BACKGROUND
cilia are cellular protrusions which have been conserved in a wide range of organisms ranging from protozoa to the digestive, reproductive and respiratory systems of vertebrates  <cit> .
mobile or immotile cilia exist on every cell of the human body  <cit>  and the insufficiently recognised importance of the cilium compartment in human physiology has been recently highlighted  <cit> .
cilia are present on most eukaryotic cell surfaces with the exception of the cells of higher plants and fungi  <cit> .
ciliary motility is important for moving fluids and particles over epithelial surfaces, and for the cell motility of vertebrate sperm and unicellular organisms.
the cilium contains a microtubule-based axoneme that extends from the cell surface into the extracellular space.
the axoneme consists of nine peripheral microtubule doublets arranged around a central core that may or may not contain two central microtubules .
cilia can be broadly classified as 9+ <dig> motile cilia or 9+ <dig> immotile sensory cilia, although there are examples of 9+ <dig> sensory cilia and 9+ <dig> motile cilia.
in mammals, motile 9+ <dig> cilia normally concentrate in large numbers on the cell surface, beat in an orchestrated wavelike fashion, and are involved in fluid and cell movement.
in contrast to motile cilia, primary cilia project as single immotile organelles from the cell surface.
primary cilia are found on nearly all cell types in mammals  <cit>  and many are highly adapted to serve specialized sensory functions.
the 9+ <dig> cilia usually have dynein arms that link the microtubule doublets and are motile, while most 9+ <dig> cilia lack dynein arms and are non-motile.
in total, eight different types of cilia has been identified to date  <cit> .
in this study, we investigate the mechanism of movement regulation for the motile type of cilia.
although each individual cilium represents a tiny hair-like protrusion of only  <dig>  μm in diameter and approximately 5- <dig> μm in length, cilia covering human airways can propel mucus with trapped particles of length up to  <dig> mm at a speed of  <dig>  mm/second  <cit> .
such efficiency can be achieved due to the coordination between cilia and stimulus-dependent regulation of the rate of cilia beat.
dysfunction of ciliary regulation gives rise to pathologic phenotypes that range from being organ specific to broadly pleiotropic  <cit> .
a link between ciliary function and human disease was discovered when individuals suffering from syndromes with symptoms including respiratory infections, anosmia, male infertility and situs inversus, were shown to have defects in ciliary structure and function  <cit> .
microscopic organisms that possess motile cilia which are used exclusively for either locomotion or to simply move liquid over their surface include paramecia, karyorelictea, tetrahymena, vorticella and others.
the human mucociliary machinery operates in at least two different modes, corresponding to a low and high rate of beating.
it has been shown that the high rate mode is mediated by second messengers  <cit> , including purinergic, adrenergic and cholinergic receptors  <cit> .
this mode enables a rapid response, which can last a significant period of time, to various stimuli by drastically increasing the ciliary beat frequency .
at the same time, several ciliary movement modes have been reported in a ciliate paramecium caudatum  <cit> .
the remarkable conservation of ciliary mechanisms  <cit>  creates grounds for the speculation that there can more than two ciliary beating modes in human tissues.
it is, therefore, reasonable to suggest, that some human diseases, associated with aberrant ciliary motility, can arise due to modifications in the beating mode.
clearly, the development of therapeutic strategies against ciliary-associated pathologies will require advanced understanding of ciliary beating regulation mechanisms.
the periodic beating of cilia is governed by the internal apparatus of the organelle  <cit> .
its core part, the axoneme, contains nine microtubule pairs encircling the central pair.
the transition at the junction of the cellular body and the ciliary axoneme is demarcated by y-shaped fibres, which extend from the microtubule outer doublets to the ciliary membrane.
the transition area, in combination with the internal structure of the basal body, is thought to function as a filter for the cilium, regulating the molecules that can pass into or out of the cilium.
ciliary motility is accomplished by dynein motor activity in a phosphorylation-dependent manner, which allows the microtubule doublets to slide relative to one another  <cit> .
the dynein phosphorylation that controls ciliary activity is regulated by the interplay of calcium  and cyclic nucleotide pathways.
the beating pattern of cilia consists of a fast effective stroke and a slower recovery stroke.
during the effective stroke cilia are in an almost upright position, generating force for mucus movement.
during the recovery stroke, the cilia are recovering from the power strike to the original position by moving in the vicinity of the cell surface.
current theories which attempt to explain the workings of the ca2+-dependent cbf regulation mechanism are incomplete and highly controversial.
elevation of intraciliary ca2+ is one of the major regulators of ciliary movement.
calcium influx regulates ciliary activity by increasing intraciliary ca2+ only, while the cytosolic bulk remains at a low level.
separate ciliary compartmentalisation for ca2+ allows prolonged activation of ciliary beating without damaging the cell through high ca2+ concentrations.
it is well known that calcium fluxes via calcium channels lead to changes in organisms' swimming behaviour  <cit> .
in mucus-transporting cilia, ca2+ mediates cbf increase  <cit> .
it has also been shown that there are some differences in the ca2+-dependent cbf regulation in single cell organisms and in humans  <cit> .
sustained cbf increase requires prolonged elevation of ca2+ levels which can be lethal to the cell  <cit> .
it has been suggested that ca2+-dependent ciliary regulation takes place locally in the vicinity or within the ciliary compartment, almost independently from intracellular ca2+ concentration  <cit> .
given that the gradient of free ca2+ in the cytosol dissipates within 1- <dig> seconds  <cit> , it appears more likely that cilia form their own compartment where ca2+ is regulated by active ca2+ transport in a similar fashion to the intracellular ca2+ regulatory system.
this hypothesis resolves the problem of maintaining physiological levels of intracellular ca2+ concentration.
a number of experimental studies have reported several controversial results relating to the ca2+-dependent mechanism of cilia regulation.
for example, it has been reported that spontaneous cilia beat does not require alterations in ca2+  <cit> , while nucleotide-dependent cbf increase requires ca2+  <cit> .
it has also been shown that uncoupling between ca2+ and cbf can be achieved by inhibition of ca2+-dependent protein calmodulin  or the cyclic nucleotide pathway  <cit> .
these findings suggest that intraciliary ca2+ does not regulate cilia beat in isolation, but instead does so as part of more complex signalling network.
although it was originally believed that ca2+, cyclic adenosine monophosphate  and guanosine monophosphate  regulate ciliary beat independently, numerous reports now strongly indicates that all three pathways are tightly interconnected  <cit> .
the camp-dependent protein kinase  phosphorylates dynein in the bases of cilia and thereby increases the forward swimming speed in paramecium  <cit> .
similar effects have been reported for pka-dependent phosphorylation of axonemal targets in mammalian respiratory cilia  <cit> .
several lines of evidence indicate that pka and cgmp-dependent kinase  both phosphorylate specific axonemal targets in a camp and cgmp-dependent manner.
a schematic diagram for the underlying biochemical machinery for cilia movement regulation is shown in figure 1c.
it is striking that, despite significant experimental characterisation of this system, there is still rather limited mechanistic understanding of how intraciliary ca2+ and nucleotide interplay relates to cbf.
another major regulator of ciliary beating is the membrane potential.
a number of studies have reported the voltage-dependent effects of ciliary beating.
the ciliate didinium nasutum has been shown to respond both to hyper- and de-polarization of the membrane  <cit> .
the transmembrane potential alterations were shown to be mediated via the potential-dependent ca2+ channels  <cit> .
electrophysiological studies in paramecium caudatum have revealed complex relationships between ciliary ca2+ currents, intraciliary ca2+ concentration and transmembrane potential in the regulation of ciliary motility  <cit> .
a number of previous computational studies have analysed various aspects of cilia movement regulation.
one earlier model assessed the degree of synchronization between small ciliary areas  <cit> .
the effects of viscosity have been investigated in mucus propelling cilia in  <cit> .
the authors found that increasing the viscosity not only decreases cbf, but also changes the degree of correlation and synchronization between cilia.
the mechanical properties of cilia motion were studied in an attempt to understand the ciliary dynamics in  <cit> .
the authors concluded that bending and twisting properties of the cilium can determine self-organized beating patterns.
while these reports offer valuable insights into the regulatory mechanisms of cilia, a number of essential questions remain unresolved.
for example, there has not been a detailed analysis of how individual ca2+ currents influence intraciliary ca2+ levels.
it also remains unclear how ca2+ modulates nucleotide levels and membrane potential, and how such regulation affects ciliary movement.
none of these reports have elucidated the underlying mechanisms governing the interplay between intraciliary ca2+ and nucleotide alterations and cbf.
in this study, we integrate the available experimental information on the molecular pathways that regulate intraciliary ca2+ concentration into a comprehensive mathematical model.
by applying systems analysis, we elucidate the mechanisms of intraciliary ca2+ spike generation, analyse the properties of such spikes and demonstrate the conditions under which the ca2+ surges can become repetitive.
we carry out detailed investigations of the individual current contributions to the regulation of the intraciliary ca2+ concentrations and elucidate both steady-state and dynamic responses of ca2+ currents and intraciliary ca2+ concentration dynamics in response to the altered transmembrane potential shift.
the model allows detailed elucidation of transmembrane potential and intraciliary ca2+ coupling.
we employ the proposed model in order to understand the underlying molecular mechanisms of the crosstalk between ca2+, membrane potential and nucleotide pathways that regulate ciliary movement.
the systems model allows detailed analysis of the individual current contributions to the intraciliary homeostatic ca2+ levels.
furthermore, we establish specific regulatory mechanisms for ca2+ and cyclic nucleotide-dependent cilia movement characteristics.
crucially, our model predicts the possibility of several ciliary beating modes and describes specific conditions that initiate them.
specifically, we describe intraciliary ca2+ dynamic modes that regulate healthy and pathologic cilia beating.
we use these findings in order to propose experimentally testable hypotheses for possible therapeutic interventions in human diseases associated with pathologic cilia motility.
