Fidelity of adaptive phototaxis

Abstract

Along the evolutionary path from single cells to multicellular organisms with a central nervous system are species of intermediate complexity that move in ways suggesting high-level coordination, yet have none. Instead, organisms of this type possess many autonomous cells endowed with programs that have evolved to achieve concerted responses to environmental stimuli. Here experiment and theory are used to develop a quantitative understanding of how cells of such organisms coordinate to achieve phototaxis, by using the colonial alga Volvox carteri as a model. It is shown that the surface somatic cells act as individuals but are orchestrated by their relative position in the spherical extracellular matrix and their common photoresponse function to achieve colony-level coordination. Analysis of models that range from the minimal to the biologically faithful shows that, because the flagellar beating displays an adaptive down-regulation in response to light, the colony needs to spin around its swimming direction and that the response kinetics and natural spinning frequency of the colony appear to be mutually tuned to give the maximum photoresponse. These models further predict that the phototactic ability decreases dramatically when the colony does not spin at its natural frequency, a result confirmed by phototaxis assays in which colony rotation was slowed by increasing the fluid viscosity.

Fig. 1.

Geometry of V. carteri and experimental setup. (A) The beating flagella, two per somatic cell (Inset), create a fluid flow from the anterior to the posterior, with a slight azimuthal component that rotates Volvox about its posterior-anterior axis at angular frequency ω_r. (Scale bar: 100 μm.) (B) Studies of the flagellar photoresponse utilize light sent down an optical fiber.

Fig. 2.

Characteristics of the adaptive photoresponse. (A) The local flagella-generated fluid speed u(t) (Blue), measured with PIV just above the flagella during a step up in light intensity, serves as a measure of flagellar activity. The baseline flow speed in the dark is u₀ = 81 μm/s for this dataset. Two time scales are evident: a short response time τ_r and a longer adaptation time τ_a. The fitted theoretical curve (Red) is from Eq. 4. (B) The times τ_r (Squares) and τ_a (Circles) vary smoothly with the stimulus light intensity, measured in terms of PAR. Error bars are standard deviations.

Fig. 3.

Photoresponse frequency dependence and colony rotation. (A) The normalized flagellar photoresponse for different frequencies of sinusoidal stimulation, with minimal and maximal light intensities of 1 and 20 μmol PAR photons m^-2 s^-1 (Blue Circles). The theoretical response function (Eq. 5, Red Line) shows quantitative agreement, using τ_r and τ_a from Fig. 2B for 16 μmol PAR photons m^-2 s^-1. (B) The rotation frequency ω_r of V. carteri as a function of colony radius R. The highly phototactic organisms for which photoresponses were measured fall within the range of R indicated by the purple box, and the distribution of R can be transformed into an approximate probability distribution function (PDF) of ω_r (Inset), by using the noisy curve of ω_r(R). The purple box in A marks the range of ω_r in this PDF (green line indicates the mean), showing that the response time scales and colony rotation frequency are mutually optimized to maximize the photoresponse.

Fig. 4.

Heuristic analysis of the phototactic fidelity. A–C illustrate simplified phototaxis models. Photoresponsive regions are colored green, the region that actually displays a photoresponse is in shades of red, and shaded regions are gray. (A) If τ_a = ∞, ω_r = 0, and the responsive region is as drawn, the posterior-anterior axis k will achieve perfect antialignment with the light direction I. The time scale for turning τ_t ∼ 3.3 s can be estimated by assuming that the fluid velocity on the illuminated side is reduced to 0.7 of its baseline value and using Eq. 8 without bottom-heaviness. (B) If τ_a < τ_t, and ω_r = 0, the photoresponse may decay before the optimal orientation has been reached. After the initial transient in A has decayed, the largest photoresponse (i.e., flagellar down-regulation) is in the region that just turned into the light. As an illustration, the configuration drawn in this panel surprisingly implies that the organism would turn away from the light, indicating that before this orientation is reached the steering is stopped at a suboptimal orientation of k with I. A remedy against this orientational limitation would be ω_r ≠ 0. (C) The best attainable orientation towards the light is drawn, if the photoresponse is localized in a small anterior region, and the eyespots display an all-or-nothing response as they move from the shaded to the illuminated side. (D) Measurements of the eyespot (Orange) placement yield κ = 57° ± 7° (see SI Text). (E) Volvox is bottom-heavy, because the center of mass (Pink) is offset from the geometric center of the colony as indicated.

Fig. 5.

Anterior-posterior asymmetry. (A) The anterior-posterior component of the fluid flow, measured 10 μm above the beating flagella, following a step up in illumination at time t = 0 s. The dashed line indicates the approximation to v₀(θ) used in the numerical model. (Inset) β(θ) is blue (with p normalized to unity), and the mean β is red. (B) The probability of flagella to respond to light correlates with the size of the somatic cell eyespots. The light-induced decrease in fluid flow occurs beyond the region of flagellar response because of the nonlocality of fluid dynamics.

Fig. 6.

Colony behavior during a phototurn. A–E show the colony axis k (Red Arrow) tipping toward the light direction I (Aqua Arrow). Colors represent the amplitude p(t) of the down-regulation of flagellar beating in a simplified model of phototactic steering. F shows the location of colonies in A–E along the swimming trajectory.

Fig. 7.

The phototactic ability decreases dramatically as ω_r is reduced by increasing the viscosity. Results from three representative populations are shown with distinct colors. Each data point represents the average phototactic ability of the population at a given viscosity. Horizontal error bars are standard deviations, whereas vertical error bars indicate the range of population mean values, when it is computed from 100 random selections of 0.1% of the data. A blue continuous line indicates the prediction of the full hydrodynamic model; the red line is obtained from the reduced model. (Inset) α(t) from the full and reduced model at the lowest viscosity.