2.1. Materials and methods

Herein, we selected Chlamydomonas reinhardtii (FACHB-265, from Freshwater Algae Culture Collection at the Institute of Hydrobiology, National Aquatic Biological Resource Center) as a model microorganism. The cells were cultivated in a sterile Tris-acetate-phosphate (TAP) medium in a dark box measuring 40 $\rm cm$ $\times$ 40 $\rm cm$ $\times$ 40 $\rm cm$ , with light intensity of light-emitting diode of 40 $\rm W$ during the daytime and 0 $\rm W$ at night for $7{-}14$  days. Before experimentation, energetic cells were chosen: 15 $\textrm {ml}$ of cell suspension was centrifuged for 10 $\rm {min}$ at 100 $\textit {g}$ force, leaving 3 $\rm {ml}$ of cell suspension at the bottom of the centrifuge tube. A standing time of 60 $\rm {min}$ was required for cell motility recovery. The cells used in this study exhibited an average length scale (major axis) of $d=19.4$ μm, an aspect ratio of $\kappa =1.27$ and a velocity of $U=3.92\times 10^{-5}$ ${\rm m\, s^{-1}}$ . These physical parameters were determined through measurements conducted on a representative sample of cells, ensuring an adequate sample size for statistical significance. The background fluid medium (TAP) possessed a density of $\rho =1.026 \times 10^6$ ${\rm g\, m^{-3}}$ and dynamic viscosity of $\mu =0.92$ ${\rm g\, (m\,s)^{-1}}$ , as measured using an Anton Paar MCR302 rheometer. Thus, the Reynolds number of a swimming cell was estimated as $Re=\rho Ud/\mu \sim 0.001$ .

Herein, as shown in figure 1, a circular microcontainer, with a height of $h=20$ μm and a radius of $r_c=150$ μm, was utilised. This microcontainer was fabricated from polydimethylsiloxane (PDMS) using a standard soft lithographic technique. It comprises a cylindrical groove in PDMS, with its top surface sealed using a glass slide after being filled with a droplet of cell suspension. To increase the likelihood of capturing individual cell movement within the microcontainer, we employed two strategies. (i) Nine uniformly distributed microcontainers were simultaneously created on the feature surface of a single PDMS device (10 $\rm mm$ $\times$ 10 $\rm mm$ ). Consequently, one droplet of cell suspension could be shared by nine microcontainers in a single experiment, thereby enhancing the probability of meeting our experimental conditions for each microcontainer. (ii) The cell suspension was diluted with TAP prior to observation. Cell movement was recorded using an inverted microscope (Nikon, Eclipse Ti) equipped with a $ \times 20$ or $ \times 40$ Nikon phase objective, a high-speed camera (Nikon, DS-Qi1Mc, 1280 $\times$ 1024 resolution, capturing at 1 ${\rm s}$ /17 frames) and microscopic image analysis software (Nikon, NIS-Elements D 4.20.03 64-bit). To mitigate the phototactic response of cells, the microscope was situated within a dark box fitted with a bandpass interference filter ( $\lambda =671\pm 2$ $\rm nm$ , with a full width at half-maximum of $10 \pm 5$ $\rm nm$ ).

Figure 1.

A schematic of the experiment.

2.2. Experimental results

As shown in figure 2(a), the trajectories of a single cell intuitively show a generally uniform distribution in the microcontainer, indicating a very weak wall-constraint effect. This pattern differs from the strong restriction-induced near-wall swimming observed for a single Chlamydomonas reinhardtii, as reported by Ostapenko et al. (Reference Ostapenko, Schwarzendahl, Böddeker, Kreis, Cammann, Mazza and Bäumchen2018). To elucidate this discrepancy, the details of a position are taken from figure 2(a) with figure 2(c–h) being the corresponding snapshots under the microscope. Accordingly, a profound observation in our experiment is the recurring rotation of Chlamydomonas reinhardtii near the bottom wall of the microcontainer. This near-wall rotational behaviour may account for the uniform trajectories compared with previous results. Notably, distinct highlight points are observed, particularly near the boundaries of the container, suggesting the occurrence of rotational behaviours in those regions (see the supplementary movies 1 and 2).