RESULTS
a new model for the interplay between ca2+ and k+ currents and transmembrane potential alterations
a new model for the regulation of ciliary movement that combines multiple ca2+ and k+ currents  <cit>  and transmembrane potential has been developed.
in this model, the intraciliary ca2+ levels are modulated by ca2+ currents through the channels of passive and active ca2+ transport, the current from the cilium into the cell body, the ca2+ leakage current, and depolarisation and hyperpolarisation-activated currents.
variable extracellular conditions have continuous impact on the transmembrane potential which is intertwined with transmembrane ion currents and intraciliary ca2+ homeostasis.
the overall network that regulates ciliary movement is divided into several functional modules .
one module combines all ca2+ and k+ currents that define intraciliary ca2+ homeostasis and the transmembrane potential.
one of the most essential intraciliary ca2+ binding proteins, cam  <cit> , selectively regulates the activities of adenylate cyclase , guanylate cyclase  and phosphodiesterases , and thereby modulates the intraciliary levels of adenosine monophosphate  and guanosine monophosphate  in a ca2+ dependent manner  <cit> .
the camp- and cgmp-dependent kinases phosphorylate dynein proteins in the bases of cilia and thereby induce the mechanical cilia movement.
the complete set of equations making up the proposed model is presented in the methods section.
below we provide a number of new insights into the mechanism of cilia regulation via a detailed investigation of the properties of this model.
the mechanism of ca2+-dependent inhibition of ca2+ channels
a subset of intraciliary ca2+ channels have been reported to operate in an intraciliary ca2+ dependent manner and have been proposed as major regulators of ciliary beat  <cit> .
it is established that ca2+ current is not inhibited by the double pulse application of depolarization impulses under voltage clamp conditions in those situations when the first transmembrane potential shift is equal to the equilibrium ca2+ potential   <cit> .
further experimental evidence reveals that ca2+ current inactivation kinetics are delayed when ca2+ ions are partially replaced by ba2+ ions  <cit> .
altogether these findings suggest that the channels are not inhibited directly by the depolarizing shift of transmembrane potential, but that instead their conductivity is dependent on the intraciliary ca2+ concentration.
some decrease of the inward current amplitude  upon transmembrane potential shift into the ca2+ equilibrium level can be explained by the fact that k+ currents can contribute to the overall current measurements.
here we consider the intraciliary ca2+ concentration-dependent ca2+ channel inhibition and employ the developed model to analyse two potential scenarios for the ca2+ channel conductivity regulation.
in one case, ca2+ ions bind to the ca2+ binding site on the channel and thereby inhibit the channel's conductivity by direct interaction.
the other possibility is that the ca2+ binding protein interacts with the ca2+ ion first and then this complex binds to the channel and inhibits its conductivity.
in both cases the conductivity dependence on transmembrane potential is assumed to be monotonic according to the experimental data  <cit> .
direct ca2+-dependent ca2+ channel conductivity inhibition
we first investigate a potential intraciliary ca2+ regulatory mechanism via direct ca2+ ion binding-dependent ca2+ channel inhibition.
the relationship between the ca2+ channel conductivity and the transmembrane potential has been experimentally characterized by an early study in paramecium species  <cit> .
the equation  in the methods section approximates the experimentally established dependence.
in the case of direct ca2+-dependent ca2+ channel inhibition, one can show that the nullclines for non-dimensional ca2+ concentration  and for the number of open channels  from the system of differential equations  intersect at one stable point for all values in the physiological range of model parameters.
the numerical solutions of the coupled differential equations  allow us to obtain the solutions for how the steady-state ca2+ levels depend on the transmembrane potential.
the model predictions for the steady-state ciliary ca2+ channel conductivity dependence on the transmembrane potential under the voltage clamp conditions are shown on figure 2a.
the extracellular conditions are subject to constant change both in the case of ciliates as well as for multicellular organisms.
the modifications in the external environment continuously shift the transmembrane potential.
in order to estimate how the transmembrane potential alterations affect the inward ca2+ current we derived the dependence for the ca2+ ion flow  in the methods section).
the experimental data-based   <cit>  model for the inward ca2+ current dependence on membrane potential  predicts a significant reduction of the inward ca2+ current amplitude as a function of membrane depolarization.
we next investigated the dynamic alterations of the intraciliary ca2+ concentrations and the inward ca2+ current in response to the transmembrane potential shifts.
equations  and  in the methods section were used for quantitative estimations of the ca2+ concentration and current responses to the normalised membrane potential shifts.
the model predicts that the rapid depolarising alterations of the ciliary transmembrane potential leads to the generation of single ca2+ spikes .
such responses can take place when ca2+ channels are inhibited by ca2+ ions and the channel's conductivity depends on the membrane potential in a monotonic manner .
the amplitude of those impulses depends on the steepness of the ca2+ channels conductivity dependence on the membrane potential.
the mechanism of the ca2+ spike generation is mainly due to the delay of the ca2+-induced inhibition with respect to the ca2+ conductivity alteration characteristic times.
the described mechanism of the ca2+ spike generation will only work if the ca2+ currents significantly alter the ca2+ concentration in the intraciliary compartments.
indirect ca2+ channel conductivity regulation
in the previous section we considered ca2+-dependent ca2+ channel regulation under the assumptions that ca2+ channels have an intracellular ca2+ binding site and ca2+ ion binding closes the channels.
however, several experimental studies have suggested that the conductivity of ca2+ channels in cilia can also be regulated indirectly, via a ca2+ binding protein.
at present, there is no direct experimental evidence that explicitly favours either direct or indirect regulatory mechanism.
we, therefore, investigated the second possibility for indirect ca2+-dependent ca2+ channels conductivity inhibition.
the model for ca2+ ion interactions followed by the interactions with the ca2+ channels  is developed in line with a previously suggested modelling methodology for ca2+-cam interactions  <cit> .
the model predictions for the number of open channels as a function of intraciliary ca2+ concentration are shown on figure  <dig> as derived by equation  in the methods section.
given that the exact nature of the ca2+ binding protein acting as a mediator between ca2+ ions and ca2+ channels is not established, equation  has been solved for different physiologically possible ratios of total number of channels to the dissociation constant for the ca2+- binding protein interactions with ca2+ channels.
according to the derived models, the comparison of direct versus indirect ca2+ channels inhibition can be carried out by setting the non dimensional ca2+ concentration to zero ).
in this case, the predictions of equation  for indirect ca2+ channels conductivity inhibition almost coincide with the model for direct ca2+-dependent inhibition.
in both models most of the ca2+ channels are open in the lower range of ca2+ concentrations.
however, the model for indirect ca2+ channels inhibition predicts that the number of open ca2+ channels would equal 1cac0+ <dig> when intraciliary ca2+ reaches high concentrations.
in other words, in the case of the indirect mechanism of inhibition, high ca2+ does not inhibit the channels completely.
the number of remaining channels in the open state would depend on the concentration of the regulatory ca2+ binding protein.
we next analysed the dynamics of the ciliary ca2+ channels inhibition in response to a change in intraciliary ca2+ concentration and in the transmembrane potential.
as mentioned earlier, the ca2+ channel conductivity is not inhibited by the membrane potential, but rather has a monotonic dependence on the transmembrane potential difference as shown on figure 2b.
we found that the characteristic time τca2+ of ca2+ channel alterations in response to step changes in ca2+ is inversely proportional to the total number of channels.
the model predictions for intraciliary ca2+ concentration and inward ca2+ current in response to depolarising transmembrane potential changes from v <dig> to v <dig>  are shown on figure  <dig>  according to the derived equation  in the methods section, the inward ca2+ current changes with a characteristic time τv, which can only be estimated by considering the k+ current contribution.
in the first instance we only consider active and passive ca2+ transport  in methods).
