Computational modelling elucidates the mechanism of ciliary regulation in health and disease
- Nikolay V Kotov^{1, 2},
- Declan G Bates^{3},
- Antonina N Gizatullina^{2},
- Bulat Gilaziev^{2},
- Rustem N Khairullin^{4},
- Michael ZQ Chen^{5, 6},
- Ignat Drozdov^{7},
- Yoshinori Umezawa^{8},
- Christian Hundhausen^{8},
- Alexey Aleksandrov^{9},
- Xing-gang Yan^{10},
- Sarah K Spurgeon^{10},
- C Mark Smales^{1} and
- Najl V Valeyev^{1}Email author
DOI: 10.1186/1752-0509-5-143
© Kotov et al; licensee BioMed Central Ltd. 2011
Received: 13 June 2011
Accepted: 15 September 2011
Published: 15 September 2011
Abstract
Background
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.
Results
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 Ca^{2+} 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.
Conclusions
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.
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 [1]. Mobile or immotile cilia exist on every cell of the human body [2] 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 [4]. 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 [5] 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 [6]. 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 [7]. 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 [3]. 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 [6].
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 [8], 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[20]. 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 [26]. 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 [12]. The dynein phosphorylation that controls ciliary activity is regulated by the interplay of calcium (Ca^{2+}) 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 Ca^{2+}-dependent CBF regulation mechanism are incomplete and highly controversial. Elevation of intraciliary Ca^{2+} is one of the major regulators of ciliary movement. Calcium influx regulates ciliary activity by increasing intraciliary Ca^{2+} only, while the cytosolic bulk remains at a low level. Separate ciliary compartmentalisation for Ca^{2+} allows prolonged activation of ciliary beating without damaging the cell through high Ca^{2+} concentrations. It is well known that calcium fluxes via calcium channels lead to changes in organisms' swimming behaviour [27–29]. In mucus-transporting cilia, Ca^{2+} mediates CBF increase [19, 30–32]. It has also been shown that there are some differences in the Ca^{2+}-dependent CBF regulation in single cell organisms and in humans [12]. Sustained CBF increase requires prolonged elevation of Ca^{2+} levels which can be lethal to the cell [33, 34]. It has been suggested that Ca^{2+}-dependent ciliary regulation takes place locally in the vicinity or within the ciliary compartment, almost independently from intracellular Ca^{2+} concentration [35]. Given that the gradient of free Ca^{2+} in the cytosol dissipates within 1-2 seconds [36], it appears more likely that cilia form their own compartment where Ca^{2+} is regulated by active Ca^{2+} transport in a similar fashion to the intracellular Ca^{2+} regulatory system. This hypothesis resolves the problem of maintaining physiological levels of intracellular Ca^{2+} concentration. A number of experimental studies have reported several controversial results relating to the Ca^{2+}-dependent mechanism of cilia regulation. For example, it has been reported that spontaneous cilia beat does not require alterations in Ca^{2+}[31, 35], while nucleotide-dependent CBF increase requires Ca^{2+}[8]. It has also been shown that uncoupling between Ca^{2+} and CBF can be achieved by inhibition of Ca^{2+}-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 [47]. The transmembrane potential alterations were shown to be mediated via the potential-dependent Ca^{2+} channels [48]. Electrophysiological studies in Paramecium caudatum have revealed complex relationships between ciliary Ca^{2+} currents, intraciliary Ca^{2+} 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 [56]. The effects of viscosity have been investigated in mucus propelling cilia in [57]. 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 [58]. 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 Ca^{2+} currents influence intraciliary Ca^{2+} levels. It also remains unclear how Ca^{2+} 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 Ca^{2+} and nucleotide alterations and CBF.
In this study, we integrate the available experimental information on the molecular pathways that regulate intraciliary Ca^{2+} concentration into a comprehensive mathematical model. By applying systems analysis, we elucidate the mechanisms of intraciliary Ca^{2+} spike generation, analyse the properties of such spikes and demonstrate the conditions under which the Ca^{2+} surges can become repetitive. We carry out detailed investigations of the individual current contributions to the regulation of the intraciliary Ca^{2+} concentrations and elucidate both steady-state and dynamic responses of Ca^{2+} currents and intraciliary Ca^{2+} concentration dynamics in response to the altered transmembrane potential shift. The model allows detailed elucidation of transmembrane potential and intraciliary Ca^{2+} coupling.