Figure 2.

Trajectories (a,b,j) and relative probability density (heat maps) (i,k) of a single Chlamydomonas reinhardtii: (a,i) experimental results; (j,k) simulation results (Re = 0.01). (b) Partially enlarged detail of the trajectories highlighted by the blue square in (a). (c–h) Snapshots of the rotational Chlamydomonas reinhardtii, corresponding to the circle trajectory (marked by a green line) in (b). Plotting scale: 3.18 px μm⁻¹ ( $\times 20$ ).

Figure 3.

Snapshots of the rotational Chlamydomonas reinhardtii with one defective flagellum. Potting scale: 13.79 px μm⁻¹ ( $\times 40$ ).

One might question the mechanism behind this rotation. We further observe the flagella of Chlamydomonas reinhardtii by using an inverted microscope with a $\times 40$ objective lens. It is found that, at the interface of fluid and solid surfaces (the boundaries of the container), an enveloped liquid film (the yellow circle) is induced, leading to the cell and one of its flagella both being captured by the fluid film (in figure 3). The defective flagellum keeps oscillating but the confinement of the liquid film leads to the characteristics of deactivation on the macrolevel. To recover normal swimming, the cell tries to escape from the confinement of the liquid film by the acute paddling of the other free flagella, leading to a rotational torque (analysed in detail in § 3). Accordingly, the Chlamydomonas reinhardtii shows rotation in a quasi-two-dimensional plane. Building upon these observations, we numerically adapt the two-dimensional squirmer model to replicating the quasi-two-dimensional rotational motion observed in our experiments. This is thoroughly discussed in § 3.

Otherwise, Chlamydomonas reinhardtii exhibits ballistic behaviour. Therefore, we further analyse mean-squared displacements (MSDs) of the cell in this experiment, as depicted in figure 4. In this section, we mainly focus on the experimental results (the black curve) and its fitting line (the red dot-dash line); other lines are analysed in § 3. On extended time scales ( $t\gt 10$ s), the MSD reaches a plateau, consistent with the available experimental results for a single Chlamydomonas reinhardtii swimming in a quasi-two-dimensional circular domain (Ostapenko et al. Reference Ostapenko, Schwarzendahl, Böddeker, Kreis, Cammann, Mazza and Bäumchen2018). Note that our single-cell experiment is repeated more than 10 times using different cells each time. On short time scales ( $t\leq 10$  s), the cell exhibits superdiffusive diffusion $\left \langle {\triangle r^2(t)} \right \rangle \sim t^{1.7}$ . This power exponent of 1.7 is smaller than that of 2.0, as reported by Ostapenko et al. (Reference Ostapenko, Schwarzendahl, Böddeker, Kreis, Cammann, Mazza and Bäumchen2018). This also is attributed to the recutting rotational behaviour of the cell (as shown in figure 2), as rotation weakens diffusion by reducing its linear travel paths.

Figure 4.

Comparison of experimental and simulation results of the MSDs for Chlamydomonas reinhardtii ( $t^{1.7}$ is the linear fitting result of the experimental results).

The heat maps for the motion of the cell are further collapsed into radial probability densities, $P(r)$ , as depicted in figure 5. We define $P(r)$ as follows:

(2.1) \begin{equation} P(r)= \dfrac {h(r)/(2\pi r \triangle r)}{{\int _{0}^{r_c}{\dfrac {h(r)}{2\pi r \triangle r} }{\rm d}r}}, \end{equation}

where $r$ is the distance from the centre of the microcontainer and $h(r)$ is the count of all the positions of the cell in a circular shell at distance $r$ with thickness $\triangle r$ . In contrast to the intuitive observation depicted in figure 2(a), $P(r)$ starts with a plateau near the centre of the container and notably increases close to the wall, consistent with the results of Ostapenko et al. (Reference Ostapenko, Schwarzendahl, Böddeker, Kreis, Cammann, Mazza and Bäumchen2018). This suggests that the rotation near the boundaries may be the cause of the contradiction between the intuitive observation and $P(r)$ , as depicted in figure 2(i).

