Re-enterant efficiency of phototaxis in Chlamydomonas reinhardtii cells
RRe-enterant efficiency of phototaxis in Chlamydomonas reinhardtii cells
Sujeet Kumar Choudhary, Aparna Baskaran, and Prerna Sharma Department of Physics, Indian Institute of Science, Bangalore, India Martin Fisher School of Physics, Brandeis University, USA (Dated: April 22, 2019)Phototaxis is one of the most fundamental stimulus-response behaviors in biology wherein motilemicro-organisms sense light gradients to swim towards the light source. Apart from single cellsurvival and growth, it plays a major role at the global scale of aquatic ecosystem and bio-reactors.We study photoaxis of single celled algae Chalmydomonas reinhardtii as a function of cell numberdensity and light stimulus using high spatio-temporal video microscopy. Surprisingly, the phototacticefficiency has a minimum at a well-defined number density, for a given light gradient, above whichthe phototaxis behaviour of collection of cells can even exceed the performance obtainable fromsingle isolated cells. We show that the origin of enhancement of performance above the criticalconcentration lies in the slowing down of the cells which enables them to sense light more effectively.We also show that this steady state phenomenology is well captured by a modelling the phototacticresponse as a density dependent torque acting on an active Brownian particle. a r X i v : . [ c ond - m a t . s o f t ] A p r INTRODUCTION
Collective behaviour is observed in biological systems at different levels of biological organization from cells in tissuesto colonies of microorganisms to flocks or herds of macroscopic animals [1–3]. Phenomena at the level of the populationin such systems cannot always be predicted by simply knowing the behaviour of individuals. For example, biofilms ofBacillus subtilis bacteria exhibit oscillatory growth rate whereas no such oscillations exist in dilute suspensions of thesame bacteria [4]. Collective behavior in microorganisms is of particular interest as it can be thought of as a precursorto multicellularity and more complex organizations of living systems. Consequently, a number of quantitative studieshave recently elucidated the origin of collective phenomena in a wide variety of micro-organisms such as E-coli [5, 6],Bacillus subtilis [4, 7, 8], Synechocystis sp. [9, 10], Pseudomonas [11, 12] and Myxococcus xanthus [13, 14].Taxis, a transport phenomenon in which organisms undergo directed movement in response to a stimulus or anutrient gradient, provides a particularly tractable context in which to explore collective behaviour. As a particularexample, phototactic cells such as algae and cyanobacteria respond to light gradients [15, 16]. Single celled eukaryoticalgae Chlamydomonas reinhardtii (CR) is a model biological organism for studying phototaxis [17]. While singlecell response of CR to light can be tuned by varying physical variables such as light intensity, fluid viscosity as wellas through chemical variables such as extracellular calcium concentration [16, 18–21]. It was shown recently thatphototaxis of dense suspensions of CR was governed by the cell number density itself revealing that collective effectscould modulate the single cell response [22]. Here, we set up quasi-two-dimensional phototaxis assay with CR to studythe cross-over from the individual to collective phototaxis and identify the mechanisms underlying the emergence ofits collective phototaxis.CR has two flagella and an eye-spot located near the cell equator. Its flagella move in breast-stroke fashion topropel the cell body through the fluid [17, 23]. The ellipsoidal shaped cell body rotates about its own axis whileswimmng enabling the eyespot to scan the incident light around the swimming path [24–26]. Under phototacticlight exposure, beating of the flagellum closest to the eyespot is inhibited whereas beating of the further away oneis enhanced resulting in aligning the cell towards the light source [27, 28]. We use a high speed camera to recordindividual trajectories of hundreds of cells under varying light intensities and cell concentrations.We find that starting from few cells per unit volume, phototatic efficiency decreases with increasing cell concentrationuntil a critical concentration is reached above which the efficiency increases with increasing concentration. Thus, thephototactic efficiency is a reentrant function of the cell density. We further show that the origin of this reentrantbehavior lies in the decrease in the swim speed of the cells as density increases beyond the critical concentration.Finally we find that the observed phenomenology is well captured by a model of active Brownian particles subject toa density dependent external torque.