we neglected by the kinetics for the channels of active transport due to the assumption that the kinetics of alterations of active transport are much faster that the characteristic alteration times of passive transport  for the ca2+ currents).
the model predictions for the steady-state dependence of transmembrane potential on intraciliary ca2+ concentration and the dynamic ca2+ current amplitude dependence on the membrane potential are shown in figure 6a and 6b, respectively.
the cilia's external environment is subject to constant change and can significantly affect the behavioural responses of ciliates and modulate ciliary beating in multicellular organisms.
in order to account for the effects of extraciliary ca2+ variations we estimated the amplitudes of intraciliary ca2+ spike generation under different external ca2+ concentrations and variable transmembrane potentials.
figure 7a shows that the increase of the ca2+ concentration in the external solution increases the amplitude of the generated intraciliary ca2+ spike.
the amplitude of the spikes goes to zero when the membrane potential equals the equilibrium potential for ca2+ ions.
the increase of membrane potential decreases the amplitude of the ca2+ current .
this effect is due to the increase of the steady-state intracilia ca2+ level and ca2+-dependent inhibition of the ca2+ channels.
we noted earlier that there is a ca2+ current in the cilia which transfers ions from the cilia into the cellular compartments.
this current can be described by equation  in methods.
the contribution of cilium-to-cell body current to the intraciliary ca2+ concentration dynamics was evaluated experimentally in  <cit> .
it was shown that under depolarized membrane potential conditions the contribution of this current is very small and the intraciliary ca2+ is mainly pumped out of the cilia into the extracellular space by the active ca2+ transport.
according to other observations, ca2+ current from cilia into the cellular compartment can be larger than the current generated by the active ca2+ transport.
in order to investigate the role and contribution of the cilia-to-cell compartment current, we introduced its contribution to the intraciliary ca2+ concentration dynamics ).
we performed qualitative analysis of the ca2+ concentration alterations in the cilia in the presence of the cilium-to-cell current and compared the ca2+ dynamics with the case when this current was not present.
we found that although the cilium-to-cell body current influences the intraciliary ca2+ concentration levels, it does not change the dynamics qualitatively when the membrane potential is depolarized and fixed.
our findings suggest that the cilium represents an excitable system with unique properties.
the ca2+-dependent inhibition of ca2+ channels inhibition allows for the generation of single impulses of variable amplitude proportional to the degree of membrane depolarisation caused by variations in the external concentrations of ions.
this system is able to generate a single spike despite unpredictable variations of ionic concentrations in the environment and is, therefore, very robust to alterations in the external conditions.
another interesting aspect of the ciliary excitation is the ability of the system to generate regulatory intraciliary ca2+ impulses proportional to the degree of membrane depolarisation .
this property can allow cells to sense and "automatically" respond to alterations in their environment.
the contribution of k+ currents
in the previous section, we analysed the dynamic properties of the intraciliary ca2+ system under voltage clamp conditions.
several lines of evidence suggest that k+ currents contribute to the currents registered in cilia under voltage clamped conditions.
the existence of k+ currents in cilia is supported by a number of experimental studies.
the experimental data shows that the measured current is not equal to zero when the membrane potential equals the equilibrium membrane potential for ca2+ ions.
instead, the current equals zero when membrane potential is about  <dig> mv while the equilibrium potential for ca2+ ions equals  <dig> mv  <cit> .
this observation suggests that both ca2+ and k+ currents contribute to the overall current measured at early stages of current registration under voltage clamp, and therefore both currents need to be taken into the consideration in order to advance understanding of the mechanisms involved in ciliary regulation.
at the same time, it has so far been impossible to register ca2+ currents by inhibiting the k+ contribution.
various compounds can only partially block the k+ current when applied from inside of the membrane.
ciliary k+ currents have also been measured separately from ca2+ currents.
in order to account for the contribution of k+ currents to the regulation of intraciliary ca2+ concentration, we developed a model for the regulation of k+ currents by membrane potential  in methods).
the dependence of k+ conductivity on the membrane potential is described by equation .
figure  <dig> shows the measured experimental values and the approximating curve calculated according to equation .
the current on figure  <dig> is normalized to  <dig> when membrane potential equals  <dig>  in order to approximate the current, we only used the experimental values obtained under the membrane depolarized conditions, and this experimental data is approximated by equation .
the predictions for intraciliary ca2+ dynamics and ca2+ impulse amplitude in response to the shift in membrane potential are shown in figure 9a and 9b, respectively.
a comparison of the membrane potential shift-induced intraciliary ca2+ spike generation in the presence of the k+ currents  with the model when k+ channels are not incorporated , suggests that while the k+ currents change the quantitative values of the ca2+ concentration dynamics, the general shape of the ca2+ response remains the same.
the transmembrane potential dynamics in the absence of voltage clamp
in the previous sections we investigated the mechanisms of the transmembrane potential shift-dependent ca2+ spike generation under voltage clamp conditions.
however, ca2+ currents themselves can alter the membrane potential.
here we incorporate the membrane potential dependence on ca2+ currents and investigate the membrane potential dynamics in the absence of voltage clamp  and ).
the non dimensional ca2+ concentration and membrane potential are described by equation .
the predictions for the individual ca2+ and k+ current responses to various membrane potential shifts are shown in figure  <dig>  one can note that the dynamics of the responses significantly differs between the ones obtained under voltage clamp conditions and the situation when the membrane potential is not fixed, but dependent on the currents .
voltage current characteristics for ca2+, k+ and full currents calculated using the current amplitudes from the currents dynamics are shown on figure 10h.
one can note that the current is an almost exponentially growing function of membrane potential.
the monotonic dependence of ca2+ current on transmembrane potential, and simultaneous ca2+-dependent inhibition of ca2+ channels, represents a classical problem of two interconnected variables: intraciliary ca2+ and membrane potential.
in this system, increasing ca2+ current with transmembrane potential depolarisation represents a positive feedback loop mechanism, whereas the intraciliary ca2+ concentration-dependent ca2+ channels inhibition represents a negative feedback loop.
we, therefore, sought to investigate the range of potential dynamical properties of the ciliary system emerging from the coupling of ca2+ current and membrane potential described by equations .
one can clearly see that there is significantly different response for different values of the inward current.
when the influx of the ions is relatively small, the dca2+dt= <dig> null cline intersects the dvdt =  <dig> null cline in the left descending area ; such a null cline crossing results in a stable solution.
in this case the system responds by the generation of a single impulse of both intraciliary ca2+ concentration and the membrane potential followed by a return to homeostatic levels .
further increasing the current causes the null cline dca2+dt= <dig> to intersect with the null cline dvdt= <dig> in the middle region of the ascending area, leading to an unstable solution with a limit cycle formed around the area that represents the oscillations.
.
however, further increase of the current causes the dca2+dt= <dig> null cline to intersect with the null cline dvdt= <dig> in the right descending area, resulting in a stable solution with a slight increase of the homeostatic ca2+ and membrane potential levels .
the key conclusion from this analysis is that the external ionic conditions can initiate essentially different dynamic properties of the system regulating ciliary movement.
one of the key factors that affect the ciliary beat cycle is the level of intraciliary ca2+.
our findings suggest that in response to the external conditions, there are several possibilities for intraciliary ca2+ upregulation.
the system can generate a single spike  of variable amplitude , permanently increase ca2+ in a dynamic fashion and maintain the high intraciliary levels , or operate in a monostable multivibrator mode  .
these three possibilities can be associated with the different modes of ciliary beat observed in human cilia as well as in various ciliates.
the dynamic properties of excitable systems with two interdependent variables are reasonably well understood at a theoretical level.
in the present case, ca2+ and membrane potential represent the slow and fast variables, respectively.
this study, therefore, establishes that the dynamic properties of ciliary systems, where the ca2+ and k+ channel conductivities represent monotonic function of membrane potential and the ca2+ channels conductivity inversely depends on intraciliary ca2+ concentration, are comparable with the properties of excitable systems based on the "n-shape" dependence of the na2+ channel conductivity on membrane potential  <cit> .
at the same time, it is essential to note that the mechanism of excitation described in motile cilia is different from the "classical" one described in most excitable cells and systems that involve ip <dig> ca2+ channels  <cit> .
the membrane hyperpolarisation-dependent currents modulate the excitatory properties of the ciliary system
the ciliary transmembrane potential can shift in two directions.
in the previous section we investigated the intraciliary ca2+ responses caused by membrane depolarisation.
here we assess the implications of the membrane hyperpolarisation which has been shown to activate the current from cilia into the cell body  <cit> .
we introduced the corresponding term into our model for the ca2+ ions movement via the membrane as a function of the corresponding membrane potential shift .
by assuming the potential independent mechanism for ca2+ and k+ ion expulsion, the system of intraciliary ca2+ and membrane potential is derived as shown in equation  in the methods section.
the model predictions for intraciliary ca2+ and membrane potential dynamics in the absence of the hyperpolarisation-induced current are shown in figure 11d, e and 11f.