Figure 5.

Comparison of experimental and simulation results of the radial probability densities $P(r)$ for the cell within the microcontainer, derived from more than 10 independent experiments.

3.1. Squirmer with one defective flagellum

The two-dimensional squirmer model, as proposed by Blake (Reference Blake1971a ), has been adopted to simulate the locomotion of microorganisms. This model defines a slip velocity at the surface of microorganisms through an infinite series expansion of the radial and polar components:

(3.1) \begin{equation} \boldsymbol {u}^s(r)=\sum _{n=0}^{\infty }A_n\cos (n\theta )\hat {\boldsymbol {r}}+\sum _{n=1}^{\infty }B_n\sin (n \theta )\hat {\boldsymbol {\theta }}, \end{equation}

where $\hat {\boldsymbol {r}}$ and $\hat {\boldsymbol {\theta }}$ refer to the radial and polar unit vectors for a given point on the surface of a squirmer, respectively, as shown in figure 6(d). Parameters $A_n$ and $B_n$ are the corresponding time-dependent amplitudes. For simplicity, the radial component can be disregarded (Blake Reference Blake1971a , Reference Blakeb ) and this simplified squirmer model is represented by the second-order truncated tangential velocity:

(3.2) \begin{equation} \boldsymbol {u}^s(\theta )=(B_1\sin \theta +2B_2\sin \theta \cos \theta )\hat {\boldsymbol {\theta }}, \end{equation}

where the coefficient $B_1$ contributes to the propulsion of the squirmer, determining the steady-state swimming velocity $U_s = B_1/2$ under Stokes flow conditions. By contrast, the coefficient $B_2$ determines the intensity of vorticity around the squirmer. Accordingly, a swimming parameter defined using $\beta = B_2/B_1$ is usually introduced to categorise squirmers into three types: pushers ( $\beta \lt 0$ ), pullers ( $\beta \gt 0$ ) and neutral squirmers ( $\beta = 0$ ). As indicated by their names, pushers (e.g. bacteria such as $\textit {E. coli}$ and $\textit {B. subtilis}$ ) propel themselves from their rear ends, while pullers (e.g. algae such as Chlamydomonas reinhardtii) generate thrust from their front ends (Koch & Subramanian Reference Koch and Subramanian2011). Neutral squirmers (e.g. $\textit {Volvox}$ ) exhibit symmetric flow without vorticity (Fadda et al. Reference Fadda, Molina and Yamamoto2020).

As previously discussed, the rotational behaviour of $\textit {Chlamydomonas reinhardtii}$ is hypothesised to result from one defective flagellum of the cell. Consequently, we numerically adapt the classic squirmer model by partially inhibiting the slip velocity on the boundaries of the squirmer for demonstrating the motion of a single microalga with one defective flagellum, as illustrated in (3.3) (also see figure 6):

(3.3) \begin{equation} \left \{ \begin{array}{ll} u^s\left (\theta \right ) = \left (B_1\sin \theta +2B_2\:\sin \theta \cos \theta \right )\hat {\boldsymbol {\theta }}, & \theta \in [\alpha_1^T,\alpha_2^T]\cup \in [\alpha_2^T, \alpha_1^T]\\ u^s\left (\theta \right ) = 0, & otherwise \end{array}, \right. \end{equation}

where the slip velocity operates within a specific angle range of $[\alpha _1^T,\alpha _2^T]$ , while being inhibited in other angle ranges. Considering the contingency of one defective flagellum, the ranges of $\alpha _1^T$ and $\alpha _2^T$ both follow a uniform distribution, i.e. $\alpha _1^T\sim U(0,2\pi )$ , $\alpha _2^T\sim U(0,2\pi )$ , at each given periodic time $T$ . Angle $\alpha =|\alpha _2^T-\alpha _1^T|$ correspondingly follows a triangular distribution, its probability density function being