EXPERIMENTAL DETAILS
CC-1690 (wild type) cells were used for the experiment. Synchronous culture of CR were grown in TAP mediaat 25 ◦ C on 12 h/12 h light/dark cycle in an orbital shaker (135 rpm). Fig. 1a shows schematic of experimentalsetup. Cell suspension was observed in rectangular quasi-two-dimensional chambers (50 mm × × µm )made of glass slide and cover slip with double sided tape as a spacer. A blue laser beam of wavelength 488 nm from the optical fiber illuminated one end of the chamber to act as a stimulus for phototaxis. Cell trajectories wereimaged using bright field imaging with red light (760 nm and above) illumination set up on an Olympus IX73 invertedmicroscope. Images were recorded at 100 frames per second at ×
10 magnification using PCO 1200hs CMOS cameracoupled to the microscope. ×
10 objective has a large depth of focus that enables us to capture 2-D projections of thecell trajectories for as long as typically ∼
20 seconds. Particle tracking was performed using image processing code inMATLAB and Python. For a given cell concentration and light intensity, 500-2500 trajectories were analysed to haverobust statistics.
RESULTS
Cells move in random directions in the absence of blue light (Fig. 1b). Presence of blue light at one end of thechamber biases the movement of a majority of cells towards the light source (Fig. 1c). However, a small but finitefraction of cells continue to move in directions other than the direction of light source (Fig. 1c). Probability densityas a function of polar angle in the plane characterizes this phenomenon quantitatively (Fig. 1d). The distribution is,naturally, peaked in the source direction with the peak height increasing with increasing light intensity (Fig. 1d). Inorder to analyze the response of the system tractably, we define phototactic efficiency, ζ , as the fraction of cells thatmove in a direction ±
15 degrees of the source direction. At low intensities of the light source, ζ is significantly lessthan 1 and approaches 1 at higher intensities (Fig. 1d inset).While the phototactic efficiency shows the anticipated increase with increasing light intensity, one expects that cellconcentration will also play a role in governing phototatxis at the population level [22]. Fig. 2 a-d show representativecell trajectories as a function of cell number density for a fixed light intensity. Starting from suspensions of few cells,the peak height of the probability density decreases with increasing concentration until a critical concentration ρ c is reached. Above ρ c , the peak height increases monotonically with the concentration (Fig. 2e). This re-entrantphototaxis behaviour can equivalently be represented by the non monotonic variation of ζ with cell concentration(Fig. 2f).It could be reasonably expected that the measured probability distribution of trajectory orientations ψ ( θ ) couldbe captured by a self-propelled particle model [29]. The simplest such model in this context would be that of non-interacting active Brownian particles subject to a polar aligning torque that tends to turn the trajectories of theparticles along some particular direction in the lab frame. Let us pick this direction to be along θ = 0. The Fokker-Planck equation governing the dynamics of the probability density ψ ( θ, t ) for the orientations of these self-propelledparticles is given by, ∂ t ψ ( θ, t ) = D R ∂ θ ψ + γξ r ∂ θ ( sinθψ ) (1)where D R is the rotational diffusion coefficient, γ is the torque strength and ξ r is the rotational friction coefficient. Thesteady state solution to this equation is the well known Von-Mises distribution function of the form, ψ ( θ ) = e κcosθ πI ( κ ) where κ = γD R and I is modified Bessel’s function of first kind. The experimentally obtained probability density as afunction of polar angle is well fit by the Von-Mises distribution (Fig. 3a). The density dependence of this probabilitydistribution can now arise either through D R , implying that the rotational diffusion and hence the characteristicdecorrelation time of the orientational autocorrelation function depends on density, or through the torque γ . Theexperimental data reveals that this decorrelation time is independent of cell concentration (Fig. 3a inset). Thereforeone can extract an effective density dependent torque acting on the cells by fitting the experimental distribution tothe Von-Mises distribution. The variation of best-fit values of γ with cell concentration (Fig. 3b) is qualitativelysimilar to that of the previously shown model independent phototactic efficiency ζ . Therefore, the reentrant behaviorof the phototactic efficiency as a function of density is reliably captured by modelling this collective phenomenon asan effective density dependent torque on each cell.While one could potentially rationalize the decrease in phototactic efficiency as the concentration increases as aneffect of cell-cell scattering, the increase in ζ at densities greater than the critical concentration is more puzzling.It may be reasonable to