Computational modelling elucidates the mechanism of ciliary regulation in health and disease

RESEARCH ARTICLE Open Access

Computational modelling elucidates themechanism of ciliary regulation in health anddiseaseNikolay V Kotov1,2, Declan G Bates3, Antonina N Gizatullina2, Bulat Gilaziev2, Rustem N Khairullin4,Michael ZQ Chen5,6, Ignat Drozdov7, Yoshinori Umezawa8, Christian Hundhausen8, Alexey Aleksandrov9,Xing-gang Yan10, Sarah K Spurgeon10, C Mark Smales1 and Najl V Valeyev1*

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 anddespite extensive experimental characterization remains poorly understood. We develop a detailed systems modelfor calcium, membrane potential and cyclic nucleotide-dependent ciliary motility regulation.

Results: The model describes the intimate relationship between calcium and potassium ionic concentrations insideand outside of cilia with membrane voltage and, for the first time, describes a novel type of ciliary excitabilitywhich plays the major role in ciliary movement regulation. Our model describes a mechanism that allows ciliaryexcitation to be robust over a wide physiological range of extracellular ionic concentrations. The model predictsthe existence of several dynamic modes of ciliary regulation, such as the generation of intraciliary Ca2+ spike withamplitude 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 intoaltered membrane voltage.

Conclusions: Computational investigation of the model offers several new insights into the underlying molecularmechanisms of ciliary pathologies. According to our analysis, the reported dynamic regulatory modes can be aphysiological 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.

BackgroundCilia are cellular protrusions which have been conservedin a wide range of organisms ranging from protozoa tothe digestive, reproductive and respiratory systems ofvertebrates [1]. Mobile or immotile cilia exist on everycell of the human body [2] and the insufficiently recog-nised importance of the cilium compartment in humanphysiology has been recently highlighted [1,3]. Cilia arepresent on most eukaryotic cell surfaces with the excep-tion of the cells of higher plants and fungi [4]. Ciliarymotility is important for moving fluids and particlesover epithelial surfaces, and for the cell motility of

vertebrate sperm and unicellular organisms. The ciliumcontains a microtubule-based axoneme that extendsfrom the cell surface into the extracellular space. Theaxoneme consists of nine peripheral microtubule doub-lets arranged around a central core that may or may notcontain two central microtubules (9+2 or 9+0 axoneme,respectively). Cilia can be broadly classified as 9+2motile cilia or 9+0 immotile sensory cilia, althoughthere are examples of 9+2 sensory cilia and 9+0 motilecilia. In mammals, motile 9+2 cilia normally concentratein large numbers on the cell surface, beat in an orche-strated wavelike fashion, and are involved in fluid andcell movement. In contrast to motile cilia, primary ciliaproject as single immotile organelles from the cell sur-face. Primary cilia are found on nearly all cell types inmammals [5] and many are highly adapted to serve

* Correspondence: [email protected] for Molecular Processing, School of Biosciences, University of Kent,Canterbury, Kent CT2 7NJ, UKFull list of author information is available at the end of the article

Kotov et al. BMC Systems Biology 2011, 5:143http://www.biomedcentral.com/1752-0509/5/143

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specialized sensory functions. The 9+2 cilia usually havedynein arms that link the microtubule doublets and aremotile, while most 9+0 cilia lack dynein arms and arenon-motile. In total, eight different types of cilia hasbeen identified to date [6]. In this study, we investigatethe mechanism of movement regulation for the motiletype of cilia.Although each individual cilium represents a tiny hair-

like protrusion of only 0.25 μm in diameter andapproximately 5-7 μm in length, cilia covering humanairways can propel mucus with trapped particles oflength up to 1 mm at a speed of 0.5 mm/second [7].Such efficiency can be achieved due to the coordinationbetween cilia and stimulus-dependent regulation of therate of cilia beat. Dysfunction of ciliary regulation givesrise to pathologic phenotypes that range from beingorgan specific to broadly pleiotropic [3]. A link betweenciliary function and human disease was discovered whenindividuals suffering from syndromes with symptomsincluding respiratory infections, anosmia, male infertilityand situs inversus, were shown to have defects in ciliarystructure and function [6].Microscopic organisms that possess motile cilia which

are used exclusively for either locomotion or to simplymove liquid over their surface include Paramecia, Kar-yorelictea, Tetrahymena, Vorticella and others. Thehuman mucociliary machinery operates in at least twodifferent modes, corresponding to a low and high rateof beating. It has been shown that the high rate mode ismediated by second messengers [8], including puriner-gic, adrenergic and cholinergic receptors [9-19]. Thismode enables a rapid response, which can last a signifi-cant period of time, to various stimuli by drasticallyincreasing the ciliary beat frequency (CBF). At the sametime, several ciliary movement modes have beenreported in a ciliate Paramecium caudatum [20]. Theremarkable conservation of ciliary mechanisms [21-25]creates grounds for the speculation that there can morethan two ciliary beating modes in human tissues. It is,therefore, reasonable to suggest, that some human dis-eases, associated with aberrant ciliary motility, can arisedue to modifications in the beating mode. Clearly, thedevelopment of therapeutic strategies against ciliary-associated pathologies will require advanced understand-ing of ciliary beating regulation mechanisms.The periodic beating of cilia is governed by the inter-

nal apparatus of the organelle [26]. Its core part, theaxoneme, contains nine microtubule pairs encircling thecentral pair. The transition at the junction of the cellularbody and the ciliary axoneme is demarcated by Y-shaped fibres, which extend from the microtubule outerdoublets to the ciliary membrane. The transition area, incombination with the internal structure of the basalbody, is thought to function as a filter for the cilium,

regulating the molecules that can pass into or out of thecilium. Ciliary motility is accomplished by dynein motoractivity in a phosphorylation-dependent manner, whichallows the microtubule doublets to slide relative to oneanother [12]. The dynein phosphorylation that controlsciliary activity is regulated by the interplay of calcium(Ca2+) and cyclic nucleotide pathways. The beating pat-tern of cilia consists of a fast effective stroke and aslower recovery stroke. During the effective stroke ciliaare in an almost upright position, generating force formucus movement. During the recovery stroke, the ciliaare recovering from the power strike to the originalposition by moving in the vicinity of the cell surface.Current theories which attempt to explain the work-

ings of the Ca2+-dependent CBF regulation mechanismare incomplete and highly controversial. Elevation ofintraciliary Ca2+ is one of the major regulators of ciliarymovement. Calcium influx regulates ciliary activity byincreasing intraciliary Ca2+ only, while the cytosolic bulkremains at a low level. Separate ciliary compartmentali-sation for Ca2+ allows prolonged activation of ciliarybeating without damaging the cell through high Ca2+

concentrations. It is well known that calcium fluxes viacalcium channels lead to changes in organisms’ swim-ming behaviour [27-29]. In mucus-transporting cilia,Ca2+ mediates CBF increase [19,30-32]. It has also beenshown that there are some differences in the Ca2+-dependent CBF regulation in single cell organisms andin humans [12]. Sustained CBF increase requires pro-longed elevation of Ca2+ levels which can be lethal tothe cell [33,34]. It has been suggested that Ca2+-depen-dent ciliary regulation takes place locally in the vicinityor within the ciliary compartment, almost independentlyfrom intracellular Ca2+ concentration [35]. Given thatthe gradient of free Ca2+ in the cytosol dissipates within1-2 seconds [36], it appears more likely that cilia formtheir own compartment where Ca2+ is regulated byactive Ca2+ transport in a similar fashion to the intracel-lular Ca2+ regulatory system. This hypothesis resolvesthe problem of maintaining physiological levels of intra-cellular Ca2+ concentration. A number of experimentalstudies have reported several controversial results relat-ing to the Ca2+-dependent mechanism of cilia regula-tion. For example, it has been reported thatspontaneous cilia beat does not require alterations inCa2+ [31,35], while nucleotide-dependent CBF increaserequires Ca2+ [8]. It has also been shown that uncou-pling between Ca2+ and CBF can be achieved by inhibi-tion of Ca2+-dependent protein calmodulin (CaM) orthe cyclic nucleotide pathway [19,32,37].These findings suggest that intraciliary Ca2+ does not

regulate cilia beat in isolation, but instead does so aspart of more complex signalling network. Although itwas originally believed that Ca2+, cyclic adenosine