(3.4) \begin{equation} p\left (\alpha \right ) =\left \{ \begin{array}{*{2}{ll}} (2\pi-\alpha)/2\pi^2, &\alpha \in [0,2\pi]\\ 0, &\textit {otherwise} \end{array}. \right. \end{equation}

Time T is defined to be negatively correlated with the translational velocity of the squirmer. According to the simulation tests, the choice of $T$ follows the relationship $T\sim 10^3/Re$ . Notably, the condition of one defective flagellum fails with $\alpha =2\pi$ , where this model degenerates into the classical squirmer. Lauga et al. (Reference Lauga, DiLuzio, Whitesides and Stone2006) investigated the hydrodynamic mechanisms of $\textit {Escherichia coli}$ swimming in circles near walls. They demonstrated that the positive rotation of the cell body generates a net viscous force on the body, while the negative rotation of its helical bundle results in a net negative viscous force on the flagella, leading to a negative torque. This torque is counterbalanced by a positive torque arising from viscous resistance to rotation. In our study, a single microalga with one defective flagellum is both force-free and torque-free, as shown in figure 6. The defective flagellum causes changes in thrust and drag (see figure 6 a,c), which are equal in magnitude but opposite in direction during steady swimming along the trajectory (force-free). However, because these forces are not collinear, they produce rotational torque. This negative torque induced by thrust and drag is counterbalanced by a positive torque resulting from the hydrodynamics of the rotation, resulting in a torque-free condition.

Figure 6.

Schematic of mapping between (b) Chlamydomonas reinhardtii cell and (d) squirmer with one defective flagellum. A single flagellum of the cell is defective by the effect of the container boundaries, leading to the ‘thrust’ and ‘drag’ change from (a) linear to (c) nonlinear and resulting in the circular trajectory.

Herein, the motion of a single microalga with one defective flagellum is numerically simulated within a lattice Boltzmann method (LBM) framework. Further details of the method can be found in Appendix A.

3.2. Model validation

To validate the feasibility of numerically adapting the squirmer model to a squirmer with one defective flagellum, simulated results are compared with the experimental findings (as shown in figure 4). Following the experimental set-up outlined in § 2.1, within the LBM framework, the squirmer with a diameter of $d$ is confined within a quasi-two-dimensional circular domain ( $d_c=15.5d$ ). The squirmer is initially released at a random location within the domain, and its trajectories are tracked. As shown in figure 2, utilising dimensional conversion, the superdiffusive diffusion of the squirmer with one defective flagellum at short times ( $t\leq 10\, \rm s$ ; $Re=0.001$ ) aligns closely with the experimental results ( $Re \approx 0.001$ ). Notably, the swimming Reynolds number for the squirmer is defined as $Re=\rho U_s d/\mu$ , where $\mu$ denotes the fluid dynamic viscosity. Furthermore, our serial code needs over 2 months to execute a calculation for low-Reynolds-number flow ( $Re=0.001$ ) while ensuring computational convergence. However, it still does not reach the desired time range of $t=10^3\, \rm s$ . Considering the computational time constraints, we opted to utilise a slightly higher Reynolds number ( $Re=0.01$ ) for validation purposes. Despite the absence of overlapping scales, the diffusion coefficient aligns with the experimental findings (figure 4), indicating similar hydrodynamics between the simulated and experimental scenarios. Furthermore, figure 4 illustrates that the radial probability densities $P(r)$ for both the cell and the squirmer ( $Re=0.01$ ) in the microcontainer exhibit exceptional agreement. Meanwhile, the heat maps depicting the relative probability density for the squirmer closely resemble those obtained from the experiments, as illustrated in figure 2(i,k). The consistency demonstrated by these findings confirms the ability of the model to accurately predict experimental outcomes. Notably, for $Re=0.01$ , a lattice time of $T = 10^5$ is employed (equation (3.3)). Hence, we find that T is not sensitive to the results of the superdiffusive behaviours for the squirmer, with lattice time of the same order of magnitude after careful testing.