postulate that at high densities its primary effect on the behavior of a single cell is that itslows down and indeed that is the case in our experiments (Fig. 4a). This has been referred to as density dependentmotility in the context of the active matter literature [30, 31] . This could potentially affect the phototactic efficiencybecause of how CR cells detect light. The cells follows a helical trajectory due to cell body rotation. The cell bodyrotation allows the cell to collect photons from all directions in space. A decrease in linear speed implies a decreasein cell body rotation rate which enables the cell to collect more photons per unit time and therefore detect the lightdirection more accurately (Fig. 4d).One way to possibly validate this postulated mechanism for the increase in ζ as density increases beyond ρ c wouldbe to slow the cells down without changing the concentration of cells. One of the simplest ways to achieve that is toadd polymer to the suspension medium which increases the drag force on the cells, thereby lowering their speed. Weuse varying concentrations of methylcellulose to tune the speed of the cells keeping cell concentration fixed (Fig. 4b).We find that ζ increases with increase in methycellulose concentration, confirming the hypothesis that the observedincrease in ζ with increasing cell concentration is mainly due to lowering of cell speed (Fig. 4c). DISCUSSION
To summarize, we find that phototactic efficiency of CR cells is re-entrant in going from low density dilute regimeto high density collective one wherein dilute suspensions have smaller efficiency than that of single cell limit anddense suspensions have the opposite trend. We have identified the mechanism of enhanced efficiency in the collectiveregime to be the decrease in linear speed of the cells as the concentration increases. We speculate that decrease inlinear speed leads to a decrease in rotational speed of the cells that enables them to sense the light direction moreaccurately.The cell speed is nearly independent of concentration below the threshold concentration that marks the crossoverbetween the individual and collective behavior. Therefore, the mechanism for decrease in efficiency with increasing cellconcentration in the dilute regime is likely to be some other form of hydrodynamics interaction or steric in nature. Italso remains to be explored how tightly the single cell response is coupled with its collective response. In other words,how chemical or genetic modifications that alter the single cell phototaxis efficiency affect the collective behavior ofsuch modified cells.Complexity is common in biological systems and often its origin is difficult to identify. Our results have demon-strated a rather simple physical and phenomenological mechanism underlying the observed complexity in the collectivephototaxis of CR cells. Apart from identifying a particular phenomenology associated with zooplanktons with a singleeye spot, this work can serve as a paradigm for analysis of collective motility and taxis in microorganisms in generaland perhaps motivate design of control algorithms in collective robotics.
Author Contributions : AB and PS designed the research, SKC carried out the experiments and associatedanalysis. All authors wrote the article.
Acknowledgement
AB acknowledges support from Brandeis Center for Bioinspired Soft Materials, NSF MRSEC,DMR-1420382 and the hospitality of IISc and IMSc where part of this work was completed. This work was supportedby the Wellcome Trust/DBT India Alliance Fellowship [grant number IA/I/16/1/502356] awarded to P. Sharma. [1] J. K. Parrish, S. V. Viscido, and D. Grnbaum, The Biological Bulletin , 296 (2002).[2] D. J. T. Sumpter,
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FOV
66 µm Double sided tape Cover slipBlack tape5mmGlass slide a d xy Directionof light b c x (µm) y ( µ m ) x (µm) p /2 p p /2 2 p Pitch Dark 0.008 W cm -2 -2 -2 P r obab ili t y den s i t y Angle (rad)
Laser intensity (W cm -2 ) ζ FIG. 1.
Experimental setup and Phototactic response. ( a ) Schematic of the experimental setup ( b ) Trajectories of Chlamydomanas reinhardtii in pitch dark condition. Each trajectory is arbitrarily coloured for visual clarity. ( c ) Trajectories inpresence of light ( I = 0 . W cm − ). Blue arrow at the bottom of Fig. 1( c ) shows the direction of stimulus light. In the pitchdark condition, cell trajectories are uniform in all directions whereas in presence of light, a large fraction of cell trajectoriesare oriented towards the light source (positive phototaxis). ( d ) Probability density P ( θ ) for different light intensities accordingto the sign convention given at the top right corner in the Fig. 1( d ). (Inset) Phototactic efficiency ζ as a function of lightintensity. The error bars correspond to the standard deviation of ζ . e f -2 -2 Cell concentration (cell cm -3 ) ζ y ( µ m ) x (µm) x (µm) x (µm) x (µm) a b c d p /2 7 p /2 11 p /2 15 p /2 19 p /20.00.51.01.52.02.5 P r obab ili t y den s i t y Angle (rad)
FIG. 2.
