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Centripetal focusing of gyrotactic phytoplankton in solid-body rotation

A suspension of gyrotactic microalgae Chlamydomonas augustae swimming in a cylindrical water vessel in solid-body rotation is studied. Our experiments show that swimming algae form an aggregate around the axis of rotation, whose intensity increases with the rotation speed. We explain this phenomenon by the centripetal orientation of the swimming direction towards the axis of rotation. This centripetal focusing is contrasted by diffusive fluxes due to stochastic reorientation of the cells. The competition of the two effects lead to a stationary distribution, which we analytically derive from a refined mathematical model of gyrotactic swimmers. The temporal evolution of the cell distribution, obtained via numerical simulations of the stochastic model, is in quantitative agreement with the experimental measurements in the range of parameters explored.


I. INTRODUCTION
Many phytoplankton species are able to swim and their motility affects several basic processes in their life and ecology [1][2][3][4]. Swimming allows phytoplankton to explore the water column, moving from the well-lit surface layers during the day to the nutrients-rich deeper layers at night [5,6] (see also [7] and references therein). In order to perform this vertical migration, many phytoplankton cells are guided by an orienting mechanism when swimming. One of the simplest mechanism leading to orientation in the vertical direction is bottom-heaviness [8]. The unbalanced distribution of mass inside the cell induces a mechanical torque, due to gravity and buoyancy forces, which orients the cell in the direction opposite to gravity. This torque competes with the viscous torque, due to the hydrodynamic shear, which tends to rotate the cell.
When the swimming direction results from the balance between this two torques, the organism is said to be gyrotactic [9][10][11].
Gyrotaxis generates remarkable spatial distributions of swimming cells. In laminar conditions, it produces beam-like accumulations in vertical pipe flows [9] and concentrated thin layers in horizontal shear flows [12]. More recently, it has been shown by theoretical analysis and numerical simulations, that gyrotaxis induces intense small-scale fractal clustering in turbulent flows [13][14][15][16], providing a mechanisms for the micro-patchiness observed in motile phytoplankton [17]. Turbulence is characterized by extreme fluctuations of fluid acceleration, which can locally exceed gravity even at moderate Reynolds numbers [18]. The origin of these fluctuations has been traced back to small-scale vortices generating intense centripetal accelerations [19].
For applications in turbulent flow, the gyrotactic model has been recently modified to take into account the effect of fluid acceleration in the mechanical torque [20]. Remarkably, numerical simulations of turbulent flows have shown that the effect of fluid acceleration is to enhance the patchiness of gyrotactic phytoplankton by gathering cells into small-scale vortices [21].
In this work, we study experimentally, analytically and numerically the behavior of gyrotactic swimming cells in a controlled environment of uniform vorticity -i.e. a cylindrical water vessel in solid-body rotation. Preliminary, qualitative, results of the experiment were described in Ref [21]. We show that phytoplankton cells, initially uniformly distributed in the container, aggregate towards the axis of rotation eventually reaching a stationary, Gaussianlike distribution. We show analytically and by means of numerical simulations that a stochastic formulation of the refined gyrotactic model [21], which takes into account the fluctuations of the swimming direction, is able to reproduce quantitatively the time evolution of a distribution of cells and the asymptotic accumulation at different rotation frequency. The remaining of the paper is organized as follows. In Section II we introduce the experimental method, the theoretical model, the different analytical techniques and the numerical method. In Section III we discuss the data analysis and we compare the experimental results with the theoretical and numerical predictions.
Finally, Section IV is devoted to conclusions.
Technical details on the analytical computations are presented in the Appendices. The experimental setup is sketched in Fig. 1.
A suspension of gyrotactic cells is placed in a small cylindrical vessel of inner radius R = 18 mm and height H = 57 mm over a plate that rotates at angular velocity Ω, digitally controlled. After a short spin-up time of order R/(νΩ) 1/2 (ν is the kinematic viscosity of wa-ter) the fluid in the vessel reaches the solid-body rotation regime [24], at which we defines the ref- The laser sheet induces fluorescence of the C. augustae cells whose emission spectrum has a peak around 685 nm (determined by a spectrophotometer) [25]. Fluorescence images are taken by a digital camera (resolution 3000×2000 pixels) with a red filter (B+W 091 Dark Red) which cuts all wavelengths below 600 nm. Samples of the fluorescence images are shown in  [20,21], an effect which can be important in turbulence [18].
Due to the small size (∼ 10µm) and the small density mismatch with the fluid (less than 5%), cells are represented as point-like, spherical and neutrally buoyant particles (for a recent refined analysis see Ref. [27]) transported by the fluid velocity u(x, t) at cell position x with a superimposed swimming velocity v ṡ The magnitude of the swimming velocity, v s , is assumed constant, while the swimming direction p evolves aṡ The first term describes the orientation towards the direction opposite to the acceleration A in the cell reference of frame with characteristic speed v o = 3ν/δ (δ is the cell center-of-mass displacement relative to the geometrical center and ν the fluid kinematic viscosity). In the original model only gravity is taken into account and Here, following [20,21], we consider the total acceleration, due to gravity and fluid ac-celeration a = du/dt (acceleration due to swimming is neglected), so that A = g − a. The second term in the rhs of (2) represents cell rotation due to the local vorticity ω = ∇ × u.
The last term Γ r represents rotational diffusion in the swimming directions, which phenomenologically models the intrinsic stochasticity in the swimming behavior [28].
We emphasize the simplicity of the above model that neglects many details, including the unsteadiness of swimming due to flagella beating, deviations from spherical shape, cell-cell interactions and the feedback of cell motion on the surrounding fluid.