3.3. Influence of parameters on the motion of a squirmer with one defective flagellum

In this section, we simulate the motion of a single microalga with one defective flagellum in an unconfined fluid to investigate the influence of finite fluid inertia, self-propelling strength ( $\beta$ ) and active angle ( $\alpha$ ) on its swimming behaviours. To provide a thorough evaluation of the hydrodynamic behaviour of the squirmer with one defective flagellum, we presume that one flagellum of each particle remains defective for an extended duration (i.e. $T=\infty$ ), like an Effrenium voratum that keeps on rotating with its lone longitudinal flagellum (Huang et al. Reference Huang, Hu, Yang, Liu and Zhang2021). The squirmer with a diameter of $d=40\triangle x$ (where $\triangle x$ is the lattice width) is released at the centre of a computational domain with dimensions of $L$ $\times$ $L$ $=$ 20 $d$ $\times$ 20 $d$ . Periodic boundary conditions are applied to all boundaries. We have carefully validated our LBM code (Appendix B). We observe that the squirmer with one defective flagellum undergoes stable circular orbit motion after transient dynamics, effectively reproducing the rotation of the cell observed in our experiment. As depicted in figure 7(a), the swimming speed of the squirmer increases with $\beta$ to varying degrees. This contrasts with the swimming of a classic squirmer at the Stokes flow limit (where $Re=0.01$ can be considered), where the speed of such a squirmer remains constant, $U_s = B_1/2$ , regardless of $\beta$ (Blake Reference Blake1971a , Reference Blakeb ; Chisholm et al. Reference Chisholm, Legendre, Lauga and Khair2016). This difference suggests that a substantial portion of the self-propelled kinetic energy of the squirmer with one defective flagellum is converted into rotational dynamics. Moreover, the increase of Re results in a decreasing speed for such a squirmer at small $\beta$ values ( $\beta \leq 2$ ). With further increases in $\beta$ , the squirmer swims fast at large Re values. Additionally, at identical Re values, an elevated $\beta$ results in a large rotational diameter, which is positively correlated with the speed, as illustrated in figure 7. To illustrate the possible mechanism, we plot the pressure and velocity distributions around the squirmer with one defective flagellum, as shown in figure 8. Around the body, the pressures are clearly not symmetrically distributed, and the combined forces cause them to rotate clockwise. For small Re values ( $Re=0.01$ ; see figure 8 a,b), the outer side of the squirmer with $\beta =0.5$ experiences predominantly positive pressure, which is evidently smaller than that for the squirmer with $\beta =5$ . This elevated pressure indicates a strong self-propelled force, subsequently resulting in a fast speed. However, it is observed that the squirmer with $\beta =0.5$ suffers higher pressure at its head than at the rear. This contrasts with the pressure distribution of the squirmer with $\beta =5$ , where the elevated pressure is observed at its rear. These two different pressure distributions may induce a greater net torque at $\beta =0.5$ than at $\beta =5$ , as the positive pressure at the rear (head) contributes to the counterclockwise (clockwise) rotation, resulting in a smaller rotational diameter at $\beta =0.5$ . Focusing on the streamlines around the bodies, an evident loop is observed inside the squirmer with $\beta =0.5$ , whereas it is outside that with $\beta =5$ . This indicates a smaller rotational diameter at $\beta =0.5$ than at $\beta =5$ intuitively. With an increase of Re (from $Re=0.01$ to 1), the pressure distribution for the squirmer with $\beta =0.5$ remains similar (compare figures 8 a and 8 c), but that for the squirmer with $\beta =5$ is distorted (compare figures 8 b and 8 d). This may be relevant to the accumulation feature of vorticity for the inertial puller ( $\beta \gt 0$ ).

Figure 7.

Swimming speed (a) and rotating diameter (b) of squirmers with one defective flagellum in an unconfined fluid with Re from 0.01 to 1.

Figure 8.