in this case the system remains in the mode of steady oscillations.
we found that such oscillatory mode can be significantly modulated by the hyperpolarisation-induced and the external conditions-dependent current.
figure 12a shows the phase diagram in the presence of the hyperpolarising current.
in contrast to the situation described on figure  <dig> when the hyperpolarising current was absent, the ciliary system generates a single spike in response to alteration of external ionic concentrations over the whole physiological range of the inward current .
under certain combinations of the hyperpolarisation-induced currents and the ca2+-dependent k+ currents the phase diagram modifies in a manner so that the system responds by generating a single spike in response to alteration of the inward current of any magnitude .
this indicates that the ciliary system under membrane hyperpolarising conditions can become a monostable multivibrator.
the role of cilia-to body ca2+ current under membrane hyperpolarisation
intraciliary ca2+ concentration has been experimentally estimated to be approximately one order of magnitude higher in comparison with the intracellular levels.
the ca2+ current generated by the ion flow from the ciliary compartment into the cell has been reported by a number of groups  <cit> , but the role this current plays in the regulation of the ciliary beat remains unclear.
in order to address this question, we introduced the term for this current into the equation that describes the intraciliary ca2+ concentration  in the methods section).
in the absence of direct measurements of the dependence of conductivity on membrane potential we set the conductivity to increase in response to hyperpolarisation of membrane potential.
a summary of the model responses for all channel conductivities as a function of membrane potential is shown in figure  <dig>  by numerically solving the coupled equations for intraciliary ca2+ and membrane potential with the cilia to the cell body contribution, we found that this current does not qualitatively change the general dynamic properties of the system.
despite the lack of a noticeable contribution to the ciliary dynamic properties, this current requires a special consideration.
experimental studies have clearly demonstrated that intraciliary ca2+ is significantly higher than intracellular ca2+ concentration.
at the same time, if the conductivity of protein structures governing the ca2+ ions movement from cilia to the body is high, most of the intraciliary ions would move from cilia into the cell body in a very short time.
a simple calculation suggests that if ca2+ could freely flow from cilia into the body, the intraciliary concentration would become equal to the intracellular ca2+ concentration in less than  <dig> μs due to the difference in the volumes of the cell body and intraciliary compartments.
experimental measurements in ciliates show that the hyperpolarisation-induced backwards movements can last longer than  <dig> μseconds.
it is also known that the avoidance reaction that requires long term elevation of intraciliary ca2+ concentration can be observed in hyperpolarizing solutions.
during all this time the intraciliary ca2+ concentration can be several orders of magnitude higher than the intraciliary concentration.
in this study, we have demonstrated that the steady-state ca2+ current under the depolarized membrane potential conditions can only be reduced by the ca2+-dependent inhibition of ca2+ channels.
all these observations suggest that the ca2+ removal from cilia to the cell body occurs in a membrane potential dependent manner.
the mechanism of ca2+ and cyclic nucleotide-dependent cbf regulation
in addition to intraciliary ca2+ and k+ potassium levels being coupled with the membrane potential modulation, cyclic nucleotides contribute to the regulation of one of the major ciliary beat parameters, frequency.
intraciliary ca2+ levels activate a variety of adenylate cyclases  and phosphodiesterases  that produce and hydrolyse cyclic nucleotides, respectively, and thereby modulate the intraciliary camp and cgmp levels.
at the same time, camp and cgmp-dependent kinases phosphorylate dynein arms  <cit>  in the bases of cilia and thereby induce the ciliary movement  <cit> .
in a previous work we showed that the ciliary beat frequency can have a "double" bell shape dependence on ca2+ concentration  <cit>  due to the differential regulation of adenylate and guanylate cyclase isoforms in a ca2+-calmodulin  dependent manner  <cit> .
one bell-shape was due to the camp production by a combination of ac and pde, whereas another one was mediated by cgmp production and degradation.
our results proposed an explanation for seemingly conflicting experimental evidence suggesting that cbf can both decrease and increase with increasing ca2+ concentration.
here we extend our previous analysis and describe the conditions when one or the other peak can be significantly reduced or even disappear.
figure 15a shows the model predictions for the ciliary beat frequency in comparison with the experimental data  <cit> .
recent studies demonstrated that hormones and pharmacological agents can regulate both function and structure of cilia by interfering with the cyclic nucleotide signalling pathways  <cit> .
these findings support the possibility for the development of novel therapeutic strategies for ciliary pathologies by modulating the dynamic mode of beating via ciliary membrane receptors  <cit> .
discussion
we develop a new computational model for ca2+ and membrane potential-dependent ciliary regulation that explains how different ciliary beating regimes are regulated.
the model describes a novel mechanism of excitability based on the membrane potential-dependence of ca2+ currents  and simultaneous intraciliary ca2+-concentration mediated inhibition of ca2+ channels .
our analysis shows that motile cilia constitute an excitable system with a novel mechanism of excitability.
the ciliary system is able to generate a ca2+ spike in response to a wide range of transmembrane depolarisation .
the major difference in the ciliary excitation described here, with respect to classical excitation mechanisms, is that ciliary excitability is robust to a wide range of ionic variations in the environment.
the excitability mechanism of cells in evolutionary advanced organisms is based on a combination of the n-shaped dependence of the quick inward cationic current on the transmembrane potential and slow alterations of the k+ conductivity  <cit> .
the ciliary voltage-current characteristic  suggests several functional dynamic modes of operation: i) single impulse generation, ii) oscillator, iii) trigger , all initiated by membrane depolarisation.
at the same time, the hyperpolarisation-induced ca2+ currents switch the system into the mode of a monostable multivibrator, when cilia can generate a ca2+ spike in response to any alteration of membrane potential.
the dynamics of such a system depends on the transmembrane potential.
in other words, any alterations in the transmembrane potential  switch functional performance of the system or make it non-excitable.
it was originally believed that ca2+, camp and cgmp each represent an independent pathway of ciliary regulation, however, there is by now a significant amount of evidence that strongly suggests that all three pathways are intimately interconnected  <cit> .
it is well established that camp and cgmp are synthesized by ac isoforms and hydrolysed by pdes in a ca2+-cam-dependent manner.
in this work we describe the mechanism of the cross talk between the three circuits and explain how cbf can be modulated via extra- and intraciliary pathways .
CONCLUSIONS
therapeutic applications of systems model for intraciliary ca2+ regulation
our detailed analysis of the effects of several ca2+ and potassium currents and membrane potential on intraciliary ca2+ levels offers a new way of interpreting ciliary motility associated pathologies.
there are two main ca2+-mediated parameters that govern the motile function of cilia: the direction and the frequency of beat.
our model shows that there can be several dynamic regimes of intraciliary ca2+ alterations during which the intraciliary ca2+ concentration can be either at low or high levels, temporarily or for a significant period of time .
experimental evidence suggests that high ca2+ reverses the direction of cilia strike , and modifies the frequency in a highly nonlinear manner  via synthesis and hydrolysis of cyclic nucleotides .
in a previous work  <cit> , we showed that genetic mutations that alter the dynamic properties of a system in a permanent manner can lead to disease.
in this study, we propose a conceptually similar mechanism for the pathologies associated with ciliary motility that can take place in some human diseases  <cit> .
our study suggests that while cbf regulation in a multicellular organism can be modulated by a number of intracellular and extracellular factors , genetic mutations that directly or indirectly, affect the ca2+-mediated cbf dependence can dramatically impair the essential processes such as clearing function in airways, male fertility, or the determination of the left-right axis during development .
the treatment of the pathologies associated with this mechanism would rely on the restoration of the original ca2+-dependent cbf dependence.
future perspective
at present there is limited understanding of the underlying biological mechanisms that govern ciliary motility.
this study describes the modes of intraciliary ca2+ dynamics in a highly detailed fashion.
it shows the conditions that switch the system between the modes of ca2+ spike generation, oscillatory dynamics and a trigger.
the interdependent influences of ca2+ and k+ currents, transmembrane potential and cyclic nucleotides modulate the ciliary beat frequency and the direction of beat in a highly nonlinear manner.
the further development of mathematical models of this system is still required to represent ciliary movements as a function of ca2+ concentration and obtain the detailed understanding of ciliary motility which will be crucial for the development of new treatments for human diseases.
while the core protein regulatory machinery involved in ciliary motility is very likely to be conserved, some variations in response to increased ca2+ between single cell ciliates and mammalian cilia have been reported  <cit> .
we would argue that those differences are not due to the change in the mechanisms of ca2+-dependent regulation but are rather caused by variations in the parameters of the regulatory circuits.
the further investigation of single cell ciliates may allow a greater degree of characterisation of ciliary movement mechanisms, because in these systems alterations of ciliary motility translate into movement trajectories which can be easily observed.