C. Deterministic motion in solid body rotation
We study analytically and numerically the motion of gyrotactic swimmers in the velocity field generated by the solid body rotation of the cylinder along its vertical axis: u = (−Ωy, Ωx, 0). Vorticity is parallel to gravity, We start by considering the dynamics in the absence of stochastic terms (i.e. Γ r = 0 in (2)) whose effects will be discussed in the following Section. In this limit it is possible to derive analytically the motion of swimming cells and show that they accumulate exponentially in time on the axis of rotation.
By introducing cylindrical coordinates with x = (r, z) and p = (p r , p z ), such that A = (Ω 2 r, −g) and u = ( where p ⊥ r = (−p y , p x ). The two parameters B ≡ v o /g and 1/γ ≡ g/Ω 2 represents the charac- When γr ≪ 1, the distance from the axis evolves asṙ = −γv s r implying that cells position moves exponentially towards the rotation axis r = 0: For general values of γr the time evolution of r(t) is more complicated but asymptotically the above results remain valid (see Appendix A for details).

D. Stationary distribution in presence of rotational diffusivity
The deterministic model predicts that swimming cells should converge on the rotation axis.
In reality, stochastic reorientation of the swimming direction, due to different biological behaviors, causes the cells to weakly deviate from the convergent trajectories predicted by (8), preventing the population from collapsing onto the rotation axis and eventually producing accumulation in a finite volume.
Because we are considering stochastic effects, which can be conveniently modeled as rotational diffusivity [11,28], in the following we will not discuss individual trajectories, rather we shall focus on the cell density n(x, t) which can be directly compared with the experimental measurements. We report here the basic ingredients and results of our approach, which is based on the so-called Generalized Taylor Dispersion theory [29], applied to gyrotactic phytoplankton [30,31]. The interested reader can find details of this approach in Refs. [30][31][32][33] (see also Appendix B).
We introduce the probability density function P(x, p, t) to find a cell at position x swimming in direction p. The evolution of P is ruled by the Fokker-Planck equation whereẋ andṗ are given by (1) and (2) and d r is the rotational diffusivity coefficient [11,26]. The quantity we are interested in is the population density, n(x, t) = dp P(x, p, t). Essentially, we seek for an effective evolution equation for the population density in terms of the advectiondiffusion equation [11,26,[30][31][32][33][34]] where V and D are the effective drift and diffusivity tensor, respectively.
Leaving the details in the Appendix B, the main idea for going from (9) to (10) is to assume that the orientation dynamics is the fastest process. Consequently, we can treat p as a random unit vector with probability density function f (p) given by the stationary solution of [11,26,34]. Within this approximation the effective drift becomes V = u + v s p = u+v s dp pf (p). The derivation of the diffusivity tensor is more complicated. The Generalized Taylor dispersion has been used to derive D in homogeneous shear flow [32,33]. The method can be extended to inhomogeneous shears provided that the relaxation in the swimming direction is sufficiently fast [30]. We are here in a similar situation: the equilibrium orientation direction depends on the distance from the axis, according to (7). We therefore assume that, at each distance r, the distribution of orientations relaxes to a stationary distribution which parametrically depends on r. Within this adiabatic approximation, we are able to obtain explicit expressions for the drift coefficient and the diffusivity tensor.
