Cellular and Molecular Bioengineering, Vol. 2, No. 2, June 2009 (
2009) pp. 244–254 DOI: 10.1007/s12195-009-0062-x

Laser Trap Characterization and Modeling of Phototaxis in Chlamydomonas reinhardtii JONATHAN A. HOLLM, RANJAN P. KHAN, ELLISHA N. MARONGELLI, and WILLIAM H. GUILFORD Department of Biomedical Engineering, University of Virginia, Box 800759, Charlottesville, VA 22908, USA (Received 17 March 2009; accepted 15 May 2009; published online 30 May 2009)

light-sensitive ‘‘eyespot.’’ The flagellum adjacent to this eyespot is referred to as the cis flagellum, and the other as the trans flagellum. The mechanism of the phototactic response of Chlamydomonas was first explained by Foster and Smyth7 as a change in cell directionality in response to varying light signal from the eye spot. It is generally agreed that the eyespot is a directional light detector.7 If the cell is swimming toward the light source, the eyespot receives a constant level of light throughout the cell body’s rotation, so the cell continues swimming along the same path. If the cell is perpendicular to the light source, the eyespot will receive varying light signals as it rotates. This rhodopsin-dependent signal triggers a change in calcium ion concentration in the flagella, causing the flagella to alter their waveforms.11,12,19,25 One flagellum then becomes ‘‘dominant,’’ causing the cell to veer toward the light source. The change in flagellar beat pattern is thought to be mediated by calcium-dependent modulation of the outer-arm dyneins,15 perhaps as a result of direct calcium regulation of dynein via its light chain 4.18 The nature and functional consequences of changes in flagellar waveform are debated. Previous studies that employed high-speed cinematography techniques concluded that in the presence of light, the trans flagellum increases its stroke amplitude while the cis flagellum decreases its amplitude, allowing for the mechanical dominance of the trans flagellum and the reorientation of the cell.12 However, high-speed cinematography is limited to relatively short durations of time, provides data only on changes in flagellar shape and is restricted to the focal plane of the microscope. To circumvent these limitations, some have used electro-optical monitors of Chlamydomonas flagellum behavior that enable long recording periods with rapid response13,16,17; these have generally supported the idea that asymmetric changes in flagellum force, frequency, or synchrony between the flagella occur in response to changes in light at the eye spot. Which of these changes is fundamental to successful phototaxis has yet to be determined.

Abstract—The unicellular protist Chlamydomonas reinhardtii swims toward light sources using an anterior pair of flagella for propulsion and an eye spot that serves as a directional photoreceptor. Free-swimming Chlamydomonas rotate about the long axis of the cell, and the varying signal of the eye spot is thought to make one flagellum dominant in its beating, re-orienting the cell toward the light source. While dominance is broadly accepted, the underlying nature of dominance—changes in force, frequency, or synchrony—has yet to be determined. It has also yet to be directly demonstrated that this model accounts for phototaxis. We used a laser trap to capture free-swimming Chlamydomonas and measure the propulsive forces and beat frequencies generated by the flagella in the presence and absence of a directional light source. A 3D computational model of Chlamydomonas motility was developed using parameters measured in the laser trap. The data suggest that the most functionally significant change in the flagella in response to light is a waveform change in the trans flagellum that increases its lateral beat strength. The computational model shows that subtle changes in lateral beat strength are necessary and sufficient for phototaxis, and explains the conserved rotational rate of Chlamydomonas. Keywords—Computational model, Spectral analysis, Laser trap, Cell motility, Flagella.

INTRODUCTION Chlamydomonas reinhardtii is a unicellular green alga with two flagella that allow swimming motility (Fig. 1). These flagella are located at one end of the organism and pull C. reinhardtii through the aqueous environment. The two flagella beat in different planes relative to each other, causing the cell to rotate approximately twice per second about its long axis as it moves forward.13,19 Chlamydomonas reinhardtii also exhibits phototaxis22; it achieves this via a

Address correspondence to William H. Guilford, Department of Biomedical Engineering, University of Virginia, Box 800759, Charlottesville, VA 22908, USA. Electronic mail: [email protected] J. A. Hollm, R. P. Khan, and E. N. Marongelli have contributed equally to this work.