Page 2 of 26

monophosphate (cAMP) and guanosine monophosphate(cGMP) regulate ciliary beat independently, numerousreports now strongly indicates that all three pathwaysare tightly interconnected [12,19,32,38-42]. The cAMP-dependent protein kinase (PKA) phosphorylates dyneinin the bases of cilia and thereby increases the forwardswimming speed in Paramecium [43-45]. Similar effectshave been reported for PKA-dependent phosphorylationof axonemal targets in mammalian respiratory cilia [46].Several lines of evidence indicate that PKA and cGMP-dependent kinase (PKG) both phosphorylate specificaxonemal targets in a cAMP and cGMP-dependentmanner. A schematic diagram for the underlying bio-chemical machinery for cilia movement regulation isshown in Figure 1C. It is striking that, despite significantexperimental characterisation of this system, there is stillrather limited mechanistic understanding of how intra-ciliary Ca2+ and nucleotide interplay relates to CBF.Another major regulator of ciliary beating is the mem-

brane potential. A number of studies have reported thevoltage-dependent effects of ciliary beating. The ciliateDidinium Nasutum has been shown to respond both tohyper- and de-polarization of the membrane [47]. Thetransmembrane potential alterations were shown to bemediated via the potential-dependent Ca2+ channels[48]. Electrophysiological studies in Paramecium cauda-tum have revealed complex relationships between ciliaryCa2+ currents, intraciliary Ca2+ concentration and trans-membrane 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 synchroniza-tion between small ciliary areas [56]. The effects of visc-osity have been investigated in mucus propelling cilia in[57]. The authors found that increasing the viscosity notonly decreases CBF, but also changes the degree of cor-relation and synchronization between cilia. The mechan-ical properties of cilia motion were studied in anattempt to understand the ciliary dynamics in [58]. Theauthors concluded that bending and twisting propertiesof the cilium can determine self-organized beating pat-terns. While these reports offer valuable insights intothe regulatory mechanisms of cilia, a number of essen-tial questions remain unresolved. For example, there hasnot been a detailed analysis of how individual Ca2+ cur-rents influence intraciliary Ca2+ levels. It also remainsunclear how Ca2+ modulates nucleotide levels and mem-brane potential, and how such regulation affects ciliarymovement. None of these reports have elucidated theunderlying mechanisms governing the interplay betweenintraciliary Ca2+ and nucleotide alterations and CBF.

Figure 1 Schematic diagram of the Ca2+-dependent regulationof motile cilium. (A) Schematic representation of the motile ciliummovement trajectory for one complete beating cycle. The cycle isdivided into two phases: effective stroke and recovery stroke shownby red and green arrows, respectively. The frequency and thedirection of cilia movement is regulated in a highly complex Ca2+

and membrane voltage-mediated manner. (B) A typical ciliumconsists of an axoneme of nine doublet microtubules. The axonemeis surrounded by a specialized ciliary membrane that is separatedfrom the cell membrane by a zone of transition fibres. Thisseparation creates an intraciliary compartment where key regulatoryevents take place somewhat independently from the cell body. Inparticular, intraciliary Ca2+ concentration can significantly differ fromintracellular levels. (C) The ciliary motion is regulated by intraciliaryCa2+ levels. The Ca2+ concentration depends on the interplay of ionchannels and membrane potential, Vm. The intraciliary Ca2+

concentration is dependent on currents via Ca2+, IICa2+, and K+

channels, IK+, the system of active, IACa2+, and passive, Ip

Ca2+, ionsremoval, Ca2+ leakage current, IU

Ca2+, hyperpolarisation-activatedcurrents IHT

Ca2+, inward current, I0, and the cilium-to-cell bodycurrent, IT

Ca2+. The conductivities of the channels are modulated bymembrane potential. The result of the cross-talk betweenmembrane potential and a variety of channels is that intraciliary Ca2+ can shift between several dynamic modes. The steady-state anddynamic Ca2+ alterations regulate the intraciliary levels of cAMP andcGMP in a Ca2+-CaM-dependent manner via the AC, GC and PDEisoforms [65]. Cyclic nucleotides, in turn, define the degree ofphosphorylation of dynein filaments in the bases of ciliary axonemevia PKA and PKG kinases. Phosphorylation of dynein filamentsregulates the relative doublet microtubules shift and therebytranslates to the overall ciliary movement.


Page 3 of 26

In this study, we integrate the available experimentalinformation on the molecular pathways that regulateintraciliary Ca2+ concentration into a comprehensivemathematical model. By applying systems analysis, weelucidate the mechanisms of intraciliary Ca2+ spike gen-eration, analyse the properties of such spikes anddemonstrate the conditions under which the Ca2+ surgescan become repetitive. We carry out detailed investiga-tions of the individual current contributions to the regu-lation of the intraciliary Ca2+ concentrations andelucidate both steady-state and dynamic responses ofCa2+ currents and intraciliary Ca2+ concentrationdynamics in response to the altered transmembranepotential shift. The model allows detailed elucidation oftransmembrane potential and intraciliary Ca2+ coupling.We employ the proposed model in order to under-

stand the underlying molecular mechanisms of thecrosstalk between Ca2+, membrane potential andnucleotide pathways that regulate ciliary movement. Thesystems model allows detailed analysis of the individualcurrent contributions to the intraciliary homeostatic Ca2+ levels. Furthermore, we establish specific regulatorymechanisms for Ca2+ and cyclic nucleotide-dependentcilia movement characteristics. Crucially, our model pre-dicts the possibility of several ciliary beating modes anddescribes specific conditions that initiate them. Specifi-cally, we describe intraciliary Ca2+ dynamic modes thatregulate healthy and pathologic cilia beating. We usethese findings in order to propose experimentally testa-ble hypotheses for possible therapeutic interventions inhuman diseases associated with pathologic cilia motility.

ResultsA new model for the interplay between Ca2+ and K+

currents and transmembrane potential alterationsA new model for the regulation of ciliary movementthat combines multiple Ca2+ and K+ currents [59-62]and transmembrane potential has been developed. Inthis model, the intraciliary Ca2+ levels are modulated byCa2+ currents through the channels of passive and activeCa2+ transport, the current from the cilium into the cellbody, the Ca2+ leakage current, and depolarisation andhyperpolarisation-activated currents. Variable extracellu-lar conditions have continuous impact on the trans-membrane potential which is intertwined withtransmembrane 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 thatdefine intraciliary Ca2+ homeostasis and the transmem-brane potential. One of the most essential intraciliaryCa2+ binding proteins, CaM [63,64], selectively regulatesthe activities of adenylate cyclase (AC), guanylate cyclase

(GC) and phosphodiesterases (PDE), and thereby modu-lates the intraciliary levels of adenosine monophosphate(cAMP) and guanosine monophosphate (cGMP) in aCa2+ dependent manner [65]. The cAMP- and cGMP-dependent kinases phosphorylate dynein proteins in thebases of cilia and thereby induce the mechanical ciliamovement. The complete set of equations making upthe proposed model is presented in the Methods sec-tion. Below we provide a number of new insights intothe mechanism of cilia regulation via a detailed investi-gation of the properties of this model.