Pressure and streamlines around squirmers with one defective flagellum: (a) $Re=0.01$ , $\beta =0.5$ ; (b) $Re=0.01$ , $\beta =5$ ; (c) $Re=1$ , $\beta =0.5$ ; (d) $Re=1$ , $\beta =5$ . Black arrows denote clockwise rotation, and pressure is normalised using $\rho U_s^2$ . In such situations, $\alpha =180^{\circ }$ ; ‘outer side’ denotes $0^{\circ }\leq \alpha \leq 180^{\circ }$ and ‘inner side’ denotes $180^{\circ }\lt \alpha \leq 360^{\circ }$ .

We further examine the effect of active angle $\alpha$ on the swimming speed and rotational diameter of the squirmers with one defective flagellum, as shown in figure 9. The speed of the squirmer with $\beta =0.5$ increases monotonically with $\alpha$ , suggesting the presence of somewhat linear superposition properties in the flows distributed around the body as $\alpha$ is increased. Consequently, the change in rotational diameters broadly coincides with the pattern of the speed (figure 9 b). Furthermore, it is observed that the speed is only slightly faster at $Re=0.01$ than at $Re=1$ , suggesting similar flow-field structures around the squirmer ( $\beta =0.5$ ) at different Re values (figure 8 a,c). Given that $\beta$ is responsible for generating vorticity around the squirmer, a small $\beta$ value indicates that only a small amount of vorticity is induced with the increase of Re. In the case of a squirmer with $\beta =5$ , its speed peak is achieved at $\alpha =180^{\circ }$ , indicating considerably nonlinear features. Furthermore, the peak of the rotational diameter coincides with that of its speed. Generally, a high $\beta$ results in a fast speed of the squirmer, as depicted in figure 9(a). This differs from the behaviour of a classic squirmer ( $\alpha =360^{\circ }$ ), where a high $\beta$ leads to a slow puller ( $\beta \gt 0$ ) owing to the inclusion of weak fluid inertia (Wang & Ardekani Reference Wang and Ardekani2012). The mechanism may be related to the rotation, which induces a more complex fluid-structure interaction affecting the rotational swimming, especially for cases beyond the Stokes flow regime.

Figure 9.

Swimming speed (a) and rotating diameter (b) of squirmers with one defective flagellum in an unconfined fluid with active angles $\alpha$ from $30 ^{\circ }$ to $300^{\circ }$ .

Figure 10.

Power expenditure for rotational swimming as a function of (a) swimming pattern $\beta$ and (b) active angle $\alpha$ in an unconfined fluid with Re from 0.01 to 1.

A swimming squirmer with one defective flagellum injects energy into the fluid through the given tangential boundary velocity. This energy is eventually dissipated within the viscous fluid. The rate of work P is hence calculated as

(3.5) \begin{equation} P=-\int _S{\boldsymbol {n}\cdot \boldsymbol {\sigma }\cdot \boldsymbol {u}_E}\,{\rm d}S, \end{equation}

where $\boldsymbol {n}$ is the unit normal outward from the surface $S$ of the squirmer and $\boldsymbol {\sigma }$ , $\boldsymbol {U}$ and $\boldsymbol {u}_E$ are the stress tensor, translational velocity and boundary velocity of the squirmer, as shown in (A7), respectively. Figure 10 shows the power expenditure of the squirmer with different $\beta$ and $\alpha$ . Notably, $P$ is the averaged results over a period (a circle), and it is normalised using $B_1^2\mu d/4$ . The pattern of energy expenditure for the squirmer with $\beta$ is akin to that of an individual classic squirmer (Magdanz et al. Reference Magdanz, Medina-Sánchez, Schwarz, Xu, Elgeti and Schmidt2017), wherein a swimmer with a high $\beta$ expends additional energy. Moreover, an increase in Re contributes to an enhancement in energy expenditure, with a notable increase observed when fluid inertia cannot be neglected (from $Re=0.1$ to 1). Additionally, with the increase in the active angle $\alpha$ , as depicted in figure 10(b), the energy expenditure increases monotonically. This contrasts with the non-monotonic trends observed in swimming speed and rotational diameter (figure 9), suggesting that the self-propelled mode, rather than the swimming state, typically determines the energy consumption.