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2009 Biomedical Engineering Society

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FIGURE 1. Schematic diagram and coordinate system for the 3D simulation of C. reinhardtii. The cell produces forces from two flagella, trans and cis (the cis flagellum is nearer the eye spot, red). Also visible are the large chloroplast (green) and the contractile vacuole (gray, center of the cell body) by which the laser trap captures swimming Chlamydomonas. The cell moves forward along its major axis (z) with velocity |v|. The cell can also rotate with velocities xx and xx about its x and z axes, respectively. The x axis is normal to a plane defined by the major axis of the cell and the minor axis through the eye spot.

The purpose of this study was to test the flagellum dominance theory of phototaxis, and specifically to determine whether changes in flagellum frequency, peak force generation, or waveform are responsible for reorientation of Chlamydomonas with respect to a light source. A laser trap transducer was used to measure the forces generated by single, ‘‘free-swimming’’ C. reinhardtii8,14 in the dark and when exposed to a directional light source. The data suggest that changes in the lateral amplitude, rather than the frequency or synchrony, of the flagella of Chlamydomonas control its phototaxis. A computational model of phototaxis confirms that the intrinsic changes in lateral force in response to light are necessary and sufficient for successful phototaxis.

MATERIALS AND METHODS Laser Trap Transducer The laser trap transducer used in these studies is described elsewhere8,9 except that the laser was replaced with a 1090 nm fiber laser (SPI, Southampton, UK). Back focal plane interferometry was used to determine the displacement of a trapped bead relative

FIGURE 2. Example calibration experiment. Chlamydomonas was fixed and trapped in 30% ethanol. The laser trap was stepped 200 nm laterally at time zero (0). The return of the cell to trap center was fitted with a single exponential (solid blue line) to determine the trap stiffness and detector sensitivity (in this example, 0.17 pN/nm and 79 nm/V, respectively). The dashed lines indicate the expected peak-to-peak variation in cell position (99% confidence intervals) for a cell in this stiffness trap undergoing Brownian motion.

to the trap center. Detector and trap stiffness calibrations were performed on ethanol-fixed cells (30% ethanol in water14) using the step response method6,23 (Fig. 2) and confirmed using power spectral density.2 We took into account the increase of viscosity of 30% ethanol (2 cP) over pure water and the proximity of the cell to the coverslip surface. Cells were illuminated using a halogen lamp attenuated to 1 mW, and a microscope objective (940, 0.65 n.a., Leica) as the condenser. From the perspective of the cell, the light will fill approximately 6% of the field of view. The illuminator was fitted with a mechanical shutter with a time constant of ~2 ms; this allowed for step changes in light intensity. A photodiode mounted under the shutter monitored the timing of changes in incident light. Laser Trap Experiments Wild-type C. reinhardtii (CC-124 mt-) and a uniflagellar mutant (CC-1926) were grown on culture plates, then transferred to liquid medium and incubated with gentle rocking. The liquid culture was added to glass flow cells coated with 2% polyvinylpyrrolidone (PVP, 360 kDa) as a blocking agent. Individual cells were trapped using ~250 mW in the specimen plane. No photodamage was evident over the period the cell was held in the trap, as evidenced by no change in the swimming force or flagellum beat frequencies. Cells were held 8–12 lm from the surface of

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the flow cell so that the flagella would not routinely interact with the surface. X and Y positions of each cell within the trap were recorded at 1 kHz during step changes in light (light to dark, or dark to light, using the aforementioned shutter in a darkened room). Each cell was observed for 30 s, with the transition in light condition occurring at 15 s.