We employ the proposed model in order to understand the underlying molecular mechanisms of the crosstalk between Ca^{2+}, membrane potential and nucleotide pathways that regulate ciliary movement. The systems model allows detailed analysis of the individual current contributions to the intraciliary homeostatic Ca^{2+} levels. Furthermore, we establish specific regulatory mechanisms for Ca^{2+} 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 Ca^{2+} 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 Ca^{2+} and K^{+} currents and transmembrane potential alterations
A new model for the regulation of ciliary movement that combines multiple Ca^{2+} and K^{+} currents [59–62] and transmembrane potential has been developed. In this model, the intraciliary Ca^{2+} levels are modulated by Ca^{2+} currents through the channels of passive and active Ca^{2+} transport, the current from the cilium into the cell body, the Ca^{2+} 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 Ca^{2+} homeostasis.
The overall network that regulates ciliary movement is divided into several functional modules (Figure 1C). One module combines all Ca^{2+} and K^{+} currents that define intraciliary Ca^{2+} homeostasis and the transmembrane potential. One of the most essential intraciliary Ca^{2+} 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 Ca^{2+} dependent manner [65]. 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 Ca^{2+}-dependent inhibition of Ca^{2+} channels
A subset of intraciliary Ca^{2+} channels have been reported to operate in an intraciliary Ca^{2+} dependent manner and have been proposed as major regulators of ciliary beat [49–51]. It is established that Ca^{2+} 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 Ca^{2+} potential (+120 mV) [66]. Further experimental evidence reveals that Ca^{2+} current inactivation kinetics are delayed when Ca^{2+} ions are partially replaced by Ba^{2+} 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 Ca^{2+} concentration. Some decrease of the inward current amplitude (by approximately 25%) upon transmembrane potential shift into the Ca^{2+} equilibrium level can be explained by the fact that K^{+} currents can contribute to the overall current measurements. Here we consider the intraciliary Ca^{2+} concentration-dependent Ca^{2+} channel inhibition and employ the developed model to analyse two potential scenarios for the Ca^{2+} channel conductivity regulation. In one case, Ca^{2+} ions bind to the Ca^{2+} binding site on the channel and thereby inhibit the channel's conductivity by direct interaction. The other possibility is that the Ca^{2+} binding protein interacts with the Ca^{2+} 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 [66].
Direct Ca^{2+}-dependent Ca^{2+} channel conductivity inhibition
Indirect Ca^{2+} channel conductivity regulation
In the previous section we considered Ca^{2+}-dependent Ca^{2+} channel regulation under the assumptions that Ca^{2+} channels have an intracellular Ca^{2+} binding site and Ca^{2+} ion binding closes the channels. However, several experimental studies have suggested that the conductivity of Ca^{2+} channels in cilia can also be regulated indirectly, via a Ca^{2+} 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 Ca^{2+}-dependent Ca^{2+} channels conductivity inhibition.
We noted earlier that there is a Ca^{2+} 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 Ca^{2+} 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 Ca^{2+} is mainly pumped out of the cilia into the extracellular space by the active Ca^{2+} transport. According to other observations, Ca^{2+} current from cilia into the cellular compartment can be larger than the current generated by the active Ca^{2+} transport. In order to investigate the role and contribution of the cilia-to-cell compartment current, we introduced its contribution to the intraciliary Ca^{2+} concentration dynamics (equation (34)). We performed qualitative analysis of the Ca^{2+} concentration alterations in the cilia in the presence of the cilium-to-cell current and compared the Ca^{2+} dynamics with the case when this current was not present. We found that although the cilium-to-cell body current influences the intraciliary Ca^{2+} 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 Ca^{2+}-dependent inhibition of Ca^{2+} 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 Ca^{2+} 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 Ca^{2+} 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 Ca^{2+} ions. Instead, the current equals zero when membrane potential is about 10 mV while the equilibrium potential for Ca^{2+} ions equals 120 mV [66]. This observation suggests that both Ca^{2+} 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 Ca^{2+} 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 Ca^{2+} 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 Ca^{2+} spike generation under voltage clamp conditions. However, Ca^{2+} currents themselves can alter the membrane potential. Here we incorporate the membrane potential dependence on Ca^{2+} currents and investigate the membrane potential dynamics in the absence of voltage clamp (equations (40) and (41)). The non dimensional Ca^{2+} concentration and membrane potential are described by equation (42).
