Computational modelling elucidates the mechanism of ciliary regulation in health and disease
© Kotov et al; licensee BioMed Central Ltd. 2011
Received: 13 June 2011
Accepted: 15 September 2011
Published: 15 September 2011
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.
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 [1, 3]. 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 (9+2 or 9+0 axoneme, respectively). Cilia can be broadly classified as 9+2 motile cilia or 9+0 immotile sensory cilia, although there are examples of 9+2 sensory cilia and 9+0 motile cilia. In mammals, motile 9+2 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+2 cilia usually have dynein arms that link the microtubule doublets and are motile, while most 9+0 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 0.25 μm in diameter and approximately 5-7 μm in length, cilia covering human airways can propel mucus with trapped particles of length up to 1 mm at a speed of 0.5 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 [9–19]. This mode enables a rapid response, which can last a significant period of time, to various stimuli by drastically increasing the ciliary beat frequency (CBF). At the same time, several ciliary movement modes have been reported in a ciliate Paramecium caudatum. The remarkable conservation of ciliary mechanisms [21–25] 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 (Ca2+) 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 [27–29]. In mucus-transporting cilia, Ca2+ mediates CBF increase [19, 30–32]. 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 [33, 34]. 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 1-2 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+[31, 35], 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 (CaM) or the cyclic nucleotide pathway [19, 32, 37].
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 [49–55].
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 [59–62] 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 (Figure 1C). 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 [63, 64], selectively regulates the activities of adenylate cyclase (AC), guanylate cyclase (GC) and phosphodiesterases (PDE), and thereby modulates the intraciliary levels of adenosine monophosphate (cAMP) and guanosine monophosphate (cGMP) 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 [49–51]. 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 (+120 mV) . Further experimental evidence reveals that Ca2+ current inactivation kinetics are delayed when Ca2+ ions are partially replaced by Ba2+ ions [67–71]. 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 (by approximately 25%) 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
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.
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 (11) in Methods. The contribution of cilium-to-cell body current to the intraciliary Ca2+ concentration dynamics was evaluated experimentally in [72, 73]. 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 (equation (34)). 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 (Figures 3 and 5). 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 10 mV while the equilibrium potential for Ca2+ ions equals 120 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.
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 (equations (40) and (41)). The non dimensional Ca2+ concentration and membrane potential are described by equation (42).
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 (42).
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 null cline intersects the = 0 null cline in the left descending area (Figure 11A); 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 (Figure 11A, B and 11C). Further increasing the current causes the null cline to intersect with the null cline 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. (Figure 11D, E and 11F). However, further increase of the current causes the null cline to intersect with the null cline in the right descending area, resulting in a stable solution with a slight increase of the homeostatic Ca2+ and membrane potential levels (Figure 11G, H and 11I). 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 (Figure 11B) of variable amplitude (data not shown), permanently increase Ca2+ in a dynamic fashion and maintain the high intraciliary levels (Figure 11E), or operate in a monostable multivibrator mode (cilia can generate a Ca2+ spike in response to any alteration of membrane potential) (Figure 11H). 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 IP3 Ca2+ channels [75, 76].
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 [77, 78]. 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 (equation 43). 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 (46) in the Methods section.
The role of cilia-to body Ca2+ current under membrane hyperpolarisation
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 100 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 100 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 (AC) and phosphodiesterases (PDE) 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 .
Figure 15B and 15C demonstrate that the "amplitude" of each peak can be significantly diminished if the activity of the AC or GC, respectively is modulated by a temporary or permanent, internal or external signal. Under such a scenario, CBF can only increase or decrease if it happens to be on one slope of the bell-shaped dependence. Therefore, according to our analysis, different organisms with the same underlying ciliary regulatory system can achieve all possible CBF regulatory modes as a function of Ca2+ concentration: the reverse bell-shaped dependence, if the "peak" values shown on Figure 15A occur at the lower and higher limits of the physiological range for Ca2+ concentration, the bell shape dependence that can be either cAMP and cGMP dependent, and either monotonic increase or decrease if the physiological range of Ca2+ concentrations occur at one of the slopes. Our model, therefore, describes the core Ca2+-dependent regulatory mechanisms of cilia beat, but also provides an explanation for the differences observed between cilia in different single cell organisms as well as tissue specific differences. It also unravels the mechanism for how various stimuli modulate the rate of CBF by signalling via Ca2+- and G-protein mediated pathways.