The X and Y axes displacement data (Fig. 3, top) were combined using the Pythagorean equation for a representation of a mean displacement magnitude. The mean displacement was then converted to units of force using the measured laser-trap stiffness and sensitivity (Fig. 3, center). When force is plotted against time, periodic signals such as cell rotation frequency and flagellar frequencies can be observed. The mean force was then displayed in a spectrogram to show changes in frequency components over time (Fig. 3, bottom). The envelope of the Fourier transforms of the first and last 10 s of the file were averaged together for reduced-noise samples of before and after light change conditions; the middle 10 s containing the light transition were excluded from analysis. These average representations of the frequency distribution in light and dark conditions were plotted on a frequency vs. magnitude scale. The main frequency peak between 30 and 100 Hz was the flagellar stroke; this peak was fitted with a Gaussian curve to determine the width, magnitude, and center frequency. Measured characteristics were compared between conditions with paired t-tests.

Free-Swimming Velocities Chlamydomonas were added to surface-blocked flow cells, as above. Movies of moving cells were captured using a CCD camera on an Olympus IX70 and a video capture card (Scion). Velocities were subsequently determined using a centroid algorithm4 with an energyminimization segmentation algorithm.5 Our implementation of this algorithm is reported elsewhere21 and is freely available upon request. Data Analysis Each 30-s file was analyzed with MATLAB and with custom software written in Borland Delphi.

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Time (s) FIGURE 3. Example data from a uniflagellate Chlamydomonas experiment. For all panels the abscissa is time in seconds. Top: Raw, uncalibrated X- and Y-displacement data within the laser trap. The slow periodicity represents cell rotation in the laser trap, while the overlayed frequency component represents flagellar beats. Center: Displacement data are converted to force, and combined to give a scalar magnitude of force. Bottom: A spectrogram is generated on short (1/4 cell rotation) time intervals. The horizontal band at ~100 Hz is the flagellar beat frequency, which is significantly higher in uniflagellate cells than in biflagellate cells. The color bar indicates the magnitude of the discrete Fourier transform in pN.

Phototaxis in Chlamydomonas

RESULTS We judged damage due to laser trapping by watching for a marked decrease in the flagellum beat frequency over a period of 60 s in the laser trap. We noted no damage to cells as a result of being trapped, provided that laser power in the specimen plain was kept below 400 mW. All data presented here were collected at trap powers of ~250 mW. The rotational period of each cell and the flagellar beating was evident in two-dimensional plots of force (Fig. 4). The mean flagellar beat frequency for all 169 biflagellate cells in the light was 72 ± 1 Hz (all errors are given as s.e.m.). Among these, 56 cells exhibited two distinct peaks in the spectrogram, which we 30

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interpreted as representing the two flagella (see below). This interpretation is supported by our observation that only one flagellar beat frequency is observed in uniflagellate cells. For those biflagellate cells that exhibit two distinct flagellar frequencies, the mean frequencies were 68 ± 2 and 84 ± 2 Hz. Rotational periods for biflagellate cells were ~0.5 s, making typical rotation frequencies ~2 Hz, in agreement with previous reports. The RMS force generated by biflagellate cells measured over several cell rotations was 13.5 ± 0.4 pN; this is not the same as the peak-to-peak force generated by a single flagellar stroke (visible in Fig. 4), and would generally be expected to be less. The change in RMS force from dark to light was only 3.5%, and not statistically significant. To validate our average force measurements, the steady-state velocity of free-swimming (i.e., not trapped) cells was measured. A histogram of these velocity data is shown in Fig. 5. We fitted a Gaussian to the peak moving velocity, disregarding adherent, nonmotile cells, and found a modal velocity of 109 nm/s. We assumed that at this velocity the RMS force generated by the cell is counterbalanced by drag, and further assumed that the cell was the form of a prolate ellipsoid with a major axis of 5.7 lm and minor axis of 3.7 lm.13 From our velocity measurements, we then estimated a RMS force of 13.0 pN, essentially identical to that measured in the laser trap (above). We employed a similar approach to measure torque about the trap axis. Cells will freely rotate about the center of the laser trap, but their rate of rotation will ultimately be limited by viscous drag on the cell. Thus,

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X Force (pN) FIGURE 4. Two-dimensional plot of Chlamydomonas swimming force in the laser trap. Top: One complete cycle of rotation of the cell about the trap center creates a ‘‘donut.’’ The radius of the red circle fitted to this trace is the RMS force directed out of the trap for this particular cell. Bottom: Magnified view showing three individual flagellum strokes, beginning with a return stroke and ending with an effective stroke. The blue arrow indicates the starting position and forward progress in time.