 cilia are cellular protrusions which have been conserved in a wide range of organisms ranging from protozoa to the digestive, reproductive and respiratory systems of vertebrates. mobile or immotile cilia exist on every cell of the human body and the insufficiently recognised importance of the cilium compartment in human physiology has been recently highlighted. cilia are present on most eukaryotic cell surfaces with the exception of the cells of higher plants and fungi. ciliary motility is important for moving fluids and particles over epithelial surfaces, and for the cell motility of vertebrate sperm and unicellular organisms. the cilium contains a microtubule-based axoneme that extends from the cell surface into the extracellular space. the axoneme consists of nine peripheral microtubule doublets arranged around a central core that may or may not contain two central microtubules. cilia can be broadly classified as 9+ motile cilia or 9+ immotile sensory cilia, although there are examples of 9+ sensory cilia and 9+ motile cilia. in mammals, motile 9+ cilia normally concentrate in large numbers on the cell surface, beat in an orchestrated wavelike fashion, and are involved in fluid and cell movement. in contrast to motile cilia, primary cilia project as single immotile organelles from the cell surface. primary cilia are found on nearly all cell types in mammals and many are highly adapted to serve specialized sensory functions. the 9+ cilia usually have dynein arms that link the microtubule doublets and are motile, while most 9+ cilia lack dynein arms and are non-motile. in total, eight different types of cilia has been identified to date. in this study, we investigate the mechanism of movement regulation for the motile type of cilia. although each individual cilium represents a tiny hair-like protrusion of only μm in diameter and approximately 5- μm in length, cilia covering human airways can propel mucus with trapped particles of length up to mm at a speed of mm/second. such efficiency can be achieved due to the coordination between cilia and stimulus-dependent regulation of the rate of cilia beat. dysfunction of ciliary regulation gives rise to pathologic phenotypes that range from being organ specific to broadly pleiotropic. a link between ciliary function and human disease was discovered when individuals suffering from syndromes with symptoms including respiratory infections, anosmia, male infertility and situs inversus, were shown to have defects in ciliary structure and function. microscopic organisms that possess motile cilia which are used exclusively for either locomotion or to simply move liquid over their surface include paramecia, karyorelictea, tetrahymena, vorticella and others. the human mucociliary machinery operates in at least two different modes, corresponding to a low and high rate of beating. it has been shown that the high rate mode is mediated by second messengers, including purinergic, adrenergic and cholinergic receptors. this mode enables a rapid response, which can last a significant period of time, to various stimuli by drastically increasing the ciliary beat frequency. at the same time, several ciliary movement modes have been reported in a ciliate paramecium caudatum. the remarkable conservation of ciliary mechanisms creates grounds for the speculation that there can more than two ciliary beating modes in human tissues. it is, therefore, reasonable to suggest, that some human diseases, associated with aberrant ciliary motility, can arise due to modifications in the beating mode. clearly, the development of therapeutic strategies against ciliary-associated pathologies will require advanced understanding of ciliary beating regulation mechanisms. the periodic beating of cilia is governed by the internal apparatus of the organelle. its core part, the axoneme, contains nine microtubule pairs encircling the central pair. the transition at the junction of the cellular body and the ciliary axoneme is demarcated by y-shaped fibres, which extend from the microtubule outer doublets to the ciliary membrane. the transition area, in combination with the internal structure of the basal body, is thought to function as a filter for the cilium, regulating the molecules that can pass into or out of the cilium. ciliary motility is accomplished by dynein motor activity in a phosphorylation-dependent manner, which allows the microtubule doublets to slide relative to one another. the dynein phosphorylation that controls ciliary activity is regulated by the interplay of calcium and cyclic nucleotide pathways. the beating pattern of cilia consists of a fast effective stroke and a slower recovery stroke. during the effective stroke cilia are in an almost upright position, generating force for mucus movement. during the recovery stroke, the cilia are recovering from the power strike to the original position by moving in the vicinity of the cell surface. current theories which attempt to explain the workings of the ca2+-dependent cbf regulation mechanism are incomplete and highly controversial. elevation of intraciliary ca2+ is one of the major regulators of ciliary movement. calcium influx regulates ciliary activity by increasing intraciliary ca2+ only, while the cytosolic bulk remains at a low level. separate ciliary compartmentalisation for ca2+ allows prolonged activation of ciliary beating without damaging the cell through high ca2+ concentrations. it is well known that calcium fluxes via calcium channels lead to changes in organisms' swimming behaviour. in mucus-transporting cilia, ca2+ mediates cbf increase. it has also been shown that there are some differences in the ca2+-dependent cbf regulation in single cell organisms and in humans. sustained cbf increase requires prolonged elevation of ca2+ levels which can be lethal to the cell. it has been suggested that ca2+-dependent ciliary regulation takes place locally in the vicinity or within the ciliary compartment, almost independently from intracellular ca2+ concentration. given that the gradient of free ca2+ in the cytosol dissipates within 1- seconds, it appears more likely that cilia form their own compartment where ca2+ is regulated by active ca2+ transport in a similar fashion to the intracellular ca2+ regulatory system. this hypothesis resolves the problem of maintaining physiological levels of intracellular ca2+ concentration. a number of experimental studies have reported several controversial results relating to the ca2+-dependent mechanism of cilia regulation. for example, it has been reported that spontaneous cilia beat does not require alterations in ca2+, while nucleotide-dependent cbf increase requires ca2+. it has also been shown that uncoupling between ca2+ and cbf can be achieved by inhibition of ca2+-dependent protein calmodulin or the cyclic nucleotide pathway. these findings suggest that intraciliary ca2+ does not regulate cilia beat in isolation, but instead does so as part of more complex signalling network. although it was originally believed that ca2+, cyclic adenosine monophosphate and guanosine monophosphate regulate ciliary beat independently, numerous reports now strongly indicates that all three pathways are tightly interconnected. the camp-dependent protein kinase phosphorylates dynein in the bases of cilia and thereby increases the forward swimming speed in paramecium. similar effects have been reported for pka-dependent phosphorylation of axonemal targets in mammalian respiratory cilia. several lines of evidence indicate that pka and cgmp-dependent kinase both phosphorylate specific axonemal targets in a camp and cgmp-dependent manner. a schematic diagram for the underlying biochemical machinery for cilia movement regulation is shown in figure 1c. it is striking that, despite significant experimental characterisation of this system, there is still rather limited mechanistic understanding of how intraciliary ca2+ and nucleotide interplay relates to cbf. another major regulator of ciliary beating is the membrane potential. a number of studies have reported the voltage-dependent effects of ciliary beating. the ciliate didinium nasutum has been shown to respond both to hyper- and de-polarization of the membrane. the transmembrane potential alterations were shown to be mediated via the potential-dependent ca2+ channels. electrophysiological studies in paramecium caudatum have revealed complex relationships between ciliary ca2+ currents, intraciliary ca2+ concentration and transmembrane potential in the regulation of ciliary motility. a number of previous computational studies have analysed various aspects of cilia movement regulation. one earlier model assessed the degree of synchronization between small ciliary areas. the effects of viscosity have been investigated in mucus propelling cilia in. the authors found that increasing the viscosity not only decreases cbf, but also changes the degree of correlation and synchronization between cilia. the mechanical properties of cilia motion were studied in an attempt to understand the ciliary dynamics in. the authors concluded that bending and twisting properties of the cilium can determine self-organized beating patterns. while these reports offer valuable insights into the regulatory mechanisms of cilia, a number of essential questions remain unresolved. for example, there has not been a detailed analysis of how individual ca2+ currents influence intraciliary ca2+ levels. it also remains unclear how ca2+ modulates nucleotide levels and membrane potential, and how such regulation affects ciliary movement. none of these reports have elucidated the underlying mechanisms governing the interplay between intraciliary ca2+ and nucleotide alterations and cbf. in this study, we integrate the available experimental information on the molecular pathways that regulate intraciliary ca2+ concentration into a comprehensive mathematical model. by applying systems analysis, we elucidate the mechanisms of intraciliary ca2+ spike generation, analyse the properties of such spikes and demonstrate the conditions under which the ca2+ surges can become repetitive. we carry out detailed investigations of the individual current contributions to the regulation of the intraciliary ca2+ concentrations and elucidate both steady-state and dynamic responses of ca2+ currents and intraciliary ca2+ concentration dynamics in response to the altered transmembrane potential shift. the model allows detailed elucidation of transmembrane potential and intraciliary ca2+ coupling. we employ the proposed model in order to understand the underlying molecular mechanisms of the crosstalk between ca2+, membrane potential and nucleotide pathways that regulate ciliary movement. the systems model allows detailed analysis of the individual current contributions to the intraciliary homeostatic ca2+ levels. furthermore, we establish specific regulatory mechanisms for ca2+ and cyclic nucleotide-dependent cilia movement characteristics. crucially, our model predicts the possibility of several ciliary beating modes and describes specific conditions that initiate them. specifically, we describe intraciliary ca2+ dynamic modes that regulate healthy and pathologic cilia beating. we use these findings in order to propose experimentally testable hypotheses for possible therapeutic interventions in human diseases associated with pathologic cilia motility.  