The mechanism of Ca2+-dependent inhibition of Ca2+

channelsA subset of intraciliary Ca2+ channels have beenreported to operate in an intraciliary Ca2+ dependentmanner and have been proposed as major regulators ofciliary beat [49-51]. It is established that Ca2+ current isnot inhibited by the double pulse application of depolar-ization impulses under voltage clamp conditions inthose situations when the first transmembrane potentialshift is equal to the equilibrium Ca2+ potential (+120mV) [66]. Further experimental evidence reveals thatCa2+ current inactivation kinetics are delayed when Ca2+

ions are partially replaced by Ba2+ ions [67-71]. Alto-gether these findings suggest that the channels are notinhibited directly by the depolarizing shift of transmem-brane potential, but that instead their conductivity isdependent on the intraciliary Ca2+ concentration. Somedecrease of the inward current amplitude (by approxi-mately 25%) upon transmembrane potential shift intothe Ca2+ equilibrium level can be explained by the factthat K+ currents can contribute to the overall currentmeasurements. Here we consider the intraciliary Ca2+

concentration-dependent Ca2+ channel inhibition andemploy the developed model to analyse two potentialscenarios for the Ca2+ channel conductivity regulation.In one case, Ca2+ ions bind to the Ca2+ binding site onthe channel and thereby inhibit the channel’s conductiv-ity by direct interaction. The other possibility is that theCa2+ binding protein interacts with the Ca2+ ion firstand then this complex binds to the channel and inhibitsits conductivity. In both cases the conductivity depen-dence on transmembrane potential is assumed to bemonotonic according to the experimental data [66].

Direct Ca2+-dependent Ca2+ channel conductivityinhibitionWe first investigate a potential intraciliary Ca2+ regula-tory mechanism via direct Ca2+ ion binding-dependentCa2+ channel inhibition. The relationship between theCa2+ channel conductivity and the transmembranepotential has been experimentally characterized by anearly study in Paramecium species [66]. The equation


Page 4 of 26

(18) in the Methods section approximates the experimen-tally established dependence. In the case of direct Ca2+-dependent Ca2+ channel inhibition, one can show thatthe nullclines for non-dimensional Ca2+ concentration

(dCa2+

dt= 0) and for the number of open channels

(dndt

= 0) from the system of differential equations (20)

intersect at one stable point for all values in the physiolo-gical range of model parameters. The numerical solutionsof the coupled differential equations (20) allow us toobtain the solutions for how the steady-state Ca2+ levelsdepend on the transmembrane potential. The model pre-dictions for the steady-state ciliary Ca2+ channel conduc-tivity dependence on the transmembrane potential underthe voltage clamp conditions are shown on Figure 2A.The extracellular conditions are subject to constantchange both in the case of ciliates as well as for multicel-lular organisms. The modifications in the external envir-onment continuously shift the transmembrane potential.In order to estimate how the transmembrane potentialalterations affect the inward Ca2+ current we derived thedependence for the Ca2+ ion flow (equation (21) in theMethods section). The experimental data-based (Figure2A) [66] model for the inward Ca2+ current dependenceon membrane potential (Figure 2B) predicts a significantreduction of the inward Ca2+ current amplitude as afunction of membrane depolarization.We next investigated the dynamic alterations of the

intraciliary Ca2+ concentrations and the inward Ca2+

current in response to the transmembrane potentialshifts. Equations (20) and (21) in the Methods sectionwere used for quantitative estimations of the Ca2+ con-centration and current responses to the normalisedmembrane potential shifts. The model predicts that therapid depolarising alterations of the ciliary transmem-brane potential leads to the generation of single Ca2+

spikes (Figure 3). Such responses can take place whenCa2+ channels are inhibited by Ca2+ ions and the chan-nel’s conductivity depends on the membrane potentialin a monotonic manner (equation 18). The amplitude ofthose impulses depends on the steepness of the Ca2+

channels conductivity dependence on the membranepotential. The mechanism of the Ca2+ spike generationis mainly due to the delay of the Ca2+-induced inhibi-tion with respect to the Ca2+ conductivity alterationcharacteristic times. The described mechanism of theCa2+ spike generation will only work if the Ca2+ cur-rents significantly alter the Ca2+ concentration in theintraciliary compartments.

Indirect Ca2+ channel conductivity regulationIn the previous section we considered Ca2+-dependentCa2+ channel regulation under the assumptions that

Ca2+ channels have an intracellular Ca2+ binding siteand Ca2+ ion binding closes the channels. However,several experimental studies have suggested that theconductivity of Ca2+ channels in cilia can also be regu-lated indirectly, via a Ca2+ binding protein. At present,there is no direct experimental evidence that explicitlyfavours either direct or indirect regulatory mechanism.We, therefore, investigated the second possibility for

Figure 2 Static and dynamic intraciliary Ca2+ concentrationlevels under membrane potential fixed conditions. (A) Themodel predictions for the relationship between the membranepotential and intraciliary Ca2+ concentration. (B) The maximumamplitude of Ca2+ current generated in response to thetransmembrane potential shift is shown as function of the appliedshift. The model predictions are obtained by setting the parametera to 2, 3, and 4 in equations 18-21. The parameter a reflects thesteepness of Ca2+ current dependence on membrane voltage. Themodel-based analysis shown here unravels the bell-shapeddependence of intraciliary Ca2+ concentration on membranepotential and the inverse relationship between the Ca2+ currentmagnitudes and the degree of membrane depolarisation.


Page 5 of 26

Figure 3 Systems model predictions for intraciliary Ca2+ concentration and current responses to the transmembrane potential shift.The intraciliary Ca2+ concentrations (A) and Ca2+ currents (B) are calculated according to the model with direct Ca2+-mediated ciliary Ca2+

channels inhibition in response to variable degree of transmembrane potential shift. The responses are colour coded according to the degree ofapplied transmembrane potential shift. The violet and red coloured lines represent dynamic responses obtained after the smallest and thelargest depolarising shift of membrane potential, respectively. The non dimensional membrane potential values following voltage shift frominitial ψ0 = -1.2 are shown in (C) and remain the same throughout the figure. The calculations suggest that ciliary membrane depolarisationinduces an intraciliary Ca2+ spike over a wide physiological range of depolarising conditions (C, E and G), whereas the current generates abiphasic response (D, F and H). The calculations were carried out for three different levels of parameter a, which reflects the steepness of theCa2+ channels conductivity dependence on membrane potential.


Page 6 of 26

indirect Ca2+-dependent Ca2+ channels conductivityinhibition.The model for Ca2+ ion interactions followed by the

interactions with the Ca2+ channels (described in theMethods section) is developed in line with a previouslysuggested modelling methodology for Ca2+-CaM inter-actions [63,64]. The model predictions for the numberof open channels as a function of intraciliary Ca2+ con-centration are shown on Figure 4 as derived by equation(25) in the Methods section. Given that the exact natureof the Ca2+ binding protein acting as a mediatorbetween Ca2+ ions and Ca2+ channels is not established,equation (25) has been solved for different physiologi-cally possible ratios of total number of channels to thedissociation constant for the Ca2+- binding proteininteractions with Ca2+ channels. According to thederived models, the comparison of direct versus indirectCa2+ channels inhibition can be carried out by settingthe non dimensional Ca2+ concentration to zero (u = 0in equation (25)). In this case, the predictions of equa-tion (25) for indirect Ca2+ channels conductivity inhibi-tion 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+ concen-trations. However, the model for indirect Ca2+ channelsinhibition predicts that the number of open Ca2+ chan-

nels would equal1

cac0 + 1when intraciliary Ca2+ reaches

high concentrations. In other words, in the case of theindirect mechanism of inhibition, high Ca2+ does notinhibit the channels completely. The number of remain-ing channels in the open state would depend on theconcentration of the regulatory Ca2+ binding protein.We next analysed the dynamics of the ciliary Ca2+

channels inhibition in response to a change in intracili-ary Ca2+ concentration and in the transmembranepotential. As mentioned earlier, the Ca2+ channel con-ductivity is not inhibited by the membrane potential,but rather has a monotonic dependence on the trans-membrane 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 chan-nels. The model predictions for intraciliary Ca2+ concen-tration and inward Ca2+ current in response todepolarising transmembrane potential changes from V0

to V1, are shown on Figure 5. According to the derivedequation (30) in the Methods section, the inward Ca2+

current changes with a characteristic time τV, which canonly be estimated by considering the K+ current contri-bution. In the first instance we only consider active andpassive Ca2+ transport (equation (31) in Methods). Weneglected by the kinetics for the channels of activetransport due to the assumption that the kinetics ofalterations of active transport are much faster that thecharacteristic alteration times of passive transport (equa-tion (32) for the Ca2+ currents). The model predictionsfor the steady-state dependence of transmembranepotential on intraciliary Ca2+ concentration and thedynamic Ca2+ current amplitude dependence on themembrane potential are shown in Figure 6A and 6B,respectively.The cilia’s external environment is subject to constant

change and can significantly affect the behaviouralresponses of ciliates and modulate ciliary beating inmulticellular organisms. In order to account for theeffects of extraciliary Ca2+ variations we estimated theamplitudes of intraciliary Ca2+ spike generation underdifferent external Ca2+ concentrations and variabletransmembrane potentials. Figure 7A shows that theincrease of the Ca2+ concentration in the external solu-tion increases the amplitude of the generated intraciliaryCa2+ spike. The amplitude of the spikes goes to zerowhen the membrane potential equals the equilibriumpotential for Ca2+ ions. The increase of membranepotential decreases the amplitude of the Ca2+ current(Figure 7B). This effect is due to the increase of thesteady-state intracilia Ca2+ level and Ca2+-dependentinhibition of the Ca2+ channels.We noted earlier that there is a Ca2+ current in the

cilia which transfers ions from the cilia into the cellularcompartments. This current can be described by