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by measuring the rate of planar rotation within the trap, we were able to estimate the changes in torque about the trap axis, and therefore the lateral force acting on the leading edge of the cell. The torque generated on the cell by the flagella, time-averaged over multiple rotations, was 22 ± 1 pNÆlm. There was a 6% variation in the rate of rotation, and therefore the torque, from dark to light, but the change was not statistically significant (p = 0.34).

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There was a statistically significant (p = 0.01) increase in force magnitude from the flagella in light when compared to dark conditions. There was also a statistically significant (p = 0.004) decrease in peak width in light conditions when compared to dark conditions. There was not, however, a significant change in the mean flagellar beat frequency between the light and dark (73 ± 1 Hz) conditions by paired t-test (p = 0.2). This suggests that in the light, one or both flagella are beating stronger and at a narrower range of frequencies. To better understand the subtle changes in frequency components, we restricted further analysis to cells that exhibited two distinct flagellar frequencies in either light or dark. There were 56 cells that exhibited two distinct beat frequencies in either light or dark, similar to the observations of Takada and Kamiya.24 There is a dominating flagellum as indicated by a difference in amplitudes of the two frequency peaks in the spectrum (Fig. 6). This dominance is present in both the dark and in the light; under both conditions, the flagellum beating at the lower frequency is the one with the higher amplitude. The two peaks were fitted with the sum of two Gaussians to aid further statistical analysis. The difference between the mean frequencies of the two peaks was higher in the dark than in the light (Df = 17.6 ± 0.8 vs. 14 ± 1 Hz, p = 0.001). The variation was due to the lower frequency component increasing in frequency by approximately 1.2%—a small but statistically significant increase—and the higher frequency component slowing by approximately 0.3% in the dark. Furthermore, both peaks were significantly lower in magnitude in the dark than in the light (p = 0.019). The magnitude of response to light was asymmetric; the dominant peak increased greater in magnitude 7% from dark to light, while the other increased by 3.8%. This is similar in magnitude to the increase in RMS force acting on the cell, measured earlier. The integrated power under each peak did not change significantly from condition to condition.

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Frequency (Hz) FIGURE 6. Force spectra at high temporal resolution of a Chlamydomonas in light and dark. These spectra show the discrete Fourier transforms of the forces on the time-scale of half-rotations of the cell (250 ms). In the dark (bottom), the magnitudes of force generation by both flagella are constant from one presumed half-rotation (red) to the next (blue). In the light (strong gradient from one side, top), the magnitude of the force spectrum in one half-rotation is greater than the other, and larger than that observed in the dark. This suggests that one flagellum (presumably the trans flagellum) dominates during the half-rotation when the eye spot faces the light source.

To analyze the transient changes in flagellar beat, average frequency distributions were calculated for each half rotation of the cell, ideally representing the eyespot either facing toward or away from the light source, by averaging 20 spectra for alternating sections of 250 ms. Since the period of cell rotation speed is ~500 ms, each averaged histogram represented a ½ cell rotation. This approach to analysis is qualitative, as we cannot determine in these experiments which direction the eye spot is facing relative to the condenser at any given moment in time. We therefore cannot consistently synchronize the starting points for generating

Phototaxis in Chlamydomonas

spectra to the absolute orientation of the cell. Instead we examined qualitatively the average frequency distributions for the alternating 250 ms time intervals (Fig. 6). When the cell is exposed to a light source, it was observed that there was a greater difference between peak heights, and presumably greater dominance, in one 250 ms time interval, presumably the half-rotation in which the eyespot is facing a light source. The other half-rotation exhibited a smaller difference in amplitude. When the cell is in the dark, there are no apparent changes in the magnitude or frequency of the flagellar beats throughout the entire rotation of the cell.

