Phototaxis of Chlamydomonas arises from a tuned adaptive photoresponse shared with multicellular Volvocine green algae

A fundamental issue in biology is the nature of evolutionary transitions from unicellular to multicellular organisms. Volvocine algae are models for this transition, as they span from the unicellular biflagellate Chlamydomonas to multicellular species of Volvox with up to 50,000 Chlamydomonas-like cells on the surface of a spherical extracellular matrix. The mechanism of phototaxis in these species is of particular interest since they lack a nervous system and intercellular connections; steering is a consequence of the response of individual cells to light. Studies of Volvox and Gonium, a 16-cell organism with a plate-like structure, have shown that the flagellar response to changing illumination of the cellular photosensor is adaptive, with a recovery time tuned to the rotation period of the colony around its primary axis. Here, combining high-resolution studies of the flagellar photoresponse of micropipette-held Chlamydomonas with 3D tracking of freely swimming cells, we show that such tuning also underlies its phototaxis. A mathematical model is developed based on the rotations around an axis perpendicular to the flagellar beat plane that occur through the adaptive response to oscillating light levels as the organism spins. Exploiting a separation of timescales between the flagellar photoresponse and phototurning, we develop an equation of motion that accurately describes the observed photoalignment. In showing that the adaptive timescales in Volvocine algae are tuned to the organisms' rotational periods across three orders of magnitude in cell number, our results suggest a unified picture of phototaxis in green algae in which the asymmetry in torques that produce phototurns arise from the individual flagella of Chlamydomonas, the flagellated edges of Gonium, and the flagellated hemispheres of Volvox.

Figure 1

Phototaxis in Chlamydomonas. Cell geometry (in box) involves the primary axis ${\widehat{\mathbf{e}}}_3$ around which the cell spins at angular frequency ${\omega }_3$, the axis ${\widehat{\mathbf{e}}}_2$ in the flagellar beat plane, and ${\widehat{\mathbf{e}}}_1={\widehat{\mathbf{e}}}_2\times {\widehat{\mathbf{e}}}_3$. As the cell swims and spins around ${\widehat{\mathbf{e}}}_3$, its eyespot (red) moves in and out of the light shining in the direction of the large blue arrows. This periodic light stimulation leads to alternating cis and trans flagella dominance, producing rotations around ${\widehat{\mathbf{e}}}_{\mathbf{1}}$ and hence a phototactic turn.

Figure 2

Master plot of adaptive timescales in Volvocine green algae. For each of Chlamydomonas (this paper), Gonium [34], and Volvox [28] the dimensionless product of the rotation frequency $f_r$ around the primary body-fixed axis and the flagellar adaptive time ${\tau }_a$ is plotted as a function of the organism radius $R$ (bottom axis) and typical cell number $N$ (top axis). Uncertainties here and below are standard deviations (SD), except where indicated to be standard errors (SE).

Figure 3

Experimental methods. Setups to measure (a) the flagellar photoresponse of cells immobilized on a micropipette, and (b) swimming trajectories of phototactic cells in a sample chamber immersed in an outer water tank to minimize thermal convection, as described in text.

Figure 4

Flagellar photoresponse of an immobilized cell after a step up in light. Light from the left (blue arrow) illuminates the eyespot. Panels show overlaid waveforms of a single beat (a) in the dark and (b) starting at $50\mathrm{ms}$ ($52.5\mathrm{ms}$) for the cis (trans) flagellum after the step up. Scale bar is $5\mu \mathrm{m}$.

Figure 5

Flagellar beat cycles. (a) Angles $\mathrm{\Theta }$ on each flagellum (red for cis, blue for trans), relative to symmetry axis ${\widehat{\mathbf{e}}}_3$ (green) of the cell, are used to define the cycle. Scale bar is $5\mu \mathrm{m}$. (b) Typical time series of the two angles.

Figure 6

Flagellar forces and torques of unstimulated cells. Propulsive force (a) and torque about the cell center (c) of cis (red) and trans (blue) flagella during beat cycle of a representative cell, with cycle averages indicated by dashed lines. (b) and (d) show boxplots of peak values (${}^{*}$) and beat-average quantities ($\langle \rangle$) computed from $n=48$ flagella in 24 movies. Standard boxplot conventions hold; the blue box indicates span of first to third quartiles, red line indicates median, whiskers denote extent of data, and statistical outliers, where present, are shown with red crosses.

