical motions being suppressed andcells attaining a new equilibrium orientation where the intrinsic torque bal-ances the combined effects of the gravitational re-orientating torque and theviscous torque so that the net swimming direction is towards the upwardsmoving fluid. This is likely to be beneficial for phototactic organisms seekingoptimum light conditions, but may also have relevance to a wide range of heli-cal swimmers such as sperm undergoing chemotaxis. We have also documentedhow helical swimming modifies transport in horizontal shear flows. Specifically,we have shown that helical swimming can reduce the ability of cells to swimvertically upwards, and consequently reduce horizontal dispersal due to differ-ential advection by the flow. Further simulations and experiments are requiredto understand what the implications are for the spatial distribution of heli-cal swimmers in more complex flow fields (Thorn and Bearon, 2010; Durhamet al, 2011). Single-celled organisms have been observed to switch gears (Polinet al, 2009), and switching between straight and helical swimming when ex-periencing a shear is a novel hypothesis that would be worth investigatingexperimentally. Gurarie et al (2011) observed significant differences betweenthe helical trajectories of strains of Heterosigma akashiwo from distinct geo-graphic regions. In particular, they found significant variation in the tangentialangle, which we predict will result in significant differences in the net verticaltransport in shear flow. In the process of adapting to the local environment,could it be that these differences in helical patterns could be due to differencesin the shear intensity experienced in their environments?

A Derivation of equations

Defining k as a unit vector in the vertical direction, the torque on the cell from intrinsicmotions and gravity is given by

L = L0p ∧ k + Tn (20)

where L0 is the magnitude of the torque due to the centre of mass offset (Pedley and Kessler,1992) and T is the magnitude of the intrinsic torque generated by the cell.

The viscous torque on a sphere rotating with angular velocity Ω in the presence ofambient vorticity ω is given by

Lv = 8πµa3(1

2ω −Ω), (21)

where µ is the viscosity and a is the radius of the sphere (Kim and Karrila, 2005).For zero Reynolds number swimming, the total torque on the sphere must be identically

zero:

Lv + L = 0, (22)

Helical swimming can provide robust upwards transport 17

thus we have the following expression for the angular velocity of the sphere:

Ω = Gp ∧ k +Rn +1

2ω, (23)

where G = L0/8πµa3, defined as 1/2B by Pedley and Kessler (1992), and R = T/8πµa3.For simplicity, we assume the flow is constrained to the x− z plane so that the vorticity isgiven by ωj.

To obtain a set of differential equations for the vectors defining the orientation of thecell, p(t) and n(t), we consider a general unit vector u rotating with angular velocity Ω:

u = Ω ∧ u

= G ((u.p)k)− (u.k)p) +Rn ∧ u +1

2ω ∧ u. (24)

Define orthonormal spherical polar co-ordinate system:

u = (sin θu cosφu, sin θu sinφu, cos θu) (25)

eθu = (cos θu cosφu, cos θu sinφu,− sin θu), (26)

eφu = (− sinφu, cosφu, 0). (27)

To obtain equations for θu etc, noting that we can write

u = θueθu + φu sin θueφu, (28)

by taking the dot product of Eq. 24 with eθu and eφu we get

θu = −G sin θp cos(φp − φu) +R sin θn sin(φn − φu) +ω

2cosφu

sin θuφu = G cos θu sin θp sin(φu − φp)−R (sin θn cos θu cos(φn − φu)− cos θn sin θu)

−ω

2cos θu sinφu

Taking these general expressions and substituting u = p and n in turn gives the requiredsystem of coupled equations (7-10).

B Equilibrium solutions

B.1 Still fluid

In still fluid (ω = 0), we can obtain a helical solution by setting θp = θn = 0 in equations(7-11):

0 = −G sin θep +R sin θen sin(φn − φp), (29)

φp =1

sin θep

(−R

(sin θen cos θep cos(φn − φp)− cos θen sin θep

)), (30)

0 = −G sin θep cos(φp − φn), (31)

φn =cos θensin θen

(G sin θep sin(φn − φp)

), (32)

cos γ = sin θep sin θen cos(φep − φen) + cos θep cos θen, (33)

where we have defined θp = θep and θn = θen as the equilibrium angles the vectors p and nmake with the vertical.