Once V(r) and D(r) are known, we can solve where F 3 (λ) a dimensionless function of the parameter λ = 1/(2Bd r ), and the coefficient N can be written in terms of the total number of cells N s as N = N s γ/(2πHv s BF 2 3 (λ)). From the distribution (11) we obtain the average distance of the population from the cylinder axis in stationary conditions which reads where the subleading contribution from the upper integration extreme R has been neglected. In particular, we will be interested in the evolution of the average radial distance r(t) = drn(r, t)r/ drn(r, t).

In this
By using the total population density, the stationary average distance (12) becomes where r s is given by (12)   Rv 0 n exp (r, t)dr (15) and looking at the plateau that can be observed at long time values (see Fig. 4). Figure 3 shows both the theoretical value in the absence of background (12), which clearly underestimates the asymptotic radius, and the expression (14) corrected with the background coefficient β = 1.1.
We remark that the theoretical line in Fig. 3 is obtained without free parameters, as β is fitted from the asymptotic distribution n s (r) for a single rotation frequency (Fig. 2).
By using the numerical method discussed in Section II E it is possible to obtain the time evolution of the radial population density n(r, t) and of the mean radius. We remark that the time More in general, understanding the interplay between swimming and fluid transport is crucial to rationalize phytoplankton ecology [2,3] and also for industrial applications. For example, many (motile) microalgae are cultured in photobioreactors to be commercially used as nutrients, for biofuels production or for cosmetic industry [40].
where I k (λ) is the modified Bessel function of the first kind and order k [42]. In the above equation and in the following we shall use the superscript and diffusion tensor where v 2 s /d r gives the dimensional contribution; ⊥ and label the direction perpendicular and parallel to the bias, respectively. For γ = 0 the swimming direction is biased towardŷ. The perpendicular component of the diffusion tensor can be found exactly [30] D (0) For the parallel one only an approximate expression (not reported here, as not needed in the following) can be found.
When γ = 0, swimming is biased toward a direction that depends on r. Our approximation consists in taking r fixed in Eq. (B4) and finding the stationary PDF of θ which will depend on r parametrically. This is a sort of adiabatic approximation, valid when r does not change much in the time scale over which the PDF of orientation becomes stationary. In this way the average swimming direction will depend on r aŝ b = −γr being at an angle −Φ(r) with Φ(r) = arctan(γr) with respect toŷ. Since r is assumed fixed, the PDF of θ will simply be (B5) with mean −Φ(r), i.e. Introducing the rotation matrix the drift can be expressed as withb given in (B9), and the diffusion tensor by As for the latter we are mainly interested in the radial component which reads Now we can approach the advection diffusion equation (10). In particular, because neither the drift nor the diffusion tensor depends on z, we can integrate over z and derive the equation for the radial dynamics, which is the one of interest for us, with Hence the stationary population takes a Gaussian form.
Then we can compute the average radial distance r s at stationarity as in very good agreement with simulations ( Fig. 5).

Analytical approximation in d = 3
The computation in d = 3 can be performed following step by step the procedure above described for the two-dimensional model. from Refs. [10,11,26]). We briefly summarize the results in the following.