Figure 7

Torque fluctuations in unstimulated Chlamydomonas cells. (a) Distribution of net torques averaged over 1,224 beats of multiple cells. Red curve is Gaussian fit to the data. Orange circles indicate the mean values of $\widehat{\mathcal{T}}$ within each of 29 contiguous in-phase subsequences of beats (offset for clarity). (b) Mean and standard error (shaded area) of normalized autocorrelation functions of torques calculated from the sample of aforementioned in-phase sequences of beats.

Figure 8

The pivoting-rod model of the power stroke.

Figure 9

Distributions of geometric quantities for flagellar beats, computed from $n=24$ movies: (a) instantaneous flagellar beat frequency ${\widehat{f}}_{\mathrm{b}}$ (b) flagellar length $L$, (c) cell-body radius $R$, (d)–(f) per-beat flagella-line initial angle ${\widehat{\phi }}_0$, sweep angle ${\widehat{\phi }}_b$, and anchor angle ${\phi }_a$.

Figure 10

Schematic of flagellar photoresponse. A step up in light at $t=0$ (green) leads to a biphasic decrease in the mean value and oscillation amplitude of the cis phototorque (red), and a biphasic increase in the magnitude of the mean value of the trans phototorque (blue). The net torque (purple), the signed sum of the two contributions, has a biphasic response in both the oscillation amplitude and its running mean value.

Figure 11

Dynamics of the flagellar photoresponse. (a) The mean (dark blue) and standard deviation (light-blue) of the proxy torque during a step-up stimulus for one cell ($n_{\mathrm{tech}}$ = 4) fitted to Eq. (26a) (red line). (b) Fitted (${\tau }_r$, ${\tau }_a$) pairs and SEs for $n_{\mathrm{cells}}$ = 4 upon step-up stimulation. (c) As in panel (b) but for the delay time ${\tau }_d$ and time of maximum photoresponse amplitude $t^{*}$.

Figure 12

Dynamics of the adaptive model. (a) Response of the variables ${\omega }_1$ (blue) and $h$ (red) to a square pulse of stimulus (green), for $\rho =0.1$. (b) Response to rectified sinusoids.

Figure 13

Dynamics of the adaptive model. (a) Gain (31a) for various values of $\rho ={\tau }_r/{\tau }_a$. (b) As in panel (a), but for the phase shift $\chi$ (31b), with ${\tau }_d=0$. (b) Comparison of peak amplitude for step-up and oscillatory forcing as a function of $\rho$.

Figure 14

Frequency response of immobilized cells. (a) Measured beat-average phototactic torque determined from RFT (for positive phototaxis) at five stimulus frequencies (0.5, 1, 2, 4, and 8 Hz) for $n_{\mathrm{cells}}=3$ (blue) fitted to Eq. (31a) (red line). (b) As in panel (a), but for the phase $\chi$ of the response, fitted to Eq. (31b). The photoresponse shown (in blue) for three different stimulus frequencies (in green): (c) 1 Hz, (d) 2 Hz, and (e) 4 Hz.

Figure 15

Helical trajectory of a cis-dominant cell which is swimming in a direction aligned with the light source located at the bottom. Principal body rotation axes are depicted as cyan, magenta and black arrows for ${\widehat{\mathbf{e}}}_1$, ${\widehat{\mathbf{e}}}_2$, and ${\widehat{\mathbf{e}}}_3$ axes, respectively. Eyespot is always located in the shading hemisphere of the cell body and is pointing outwards from the helix, as is the cis flagellum (not shown).

Figure 16

Illustration of the effect of torque fluctuations on eyespot orientation. A fluctuating torque imbalance between the two flagella induces transient rotations about ${\widehat{\mathbf{e}}}_1$, tilting the ${\widehat{\mathbf{e}}}_3$ axis to new directions ${\widehat{\mathbf{e}}}_3^{\prime }$ or ${\widehat{\mathbf{e}}}_3^{″}$, depending on the sign of the tilt angle ${\theta }_1$. The eyespot, oriented as shown by the dark orange oval, then points up or down (lighter orange ovals).