a new model for the interplay between ca2+ and k+ currents and transmembrane potential alterations a new model for the regulation of ciliary movement that combines multiple ca2+ and k+ currents and transmembrane potential has been developed. in this model, the intraciliary ca2+ levels are modulated by ca2+ currents through the channels of passive and active ca2+ transport, the current from the cilium into the cell body, the ca2+ leakage current, and depolarisation and hyperpolarisation-activated currents. variable extracellular conditions have continuous impact on the transmembrane potential which is intertwined with transmembrane ion currents and intraciliary ca2+ homeostasis. the overall network that regulates ciliary movement is divided into several functional modules. one module combines all ca2+ and k+ currents that define intraciliary ca2+ homeostasis and the transmembrane potential. one of the most essential intraciliary ca2+ binding proteins, cam, selectively regulates the activities of adenylate cyclase, guanylate cyclase and phosphodiesterases, and thereby modulates the intraciliary levels of adenosine monophosphate and guanosine monophosphate in a ca2+ dependent manner. the camp- and cgmp-dependent kinases phosphorylate dynein proteins in the bases of cilia and thereby induce the mechanical cilia movement. the complete set of equations making up the proposed model is presented in the methods section. below we provide a number of new insights into the mechanism of cilia regulation via a detailed investigation of the properties of this model. the mechanism of ca2+-dependent inhibition of ca2+ channels a subset of intraciliary ca2+ channels have been reported to operate in an intraciliary ca2+ dependent manner and have been proposed as major regulators of ciliary beat. it is established that ca2+ current is not inhibited by the double pulse application of depolarization impulses under voltage clamp conditions in those situations when the first transmembrane potential shift is equal to the equilibrium ca2+ potential. further experimental evidence reveals that ca2+ current inactivation kinetics are delayed when ca2+ ions are partially replaced by ba2+ ions. altogether these findings suggest that the channels are not inhibited directly by the depolarizing shift of transmembrane potential, but that instead their conductivity is dependent on the intraciliary ca2+ concentration. some decrease of the inward current amplitude upon transmembrane potential shift into the ca2+ equilibrium level can be explained by the fact that k+ currents can contribute to the overall current measurements. here we consider the intraciliary ca2+ concentration-dependent ca2+ channel inhibition and employ the developed model to analyse two potential scenarios for the ca2+ channel conductivity regulation. in one case, ca2+ ions bind to the ca2+ binding site on the channel and thereby inhibit the channel's conductivity by direct interaction. the other possibility is that the ca2+ binding protein interacts with the ca2+ ion first and then this complex binds to the channel and inhibits its conductivity. in both cases the conductivity dependence on transmembrane potential is assumed to be monotonic according to the experimental data. direct ca2+-dependent ca2+ channel conductivity inhibition we first investigate a potential intraciliary ca2+ regulatory mechanism via direct ca2+ ion binding-dependent ca2+ channel inhibition. the relationship between the ca2+ channel conductivity and the transmembrane potential has been experimentally characterized by an early study in paramecium species. the equation in the methods section approximates the experimentally established dependence. in the case of direct ca2+-dependent ca2+ channel inhibition, one can show that the nullclines for non-dimensional ca2+ concentration and for the number of open channels from the system of differential equations intersect at one stable point for all values in the physiological range of model parameters. the numerical solutions of the coupled differential equations allow us to obtain the solutions for how the steady-state ca2+ levels depend on the transmembrane potential. the model predictions for the steady-state ciliary ca2+ channel conductivity dependence on the transmembrane potential under the voltage clamp conditions are shown on figure 2a. the extracellular conditions are subject to constant change both in the case of ciliates as well as for multicellular organisms. the modifications in the external environment continuously shift the transmembrane potential. in order to estimate how the transmembrane potential alterations affect the inward ca2+ current we derived the dependence for the ca2+ ion flow in the methods section). the experimental data-based model for the inward ca2+ current dependence on membrane potential predicts a significant reduction of the inward ca2+ current amplitude as a function of membrane depolarization. we next investigated the dynamic alterations of the intraciliary ca2+ concentrations and the inward ca2+ current in response to the transmembrane potential shifts. equations and in the methods section were used for quantitative estimations of the ca2+ concentration and current responses to the normalised membrane potential shifts. the model predicts that the rapid depolarising alterations of the ciliary transmembrane potential leads to the generation of single ca2+ spikes. such responses can take place when ca2+ channels are inhibited by ca2+ ions and the channel's conductivity depends on the membrane potential in a monotonic manner. the amplitude of those impulses depends on the steepness of the ca2+ channels conductivity dependence on the membrane potential. the mechanism of the ca2+ spike generation is mainly due to the delay of the ca2+-induced inhibition with respect to the ca2+ conductivity alteration characteristic times. the described mechanism of the ca2+ spike generation will only work if the ca2+ currents significantly alter the ca2+ concentration in the intraciliary compartments. indirect ca2+ channel conductivity regulation in the previous section we considered ca2+-dependent ca2+ channel regulation under the assumptions that ca2+ channels have an intracellular ca2+ binding site and ca2+ ion binding closes the channels. however, several experimental studies have suggested that the conductivity of ca2+ channels in cilia can also be regulated indirectly, via a ca2+ binding protein. at present, there is no direct experimental evidence that explicitly favours either direct or indirect regulatory mechanism. we, therefore, investigated the second possibility for indirect ca2+-dependent ca2+ channels conductivity inhibition. the model for ca2+ ion interactions followed by the interactions with the ca2+ channels is developed in line with a previously suggested modelling methodology for ca2+-cam interactions. the model predictions for the number of open channels as a function of intraciliary ca2+ concentration are shown on figure as derived by equation in the methods section. given that the exact nature of the ca2+ binding protein acting as a mediator between ca2+ ions and ca2+ channels is not established, equation has been solved for different physiologically possible ratios of total number of channels to the dissociation constant for the ca2+- binding protein interactions with ca2+ channels. according to the derived models, the comparison of direct versus indirect ca2+ channels inhibition can be carried out by setting the non dimensional ca2+ concentration to zero ). in this case, the predictions of equation for indirect ca2+ channels conductivity inhibition almost coincide with the model for direct ca2+-dependent inhibition. in both models most of the ca2+ channels are open in the lower range of ca2+ concentrations. however, the model for indirect ca2+ channels inhibition predicts that the number of open ca2+ channels would equal 1cac0+ when intraciliary ca2+ reaches high concentrations. in other words, in the case of the indirect mechanism of inhibition, high ca2+ does not inhibit the channels completely. the number of remaining channels in the open state would depend on the concentration of the regulatory ca2+ binding protein. we next analysed the dynamics of the ciliary ca2+ channels inhibition in response to a change in intraciliary ca2+ concentration and in the transmembrane potential. as mentioned earlier, the ca2+ channel conductivity is not inhibited by the membrane potential, but rather has a monotonic dependence on the transmembrane potential difference as shown on figure 2b. we found that the characteristic time τca2+ of ca2+ channel alterations in response to step changes in ca2+ is inversely proportional to the total number of channels. the model predictions for intraciliary ca2+ concentration and inward ca2+ current in response to depolarising transmembrane potential changes from v to v are shown on figure according to the derived equation in the methods section, the inward ca2+ current changes with a characteristic time τv, which can only be estimated by considering the k+ current contribution. in the first instance we only consider active and passive ca2+ transport in methods). we neglected by the kinetics for the channels of active transport due to the assumption that the kinetics of alterations of active transport are much faster that the characteristic alteration times of passive transport for the ca2+ currents). the model predictions for the steady-state dependence of transmembrane potential on intraciliary ca2+ concentration and the dynamic ca2+ current amplitude dependence on the membrane potential are shown in figure 6a and 6b, respectively. the cilia's external environment is subject to constant change and can significantly affect the behavioural responses of ciliates and modulate ciliary beating in multicellular organisms. in order to account for the effects of extraciliary ca2+ variations we estimated the amplitudes of intraciliary ca2+ spike generation under different external ca2+ concentrations and variable transmembrane potentials. figure 7a shows that the increase of the ca2+ concentration in the external solution increases the amplitude of the generated intraciliary ca2+ spike. the amplitude of the spikes goes to zero when the membrane potential equals the equilibrium potential for ca2+ ions. the increase of membrane potential decreases the amplitude of the ca2+ current. this effect is due to the increase of the steady-state intracilia ca2+ level and ca2+-dependent inhibition of the ca2+ channels. we noted earlier that there is a ca2+ current in the cilia which transfers ions from the cilia into the cellular compartments. this current can be described by equation in methods. the contribution of cilium-to-cell body current to the intraciliary ca2+ concentration dynamics was evaluated experimentally in. it was shown that under depolarized membrane potential conditions the contribution of this current is very small and the intraciliary ca2+ is mainly pumped out of the cilia into the extracellular space by the active ca2+ transport. according to other observations, ca2+ current from cilia into the cellular compartment can be larger than the current generated by the active ca2+ transport. in order to investigate the role