The monotonic dependence of Ca^{2+} current on transmembrane potential, and simultaneous Ca^{2+}-dependent inhibition of Ca^{2+} channels, represents a classical problem of two interconnected variables: intraciliary Ca^{2+} and membrane potential. In this system, increasing Ca^{2+} current with transmembrane potential depolarisation represents a positive feedback loop mechanism, whereas the intraciliary Ca^{2+} concentration-dependent Ca^{2+} 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 Ca^{2+} 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 $\frac{dC{a}^{2+}}{dt}=0$ null cline intersects the $\frac{dV}{dt}$ = 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 Ca^{2+} 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 $\frac{dC{a}^{2+}}{dt}=0$ to intersect with the null cline $\frac{dV}{dt}=0$ 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 $\frac{dC{a}^{2+}}{dt}=0$ null cline to intersect with the null cline $\frac{dV}{dt}=0$ in the right descending area, resulting in a stable solution with a slight increase of the homeostatic Ca^{2+} 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 Ca^{2+}. Our findings suggest that in response to the external conditions, there are several possibilities for intraciliary Ca^{2+} upregulation. The system can generate a single spike (Figure 11B) of variable amplitude (data not shown), permanently increase Ca^{2+} in a dynamic fashion and maintain the high intraciliary levels (Figure 11E), or operate in a monostable multivibrator mode (cilia can generate a Ca^{2+} 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, Ca^{2+} and membrane potential represent the slow and fast variables, respectively. This study, therefore, establishes that the dynamic properties of ciliary systems, where the Ca^{2+} and K^{+} channel conductivities represent monotonic function of membrane potential and the Ca^{2+} channels conductivity inversely depends on intraciliary Ca^{2+} concentration, are comparable with the properties of excitable systems based on the "N-shape" dependence of the Na^{2+} channel conductivity on membrane potential [74]. 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_{3} Ca^{2+} 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 Ca^{2+} 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 Ca^{2+} ions movement via the membrane as a function of the corresponding membrane potential shift (equation 43). By assuming the potential independent mechanism for Ca^{2+} and K^{+} ion expulsion, the system of intraciliary Ca^{2+} and membrane potential is derived as shown in equation (46) in the Methods section.
The role of cilia-to body Ca^{2+} 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 Ca^{2+} is significantly higher than intracellular Ca^{2+} concentration. At the same time, if the conductivity of protein structures governing the Ca^{2+} 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 Ca^{2+} could freely flow from cilia into the body, the intraciliary concentration would become equal to the intracellular Ca^{2+} 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 Ca^{2+} concentration can be observed in hyperpolarizing solutions. During all this time the intraciliary Ca^{2+} concentration can be several orders of magnitude higher than the intraciliary concentration. In this study, we have demonstrated that the steady-state Ca^{2+} current under the depolarized membrane potential conditions can only be reduced by the Ca^{2+}-dependent inhibition of Ca^{2+} channels. All these observations suggest that the Ca^{2+} removal from cilia to the cell body occurs in a membrane potential dependent manner.
The mechanism of Ca^{2+} and cyclic nucleotide-dependent CBF regulation
In addition to intraciliary Ca^{2+} 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 Ca^{2+} 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 [45] in the bases of cilia and thereby induce the ciliary movement [79].
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 Ca^{2+} 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 Ca^{2+} concentration, the bell shape dependence that can be either cAMP and cGMP dependent, and either monotonic increase or decrease if the physiological range of Ca^{2+} concentrations occur at one of the slopes. Our model, therefore, describes the core Ca^{2+}-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 Ca^{2+}- and G-protein mediated pathways.
Discussion
We develop a new computational model for Ca^{2+} 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 Ca^{2+} currents (Figure 2) and simultaneous intraciliary Ca^{2+}-concentration mediated inhibition of Ca^{2+} 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 Ca^{2+} 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 Ca^{2+} currents switch the system into the mode of a monostable multivibrator, when cilia can generate a Ca^{2+} 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 Ca^{2+}, 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 [88]. It is well established that cAMP and cGMP are synthesized by AC isoforms and hydrolysed by PDEs in a Ca^{2+}-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).
Conclusions
Therapeutic applications of systems model for intraciliary Ca^{2+} regulation
Future perspective
At present there is limited understanding of the underlying biological mechanisms that govern ciliary motility. This study describes the modes of intraciliary Ca^{2+} dynamics in a highly detailed fashion. It shows the conditions that switch the system between the modes of Ca^{2+} spike generation, oscillatory dynamics and a trigger. The interdependent influences of Ca^{2+} 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 Ca^{2+} 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 Ca^{2+} between single cell ciliates and mammalian cilia have been reported [90]. We would argue that those differences are not due to the change in the mechanisms of Ca^{2+}-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.