Figure 17

Positive phototaxis from model for nonhelical swimming. (a)–(c) Time evolution of adaptive variables, Euler angles, and eyespot signal during a phototurn, i.e., from swimming orthogonal to the light to moving directly toward it. (d) Trajectory of the turn showing initial (S) and final (T) orientations of the cell (also in magnified insets). (e)–(h) Analogous to panels above but with dynamics starting from an orientation facing nearly away from the light. Parameters are $\widehat{P}=0$, $P^{*}=-0.4$, $\alpha =7$, $\beta =0.14$, and $U=2$, with the eyespot along ${\widehat{\mathbf{e}}}_2$ (i.e., $\kappa =0$ but shown in canonical position), and ${\tau }_d=0$. Color scheme of cell's principal axes is same as in Fig. 15.

Figure 18

Positive phototaxis with helical swimming. As in Fig. 17 but with $\widehat{P}=0.3$.

Figure 19

Eyespot shading during phototurns with underlying cis and trans dominance. Evolution of (a) the adaptive photoresponse variables $P$ and $H$ and light projection (b) during positive phototaxis with unstimulated cis dominance, with $\widehat{P}=0.3$ and eyespot angle $\kappa =\pi /4$. (c), (d) As in panels (a) and (b), but for trans dominance, with $\widehat{P}=-0.3$.

Figure 20

Iterated map of the reorientation model. (a) Cobwebbing of iterations starting from $\varphi$ near $-\pi /2$ for positive phototaxis (upper branch in yellow) and near $+\pi /2$ for negative phototaxis (lower branch in purple), as indicated by triangles, for light shining toward $-{\widehat{\mathbf{e}}}_x$. Values of $\xi =\mp 0.29$ are used which are calculated based on Eq. (58) using values from Table 3. (b) Response factor $1/Q$ in Eq. (59) for number of half-turns needed for alignment as a function of tuning parameter, for various values of $\rho$ as indicated.

Figure 21

Phototactic swimmers tracked in three dimensions. (a) A U-turn: trajectory in black is prior to light stimulation, that in green is afterwords. Blue arrows indicate direction of light. The cropped trajectory used for fitting the reorientation dynamics is bounded by the points S and T. (b) Dynamics of the reorientation angle $\mathrm{\Phi }$ (blue) for the cropped trajectory fitted using Eq. (61). (c) Box plot of the distribution of fitted ${\tau }_x$, along with steady-state (green) and nonequilibrium (purple) estimates and SEs derived from micropipette experiments.

Figure 22

Response functions. For $\kappa ={\tau }_d=0$, the graph compares the steady-state response function $(\pi /4)G\left({\omega }_3\right)cos{\chi }_0$ in Eq. (31) (dashed blue), the initial average response $\overline{\mathrm{\Xi }}\left({\omega }_3\right)$ in Eq. (A9) (green) and the $n\to \infty$ limit of the transient response ${\overline{\mathrm{\Xi }}}_{\infty }\left({\omega }_3\right)$ (dashed red), as functions of $f_r={\omega }_3/2\pi$.

Figure 23

The functions $Q\left(T\right)$. The function $Q^{\left(n\right)}\left(T\right)$ (blue), defined in a piecewise manner between endpoints (blue circles), can be approximated as a straight line $Q_{\mathrm{approx}}\left(T\right)=\overline{\mathcal{Q}}T$ (red). Both $Q^{\left(n\right)}\left(T\right)$ and $\overline{\mathcal{Q}}$ were computed with the experimentally derived ${\tau }_r=0.009$ s and ${\tau }_a=0.524$ s. The value of $f_r$ was taken to be 1.67 Hz. The value of $\overline{\mathcal{Q}}({\omega }_3,{\tau }_r,{\tau }_a)$ was calculated to be 0.268.

Figure 24

Contour plot maps of $\overline{\mathcal{Q}}$ (a) and $\mathcal{Y}$ (b) with ($\alpha ,\beta$) pairs acquired from micropipette experiments shown with red markers (step up as “x” and frequency as “o”). (c)–(e) Plots of $Q^{\left(n\right)}\left(T\right)$ (blue line) and linear approximation $Q_{\mathrm{approx}}\left(T\right)=\overline{\mathcal{Q}}T$ (red line) for three ($\alpha$, $\beta$) value pairs shown in panels (a) and (b) as circle, square, and triangle, respectively. (c) is based on experimental data (open red circle), (d) corresponds to $\mathcal{Y}\approx 0.7$ (solid purple square), and (e) corresponds to $\mathcal{Y}\approx 0$ (solid purple triangle).