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ellipsoid with a major axis of b = 5.7 lm and minor axis of a = 3.7 lm.13 The cell was assumed to be neutrally buoyant and of negligible mass. This allowed for instantaneous balancing of flagellar forces and translational and rotational drag forces. Figure 1 shows the coordinate system internal to the cell. The cis and trans flagella generate force vectors FT and FC, respectively, at given moments in time. The directional component of the cell’s velocity is assumed to always be parallel with the cell’s major (z) axis; this means that the only translational drag force acting upon the cell is along its major axis. We allowed the cell to rotate about the X and Z axes as well, which result in rotational drag forces about these axes. The drag coefficients are those of a prolate ellipsoid,

UNIFLAGELLATE CELLS The uniflagellar mutant (CC-1926) of Chlamydomonas was also tested in the laser trap. Ninety-five percent of CC-1926 cells possess only the trans flagellum.10 These cells undergo planar rotation in the laser trap at ~3.5 Hz, since there is no counterbalancing torque from the cis flagellum. During this planar rotation, the eye spot will presumably always be facing the same direction relative to the light source; thus, subdividing the rotation into short segments, as was done above, is fruitless. A single main frequency component was observed, suggesting that we correctly interpret two peaks as representing the two separate flagella in wild-type cells. The change in flagellar beat frequency was significantly higher in uniflagellate cells than in the biflagellate cells, but did not differ significantly between light and dark conditions (84 ± 5 and 89 ± 4 Hz, respectively). However, by measuring the rate of planar rotation, we estimated the torque about the trap axis generated on the cell by the trans flagellum alone to be 95 ± 10 pNÆlm. This is larger than the torque in biflagellate cells, presumably because the two flagella were generating torque in opposition. The lateral force acting on the leading end of the cell due to a single flagellum would be ~16 pN. On transition from dark to light, the lateral force generated by the trans flagellum increased by 16 ± 5%, as reflected in increased rate of planar rotation. COMPUTATIONAL MODEL A computational model of Chlamydomonas was created to quantitatively test the contributions of observed variations in force on phototaxis. The model was realized in Matlab. The computational model is a simple threedimensional representation of Chlamydomonas (Fig. 6). The cell body was modeled as a prolate

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ðb þ lF Þ  bRtX xx ¼ 0 where lF is the half-length of a flagellum (assumed to be 5 lm), and xx and xz are the angular velocities of rotation about the x and z axes, respectively. We did not explicitly model the waveforms of the flagella. Rather, in this initial study, the flagella were represented as periodic force magnitudes at 70 Hz. The thrust (z-axis) force driving the cell forward was 33 pN, and the lateral (y-axis) force was 8 pN. These values were estimated from the two-dimensional plots of biflagellate and uniflagellate cells (Fig. 4), though it is difficult in our measurements to determine where true zero lies. Thus, the force vectors from the two flagella (FT and FC) evolved as ellipses with 4.125 (33/ 8) eccentricity. This eccentricity reflects the experimental data, and the fact that the flagellar stroke is nonlinear and not perfectly colinear with the major axis of the cell. Because the lateral force component acts at the anterior tip of the ellipsoidal cell, it generates a rotational torque about the X-axis; a dominant trans flagellum lateral force will therefore rotate the cell toward the eye spot. The ellipses were set to dip

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below the xy- and xz-planes, which are representative of the recovery strokes of the flagella which will generate small negative forces. The ellipses were also tilted out of the yz-plane by 0.1 radians; the resulting torque about the Z-axis resulted in the 2 Hz rotational motion of the cell observed in vivo. The eyespot is directional and spins with the cell, as just described. We assumed a nondecaying point source of light at the origin of a global rectangular coordinate system (0,0,0). The response of the eye spot to the light was modeled as the dot product of a unit vector normal to the eye spot with a unit vector directed toward the light source. The scalar projection of the light source onto the eye spot is adjusted so that it ranges from 0 (when the eye spot is directed exactly away from the light source) to 1 (facing the light source). We initially assumed instantaneous coupling between the eye spot and changes in the trans flagellar beat, and allowed the eye spot signal to proportionately vary the magnitudes of the thrust force, the lateral force, or both. The simulation was conducted on 1 ms time intervals. The new position of the cell was determined by moving it along its current heading by |v| Æ (1 ms). Likewise, the cell’s new heading was determined by rotating the cell about its x-axis by xx Æ (1 ms) radians, and the new direction of eyespot was determined by