Methods
Model Description
Figure 1 provides a schematic outline of the network regulating intraciliary Ca^{2+} concentration that is considered in our model. Intraciliary Ca^{2+} concentration is regulated by the currents of passive and active Ca^{2+} transport, as well as by Ca^{2+} leak into the extracellular space and into the cell body.
A basic mathematical model for intraciliary Ca^{2+} concentration and its relationship to transmembrane potential was proposed for the first time in [91]. 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 Ca^{2+} regulation developed in this study.
where V_{ R } - is the cilium volume, S_{R} - is the cilium surface area, and ${I}_{C{a}^{2+}}^{P}$ and ${I}_{C{a}^{2+}}^{A}$-are the Ca^{2+} currents through the channels of passive and active Ca^{2+} transport, respectively. ${I}_{C{a}^{2+}}^{T}$ is the current from the cilium into the cell body. ${I}_{C{a}^{2+}}^{u}$-is the Ca^{2+} leakage current. $J\left(\left[C{a}^{2+}\right],\left[Ca{M}_{0}\right]\right)$ is the function that encounters Ca^{2+} binding to and release from CaM, the main Ca^{2+} binding protein in cilia, z = 2 is the Ca^{2+} ions charge, and F is the Faraday constant.
where ${g}_{C{a}^{2+}}^{i}\left({V}_{m},C{a}^{2+}\right)$ is the conductivity of a single channel in the state i (in the most general state Ca^{2+} channels can have a number of states with different degrees of conductivity), ${E}_{C{a}^{2+}}=\left(\frac{R\cdot T}{2\cdot F}\right)\cdot ln\left(\frac{C{a}_{out}^{2+}}{C{a}_{in}^{2+}}\right)$ is the Ca^{2+} potential in the equilibrium, V_{ m } is the transmembrane potential of the cilia membrane.
where ${k}_{A}=\frac{{k}_{A}^{m}+{k}_{A}^{p}}{{k}_{A}^{p}\cdot {K}_{CaM}},\tau ={i}_{A}\cdot {N}_{C{a}^{2+}}^{00},u=\frac{C{a}^{2+}}{{K}_{CaM}}$.
where ${\beta}_{1}=\frac{{g}_{t}\left({V}_{m}\right)\cdot R\cdot T}{F},{\psi}_{rt}=\frac{{V}_{rt}\cdot F}{R\cdot T},u=\frac{C{a}_{r}^{2+}}{{K}_{CaM}},{u}_{t}=\frac{C{a}_{t}^{2+}}{{K}_{CaM}}$.
In the following sections we derive the models and analyse the individual contributions of the different types of Ca^{2+} currents to the intraciliary Ca^{2+} homeostasis.
Model for intraciliary Ca^{2+}-dependent Ca^{2+} channel conductivity inhibition
where $n=\frac{\left[N\right]}{\left[{N}_{0}\right]},{k}_{C}=\frac{{K}_{C}}{{K}_{CaM}},u=\frac{C{a}^{2+}}{{K}_{CaM}}$.
where τ _{ p } is the characteristic time of the transmembrane potential alteration from V_{0} to V_{1}.
where $\psi =\frac{V\cdot F}{R\cdot T}$, v(ψ, t) is the Ca^{2+} 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.
where $u=\frac{C{a}^{2+}}{{K}_{CaM}},{u}_{out}=\frac{C{a}_{out}^{2+}}{{K}_{CaM}},{k}_{A}=\frac{{K}_{A}}{{K}_{CaM}},\psi =\frac{V\cdot F}{R\cdot T},\nu \left(\psi \right)=\frac{g\left(\psi \right)}{{g}_{0}},{g}_{0}=max\left(g\left(\psi \right)\right),n\left(\eta \right)=\frac{\left[N\left(\eta \right)\right]}{\left[{N}_{0}\right]},\eta ={n}^{m}\cdot t,{\tau}_{0}=\tau \cdot {n}^{m},s=\frac{{S}_{R}\cdot \beta}{z\cdot F\cdot {V}_{R}\cdot {K}_{CaM}\cdot {n}^{m}},b=\frac{\left[{N}_{0}\right]\cdot {g}_{0}\cdot R\cdot T}{\beta \cdot F},k=\frac{{K}_{CaM}}{{K}_{C}}$
Indirect Ca^{2+} channel conductivity regulation
where [CaC_{0}] is the total concentration of the Ca^{2+} binding protein and ${K}_{C}=\frac{{k}^{m}}{{k}^{p}}$ is the equilibrium dissociation constant.
