Fig 1.
A, Scanning electron microscopy image (left) [12], and bright field microscopy image (right) of Paramecium tetraurelia (scale bars: 10 μm). B, Typical helicoidal swim of Paramecium [13]. C, Accumulation of paramecia in a drop of acid [13]. D, Trajectory of a single cell in the acid drop, showing directional changes at the boundary [13]. E, Avoiding reaction against an obstacle, showing ciliary reversal followed by reorientation [14].
Fig 2.
The avoiding reaction of Paramecium.
A, Typical spontaneous avoiding reaction: the ciliate swims backward, then turns and eventually resumes forward swimming, while spinning around its main axis during the entire movement. Images are separated by 150 ms, with intermediate shaded frames every 37 ms. The cell was placed in 20 mM NaCl and 0.3 mM CaCl2 to induce spontaneous avoiding reactions [21]. B, Intracellular recording of a voltage response (bottom right) to a square current pulse of amplitude 300 pA (top right) in an immobilized cell (left; A: anterior end; P: posterior end), showing a small action potential (in the standard extracellular solution, see Methods). The arrow points at a small upward inflexion due to the calcium current. Inset: Paramecium immobilized on a filter (background) with two electrodes. C, Velocity field of the fluid on a plane ~30 μm above the cell, calculated over the three shaded intervals shown in B. The blue arrow indicates mean velocity in the whole field, represented twice larger for clarity. The red arrows highlight the area neighboring the cell. C1, The fluid moves backward, which would make the cell swim forward. C2, The fluid moves forward. C3, The flow direction reverts on the posterior end, but not on the anterior right end, resulting in a swirling pattern.
Fig 3.
Summary of the model, showing a cilium attached to the base, and the movements of the two main ions, Ca2+ (red) and K+ (blue).
There is more Ca2+ outside than inside, and more K+ inside than outside. Calcium enters through ciliary voltage-gated channels as a current ICa. It then quickly inactivates these channels, forming a negative feedback loop. Calcium activates motor proteins, triggering ciliary reversal, as well as a ciliary K+ channel, producing an outward K+ current IK(Ca). The motor activation results in calcium concentration [Ca2+]i modulating kinematic parameters. Calcium is then expelled, in particular by plasma membrane Ca2+ ATPases (PMCA). Depolarization opens voltage-gated K+ channels in the basilar membrane, presumably near the cilium, creating a current IKd (delayed rectifier). A linear leak current is also present. Finally, an inward rectifier current IKir opens at very hyperpolarized voltages, which has little impact on the avoiding reaction.
Fig 4.
Outline of the modeling strategy.
Each number indicates the corresponding figure. The left column (5–7) is the electrophysiological modeling. The right column (8–9) is the modeling of couplings between electrophysiology and kinematic variables. Put together, we obtain a model of a freely swimming Paramecium (9), which is then augmented with elementary sensory transduction models to yield model simulations in structured environments (10–11).
Fig 5.
Passive properties and inward rectifier current.
A, Top: voltage responses of one cell to negative current pulses (I = 0 to -4 nA in 300 pA increments; dashed lines: start and end of pulses), in the standard extracellular solution (4 mM KCl and 1 mM CaCl2). The arrow points at an inflexion due to the inward rectifier current IKir. Bottom: model responses fitted to the data, showing the inferred reversal potential of K+ (EK) and the half-activation voltage VKir of the inward rectifier current. B, Current-voltage relationship over all cells (mean ± standard deviation, measured at pulse end) in 4 mM KCl (grey) and 0 mM KCl (blue). Removing K+ from the extracellular solution largely suppresses the inward current. C, Activation curve of the inward rectifier current in the fitted models. The current activates below EK (EL is leak reversal potential). The solid curve is the activation function with median parameters, the dashed curve is the activation function of the cell shown in A. D, Fitted parameters over n = 28 cells, grouped in passive parameters and inward rectifier parameters.
Fig 6.
The delayed rectifier current measured in deciliated cells.
A, Current-voltage relationship in deciliated cells, showing a strong delayed rectifier current for depolarized voltages, and the inward rectifier current for hyperpolarized voltages. B, Top: voltage responses of one cell to positive current pulses (I = 0 to 4 nA in 300 pA increments). Bottom: responses of the two-gate Boltzmann model fitted to the data, showing the inferred half-activation voltage of the delayed rectifier current (dashed). C, Activation and time constant of the delayed rectifier current as a function of voltage in fitted models, with median parameters (solid) and for the cell shown in B (dashed). D, Statistics of fitted parameters (n = 16).
Fig 7.
Fitting the action potential of Paramecium.
A, Voltage responses of a cell (top) to two sets of current pulses (bottom), from 0 to 5 nA (in 300 pA increments) and from 0 to 300 pA (in 25 pA increments). B, Ciliary response to the same currents, measured as the cosine of the mean angle of the velocity field, relative to the anteroposterior axis. C, Close up of an action potential triggered by a 1.5 nA current pulse, with the model fit (dashed). The arrow points at an upward deflection due to the calcium current. D, Ionic current calculated by subtracting the estimated leak current from the capacitive current. The inward current (I<0, shaded) corresponds to the calcium current. Integrating this current yields a calcium entry corresponding to a 10 μM increase in intraciliary calcium concentration. E, Responses of the fitted model. E1, Voltage responses. E2, Ciliary responses. E3, Voltage-gated calcium current ICa (top, negative traces), delayed rectifier K+ current IKd (bottom) and calcium-activated K+ current IK(Ca) (top, positive traces) in the fitted model. Currents are shown with the electrophysiological convention, i.e., I<0 is inward. E4, Intraciliary calcium concentration in the fitted model. The dashed lines show the ciliary reversal threshold and the half-inactivation concentration. F, Ionic currents inferred by the model for the action potential shown in C. G, Statistics of fitted parameters (n = 18).