Phototactic efficiency is reentrant with the cell concentration. ( a-d ) Typical cell trajectories of Chlamydo-manas reinhardtii under fixed light intensity ( I = 0 . W cm − ) with varying cell concentration (Legend in unit of cells cm − ).( e ) Probability density P ( θ ) under fixed light intensity ( I = 0 . W cm − ) for increasing cell concentration. The angle θ hasbeen offset by multiple of 2 π to shift the peak position for clarity. The height of the peaks quantify the reentrant behaviour ofphototactic efficiency (Legend in unit of cells cm − ). ( f ) Phototactic efficiency ζ as a function of cell concentration correspond-ing to two different light intensities. The error bars correspond to the standard deviation of ζ . Phototactic efficiency decreaseswith increasing cell concentration until a critical concentration ( ρ c ) reached, above which phototactic efficiency increases withcell concentration. The dependence of phototactic efficiency on cell concentration is stronger at the lower light intensities. a b p /2 7 p /2 11 p /2 15 p /2 19 p /20.00.51.01.52.02.53.03.54.0 von Mises-distribution fit P r obab ili t y den s i t y Angle (rad) D R (r ad s - ) Cell concentration (cell cm -3 ) Cell concentration (cell cm -3 ) g (r ad s - ) FIG. 3.
Self propelled particle model for collective phototaxis ( a ) Fit of the experimental probability densities (Fig2( e )) to von Mises distribution, ψ ( θ ) = e κcosθ πI ( κ ) where κ = γD R with rotational diffusion coefficient D R , 0 . rad s − . Theangle θ has been offset by multiple of 2 π to shift the peak position for clarity (Legend in units of cell cm − ). ( Inset ) Plot ofrotation diffusion coefficient, D R with the cell concentration shows that, D R is independent of the cell concentration. ( b ) ALog-log plot of torque strength γ as a function of cell concentration under two different light intensities I = 0 . W cm − (Open circle) and I = 0 . W cm − (Closed circle). The error bars correspond to the standard deviation of γ . The reentrantphototaxis behaviour observed in the experiment can be effectively captured by a density dependent aligning torque. ( Inset )A linear representation of the same data. a bc -2 -2 Cell concentration (cell cm -3 ) M ean S peed ( μ m s - ) -2 -2 Methyl cellulose concentration (% w/v) M ean S peed ( μ m s - )
60 80 100 120 140 1600.00.20.40.60.81.0
With MC Without MC
Mean Speed (μm s -1 ) ζ
60 80 100 120 140 1600.20.40.60.81.0 ω V
ZY X p r ω ω ω ijk V d FIG. 4.
Physical origin of reentrant phototactic efficiency ( a ) Mean speed as a function of cell concentration abovecritical concentration ( ρ c ). As cell concentration increases, cells slow down. ( b ) Mean speed as a function of Methyl celluloseconcentration at a constant cell concentration 2 . × cell cm − . Cell’s speed was slowed down using Methyl cellulose inthe suspension medium. ( c ) Phototactic efficiency ζ as a function of cell’s mean speed at the intensity I = 0 . W cm − .Pink open circle (colour online) correspond to the phototactic efficiency when cells are slowed down using methyl cellulose andgreen filled circle (colour online) correspond to the phototactic efficiency when cell’s speed was varied by cell concentration. Inboth the cases phototactic efficiency decreases as the cell’s speed increases. Phototactic efficiency is controlled by mean speed.( Inset ) Phototactic efficiency ζ as a function of cell’s mean speed at the intensity I = 0 . W cm − . ( d ) Schematic of a Chlamydomonas cell trajectory illustrating V = | (cid:126)ω | π p . Slower cells turns slowly. ( Inset ) Definition of angular and linear velocitycomponent along the body axes of the cell. Helical trajectory results whenever (cid:126)ω is neither parallel nor perpendicular to (cid:126)V ( ω (cid:54) = 0 , √ ω + ω (cid:54)(cid:54)