where $c=\frac{\left[C\right]}{\left[{C}_{0}\right]},u=\frac{C{a}^{2+}}{{K}_{CaM}},ca{c}_{0}=\frac{Ca{C}_{0}}{{K}_{CC}},{k}_{C}=\frac{{K}_{C}}{{K}_{CaM}}$,
where $\eta ={n}^{m}\cdot t,c=\frac{\left[C\right]}{\left[{C}_{0}\right]},{k}_{C}=\frac{{K}_{C}}{{K}_{CaM}},u=\frac{C{a}_{2+}}{{K}_{CaM}},ca{c}_{0}=\frac{\left[Ca{C}_{0}\right]}{{K}_{CC}}$,.
where ${c}^{\infty}=\frac{{k}_{C}+{u}_{1}}{{k}_{C}+\left(ca{c}_{0}+1\right)\cdot {u}_{1}},{c}^{0}=\frac{{k}_{C}+{u}_{0}}{{k}_{C}+\left(ca{c}_{0}+1\right)\cdot {u}_{0}}$,.
For cases when cac_{0} > > 1 and u > > 1, the characteristic time approximately equals ${\tau}_{C{a}^{2+}}\approx \frac{1}{\left[Ca{C}_{0}\right]\cdot {n}^{p}}.$
In this equation, we include the kinetics for the active Ca^{2+} channels due to the assumption that the dynamics of currents via the active Ca^{2+} channels is much faster than the dynamics of currents through the passive Ca^{2+} transport.
where $u=\frac{C{a}^{2+}}{{K}_{CaM}},{u}_{out}=\frac{C{a}_{out}^{2+}}{{K}_{CaM}},{k}_{C}=\frac{{K}_{C}}{{K}_{CaM}},{k}_{A}=\frac{{K}_{A}}{{K}_{CaM}},\psi =\frac{{V}_{m}\cdot F}{R\cdot T},a=\frac{{S}_{R}\cdot \beta}{z\cdot F\cdot {V}_{R}\cdot {K}_{CaM}\cdot {n}^{m}},\nu \left(\psi \right)=\frac{g\left(\psi \right)}{{g}_{0}},{g}_{0}=max\left(g\left(\psi \right)\right),c\left(\eta \right)=\frac{C\left(\eta \right)}{\left[{C}_{0}\right]},\eta ={n}^{m}\cdot t,{\tau}_{0}=\tau \cdot {n}^{m},b=\frac{\left[{C}_{0}\right]\cdot {g}_{0}\cdot R\cdot T}{\beta \cdot F},k=\frac{{K}_{CaM}}{{K}_{C}}$
where $u=\frac{C{a}^{2+}}{{K}_{CaM}},{u}_{out}=\frac{C{a}_{out}^{2+}}{{K}_{CaM}},{k}_{A}=\frac{{K}_{A}}{{K}_{CaM}},\psi =\frac{{V}_{m}\cdot F}{R\cdot T},\nu \left(\psi \right)=\frac{g\left(\psi \right)}{{g}_{0}},{g}_{0}=max\left(g\left(\psi \right)\right),c\left(\eta ,u\right)=\frac{\left[C\left(\eta ,u\right)\right]}{\left[{C}_{0}\right]},\eta ={n}^{m}\cdot t,{\tau}_{0}=\tau \cdot {n}^{m},a=\frac{{S}_{R}\cdot \beta}{z\cdot F\cdot {V}_{R}\cdot {K}_{CaM}\cdot {n}^{m}},b=\frac{\left[{C}_{0}\right]\cdot {g}_{0}\cdot R\cdot T}{\beta \cdot F},k=\frac{{K}_{CaM}}{{K}_{C}},{\nu}_{t}=\frac{\beta 1}{\beta}$.
Potassium current
where ${N}_{{K}^{+}}$ is the number of open K^{+} channels, ${g}_{{K}^{+}}^{0}$ is the maximal conductivity, and ${E}_{{K}^{+}}$ is the equilibrium K^{+} potential.
where ${b}_{1}={N}_{{K}^{+}}\cdot \frac{{g}_{{K}^{+}}^{0}}{{g}_{0}}$.
The transmembrane potential dynamics
where $\psi =\frac{{V}_{m}\cdot F}{R\cdot T},u=\frac{\left[C{a}^{2+}\right]}{{K}_{CaM}},{\nu}_{C{a}^{2+}}^{st}$, and ${\nu}_{{K}^{+}}^{st}$ are the steady-state Ca^{2+} and K^{+} channel conductivities, respectively, $\eta ={n}^{m}\cdot t,\rho =\frac{{g}^{0}}{{C}_{m}\cdot {n}^{m}}$,.
where I_{0} is the non dimensional inward current.
