Biol. Bull. 210: 78-80. (April 2006)
© 2006 Marine Biological Laboratory
Mechanisms of Gravitaxis in Chlamydomonas
A. M. Roberts
Department of Applied Science, London South Bank University, London SE1 0AA, UK
The mechanisms by which many free-swimming microorganisms are able to swim preferentially upward despite the sedimenting effects of gravity (termed gravitaxis) have long been the subject of debate. Early suggestions were that upward swimming could be caused by cells either being back-heavy (1,2,3) or possessing fore-aft body asymmetry (4). However, there is a widespread perception that simple physical mechanisms cannot explain how physiologically active agents can influence gravitaxis, or how non-gravitactic mutants can exist; it is therefore argued that cells must contain a biological gravity sensor which somehow directs active swimming (5,6). Published quantitative data on the unicellular biflagellated green alga Chlamydomonas reinhardtii are used here to show that shape orientation is primarily responsible for upward orientation in this organism. It is also shown that the resulting gravitational response is governed by the cells’ own swimming patterns. It is concluded that gravitaxis in Chlamydomonas is explicable in purely physical terms, and that no biosensor need be involved.
Chlamydomonas cells whose flagella have been immobilized by chemical agents are found to settle in suspension with their flagella uppermost (7, 8). There are only two possible explanations. One is that the cell is back-heavy, resulting from a longitudinal density gradient that displaces the center of gravity behind the center of buoyancy, thereby producing a gravitational torque on the cell that causes orientation. The other is that the asymmetry of cell body and flagella, even when of uniform density, causes shape orientation during sedimentation because the larger cell body sediments downward faster than the flagella. Both mechanisms result in orientation towards the vertical at a rate described by the relation
 | (1) |
where 
is the instantaneous angle of inclination of the cell axis to the upward vertical at time t, and ß, a constant for a given geometry, is the maximum orientation rate (measured in rad s–1 or deg s–1) which occurs when the long axis of the cell is horizontal (at = 90 deg). Equation 1 implies that a cell, lying in a near-vertical orientation with its flagella beneath it, will reorientate upward at an angular rate that first increases to a maximum value of ß when lying horizontally, and then gradually decreases at it approaches the upward vertical, with its flagella now positioned above the cell body. ß is thus a useful measure of the size of the orientating mechanism.
These two physical mechanisms can in principle be distinguished by immersing immobilized cells in a fluid medium of the same density as the cells themselves. Neither sedimentation nor shape orientation can now occur, whereas back-heavy orientation is unaffected. Kessler (3) found that immobilized specimens of Dunaliella tertiolecta, a small green biflagellate similar to Chlamydomonas, continued to orientate vertically under these conditions, but did so up to twice as rapidly when the cells were allowed to sediment in pure water. Both mechanisms therefore seem to be involved, though the relative magnitudes are unclear.
Not enough is known about the density distribution within Chlamydomonas to calculate the effects of back-heaviness, but the effects of shape orientation can be estimated directly using the hydrodynamic model described by Jones et al. (9). Chlamydomonas is represented as a sphere of diameter 8 µm and density 1040 kg m–3, with two flagella each of length 9 µm and diameter 0.2 µm. The viscous forces and torques on the sphere are given by the standard Stokes equations for translation and rotation, and those on the flagella are estimated using resistive force theory (10). The calculation of flagellar forces includes the effects of the fluid flow caused by linear and angular motions of the sphere and by its sedimentation under gravity. The equations are solved simultaneously subject to the conditions that the total force and torque on the cell are always zero. No analytical solution is possible, and the equations are solved numerically. Assuming that the flagella are straight, stationary, and at a typical angle apart of 20 deg, the theory predicts an orientation rate of ß 
3.8 deg s–1. This estimate also agrees with the value inferred from physical scale model experiments (11). The experimentally determined average value for ß in C. reinhardtii has recently been estimated by Yoshimura et al. (8) to be about 4.4 deg s–1. The agreement between theoretical and measured values of ß for immobilized cells is good, bearing in mind that real cells vary considerably in both size and density (8), making exact comparisons difficult. It seems reasonable to conclude that shape asymmetry is the principal mechanism for orientation in Chlamydomonas, although a rather smaller contribution to ß from a back-heavy mechanism seems likely.
Orientation must occur in both immobilized and swimming cells. In the latter case, the hydrodynamic model shows that the shape orientation rate decreases as the two straight flagella sweep backward with their normal breast-stroke movement toward the cell body during the beat cycle, with the result that cells moving flagellum-first will on average orientate upward at a reduced average rate 
of about 2 deg s–1. However, any contribution to ß from back-heaviness will increase as the flagella sweep back towards the cell body, with the result that may well be somewhat greater than estimated above.