rotating the eyespot about the cell’s heading by xz Æ (1 ms) radians. We ignored motion due to diffusion of the cell, as over the 1 ms time interval it would amount to a displacement less than 1/20th that generated by the flagella. An example simulation is illustrated in Fig. 7. Our measurements of longitudinal and lateral force generated by the flagella result in a cell speed that is slightly higher than those measured in free-swimming cells (Fig. 5). During its forward progress, the cell spirals about its Z-axis at approximately 2 Hz, the same as we observed in trapped cells. This rotates the eye spot about the cell, alternately facing toward and away from the light, creating an eye spot signal that varies at 2 Hz and highest in magnitude when the Z-axis of the cell is normal to the light source. In the example shown, the longitudinal force of the trans flagellum was allowed to vary to a maximum of 1.07, and the lateral force to 1.16 over the cis flagellum, mimicking the native cell. The simulated cell is in this instance clearly phototactic, always circulating about the point source of light. To better quantify the degree to which a given set of conditions resulted in phototaxis, we defined a ‘‘phototactic factor’’ (PF) PF ¼ ðhd0 i  hdiÞ=hd0 i

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Phototaxis in Chlamydomonas

where hdi is the mean distance of the cell from the light source over a given time interval, and hd0 i is the mean distance over the same interval when no changes in longitudinal or lateral force are allowed (absolute nonphototaxis). PF will vary from 1 (perfect phototaxis, stationary cell colocalized with the light source) to 0 (non-phototactic). For each set of conditions, the cell was started 1 mm from the light source and given a random initial orientation. Simulations were run for 30 s (30,000 time intervals), and the mean distances from the light source recorded for the latter 15 s; the first 15 s allowed the cell to reach steady-state behavior, and those data were discarded. We first tested the relative contributions of variations in longitudinal and lateral force from the trans flagellum to phototaxis. The results are shown in Fig. 8. Variations in lateral force alone are sufficient for phototaxis; indeed the relationship between the degree of lateral force variation and PT is quite steep, with half maximal response at 8%. Variations in longitudinal force have a modest effect since a fractional variation by x will only generate a change in lateral force of x/4.125 (due to the eccentricity of the ellipse describing cyclic force generation by the flagella). The variations measured for the native cell are also shown on the graph. It is clear that changes in lateral force development are vital to successful phototaxis.

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The torque about the cell that accompanies an increase in forward thrust is probably not sufficient. As mentioned above, we assumed instantaneous coupling of the eye spot signal to changes in the flagellum waveform. To further investigate the importance of eye spot-to-flagellum signaling, we varied the time constant (s) coupling the eye spot signal to the change in flagellum force output. The results (Fig. 9, bottom) show the response is essentially flat until s approaches the half-rotational period of the cell (0.25 s). It is interesting to note that at about s  10 ms the simulated cell swims in short ‘‘spiral’’ patterns (Fig. 9, top center); native free-swimming cells frequently exhibit this behavior.

DISCUSSION These data expand upon and clarify the dominance theory of phototaxis in C. reinhardtii. The data suggest that the largest and most functionally significant change in the flagella in response to light changes is a waveform change in the trans flagellum that increases its lateral beat strength more so than its longitudinal beat strength. As a result, when the eye spot faces light, the cell body rotates slightly toward the light. Variations in longitudinal beat strength (thrust) play a role, but a much smaller role than lateral variations. This is actually a simplification of the dominant flagellum theory, since presumably only small variations in flagellum waveform are necessary, rather than much larger changes in stroke magnitude or frequency. Our data also show that the flagella change their beat frequencies in the light, as have others.1,13 However, the magnitudes of the changes measured by us suggest that frequency alterations are not a viable mechanism for Chlamydomonas phototaxis, yet the cells are evidently phototactic, as cells congregate within the focal illuminated area during an experiment. Transitioning from the dark (no directional light source) to a directional light source causes a change in the fundamental frequency of flagellum beating of only about 1% on average. This implies a 1% change in flagellum velocity, and a concomitant 1% increase in generated force at low Reynolds numbers. This is less than the 3–7% increase in mean thrust force measured in biflagellate cells, and far less than the 16% increase in lateral force measured in trans uniflagellate mutants. Thus, while beat frequency and force are related, changes in beat frequency cannot in and of themselves explain phototaxis and were therefore not included in our final model. Furthermore, we found no evidence in our model that de-synchronization of the flagella (changes in frequency or phase) play a role in phototaxis,