II2C

0.8

1.0

Cac0=10

chan

nels

2CaI2Ca

0.2

0.4

0.6 Cac0=50

Cac0=5

tio o

f ope

n c

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.00.0

0.2Cac0=100

The

rat

Log(Ca2+), uFigure 4 The predicted probabilities for Ca2+ channels to be inthe open state. The indirect Ca2+-mediated inhibition of Ca2+

channels via a Ca2+-sensor protein, Cac0, is shown for a range ofCa2+-binding protein concentrations. The higher levels of Ca2+-sensor contributes to a greater degree of inhibition, although thegeneral mechanism remains the same.


Page 7 of 26

A B2

ICaI2CamV mV

150

200

V = 5 mVVm= 0 mV

Vm= -12.5 mV

on,

M

0.0

t, nA

0 2CacC D

0

50

100

Vm= 20 mV

Vm 5 mV

Vm= -5 mVVm= -22.5 mV

+ con

cent

ratio

-4.5

-3.0

-1.5

Ca2+

cur

ren

0 2Cac

0.0 0.2 0.4 0.6 0.8 1.00

Vm= 12.5 mVCa2+

Time, sec

0.0 0.2 0.4 0.6 0.8 1.0Time, sec

60M 0.2A

E F

15

30

45

once

ntra

tion,

-0.4

-0.2

0.0

a2+ c

urre

nt, n

A

0 5Cac

0 5Cac

0.0 0.2 0.4 0.6 0.8 1.00

Ca2+

co

Time, sec0.0 0.2 0.4 0.6 0.8 1.0

-0.6

C

Time, sec

10M

G H

4

6

8

0

entra

tion,

M

-0.5

0.0

0.5

curr

ent,

nA

0 20Cac

0.0 0.1 0.2 0.3

2

Ca2+

con

ce

Time, sec0.0 0.1 0.2 0.3

-1.0Ca2+

Time, sec

0 20Cac

Figure 5 The responses of Ca2+ current and concentration to the potential shift via indirect Ca2+ inhibition. The intraciliary Ca2+

concentration (A) and Ca2+ current (B) responses to membrane depolarisation are computed under the assumption of indirect Ca2+ channelinhibition for different concentrations of Ca2+-sensor protein, Cac0. The model predicts that in response to fast membrane depolarisation, theciliary coupled Ca2+ and membrane potential system generates an intraciliary Ca2+ concentration spike similar to the one observed under thedirect Ca2+-inhibition assumptions. The non dimensional membrane potential values following voltage shift from the initial ψ0 = -1.2 are shownin (C) and are the same throughout the figure. The major difference between direct (Figure 3) and indirect mechanisms of inhibition occurs inthe steady-state levels of Ca2+ concentration that takes place after the transitional process. The comparison of intraciliary Ca2+ concentrationresponses for increasing levels of the Ca2+ sensor protein (C, E and G) predicts that intraciliary Ca2+ concentrations following the voltage shift areinversely dependent on the Ca2+ sensor concentration, Cac0. Different concentrations of Ca

2+ sensor protein do not affect the membranedepolarisation-induced Ca2+ current (D, F and H).


Page 8 of 26

equation (11) in Methods. The contribution of cilium-to-cell body current to the intraciliary Ca2+ concentra-tion dynamics was evaluated experimentally in [72,73].It was shown that under depolarized membrane poten-tial conditions the contribution of this current is verysmall and the intraciliary Ca2+ is mainly pumped out ofthe cilia into the extracellular space by the active Ca2+

transport. According to other observations, Ca2+ currentfrom cilia into the cellular compartment can be largerthan the current generated by the active Ca2+ transport.In order to investigate the role and contribution of thecilia-to-cell compartment current, we introduced itscontribution to the intraciliary Ca2+ concentration

dynamics (equation (34)). We performed qualitative ana-lysis of the Ca2+ concentration alterations in the cilia inthe presence of the cilium-to-cell current and comparedthe Ca2+ dynamics with the case when this current wasnot present. We found that although the cilium-to-cellbody current influences the intraciliary Ca2+ concentra-tion levels, it does not change the dynamics qualitativelywhen 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

0 06V

AV

0.02

0.04

0.06

Cac0 = 2

oten

tial,

V

mV

0 04

-0.02

0.00

Cac0 = 5

Cac0 = 20

mbr

ane

po

2Ca

0.1 1 10 100

-0.04

mem

Ca2+ concentration, M

-1

0 Cac0 = 20

nA

B

-3

-2

-1

Cac0 = 5

Cac = 2ax (I

Ca2+

), n

0.000 0.005 0.010 0.015

-4

Cac0 = 2ma

b t ti l lt ti Vmembrane potential alteration, VmFigure 6 The membrane potential effects on Ca2+ currentunder fixed voltage. (A) The relationship between membranepotential and intraciliary Ca2+ concentration is shown under voltageclamp conditions. (B) The inward current amplitude dependence asa function of a fast shift in the holding membrane potential.Calculations were performed for different concentrations of Ca2+

sensor protein, CaC0. The model predicts qualitatively similarresponses for a relatively wide range of Cac0 concentrations.

2A

-3

0Ca2+

out = 0.1 mM

Ca2+ = 1 mMCa2+

), nA

0 04 0 02 0 00 0 02 0 04

-6

Ca out 1 mM

Ca2+out = 5 mMm

ax (I

C

-0.04 -0.02 0.00 0.02 0.04

Membrane potential alteration, V

V 2 5 VB

-2

0Vm= 2.5 mV

V 30 V

Vm= 12.5 mV

Ca2+

), nA

B

-6

-4

Vm= 37.5 mV

Vm= 30 mV

max

(I

-0.02 0.00 0.02 0.04

Membrane potential alteration, VFigure 7 The model predictions for Ca2+ current in response tomembrane potential shift. The model predictions for Ca2+ currentamplitude shown as a function of extracellular Ca2+ concentrationand the initial membrane potential values before the voltage shiftwas applied. (A) The dependence of Ca2+ current amplitude on thetransmembrane potential alterations is calculated for differentextracellular Ca2+ concentrations. (B) The graph shows the Ca2+

current amplitude dependence on the initial values of the potentialwhen the depolarising voltage shift was applied. The model showsthat the ciliary system properties are reasonably conserved undersignificant variations of extracellular Ca2+ and over a wide range ofmembrane potential values.


Page 9 of 26

allows for the generation of single impulses of variableamplitude proportional to the degree of membranedepolarisation caused by variations in the external con-centrations of ions. This system is able to generate asingle spike despite unpredictable variations of ionicconcentrations in the environment and is, therefore,very robust to alterations in the external conditions.Another interesting aspect of the ciliary excitation is theability of the system to generate regulatory intraciliaryCa2+ impulses proportional to the degree of membranedepolarisation (Figures 3 and 5). This property canallow cells to sense and “automatically” respond toalterations in their environment.