To assess the effect of upward orientation on the resulting vertical distribution of cells, it is necessary to adopt some overall description of their movement. The following discussion is not mechanism-specific: it applies to any mechanism that causes cells to orientate according to Equation 1. Observation of Chlamydomonas trajectories shows that individual cells swim in approximately straight lines, periodically making sudden, small angular changes in direction of travel. Two parameters characterize this motion. First, swimming velocity U can readily be measured. Second, the rate at which cells change direction is described by the rotational diffusivity, 
2, a measure of the rate of increase of variance of projected track direction with time (12), which can also be deduced from recorded microscope images of cell tracks. If cells swim in straight lines for some mean time, 
, between turns through some small angle, 
, the assembly of cells will, in the absence of gravity, diffuse equally in all directions (13) with rotational diffusivity 2 given by
 | (2) |
Although both and are governed by physiological processes within the cell, it is explicitly assumed here that neither parameter is affected in any way by gravity.
The effects of gravity on the cell population are twofold. Downward sedimentation of individual cells with velocity V (which is almost independent of ) causes a net downward population drift, also at velocity V. Upward track curvature described by Equation 1 causes an active upward drift at velocity Vß that can be calculated for the present model in terms of the Langevin function (14). For small the result simplifies to
 | (3) |
provided that Vß << U, as is usually the case. The condition for net upward swimming (negative gravitaxis) is that Vß > V, or
 | (4) |
This result implies that the magnitude of is not really critical; as long as is positive, up-swimming can always be achieved by reducing 2 below the critical value, either by increasing , decreasing , or by a combination of both. In C. reinhardtii the maximum value of 2 that permits up-swimming, 
max2, can be estimated from Equation 4; taking 2 deg s–1 (as surmised earlier), U 50 µm s–1 and V 1.3 µm s–1 yields max2 0.9 rad2 s–1.
Although no direct determinations of rotational diffusivity 2 have been reported in C. reinhardtii, two indirect estimates can be made. Measurements on the rates of accumulation of cells in the upper halves of vertical tubes containing initially uniform cell suspensions suggest values of Vß varying between 4 and 8 µm s–1 from one cell culture to another (15); taking 2 deg s–1, Equation 3 implies that 2 lies between 0.15 and 0.3 rad2 s–1. A second estimate can be made from measured K-values (16). K is a measure of the strength of gravitaxis, defined as
 | (5) |
where r+ and r– are, respectively, the number of trajectories oriented upward ( < 90 deg) and downward ( > 90 deg), in a population of cells at any instant. K-values in resting cultures of C. reinhardtii generally lie between 0.2 and 0.6, and tend to increase with age of culture and decreasing ionic strength of the medium (16). Assuming that trajectories are described by Equation 1 and that steady downward sedimentation at velocity V occurs at the same time, Equation 5 leads to the following expression for K:
 | (6) |
once the steady state is achieved. This range of K-values corresponds to values of 2 lying in the range 0.05 to 0.15 rad2 s–1. Equation 6 implies that K-values are not very dependent on swimming speed U, and experiments show that there is no direct correlation between K and U when cells are subjected to various light beam combinations (16). Direct measurements of rotational diffusivity in a different strain, C. nivalis, give 2 0.035 to 0.14 rad2 s–1 (11), a similar range of values. These values of 2 are significantly smaller than the previously calculated critical value max2, thus confirming the ability of this microorganism to swim upward purely as a result of passive orientation under gravity.
Mutant cells that swim backward are reported to sink at about the same rate as immobilized cells (8), and this is to be expected since backward-beating flagella cause (and so Vß) to be reduced to near-zero. Other motility mutants that show reduced up-swimming even with normal swimming velocities can be explained by supposing that they have increased values of 2 for physiological reasons, although there is as yet no experimental confirmation of this.
This analysis suggests that gravitaxis in Chlamydomonas reinhardtii has a purely mechanical basis. Shape orientation appears to be the main orienting mechanism, although back-heaviness may also make a contribution. It is perhaps surprising that a simple physical mechanism can give rise to such relatively complex behavior. This complexity arises because the effects of orientation are determined by the swimming velocity U and the rotational diffusivity 2, which in turn depend on cell physiology. Any physical or chemical agent that affects these parameters may also alter the gravitactic response. Although the present study cannot disprove the existence of a biological gravisensor in Chlamydomonas, it does show that it is unnecessary to suppose that one exists in order to account for the available experimental evidence. Mechanical orientation must occur in microorganisms whether or not they possess a gravisensor, and any effects of such a sensor would be superimposed upon and in addition to those described here. Evolutionary pressure for the development of an internal receptor seems unlikely given that such a simple and effective orientational mechanism already exists in cells as small as Chlamydomonas. Since many larger flagellates and ciliates seem to have the correct shape asymmetry to produce passive upward orientation (17), and recent experiments suggest that shape orientation is the principal orienting mechanism in the ciliate Paramecium (18), there remains the distinct possibility that mechanical orientation alone may also be the cause of gravitaxis in these microorganisms.
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Footnotes
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Received 17 October 2005: accepted 24 January 2006.
Current address: S C Associates, 32 Sixth Cross Road, Twickenham, TW2 5PB, UK. E-mail: amr{at}physics.org
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