Fig 8.
A, Direction of fluid motion during forward swimming (blue) and backward swimming (red), relative to the anteroposterior axis. Averages are shown by arrows. B, Example of helicoidal motion of Paramecium, with the oral groove facing the axis. Highlighted frames are spaced by 750 ms. C, The translational velocity vector v is oriented along the anteroposterior axis. The rotation vector ω is in the dorsoventral plane (including the oral groove), making an angle θ with the anteroposterior axis. D, Rotating movement of the cell at the end of avoiding reactions of increasing strength [27]. E, Organization of ciliary basal bodies on the oral (ventral) side [10]. The oral apparatus (oa) is in the center (R: right; L: left; A: anterior; P: posterior; as: anterior suture; ps: posterior suture; cy: cytoproct). F, Calculation of kinematic parameters v, θ and ω in a spherical model of radius 60 μm, during successive phases of the avoiding reaction. First column: cilia beat to the rear and right, producing an axisymmetric force field pushing the organism forward while spinning around its axis. Local force amplitude is adjusted for a velocity of 500 μm/s. Second column: cilia revert and now beat to the front and right, pushing the organism backward. Third column: anterior left cilia revert back to the initial direction while anterior right cilia still beat towards the front, and posterior cilia partially revert, beating to the right. Translational velocity is now 0 and the rotation axis tilts to about 34°. Spinning speed ω also increases by a factor four. Fourth column: all cilia revert back to the initial beating direction. G, Measurement of fluid velocity in a sample cell beyond the anterior end (top) and beyond the posterior end (bottom), in response to 100 ms positive current pulses (1–5 nA), relative to the anteroposterior axis. H, Over n = 9 cells, the direction of posterior motion reverts back about 30 ms after anterior fluid motion (dashed line: linear regression; solid line: identity). Reversal duration is calculated as the time when cos(α) crosses 0, relative to the pulse end time.
Fig 9.
Simulation of the avoiding reaction.
A, Velocity v as a function of intraciliary calcium concentration [Ca2+] in the model. B, Angle θ of the rotation axis as a function of [Ca2+] in the model. C, Spinning speed ω as a function of [Ca2+] in the model. Angle and spinning speed increase at intermediate Ca2+ concentration, as implied by the spherical model in Fig 8F and 8D, Simulated model trajectory with three 2 ms current pulse stimulations of increasing amplitude. Images are shown at 400 ms intervals. Without stimulation, the organism swims in spiral, with the oral groove facing the spiral axis. A very small stimulation deviates the trajectory. Stronger stimulations produce avoiding reactions, with backward swimming and turning. E, Example of an observed Paramecium trajectory showing a directional change without backward swimming (right), followed by a full avoiding reaction (left). Images are shown at 400 ms intervals, starting on the right. F, Backward swimming duration (F1) and reorientation angle (F2) as a function of current amplitude for 2 ms pulses. Red and black curves show results for the same model but different initial positions of the oral groove, differing by a quarter of a cycle. G, Backward swimming duration (G1) and reorientation angle (G2) as a function of current pulse duration T with 100 pA amplitude. H, Reorientation angle vs. backward swimming duration in n = 1138 spontaneous avoiding reactions of Paramecium, showing a positive correlation (linear regression r = 0.2, p ≈ 10−11). About 15% of data points are not represented (larger angle or duration). Colors represent contour lines of the distribution.
Fig 10.
Interaction of a model Paramecium with a generic stimulus, modelled as a positive current proportional to the cell area within the stimulus area.
A, Trajectory of the model doing several avoiding reactions against the stimulus. B, Membrane potential (top), stimulus current (middle) and intraciliary calcium concentration (bottom) at the first contact. Contact occurs at the boundary with the shaded region. Orange curves indicate backward swimming. Several weak avoiding reactions occur in succession. C, Trajectory of the model where sensory transduction has a 40 ms activation/deactivation time constant. In red, the stimulus is placed 300 μm further away. D, Same as B, for the black trajectory in C. The stimulus current lasts longer and peaks after the organism has started reacting, resulting in a stronger avoiding reaction.
Fig 11.
Closed-loop behavior of the Paramecium model.
A, Top: trajectories of 100 models swimming for 20 s in a torus with a depolarizing circular stimulus, modelled as in Fig 10C. The proportion of cells in the disk quickly decays (below). Membrane potential, stimulus current, and intraciliary calcium concentration are shown for the trajectory highlighted in red, which does an avoiding reaction against the disk after a number of spontaneous avoiding reactions. B, 100 model trajectories with a circular stimulus triggering an adapting hyperpolarizing current. Organisms tend to make avoiding reactions on the inner boundary of the disk. The proportion of cells in the disk increases over time. The highlighted trajectory enters the disk around t = 5 s with a large hyperpolarization, then displays several avoiding reactions against the boundary of the disk before exiting. C, Paramecia swimming in a linear stimulus gradient, modelled as in B. The position of 200 cells starting at position x = 5 mm is displayed every 5 s. D, Collective behavior in model paramecia induced by respiration and chemosensitivity. CO2 produced by cells is displayed in shades of grey (normalized to the spatial peak), and diffusion is simulated. CO2 concentration represents an attracting stimulus modelled as in B and C.
Fig 12.
Conventions on the spherical model, with spherical coordinates and a local force F on the surface of the sphere.
Table 1.
Statistics of fitted parameters (n = 28) for hyperpolarized responses (model with two gates).
Table 2.
Statistics of fitted parameters (n = 16) for deciliated cells (Boltzmann model with two gates).
Table 3.
Statistics of fitted parameters (n = 16) for depolarized ciliated cells.