The contribution of K+ currentsIn the previous section, we analysed the dynamic prop-erties of the intraciliary Ca2+ system under voltageclamp conditions. Several lines of evidence suggest thatK+ currents contribute to the currents registered in ciliaunder voltage clamped conditions. The existence of K+

currents in cilia is supported by a number of experimen-tal studies. The experimental data shows that the mea-sured current is not equal to zero when the membranepotential equals the equilibrium membrane potential forCa2+ ions. Instead, the current equals zero when mem-brane potential is about 10 mV while the equilibriumpotential for Ca2+ ions equals 120 mV [66]. This obser-vation suggests that both Ca2+ and K+ currents contri-bute to the overall current measured at early stages ofcurrent registration under voltage clamp, and thereforeboth currents need to be taken into the consideration inorder to advance understanding of the mechanismsinvolved in ciliary regulation. At the same time, it hasso far been impossible to register Ca2+ currents by inhi-biting the K+ contribution. Various compounds can onlypartially block the K+ current when applied from insideof the membrane. Ciliary K+ currents have also beenmeasured separately from Ca2+ currents.In order to account for the contribution of K+ cur-

rents to the regulation of intraciliary Ca2+ concentration,we developed a model for the regulation of K+ currentsby membrane potential (equation (35) in Methods). Thedependence of K+ conductivity on the membrane poten-tial is described by equation (36). Figure 8 shows themeasured experimental values and the approximatingcurve calculated according to equation (36). The currenton Figure 8 is normalized to 1 when membrane poten-tial equals 0. In order to approximate the current, weonly used the experimental values obtained under themembrane depolarized conditions, and this experimentaldata is approximated by equation (36). The predictionsfor intraciliary Ca2+ dynamics and Ca2+ impulse ampli-tude in response to the shift in membrane potential areshown 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(Figure 9A) with the model when K+ channels are notincorporated (Figure 3 and 5), 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 ofvoltage clampIn the previous sections we investigated the mechanismsof the transmembrane potential shift-dependent Ca2+

spike generation under voltage clamp conditions. How-ever, Ca2+ currents themselves can alter the membranepotential. Here we incorporate the membrane potentialdependence on Ca2+ currents and investigate the mem-brane potential dynamics in the absence of voltageclamp (equations (40) and (41)). The non dimensionalCa2+ concentration and membrane potential aredescribed by equation (42).The predictions for the individual Ca2+ and K+ current

responses to various membrane potential shifts areshown in Figure 10. One can note that the dynamics ofthe responses significantly differs between the onesobtained under voltage clamp conditions and the situa-tion when the membrane potential is not fixed, butdependent on the currents (Figures 3, 5 and 9). Voltagecurrent characteristics for Ca2+, K+ and full currents cal-culated using the current amplitudes from the currentsdynamics are shown on Figure 10H. One can note that

4

6

nt, n

A

KI

mV

0

2

Ca2+

cur

ren K m

0.00 0.02 0.04 0.06-2

C

Membrane potential, VFigure 8 Current-voltage characteristic for the steady-state K+

current. The model predictions for the K+ current as a function ofmembrane potential are compared with the experimentallymeasured voltage-current. The model elicits accurate agreementwith the experimental data for the physiologically relevant range ofmembrane potential alterations.


Page 10 of 26

the current is an almost exponentially growing functionof membrane potential.The monotonic dependence of Ca2+ current on trans-

membrane potential, and simultaneous Ca2+-dependentinhibition of Ca2+ channels, represents a classical pro-blem of two interconnected variables: intraciliary Ca2+

and membrane potential. In this system, increasing Ca2+

current with transmembrane potential depolarisationrepresents 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 poten-tial dynamical properties of the ciliary system emergingfrom the coupling of Ca2+ current and membranepotential described by equations (42).Figure 11 shows the Ca2+ current and membrane

potential dynamics and the phase diagrams for anincreasing range of inward current. We found that theinward current into the cilium can modify the dynamicproperties of the Ca2+-membrane potential system. In

all cases, the null clinedVdt

= 0 represents the N-shaped

curve. In the physiological range of non-dimensionalmembrane potential V (50 mV, -0.025 mV) and intracili-ary Ca2+ (u) from 0.04 to 40 μM, the null clinedCa2+

dt= 0 shows a monotonic growth.

One can clearly see that there is significantly differentresponse for different values of the inward current.When the influx of the ions is relatively small, thedCa2+

dt= 0 null cline intersects the

dVdt

= 0 null cline in

the left descending area (Figure 11A); such a null clinecrossing results in a stable solution. In this case the sys-tem responds by the generation of a single impulse ofboth intraciliary Ca2+ concentration and the membranepotential followed by a return to homeostatic levels (Fig-ure 11A, B and 11C). Further increasing the current

causes the null cline dCa2+

dt= 0 to intersect with the null

clinedVdt

= 0 in the middle region of the ascending area,

leading to an unstable solution with a limit cycle formedaround the area that represents the oscillations. (Figure11D, E and 11F). However, further increase of the cur-

rent causes the dCa2+

dt= 0 null cline to intersect with

the null clinedVdt

= 0 in the right descending area,

resulting in a stable solution with a slight increase ofthe homeostatic Ca2+ and membrane potential levels(Figure 11G, H and 11I). The key conclusion from thisanalysis is that the external ionic conditions can initiateessentially different dynamic properties of the systemregulating ciliary movement. One of the key factors thataffect the ciliary beat cycle is the level of intraciliary Ca2+. Our findings suggest that in response to the externalconditions, there are several possibilities for intraciliaryCa2+ upregulation. The system can generate a singlespike (Figure 11B) of variable amplitude (data notshown), permanently increase Ca2+ in a dynamic fashionand maintain the high intraciliary levels (Figure 11E), oroperate in a monostable multivibrator mode (cilia can

A

45

60

Vm= 12.5 mV

Vm= 0 mV

Vm= -12.5 mV

atio

n,M

15

30

Vm= 25 mV

Vm= 17.5 mV

m

a2+ c

once

ntra

0 2 4 6 8 100C

a

Time, secB

30

45

60

a2+),

M

0 01 0 00 0 01 0 02 0 030

15

30

max

(Ca

-0.01 0.00 0.01 0.02 0.03

Membrane potential alteration, VFigure 9 Ca2+ concentration dynamics response totransmembrane potential shift in the presence of K+ current.(A) The intraciliary Ca2+ dynamics is computed in response totransmembrane potential shift by incorporating the contributionfrom K+ channels to the overall current. Comparison of these resultswith Figures 3 and 5 suggests that while K+ currents introducesome quantitative changes, the overall qualitative characteristics ofthe response remain the same. The non dimensional membranepotential values following voltage shift are indicated for eachpredicted response. (B) The amplitude of the generated Ca2+

concentration response is shown as a function of membranepotential alteration. The amplitude of the current is approximatelylinearly inversely proportional to the magnitude of the appliedmembrane potential shift.


