The internal mechanism of cilia is among the most ancient biological motors on an evolutionary scale. It produces beat patterns that consist of two phases: during the effective stroke, the cilium moves approximately as a straight rod, and during the recovery stroke, it rolls close to the surface in a tangential motion. It is commonly agreed that these two phases are designed for efficient functioning: the effective stroke encounters strong viscous resistance and generates thrust, whereas the recovery stroke returns the cilium to starting position while avoiding viscous resistance. Metachronal coordination between cilia, which occurs when many of them beat close to each other, is believed to be an autonomous result of the hydrodynamical interactions in the system. Qualitatively, metachronism is perceived as a way for reducing the energy expenditure required for beating. This paper presents a quantitative study of the energy expenditure of beating cilia, and of the energetic significance of metachronism. We develop a method for computing the work done by model cilia that beat in a viscous fluid. We demonstrate that for a single cilium, beating in water, the mechanical work done during the effective stroke is approximately five times the amount of work done during the recovery stroke. Investigation of multicilia configurations shows that having neighboring cilia beat metachronally is energetically advantageous and perhaps even crucial for multiciliary functioning. Finally, the model is used to approximate the number of dynein arm attachments that are likely to occur during the effective and recovery strokes of a beat cycle, predicting that almost all of the available dynein arms should participate in generating the motion.

We present a simple but realistic model for the internal bend-generating mechanism of cilia, using parameters obtained from the analysis of data of the beat of a single cilium, and incorporate it into a recently developed dynamical model. Comparing the results to experimental data for two-dimensional beats, we demonstrate that the model captures the essential features of the motion, including many properties that are not built in explicitly. The beat pattern and frequency change in response to increased viscosity and the presence of neighboring cilia in a realistic fashion. Using the model, we are able to investigate multicilia configurations such as rows of cilia and two-dimensional arrays of cilia. When two adjacent model cilia start beating at different phase, they synchronize within two cycles, as observed in experiments in which two flagella beating out of phase are brought close together. Examination of various multicilia configurations shows that metachronal patterns (i.e., beats with a constant phase difference between neighboring cilia) evolve autonomously. This provides modeling evidence in support of the conjecture that metachronism may occur as a self-organized phenomenon due to hydrodynamical interactions between the cilia.

1. The lateral pyloric (LP) neuron is a component of the 14-neuron pyloric central pattern generator in the stomatogastric ganglion of the spiny lobster, Panulirus interruptus. In the pyloric rhythm, this neuron fires rhythmic bursts of action potentials whose phasing depends on the pattern of synaptic inhibition from other network neurons and on the intrinsic postinhibitory rebound properties of the LP cell itself. Bath-applied dopamine excites the LP cell and causes its activity to be phase advanced in the pyloric motor pattern. At least part of this modulatory effect is due to dopaminergic modulation of the intrinsic rate of postinhibitory rebound in the LP cell. 2. The LP neuron was isolated from all detectable synaptic input. We measured the rate of recovery after 1-s hyperpolarizing current injections of varying amplitudes, quantifying the latency to the first spike following the hyperpolarizing prepulse and the interval between the first and second action potentials. Dopamine reduced both the first spike latency and the first interspike interval (ISI) in the isolated LP neuron. During the hyperpolarizating pre-steps, the LP cell showed a slow depolarizing sag voltage that was enhanced by dopamine. 3. We used voltage clamp to analyze dopamine modulation of subthreshold ionic currents whose activity is affected by hyperpolarizing prepulses. Dopamine modulated the transient potassium current IA by reducing its maximal conductance and shifting its voltage dependence for activation and inactivation to more depolarized voltages. This outward current is normally transiently activated after hyperpolarization of the LP cell, and delays the rate of postinhibitory rebound; by reducing IA, dopamine thus accelerates the rate of rebound of the LP neuron. 4. Dopamine also modulated the hyperpolarization-activated inward current Ih by shifting its voltage dependence for activation 20 mV in the depolarizing direction and accelerating its rate of activation. This enhanced inward current helps accelerate the rate of rebound in the LP cell after inhibition. 5. The relative roles of Ih and IA in determining the first spike latency and first ISI were explored using pharmacological blockers of Ih (Cs+) and IA [4-aminopyridine (4-AP)]. Blockade of Ih prolonged the first spike latency and first ISI, but only slightly reduced the net effect of dopamine. In the continued presence of Cs+, blockade of IA with 4-AP greatly shortened the first spike latency and first ISI. Under conditions where both Ih and IA were blocked, dopamine had no additional effect on the LP cell. 6. We used the dynamic clamp technique to further study the relative roles of IA and Ih modulation in dopamine's phase advance of the LP cell. We blocked the endogenous Ih with Cs+ and replaced it with a simulated current generated by a computer model of Ih. The neuron with simulated Ih gave curves relating the hyperpolarizing prepulse amplitude to first spike latency that were the same as in the untreated cell. Changing the computer parameters of the simulated Ih to those induced by dopamine without changing IA caused only a slight reduction in first spike latency, which was approximately 20% of the total reduction caused by dopamine in an untreated cell. Bath application of dopamine in the presence of Cs+ and simulated Ih (with control parameters) allowed us to determine the effect of altering IA but not Ih: this caused a significant reduction in first spike latency, but it was still only approximately 70% of the effect of dopamine in the untreated cell. Finally, in the continued presence of dopamine, changing the parameters of the simulated Ih to those observed with dopamine reduced the first spike latency to that seen with dopamine in the untreated cell. 7. We generated a mathematical model of the lobster LP neuron, based on the model of Buchholtz et al. for the crab LP neuron.

Bath application of dopamine modifies the rhythmic motor pattern generated by the 14 neuron pyloric network in the stomatogastric ganglion of the spiny lobster, Panulirus interruptus. Among other effects, dopamine excites many of the pyloric constrictor (PY) neurons to fire at high frequency and phase-advances the timing of their activity in the motor pattern. These responses arise in part from direct actions of dopamine to modulate the intrinsic electrophysiological properties of the PY cells, and can be studied in synaptically isolated neurons. The rate of rebound following a hyperpolarizing prestep and the spike frequency during a subsequent depolarization are both accelerated by dopamine. Based on theoretical simulations, Hartline (1979) suggested that the rate of postinhibitory rebound in stomatogastric neurons could vary with the amount of voltage- sensitive transient potassium current (IA). Consistent with this prediction, we found that dopamine evokes a net conductance decrease in synaptically isolated PY neurons. In voltage clamp, dopamine reduces IA, specifically by reducing the amplitude of the slowly inactivating component of the current and shifting its voltage activation curve in the depolarized direction. 4-Aminopyridine, a selective blocker of IA in stomatogastric neurons, mimics and occludes the effects of dopamine on isolated PY neurons. A conductance-based mathematical model of the PY neuron shows appropriate changes in activity upon quantitative modification of the IA parameters affected by dopamine. These results demonstrate that dopamine excites and phase-advances the PY neurons in the rhythmic pyloric motor pattern at least in part by reducing the transient K+ current, IA.