18 R. N. Bearon

B.1.1 Straight upwards swimming

If p and n are identical, so that γ = 0, we obtain the trivial equilibrium solution of the cellswimming directly upwards with θep = θen = 0. Similarly, if we set R = 0 we can obtain thetrivial solution of the cell swimming directly upwards with θep = 0 for any value of θen.

B.1.2 Helical swimming

We now consider cells which are not swimming directly upwards, that is seek genuinelyhelical solutions with θep 6= 0. From equation 31, we have that cos(φp − φn) = 0, and thusfrom equations 29 and 30 we have that

sin2 θep = (R/G)2 sin2 θen (34)

φp = −R cos θen. (35)

Finally, making use of equation (33) and the fact that cos(φp − φn) = 0, we have that

cos θep cos θen = cos γ. (36)

These equilibrium expressions can be re-arranged to give equations (12, 13).

B.2 Shear flow

In the presence of shear (ω 6= 0), when R = 0, the equations (7-8) describing the evolution ofswimming direction have a stable equilibrium solution found previously (Pedley and Kessler,1992) of sin θp = ω/2G,φp = 0. For this equilibrium solution, the equations describing theevolution of n, equations (9-10) are automatically satisfied and thus we are free to make anychoice of γ. However, we note that when R = 0, the equations describing the evolution of ndo not influence the cell trajectory.

An alternative equilibrium solution for more general values of R can be found by con-sidering equilibrium solutions of equations (9-10):

0 = −G sin θp cos(φp − φn) +ω

2cosφn, (37)

0 =cos θn

sin θn

(G sin θp sin(φn − φp)−

ω

2sinφn

). (38)

In particular if 2Gω

cos γ < 1, we can obtain an equilibrium solution θen = π/2, cosφen =2Gω

cos γ. From numerical simulations, we have found upswimming solutions, that is solutionswith 0 ≤ θp < π/2 to be stable. Furthermore, for solutions with γ ≤ π/2, we have foundthat π ≤ φen < 2π is also required for stability. With this choice of n, in order to satisfyθp = φp = 0, neglecting the trivial solution of p = n, from equations (7-8) we require:

sin θep =R

Gsin(φen − φep) +

ω

2Gcosφep, (39)

0 = −R

Gcos(φen − φep)−

ω

2Gsinφep. (40)

These equations can be solved for φp and θp in certain regions of R/G-ω/2G parameterspace. Specifically, equation (40) provides an implicit expression for φep, the solution ofwhich can be inserted into equation (39) to obtain an expression for sin θep which we requireto be less than 1 for the equilibrium solution to exist. Explicit constraints on the parameterscan be found for special cases. For example, if we take γ = 0, for which θen = θep = π/2, and

φen = φep = cos−1 2Gω

we can only obtain a stable equilibrium solution when

R

G=

ω

2G

√1−

4G2

ω2. (41)

Helical swimming can provide robust upwards transport 19

Fig. 9 Regions of R/G-ω/2G parameter space for which equilibrium solutionexists for example values of γ. Solution as given by equation 41 for γ = 0 (solid blueline with squares) and equation 42 for γ = π/2 (solid cyan line with circles) . Numericallycalculated solutions for γ = π/4 (green solid line) and γ = 3π/8 (red dashed line). Thehorizontal lines represents an additional lower bound on shear for equilibrium to exist:2Gω

cos γ = 1.

If we take γ = π/2, so that cosφen = 0 we have the solution sinφep = 0. Taking φen = 3π/2

we can obtain equilibrium solutions φep = 0, π yielding sin θep = ±(RG− ω

2G

)which gives

the constraint

R

G− 1 ≤

ω

2G≤R

G+ 1. (42)

As shown in figure 9, the region of space where solutions exist smoothly expands withincreasing γ from the line of solutions given by equation 41 when γ = 0 to the region givenby equation 42 when γ = π/2.

For general γ, we can combine equations (39) and (40) with the defining equation forγ, equation (11), to obtain an expression for the swimming orientation:

px = sin θep cosφep =ω

2G−R

G

√1−

4G2

ω2cos2 γ, py = −

2R

ωcos γ. (43)

Acknowledgements The model was developed through helpful discussions with Eli Gu-rarie.

20 R. N. Bearon

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