Page 11 of 26

5

10

Vm = -12.5 mV

IK+

IFullCa2+

rent

, nA

5

10 Vm = -5 mVIK+

IFullCa2+

ent,

nA

A B

0.0 0.2 0.4 0.6 0.8 1.0-5

0ICa2+

Cur

r

Time, sec0.0 0.2 0.4 0.6 0.8 1.0

-5

0ICa2+C

urre

Time, secC D ,

0

5

10

IFullCa2+

Vm = 0 mVIK+

rent

, nA

0.0

2.5

5.0IFull

Ca2+IK+

rrent

, nA

C D

0.0 0.2 0.4 0.6 0.8 1.0-5

0

ICa2+Cur

r

Time, sec0.0 0.2 0.4 0.6 0.8 1.0

-5.0

-2.5 ICa2+Vm = 5 mVC

ur

Time, sec5 0E F

2 5

0.0

2.5

5.0

ICa2+

IFullCa2+IK+

V = 12 5 mVCur

rent

, nA

1

0

1

2

I

IFullCa2+IK+

Vm = 20 mV

rrent

, nA

E F

0.0 0.2 0.4 0.6 0.8 1.0-5.0

-2.5 Vm = 12.5 mVC

Time, sec0.0 0.2 0.4 0.6 0.8 1.0

-2

-1 ICa2+

Cu

Time, sec5 0G H

-1

0

1

2

I

IFullCa2+IK+

Vm = 25 mV

Cur

rent

, nA

0.0

2.5

5.0

IFull

IPS

rrent

, nA

G H

0.0 0.2 0.4 0.6 0.8 1.0

-2ICa2+C

Time, sec 0.00 0.02-5.0

-2.5ICa2+

Cu

Vm, VFigure 10 The dynamics of current alterations in the absence of voltage clamp. The Ca2+ and K+ current dynamics were calculated inresponse to the membrane potential shift in the absence of voltage holding conditions (A-G). The membrane potential was depolarised fromthe normalised value V0 = -30 mV to -12.5 mV (A) and 25 mV (G), which represent the smallest and the largest change shift, respectively. Thecurrents elicited by the membrane potential alteration within the range are shown in (B)-(F). ICa2+, IK+ and IFull represent calcium, potassium andfull currents, respectively. The comparison with the Ca2+ current responses to membrane potential shift obtained under the voltage clampedconditions (Figure 3, 5 and 9) reveals significant differences in the dynamics. (H) Voltage current characteristic is calculated using the currentamplitudes from the currents dynamics for the full (IFull) and Ca2+ (ICa2+) currents. The steady-state voltage current relationship is calculatedaccording to the stationary current values, (IPS).


Page 12 of 26

generate a Ca2+ spike in response to any alteration ofmembrane potential) (Figure 11H). These three possibi-lities can be associated with the different modes of cili-ary beat observed in human cilia as well as in variousciliates.The dynamic properties of excitable systems with two

interdependent variables are reasonably well understoodat a theoretical level. In the present case, Ca2+ and

membrane potential represent the slow and fast vari-ables, respectively. This study, therefore, establishes thatthe dynamic properties of ciliary systems, where the Ca2+ and K+ channel conductivities represent monotonicfunction of membrane potential and the Ca2+ channelsconductivity inversely depends on intraciliary Ca2+ con-centration, are comparable with the properties of excita-ble systems based on the “N-shape” dependence of the

30dVm/dt = 0 dCa2+/dt = 0

on,

M

A

30

, M

B0.06

ntia

l, V

C

10

20

once

ntra

tio

10

20

ncen

tratio

n,

0.02

0.04

rane

pot

en

0 0.05I 0 0.05I

0 0.05I

0.00 0.02 0.04 0.06Ca2+

co

membrane potential, V0 20 40 60 80 100

10

Ca2+

con

Time, sec0 20 40 60 80 100

0.00

mem

br

Time, sec

30

Limit cycle

dVm/dt = 0 dCa2+/dt = 0

on,

M

D E F

30

ion,

M

0 04

0.06

entia

l, V

0 0.07I0 0.07I

10

20Limit cycle

conc

entra

tio

10

20

conc

entra

ti

0.02

0.04

bran

e po

te

0 0.07I

0.00 0.02 0.04 0.06

Ca2+

c

membrane potential, V0 40 80 120 160C

a2+ c

Time, sec0 40 80 120 160

0.00

mem

b

Time, sec

G H

30

40

dVm/dt = 0 dCa2+/dt = 0

tion,

M

G H I

0.04

0.06nt

ial,

V

30

40

on,

M

0 15I

10

20

30

conc

entra

0 00

0.02

bran

e po

te

10

20

30

conc

entra

tio

0 0.15I

0 0.15I

0 0.15I

0.00 0.02 0.04 0.06Ca2+

membrane potential, Vm

0 40 80 120 160

0.00

mem

b

Time, sec0 40 80 120 160

Ca2+

c

Time, secFigure 11 The model predictions for coupled intraciliary Ca2+ concentration and membrane potential alterations. The systems modelfor intraciliary Ca2+ concentrations, Ca2+ and K+ currents coupled with membrane potential responds to applied inward current in a dynamicallydiverse fashion. The dynamic mode of the ciliary system is defined by the intersection of nullclines shown on phase diagrams for threerepresentative values of inward current, I0. A. A small inward current, I0, initiates the generation of a small pulse of intraciliary Ca2+ (B) andmembrane potential (C). D. A larger inward current can shift the system into the mode of sustained oscillations of both intraciliary Ca2+ (E) andmembrane potential (F). G. Significantly higher values of inward current shift the intersection of the null clines to a higher region with a stablesolution. This leads to a switch type of response to a higher steady-state level of Ca2+ (H) and membrane potential (I). The model describes anovel mechanism of ciliary excitability and predicts that the ciliary system can generate either a single impulse, generate sustainable oscillations,or operate as a switch between lower and higher Ca2+ and membrane potential levels.


Page 13 of 26

Na2+ channel conductivity on membrane potential [74].At the same time, it is essential to note that themechanism of excitation described in motile cilia is dif-ferent from the “classical” one described in most excita-ble cells and systems that involve IP3 Ca2+ channels[75,76].

The membrane hyperpolarisation-dependent currentsmodulate the excitatory properties of the ciliary systemThe ciliary transmembrane potential can shift in twodirections. In the previous section we investigated theintraciliary Ca2+ responses caused by membrane depo-larisation. Here we assess the implications of the mem-brane hyperpolarisation which has been shown toactivate the current from cilia into the cell body [77,78].We introduced the corresponding term into our modelfor the Ca2+ ions movement via the membrane as afunction of the corresponding membrane potential shift(equation 43). By assuming the potential independentmechanism for Ca2+ and K+ ion expulsion, the systemof intraciliary Ca2+ and membrane potential is derivedas shown in equation (46) in the Methods section.The model predictions for intraciliary Ca2+ and mem-

brane potential dynamics in the absence of the hyperpo-larisation-induced current are shown in Figure 11D, Eand 11F. In this case the system remains in the mode ofsteady oscillations. We found that such oscillatory modecan be significantly modulated by the hyperpolarisation-induced and the external conditions-dependent current.Figure 12A shows the phase diagram in the presence ofthe hyperpolarising current. In contrast to the situationdescribed on Figure 11 when the hyperpolarising currentwas absent, the ciliary system generates a single spike inresponse to alteration of external ionic concentrationsover the whole physiological range of the inward current(Figure 12B and 12C). Under certain combinations ofthe hyperpolarisation-induced currents and the Ca2+-dependent K+ currents the phase diagram modifies ina manner so that the system responds by generating asingle spike in response to alteration of the inward cur-rent of any magnitude (Figure 13). This indicates thatthe ciliary system under membrane hyperpolarising con-ditions can become a monostable multivibrator.

The role of cilia-to body Ca2+ current under membranehyperpolarisationIntraciliary Ca2+ concentration has been experimentallyestimated to be approximately one order of magnitudehigher in comparison with the intracellular levels. TheCa2+ current generated by the ion flow from the ciliarycompartment into the cell has been reported by a num-ber of groups [28,52,53], but the role this current playsin the regulation of the ciliary beat remains unclear. Inorder to address this question, we introduced the term

for this current into the equation that describes theintraciliary Ca2+ concentration (equation (47) in theMethods section). In the absence of direct measure-ments of the dependence of conductivity on membranepotential we set the conductivity to increase in responseto hyperpolarisation of membrane potential. A summaryof the model responses for all channel conductivities asa function of membrane potential is shown in Figure 14.By numerically solving the coupled equations for intra-ciliary Ca2+ and membrane potential with the cilia tothe cell body contribution, we found that this current

80

I 0 AI0 = 27.5 nA

dCa2+/dt = 0 dVm/dt = 0

on,

M

A

20

40

60

I0 = 5 nA

I0 = 0 nA

I0 = 15 nA

0

2+ c

once

ntra

tio

-0.02 0.00 0.02 0.04 0.060C

a2

membrane potential, V

B

40

60

80

I0 = 27.5 nA

ntra

tion,

M

0 50 100 150 200

20

40

I0 = 5 nAI0 = 15 nA

Ca2+

con

ce

TiTime, secFigure 12 The “monostable multivibrator” mode occurs inresponse to membrane hyperpolarisation. The model predictionsare shown for the ciliary response to membrane hyperpolarisation.The phase diagram analysis (A) suggests that the nullclines for Ca2+

and membrane potential V always intersect at a single point thatcorrespond to a steady-state solution in the whole physiologicalrange of possible inward current values, I0. The model shows that thistype of system always generates a Ca2+ impulse in response to anyalteration of inward current, I0, within the physiological range (B).