and contribution of the cilia-to-cell compartment current, we introduced its contribution to the intraciliary ca2+ concentration dynamics ). we performed qualitative analysis of the ca2+ concentration alterations in the cilia in the presence of the cilium-to-cell current and compared the ca2+ dynamics with the case when this current was not present. we found that although the cilium-to-cell body current influences the intraciliary ca2+ concentration levels, it does not change the dynamics qualitatively when the membrane potential is depolarized and fixed. our findings suggest that the cilium represents an excitable system with unique properties. the ca2+-dependent inhibition of ca2+ channels inhibition allows for the generation of single impulses of variable amplitude proportional to the degree of membrane depolarisation caused by variations in the external concentrations of ions. this system is able to generate a single spike despite unpredictable variations of ionic concentrations in the environment and is, therefore, very robust to alterations in the external conditions. another interesting aspect of the ciliary excitation is the ability of the system to generate regulatory intraciliary ca2+ impulses proportional to the degree of membrane depolarisation. this property can allow cells to sense and "automatically" respond to alterations in their environment. the contribution of k+ currents in the previous section, we analysed the dynamic properties of the intraciliary ca2+ system under voltage clamp conditions. several lines of evidence suggest that k+ currents contribute to the currents registered in cilia under voltage clamped conditions. the existence of k+ currents in cilia is supported by a number of experimental studies. the experimental data shows that the measured current is not equal to zero when the membrane potential equals the equilibrium membrane potential for ca2+ ions. instead, the current equals zero when membrane potential is about mv while the equilibrium potential for ca2+ ions equals mv. this observation suggests that both ca2+ and k+ currents contribute to the overall current measured at early stages of current registration under voltage clamp, and therefore both currents need to be taken into the consideration in order to advance understanding of the mechanisms involved in ciliary regulation. at the same time, it has so far been impossible to register ca2+ currents by inhibiting the k+ contribution. various compounds can only partially block the k+ current when applied from inside of the membrane. ciliary k+ currents have also been measured separately from ca2+ currents. in order to account for the contribution of k+ currents to the regulation of intraciliary ca2+ concentration, we developed a model for the regulation of k+ currents by membrane potential in methods). the dependence of k+ conductivity on the membrane potential is described by equation. figure shows the measured experimental values and the approximating curve calculated according to equation. the current on figure is normalized to when membrane potential equals in order to approximate the current, we only used the experimental values obtained under the membrane depolarized conditions, and this experimental data is approximated by equation. the predictions for intraciliary ca2+ dynamics and ca2+ impulse amplitude in response to the shift in membrane potential are shown in figure 9a and 9b, respectively. a comparison of the membrane potential shift-induced intraciliary ca2+ spike generation in the presence of the k+ currents with the model when k+ channels are not incorporated, suggests that while the k+ currents change the quantitative values of the ca2+ concentration dynamics, the general shape of the ca2+ response remains the same. the transmembrane potential dynamics in the absence of voltage clamp in the previous sections we investigated the mechanisms of the transmembrane potential shift-dependent ca2+ spike generation under voltage clamp conditions. however, ca2+ currents themselves can alter the membrane potential. here we incorporate the membrane potential dependence on ca2+ currents and investigate the membrane potential dynamics in the absence of voltage clamp and ). the non dimensional ca2+ concentration and membrane potential are described by equation. the predictions for the individual ca2+ and k+ current responses to various membrane potential shifts are shown in figure one can note that the dynamics of the responses significantly differs between the ones obtained under voltage clamp conditions and the situation when the membrane potential is not fixed, but dependent on the currents. voltage current characteristics for ca2+, k+ and full currents calculated using the current amplitudes from the currents dynamics are shown on figure 10h. one can note that the current is an almost exponentially growing function of membrane potential. the monotonic dependence of ca2+ current on transmembrane potential, and simultaneous ca2+-dependent inhibition of ca2+ channels, represents a classical problem of two interconnected variables: intraciliary ca2+ and membrane potential. in this system, increasing ca2+ current with transmembrane potential depolarisation represents a positive feedback loop mechanism, whereas the intraciliary ca2+ concentration-dependent ca2+ channels inhibition represents a negative feedback loop. we, therefore, sought to investigate the range of potential dynamical properties of the ciliary system emerging from the coupling of ca2+ current and membrane potential described by equations. one can clearly see that there is significantly different response for different values of the inward current. when the influx of the ions is relatively small, the dca2+dt= null cline intersects the dvdt = null cline in the left descending area; such a null cline crossing results in a stable solution. in this case the system responds by the generation of a single impulse of both intraciliary ca2+ concentration and the membrane potential followed by a return to homeostatic levels. further increasing the current causes the null cline dca2+dt= to intersect with the null cline dvdt= in the middle region of the ascending area, leading to an unstable solution with a limit cycle formed around the area that represents the oscillations.. however, further increase of the current causes the dca2+dt= null cline to intersect with the null cline dvdt= in the right descending area, resulting in a stable solution with a slight increase of the homeostatic ca2+ and membrane potential levels. the key conclusion from this analysis is that the external ionic conditions can initiate essentially different dynamic properties of the system regulating ciliary movement. one of the key factors that affect the ciliary beat cycle is the level of intraciliary ca2+. our findings suggest that in response to the external conditions, there are several possibilities for intraciliary ca2+ upregulation. the system can generate a single spike of variable amplitude, permanently increase ca2+ in a dynamic fashion and maintain the high intraciliary levels, or operate in a monostable multivibrator mode. these three possibilities can be associated with the different modes of ciliary beat observed in human cilia as well as in various ciliates. the dynamic properties of excitable systems with two interdependent variables are reasonably well understood at a theoretical level. in the present case, ca2+ and membrane potential represent the slow and fast variables, respectively. this study, therefore, establishes that the dynamic properties of ciliary systems, where the ca2+ and k+ channel conductivities represent monotonic function of membrane potential and the ca2+ channels conductivity inversely depends on intraciliary ca2+ concentration, are comparable with the properties of excitable systems based on the "n-shape" dependence of the na2+ channel conductivity on membrane potential. at the same time, it is essential to note that the mechanism of excitation described in motile cilia is different from the "classical" one described in most excitable cells and systems that involve ip ca2+ channels. the membrane hyperpolarisation-dependent currents modulate the excitatory properties of the ciliary system the ciliary transmembrane potential can shift in two directions. in the previous section we investigated the intraciliary ca2+ responses caused by membrane depolarisation. here we assess the implications of the membrane hyperpolarisation which has been shown to activate the current from cilia into the cell body. we introduced the corresponding term into our model for the ca2+ ions movement via the membrane as a function of the corresponding membrane potential shift. by assuming the potential independent mechanism for ca2+ and k+ ion expulsion, the system of intraciliary ca2+ and membrane potential is derived as shown in equation in the methods section. the model predictions for intraciliary ca2+ and membrane potential dynamics in the absence of the hyperpolarisation-induced current are shown in figure 11d, e and 11f. in this case the system remains in the mode of steady oscillations. we found that such oscillatory mode can be significantly modulated by the hyperpolarisation-induced and the external conditions-dependent current. figure 12a shows the phase diagram in the presence of the hyperpolarising current. in contrast to the situation described on figure when the hyperpolarising current was absent, the ciliary system generates a single spike in response to alteration of external ionic concentrations over the whole physiological range of the inward current. under certain combinations of the hyperpolarisation-induced currents and the ca2+-dependent k+ currents the phase diagram modifies in a manner so that the system responds by generating a single spike in response to alteration of the inward current of any magnitude. this indicates that the ciliary system under membrane hyperpolarising conditions can become a monostable multivibrator. the role of cilia-to body ca2+ current under membrane hyperpolarisation intraciliary ca2+ concentration has been experimentally estimated to be approximately one order of magnitude higher in comparison with the intracellular levels. the ca2+ current generated by the ion flow from the ciliary compartment into the cell has been reported by a number of groups, but the role this current plays in the regulation of the ciliary beat remains unclear. in order to address this question, we introduced the term for this current into the equation that describes the intraciliary ca2+ concentration in the methods section). in the absence of direct measurements of the dependence of conductivity on membrane potential we set the conductivity to increase in response to hyperpolarisation of membrane potential. a summary of the model responses for all channel conductivities as a function of membrane potential is shown in figure by numerically solving the coupled equations for intraciliary ca2+ and membrane potential with the cilia to the cell body contribution, we found that this current does not qualitatively change the general dynamic properties of the system. despite the lack of a noticeable contribution to the ciliary dynamic properties, this current requires a special consideration. experimental studies have clearly demonstrated