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τ (sec) FIGURE 9. The effect of time constant (s) for eye-spot to flagellum response on phototaxis. Top: Example 3D swimming patterns for three different s. Note the small, spiraling swimming patterns at 10 ms, and the transition to a non-phototactic spiraling pattern by 100 ms. The color code represents forward progression in time (s). Bottom: The effect of time constant (s) for eye-spot to flagellum response on phototactic factor for a cell rotating at 2 Hz. s in living cells is uncertain, but is probably on the order of 1022 s (see text).

as changes in frequency in the absence of a change in force have no effect on overall cell trajectory. Our computational model suggests that changes in force by the native cell are conservative; that is, the changes in lateral force are close to the minimum necessary to achieve the best possible degree of phototaxis (see Fig. 8). Furthermore, the rotational speed of the cell appears to match that for the maximum degree of phototaxis given the likely rate of reaction in the flagella in response to light. There is a 6.5–9 ms delay from the photocurrent at the eye spot to the calcium current in the flagellum,3 and the rate at which dynein responds to calcium is to our knowledge undetermined. Still, 100 ‡ s ‡ 9 ms is consistent with the observed spiraling paths taken by native

Chlamydomonas and with optimal phototaxis with a 2 Hz cell rotation. Foster and Smyth7 posed the question of why 2 Hz was evolutionarily selected as the cell rotational rate. Our data suggest that it optimizes phototaxis for a fixed time constant for propagation of the photocurrent into the flagellum and subsequent delayed response of dynein.

LIMITATIONS While there are many advantages to studying ‘‘freeswimming’’ cells in the laser trap, there are also limitations to our experimental approach. The cell is a

Phototaxis in Chlamydomonas

complex environment with a number of refractive structures that limit the spatial resolution of our data. Further, it has proven impossible to calibrate each experimental cell separately, as none of our calibration schema work on actively motile cells; aggregate calibrations from ethanol-fixed cells must be used, and the cell-to-cell variability is significant. This decreases the statistical confidence in any given experiment. The orientation of the cell in the laser trap cannot be directly controlled, meaning that we do not know the orientation of the flagella or the eye spot relative to the trap or the light source. Finally, the wavelength of the laser light used here (1090 nm) is approximately double that of the wavelength that results in maximum phototaxis (500 nm19,20). It is therefore possible that a small amount of two-photon excitation of the eye-spot is occurring. It is unlikely, however, that this would result in directional phototaxis, as cells are otherwise mostly transparent to light at this wavelength. Furthermore, the low efficiency of two-photon excitation makes it unlikely that the eye-spot would be ‘‘blinded’’ (i.e., saturated) by the trapping laser. The trap stiffness too is a limiting factor, as a large number of cells escape the laser trap, biasing our sample. One approach to overcoming this limitation is to trap the cells nearer the coverslip surface, but this inevitably leads to transient interactions of the flagella with the surface, contaminating the force trace. An alternative approach is to work further from the surface, and significantly raise laser power to increase trap stiffness. This, however, is not a viable option, as doing so leads to aberrant behavior of the trapped cell, presumably due to overheating. Our computational model also has intrinsic limitations. Flagellar waveforms were not explicitly described in our computational model; this an obvious area for improvement that will more fully test our hypothesis that changes in lateral force generation rather than flagellar beat amplitude or frequency are responsible for phototaxis. Explicit modeling of the flagella will also re-introduce the effects of beat frequency on phototaxis, though the effects are expected to be small. Finally, explicit modeling of the signaling pathways should prove fruitful for understanding the mechanical effects of the photocurrent at the level of the flagellar stroke.

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