Page 14 of 26

does not qualitatively change the general dynamic prop-erties of the system.Despite the lack of a noticeable contribution to the cili-

ary dynamic properties, this current requires a specialconsideration. Experimental studies have clearly demon-strated that intraciliary Ca2+ is significantly higher thanintracellular Ca2+ concentration. At the same time, if theconductivity of protein structures governing the Ca2+

ions movement from cilia to the body is high, most ofthe intraciliary ions would move from cilia into the cellbody in a very short time. A simple calculation suggeststhat if Ca2+ could freely flow from cilia into the body, theintraciliary concentration would become equal to theintracellular Ca2+ concentration in less than 100 μs dueto the difference in the volumes of the cell body andintraciliary compartments. Experimental measurementsin ciliates show that the hyperpolarisation-induced back-wards movements can last longer than 100 μseconds. It

is also known that the avoidance reaction that requireslong term elevation of intraciliary Ca2+ concentration canbe observed in hyperpolarizing solutions. During all thistime the intraciliary Ca2+ concentration can be severalorders of magnitude higher than the intraciliary concen-tration. In this study, we have demonstrated that thesteady-state Ca2+ current under the depolarized mem-brane potential conditions can only be reduced by theCa2+-dependent inhibition of Ca2+ channels. All theseobservations suggest that the Ca2+ removal from cilia tothe cell body occurs in a membrane potential dependentmanner.

The mechanism of Ca2+ and cyclic nucleotide-dependentCBF regulationIn addition to intraciliary Ca2+ and K+ potassium levelsbeing coupled with the membrane potential modulation,cyclic nucleotides contribute to the regulation of one of

80 I0 = 5 nAM 4I = -2.5 nAM

A B

60

0

I = 25 nAtratio

n,

3

I = -5 nA

I0 2.5 nA

ratio

n,

20

40

I 10 AI0 = 15 nAI0 = 20 nA

I0 = 25 nA

conc

ent

1

2

I0 = -15 nA

I0 = -5 nA

conc

ent

0 20 40 60 80 1000

I0 = 10 nA

Ca2+

c

Ti0 20 40 60 80 100

0I0 = -25 nA

Ca2+

c

Time, sec Time, sec

dCa2+/dt = 0 dV /d = 0+ 50M

C D

40

50

Cah = 0 9 nA

dCa /dt 0 dVm/d = 0

on, C

a2+

40

50dCa2+/dt = 0 dVm/dt = 0

ion,

M

20

30Cah 0.9 nA

I 0 AI = 15 nAentra

tio

20

30

Cah = 0 nAcent

rat

0 02 0 00 0 02 0 04 0 060

10I0 = 0 nAI0 = 15 nA

2+ c

once

0

10Cah = 0 nA

a2+ c

on

-0.02 0.00 0.02 0.04 0.06

Ca

membrane potential, V0.00 0.04 0.08C

membrane potential, VFigure 13 The hyperpolarisation-mediated Ca2+ spike is proportional to inward current. The effects of hyperpolarisation-mediated currentsare described as a function of inward current magnitude. The computational calculations predict that in the presence of hyperpolarisationinduced currents, the ciliary system always generates Ca2+ impulses proportional to the applied inward current in both positive (A) and negative(B) ranges of values. The comparison of phase diagrams in the presence (C) and absence (D) of the hyperpolarisation-induced current vCahsuggests that the Ca2+ spike generation in response to inward current alteration is due to the hyperpolarisation-mediated currents. In theabsence of hyperpolarisation the solution becomes unstable with a limit cycle and the system undergoes sustained oscillations (D).


Page 15 of 26

the major ciliary beat parameters, frequency. IntraciliaryCa2+ levels activate a variety of adenylate cyclases (AC)and phosphodiesterases (PDE) that produce and hydro-lyse cyclic nucleotides, respectively, and thereby modu-late the intraciliary cAMP and cGMP levels. At thesame time, cAMP and cGMP-dependent kinases phos-phorylate dynein arms [45] in the bases of cilia andthereby induce the ciliary movement [79].In a previous work we showed that the ciliary beat

frequency can have a “double” bell shape dependenceon Ca2+ concentration [65] due to the differential reg-ulation of adenylate and guanylate cyclase isoforms ina Ca2+-calmodulin (CaM) dependent manner [63,64].One bell-shape was due to the cAMP production by acombination of AC and PDE, whereas another one wasmediated by cGMP production and degradation. Ourresults proposed an explanation for seemingly conflict-ing experimental evidence suggesting that CBF canboth decrease and increase with increasing Ca2+ con-centration. Here we extend our previous analysis anddescribe the conditions when one or the other peakcan be significantly reduced or even disappear. Figure15A shows the model predictions for the ciliary beatfrequency in comparison with the experimental data[65,80]. Recent studies demonstrated that hormonesand pharmacological agents can regulate both functionand structure of cilia by interfering with the cyclicnucleotide signalling pathways [81,82]. These findings

support the possibility for the development of noveltherapeutic strategies for ciliary pathologies by modu-lating the dynamic mode of beating via ciliary mem-brane receptors [83].Figure 15B and 15C demonstrate that the “amplitude”

of each peak can be significantly diminished if the activ-ity of the AC or GC, respectively is modulated by a tem-porary or permanent, internal or external signal. Undersuch a scenario, CBF can only increase or decrease if ithappens to be on one slope of the bell-shaped depen-dence. Therefore, according to our analysis, differentorganisms with the same underlying ciliary regulatorysystem can achieve all possible CBF regulatory modes asa function of Ca2+ concentration: the reverse bell-shapeddependence, if the “peak” values shown on Figure 15Aoccur at the lower and higher limits of the physiologicalrange for Ca2+ concentration, the bell shape dependencethat can be either cAMP and cGMP dependent, andeither monotonic increase or decrease if the physiologi-cal range of Ca2+ concentrations occur at one of theslopes. Our model, therefore, describes the core Ca2+-dependent regulatory mechanisms of cilia beat, butalso provides an explanation for the differences observedbetween cilia in different single cell organisms as well astissue specific differences. It also unravels the mechan-ism for how various stimuli modulate the rate of CBFby signalling via Ca2+- and G-protein mediatedpathways.

DiscussionWe develop a new computational model for Ca2+ andmembrane potential-dependent ciliary regulation thatexplains how different ciliary beating regimes are regu-lated. The model describes a novel mechanism of excit-ability based on the membrane potential-dependence ofCa2+ currents (Figure 2) and simultaneous intraciliaryCa2+-concentration mediated inhibition of Ca2+ channels(Figure 4). Our analysis shows that motile cilia consti-tute an excitable system with a novel mechanism ofexcitability. The ciliary system is able to generate a Ca2+

spike in response to a wide range of transmembranedepolarisation (Figure 3, 5 and 9). The major differencein the ciliary excitation described here, with respect toclassical excitation mechanisms, is that ciliary excitabil-ity is robust to a wide range of ionic variations in theenvironment.The excitability mechanism of cells in evolutionary

advanced organisms is based on a combination of theN-shaped dependence of the quick inward cationic cur-rent on the transmembrane potential and slow altera-tions of the K+ conductivity [84-87]. The ciliary voltage-current characteristic (Figure 10H) suggests severalfunctional dynamic modes of operation: i) single impulsegeneration, ii) oscillator, iii) trigger (Figure 11), all

1 0 (V )

0.6

0.8

1.0

(V )

h(Vm)(Vm)

ctiv

ity,

0.2

0.4t(Vm)

K+(V )ve c

ondu

c

-0.04 0.00 0.04 0.080.0

K ( m)

rela

tiv

Membrane potential, VFigure 14 Key ciliary ion channels conductivity dependence onmembrane potential. The overall ciliary excitability properties aredue to the unique combination of ion channel conductivitydependence on membrane potential. The normalised ion channelconductivities are shown for Ca2+ currents v(V), K+ currents νK+(V),hyperpolarisation-induced currents vh(V), and the cilia-to-cellcurrents vt(V). The hyperpolarisation and cilia-to-cell currents areactivated by membrane hyperpolarisation, whereas the activation ofCa2+ and K+ channels takes place under depolarised conditions.