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 (Figure 2) and simultaneous intraciliary Ca2+-concentration mediated inhibition of Ca2+ channels (Figure 4). 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 (Figure 3, 5 and 9). 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 [84–87]. The ciliary voltage-current characteristic (Figure 10H) suggests several functional dynamic modes of operation: i) single impulse generation, ii) oscillator, iii) trigger (Figure 11), 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 (for example, initiated by variations of the external ion concentrations) 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 (Figure 15).
Therapeutic applications of systems model for intraciliary Ca2+ regulation
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.
Figure 1 provides a schematic outline of the network regulating intraciliary Ca2+ concentration that is considered in our model. Intraciliary Ca2+ concentration is regulated by the currents of passive and active Ca2+ transport, as well as by Ca2+ leak into the extracellular space and into the cell body.
A basic mathematical model for intraciliary Ca2+ concentration and its relationship to transmembrane potential was proposed for the first time in . A large number of recent experimental findings now allow the formulation of a more advanced model that includes the crucial aspects of the molecular mechanisms governing cilia movement. Below we describe the complete model for intraciliary Ca2+ regulation developed in this study.
where V R - is the cilium volume, SR - is the cilium surface area, and and -are the Ca2+ currents through the channels of passive and active Ca2+ transport, respectively. is the current from the cilium into the cell body. -is the Ca2+ leakage current. is the function that encounters Ca2+ binding to and release from CaM, the main Ca2+ binding protein in cilia, z = 2 is the Ca2+ ions charge, and F is the Faraday constant.
where is the conductivity of a single channel in the state i (in the most general state Ca2+ channels can have a number of states with different degrees of conductivity), is the Ca2+ potential in the equilibrium, V m is the transmembrane potential of the cilia membrane.
In the following sections we derive the models and analyse the individual contributions of the different types of Ca2+ currents to the intraciliary Ca2+ homeostasis.
Model for intraciliary Ca2+-dependent Ca2+ channel conductivity inhibition
where τ p is the characteristic time of the transmembrane potential alteration from V0 to V1.
where , v(ψ, t) is the Ca2+ channel conductivity dependence on the transmembrane potential and on time.
where α, d and λ are the parameter values that allow the best representation of the available experimental data. In this model, the steepness of the dependence of the conductivity on membrane potential is represented by the parameter α.
Indirect Ca2+ channel conductivity regulation
where [CaC0] is the total concentration of the Ca2+ binding protein and is the equilibrium dissociation constant.
For cases when cac0 > > 1 and u > > 1, the characteristic time approximately equals
In this equation, we include the kinetics for the active Ca2+ channels due to the assumption that the dynamics of currents via the active Ca2+ channels is much faster than the dynamics of currents through the passive Ca2+ transport.
where is the number of open K+ channels, is the maximal conductivity, and is the equilibrium K+ potential.
The transmembrane potential dynamics
where , and are the steady-state Ca2+ and K+ channel conductivities, respectively, ,.
where I0 is the non dimensional inward current.
Currents activated by the membrane hyperpolarisation
where v h (ψ) is the Ca2+ current contribution, activated by membrane depolarization, and is the Ca2+-dependent K+ current contribution.
Cilia-to body Ca2+ current
Parameter values employed in the systems model for the ciliary excitation
Value (dimensionless unless
2B, 3, 5, 6, 7, 9, 10
20, 21, 34, 39
2B, 3, 5, 6, 7, 11
20, 34, 42
2B, 3, 5, 6, 7
20, 21, 34
2B, 3, 5, 6, 7, 8
20, 21, 34, 39
2B, 3, 5, 6, 7, 8
20, 21, 34, 39
-1, -0.8, -0.5, -0.2, 0, 0.2, 0.5
2B, 3, 5, 6, 7, 8
20, 21, 34, 39
25, 20, 12.5, 5, 0, -5, -12.5 mV
2B, 3, 5, 6, 7, 8
20, 21, 34, 39
2B, 3, 5, 6, 7
2B, 3, 5, 6, 7, 9, 10
20, 34, 39
2B, 3, 5, 6, 7
4, 5, 6, 7
5, 10, 50, 100
4, 5, 6, 7
11, 12, 13, 14
11, 12, 13, 14
11, 12, 13, 14
11, 12, 13, 14
11, 12, 13, 14
11, 12, 13, 14
12, 13, 14
12, 13, 14
12, 13, 14
12, 13, 14
The relationship between dimensional and non-dimensional quantities for Ca2+ concentration and membrane potential
K CaM = 4 μM
This work was supported by the Strategic Research Development Award from Faculty of Sciences, University of Kent (NVV) and the Russian Fund for Basic Research (NVK).
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