Currents activated by the membrane hyperpolarisation
where ${g}^{0}=max\left(\underset{C{a}^{2+}}{\overset{ht}{g}}\left(\psi ,t\right)\right),{\nu}_{h}\left(\psi \right)=\frac{exp\left({\alpha}_{h}\cdot \left(\psi +{d}_{h}\right)\right)}{{\lambda}_{h}+exp\left({\alpha}_{h}\cdot \left(\psi +{d}_{h}\right)\right)}$
where v_{ h } (ψ) is the Ca^{2+} current contribution, activated by membrane depolarization, and ${\nu}_{{K}^{+}}\left(\psi ,u\right)$ is the Ca^{2+}-dependent K^{+} current contribution.
Cilia-to body Ca^{2+} current
Parameter values employed in the systems model for the ciliary excitation
Parameter | Value (dimensionless unless otherwise stated) | Figure No | Equation |
---|---|---|---|
α | 4 | 2A | 18 |
d | 0.4 | 2A | 18 |
λ | 0.5 | 2A | 18 |
b | 2 | 2B, 3, 5, 6, 7, 9, 10 | 20, 21, 34, 39 |
k _{ A } | 1 | 2B, 3, 5, 6, 7, 11 | 20, 34, 42 |
u _{ out } | 1000 | 2B, 3, 5, 6, 7 | 20, 21, 34 |
ψ _{0} | -1.2 | 2B, 3, 5, 6, 7, 8 | 20, 21, 34, 39 |
V _{0} | 30 mV | 2B, 3, 5, 6, 7, 8 | 20, 21, 34, 39 |
ψ _{1} | -1, -0.8, -0.5, -0.2, 0, 0.2, 0.5 | 2B, 3, 5, 6, 7, 8 | 20, 21, 34, 39 |
V _{1} | 25, 20, 12.5, 5, 0, -5, -12.5 mV | 2B, 3, 5, 6, 7, 8 | 20, 21, 34, 39 |
s | 0.5 | 2B, 3, 5, 6, 7 | 20 |
τ _{0} | 0.02 | 2B, 3, 5, 6, 7, 9, 10 | 20, 34, 39 |
k | 2 | 2B, 3, 5, 6, 7 | 20 |
K _{ C } | 1 | 4, 5, 6, 7 | 25 |
CaC _{0} | 5, 10, 50, 100 | 4, 5, 6, 7 | 25, 34 |
a | 4 | 7, 11 | 34, 42 |
${\alpha}_{{K}^{+}}$ | 0.5 | 8, 10 | 36 |
${\lambda}_{{K}^{+}}$ | 0.005 | 8, 10 | 36 |
${d}_{{K}^{+}}$ | 0.5 | 8, 10 | 36 |
b _{1} | 1 | 9 | 39 |
b | 8 | 11, 12, 13, 14 | 42, 46 |
cac _{0} | 20 | 11, 12, 13, 14 | 42 |
${\nu}_{C{a}^{2+}}^{st}$ | 0.01 | 11, 12, 13, 14 | 42, 46 |
${\nu}_{{K}^{+}}^{st}$ | 0.01 | 11, 12, 13, 14 | 42, 46 |
ρ | 10 | 11, 12, 13, 14 | 42, 46 |
s | 0.5 | 11, 12, 13, 14 | 42, 46 |
vCah | 0.9 | 12, 13, 14 | 46 |
λ _{ h } | 5 | 12, 13, 14 | 44 |
d _{ h } | 1 | 12, 13, 14 | 44 |
α _{ h } | 4 | 12, 13, 14 | 44 |
The relationship between dimensional and non-dimensional quantities for Ca^{2}^{+} concentration and membrane potential
Variables | Dimensional variables | Non-dimensional variables | Coefficient value |
---|---|---|---|
Calcium | Ca^{2+} (M/L) | $u=\frac{C{a}^{2+}}{{K}_{CaM}}$ | K_{ CaM } = 4 μM |
Transmembrane potential | V _{ m } | ${\psi}_{m}=\frac{F\cdot {V}_{m}}{R\cdot T}$ | $\frac{RT}{F}=-0.025\phantom{\rule{0.3em}{0ex}}V$ |
Declarations
Acknowledgements
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).
Authors’ Affiliations
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