that intraciliary ca2+ is significantly higher than intracellular ca2+ concentration. at the same time, if the conductivity of protein structures governing the ca2+ ions movement from cilia to the body is high, most of the intraciliary ions would move from cilia into the cell body in a very short time. a simple calculation suggests that if ca2+ could freely flow from cilia into the body, the intraciliary concentration would become equal to the intracellular ca2+ concentration in less than μs due to the difference in the volumes of the cell body and intraciliary compartments. experimental measurements in ciliates show that the hyperpolarisation-induced backwards movements can last longer than μseconds. it is also known that the avoidance reaction that requires long term elevation of intraciliary ca2+ concentration can be observed in hyperpolarizing solutions. during all this time the intraciliary ca2+ concentration can be several orders of magnitude higher than the intraciliary concentration. in this study, we have demonstrated that the steady-state ca2+ current under the depolarized membrane potential conditions can only be reduced by the ca2+-dependent inhibition of ca2+ channels. all these observations suggest that the ca2+ removal from cilia to the cell body occurs in a membrane potential dependent manner. the mechanism of ca2+ and cyclic nucleotide-dependent cbf regulation in addition to intraciliary ca2+ and k+ potassium levels being coupled with the membrane potential modulation, cyclic nucleotides contribute to the regulation of one of the major ciliary beat parameters, frequency. intraciliary ca2+ levels activate a variety of adenylate cyclases and phosphodiesterases that produce and hydrolyse cyclic nucleotides, respectively, and thereby modulate the intraciliary camp and cgmp levels. at the same time, camp and cgmp-dependent kinases phosphorylate dynein arms in the bases of cilia and thereby induce the ciliary movement. in a previous work we showed that the ciliary beat frequency can have a "double" bell shape dependence on ca2+ concentration due to the differential regulation of adenylate and guanylate cyclase isoforms in a ca2+-calmodulin dependent manner. one bell-shape was due to the camp production by a combination of ac and pde, whereas another one was mediated by cgmp production and degradation. our results proposed an explanation for seemingly conflicting experimental evidence suggesting that cbf can both decrease and increase with increasing ca2+ concentration. here we extend our previous analysis and describe the conditions when one or the other peak can be significantly reduced or even disappear. figure 15a shows the model predictions for the ciliary beat frequency in comparison with the experimental data. recent studies demonstrated that hormones and pharmacological agents can regulate both function and structure of cilia by interfering with the cyclic nucleotide signalling pathways. these findings support the possibility for the development of novel therapeutic strategies for ciliary pathologies by modulating the dynamic mode of beating via ciliary membrane receptors. discussion we develop a new computational model for ca2+ and membrane potential-dependent ciliary regulation that explains how different ciliary beating regimes are regulated. the model describes a novel mechanism of excitability based on the membrane potential-dependence of ca2+ currents and simultaneous intraciliary ca2+-concentration mediated inhibition of ca2+ channels. our analysis shows that motile cilia constitute an excitable system with a novel mechanism of excitability. the ciliary system is able to generate a ca2+ spike in response to a wide range of transmembrane depolarisation. the major difference in the ciliary excitation described here, with respect to classical excitation mechanisms, is that ciliary excitability is robust to a wide range of ionic variations in the environment. the excitability mechanism of cells in evolutionary advanced organisms is based on a combination of the n-shaped dependence of the quick inward cationic current on the transmembrane potential and slow alterations of the k+ conductivity. the ciliary voltage-current characteristic suggests several functional dynamic modes of operation: i) single impulse generation, ii) oscillator, iii) trigger, all initiated by membrane depolarisation. at the same time, the hyperpolarisation-induced ca2+ currents switch the system into the mode of a monostable multivibrator, when cilia can generate a ca2+ spike in response to any alteration of membrane potential. the dynamics of such a system depends on the transmembrane potential. in other words, any alterations in the transmembrane potential switch functional performance of the system or make it non-excitable. it was originally believed that ca2+, camp and cgmp each represent an independent pathway of ciliary regulation, however, there is by now a significant amount of evidence that strongly suggests that all three pathways are intimately interconnected. it is well established that camp and cgmp are synthesized by ac isoforms and hydrolysed by pdes in a ca2+-cam-dependent manner. in this work we describe the mechanism of the cross talk between the three circuits and explain how cbf can be modulated via extra- and intraciliary pathways.  therapeutic applications of systems model for intraciliary ca2+ regulation our detailed analysis of the effects of several ca2+ and potassium currents and membrane potential on intraciliary ca2+ levels offers a new way of interpreting ciliary motility associated pathologies. there are two main ca2+-mediated parameters that govern the motile function of cilia: the direction and the frequency of beat. our model shows that there can be several dynamic regimes of intraciliary ca2+ alterations during which the intraciliary ca2+ concentration can be either at low or high levels, temporarily or for a significant period of time. experimental evidence suggests that high ca2+ reverses the direction of cilia strike, and modifies the frequency in a highly nonlinear manner via synthesis and hydrolysis of cyclic nucleotides. in a previous work, we showed that genetic mutations that alter the dynamic properties of a system in a permanent manner can lead to disease. in this study, we propose a conceptually similar mechanism for the pathologies associated with ciliary motility that can take place in some human diseases. our study suggests that while cbf regulation in a multicellular organism can be modulated by a number of intracellular and extracellular factors, genetic mutations that directly or indirectly, affect the ca2+-mediated cbf dependence can dramatically impair the essential processes such as clearing function in airways, male fertility, or the determination of the left-right axis during development. the treatment of the pathologies associated with this mechanism would rely on the restoration of the original ca2+-dependent cbf dependence. future perspective at present there is limited understanding of the underlying biological mechanisms that govern ciliary motility. this study describes the modes of intraciliary ca2+ dynamics in a highly detailed fashion. it shows the conditions that switch the system between the modes of ca2+ spike generation, oscillatory dynamics and a trigger. the interdependent influences of ca2+ and k+ currents, transmembrane potential and cyclic nucleotides modulate the ciliary beat frequency and the direction of beat in a highly nonlinear manner. the further development of mathematical models of this system is still required to represent ciliary movements as a function of ca2+ concentration and obtain the detailed understanding of ciliary motility which will be crucial for the development of new treatments for human diseases. while the core protein regulatory machinery involved in ciliary motility is very likely to be conserved, some variations in response to increased ca2+ between single cell ciliates and mammalian cilia have been reported. we would argue that those differences are not due to the change in the mechanisms of ca2+-dependent regulation but are rather caused by variations in the parameters of the regulatory circuits. the further investigation of single cell ciliates may allow a greater degree of characterisation of ciliary movement mechanisms, because in these systems alterations of ciliary motility translate into movement trajectories which can be easily observed. 
the model describes the intimate relationship between calcium and potassium ionic concentrations inside and outside of cilia with membrane voltage and, for the first time, describes a novel type of ciliary excitability which plays the major role in ciliary movement regulation. our model describes a mechanism that allows ciliary excitation to be robust over a wide physiological range of extracellular ionic concentrations. the model predicts the existence of several dynamic modes of ciliary regulation, such as the generation of intraciliary ca2+ spike with amplitude proportional to the degree of membrane depolarization, the ability to maintain stable oscillations, monostable multivibrator regimes, all of which are initiated by variability in ionic concentrations that translate into altered membrane voltage. computational investigation of the model offers several new insights into the underlying molecular mechanisms of ciliary pathologies. we develop a detailed systems model for calcium, membrane potential and cyclic nucleotide-dependent ciliary motility regulation. however, modification of the dynamic modes, as a result of genetic mutations or environmental conditions, can cause a life threatening pathology.
 ciliary dysfunction leads to a number of human pathologies, including primary ciliary dyskinesia, nephronophthisis, situs inversus pathology or infertility. the mechanism of cilia beating regulation is complex and despite extensive experimental characterization remains poorly understood. we develop a detailed systems model for calcium, membrane potential and cyclic nucleotide-dependent ciliary motility regulation.  the model describes the intimate relationship between calcium and potassium ionic concentrations inside and outside of cilia with membrane voltage and, for the first time, describes a novel type of ciliary excitability which plays the major role in ciliary movement regulation. our model describes a mechanism that allows ciliary excitation to be robust over a wide physiological range of extracellular ionic concentrations. the model predicts the existence of several dynamic modes of ciliary regulation, such as the generation of intraciliary ca2+ spike with amplitude proportional to the degree of membrane depolarization, the ability to maintain stable oscillations, monostable multivibrator regimes, all of which are initiated by variability in ionic concentrations that translate into altered membrane voltage.  computational investigation of the model offers several new insights into the underlying molecular mechanisms of ciliary pathologies. according to our analysis, the reported dynamic regulatory modes can be a physiological reaction to alterations in the extracellular environment. however, modification of the dynamic modes, as a result of genetic mutations or environmental conditions, can cause a life threatening pathology. 