Page 16 of 26

initiated by membrane depolarisation. At the same time,the hyperpolarisation-induced Ca2+ currents switch thesystem into the mode of a monostable multivibrator,when cilia can generate a Ca2+ spike in response to anyalteration of membrane potential. The dynamics of sucha system depends on the transmembrane potential. Inother words, any alterations in the transmembranepotential (for example, initiated by variations of theexternal ion concentrations) switch functional perfor-mance of the system or make it non-excitable.It was originally believed that Ca2+, cAMP and cGMP

each represent an independent pathway of ciliary regula-tion, however, there is by now a significant amount ofevidence that strongly suggests that all three pathwaysare intimately interconnected [88]. It is well establishedthat cAMP and cGMP are synthesized by AC isoformsand hydrolysed by PDEs in a Ca2+-CaM-dependentmanner. In this work we describe the mechanism of thecross talk between the three circuits and explain howCBF can be modulated via extra- and intraciliary path-ways (Figure 15).

ConclusionsTherapeutic applications of systems model for intraciliaryCa2+ regulationOur detailed analysis of the effects of several Ca2+ andpotassium currents and membrane potential on intracili-ary Ca2+ levels offers a new way of interpreting ciliarymotility associated pathologies. There are two main Ca2+-mediated parameters that govern the motile functionof cilia: the direction and the frequency of beat. Ourmodel shows that there can be several dynamic regimesof intraciliary Ca2+ alterations during which the intracili-ary Ca2+ concentration can be either at low or highlevels, temporarily or for a significant period of time(Figure 16A). Experimental evidence suggests that highCa2+ reverses the direction of cilia strike (Figure 16B),and modifies the frequency in a highly nonlinear man-ner (Figure 15) via synthesis and hydrolysis of cyclicnucleotides (Figure 15). In a previous work [89], weshowed that genetic mutations that alter the dynamicproperties of a system in a permanent manner can leadto disease. In this study, we propose a conceptuallysimilar mechanism for the pathologies associated withciliary motility that can take place in some human dis-eases [6]. Our study suggests that while CBF regulationin a multicellular organism can be modulated by a num-ber of intracellular and extracellular factors (Figure 15),genetic mutations that directly or indirectly, affect theCa2+-mediated CBF dependence can dramatically impairthe essential processes such as clearing function in air-ways, male fertility, or the determination of the left-rightaxis during development (Figure 16C and 16D). Thetreatment of the pathologies associated with this

0.8

1.0

eque

ncy

A

2 1 0 1 20.0

0.2

0.4

0.6

Cili

a be

at fr

e

-2 -1 0 1 2C

Ca2+ concentration, log(u)

y

B

0 4

0.6

0.8

1.0

at fr

eque

ncy

cGMP

-2 -1 0 1 20.0

0.2

0.4

Cilia

bea

2+Ca2+ concentration, log(u)

1.0

ncy

G

C

0 2

0.4

0.6

0.8

beat

freq

uen G

-2 -1 0 1 20.0

0.2

Cili

a b

Ca2+ concentration, log(u)Figure 15 Predictions for ciliary beat frequency modulation. (A)The comparison of experimental data and model predictions for theCBF dependence on Ca2+ concentrations. The experimental data forCBF [80] is shown as circles whereas the model prediction isrepresented as continuous blue line [65]. (B) GC activity depends ona number of intracellular mediators. The model describes thecorrelation of the right “peak” with the GC activity levels. (C) Themodel predicts that extracellular signals via G-protein pathway cansignificantly modulate the left “peak” of the plot of CBF dependenceon Ca2+ concentration. In both cases (B and C) the diminishingamplitude is shown as green and red lines. The model describesthe distinct contribution of both internal and external signals to CBFmodulation.


Page 17 of 26

mechanism would rely on the restoration of the originalCa2+-dependent CBF dependence.

Future perspectiveAt present there is limited understanding of the underly-ing biological mechanisms that govern ciliary motility.This study describes the modes of intraciliary Ca2+

dynamics in a highly detailed fashion. It shows the condi-tions that switch the system between the modes of Ca2+

spike generation, oscillatory dynamics and a trigger. Theinterdependent influences of Ca2+ and K+ currents, trans-membrane potential and cyclic nucleotides modulate the

ciliary beat frequency and the direction of beat in a highlynonlinear manner. The further development of mathe-matical models of this system is still required to representciliary movements as a function of Ca2+ concentrationand obtain the detailed understanding of ciliary motilitywhich will be crucial for the development of new treat-ments for human diseases. While the core protein regula-tory machinery involved in ciliary motility is very likely tobe conserved, some variations in response to increasedCa2+ between single cell ciliates and mammalian ciliahave been reported [90]. We would argue that those dif-ferences are not due to the change in the mechanisms ofCa2+-dependent regulation but are rather caused by var-iations in the parameters of the regulatory circuits. Thefurther investigation of single cell ciliates may allow agreater degree of characterisation of ciliary movementmechanisms, because in these systems alterations of cili-ary motility translate into movement trajectories whichcan be easily observed.

MethodsModel DescriptionFigure 1 provides a schematic outline of the networkregulating intraciliary Ca2+ concentration that is consid-ered in our model. Intraciliary Ca2+ concentration isregulated by the currents of passive and active Ca2+

transport, as well as by Ca2+ leak into the extracellularspace and into the cell body.A basic mathematical model for intraciliary Ca2+ con-

centration and its relationship to transmembrane poten-tial was proposed for the first time in [91]. A largenumber of recent experimental findings now allow theformulation of a more advanced model that includes thecrucial aspects of the molecular mechanisms governingcilia movement. Below we describe the complete modelfor intraciliary Ca2+ regulation developed in this study.The dynamics of intraciliary Ca2+ alteration are given

by:

VR · d[Ca2+

]dt

=SR

z · F· (IP

Ca2+ + IACa2+ + Iu

Ca2+

)+ IT

Ca2+ + J([

Ca2+] , [CaM0])

, (1)

where VR - is the cilium volume, SR - is the ciliumsurface area, and IP

Ca2+ and IACa2+-are the Ca2+ currents

through the channels of passive and active Ca2+ trans-port, respectively. IT

Ca2+ is the current from the ciliuminto the cell body. Iu

Ca2+-is the Ca2+ leakage current.J([

Ca2+]

, [CaM0])is the function that encounters Ca2+

binding to and release from CaM, the main Ca2+ bind-ing protein in cilia, z = 2 is the Ca2+ ions charge, and Fis the Faraday constant.The dynamics of Ca2+ concentration alterations within

the cilium are defined by the individual contribution of

A B

low Ca2+ concentrationhigh Ca2+ concentration

modifies the direction of beat

healthy body development situs inversus pathology

CiliaryC D

healthy body development

CiliaryPathology 1

CiliaryPathology 2

Figure 16 Systems model for ciliary excitation offers newavenues for mechanistic interpretation of ciliary pathologies.The systems model for intraciliary Ca2+ regulation offers newstrategies for interpretation of experimental data and developmentof pharmaceutical interventions for ciliary motility-associatedpathologies. The ciliary system is predicted to maintain either low(A) or high (B) levels of intraciliary Ca2+. The extracellular conditionscan shift the functional modes of ciliary activity and cause atemporal, repetitive or a long term Ca2+ increase which causes ciliato reverse the direction of beat. The long term reversal of thedirection of beat can explain the mechanism of the situs inversusdisease, which is a congenital condition in which the major organsare mirrored with respect to their normal positions. Intraciliary Ca2+

levels modulate ciliary beat frequency [65] via either an externalsignal through the G-protein mediated pathways or by parametricregulation of GC activity. Such alterations can represent aphysiological response to external and intracellular signals, but canalso occur as a result of genetic mutations. According to our model,the latter case represents a potential pathology and in either caseof permanent Ca2+-dependent CBF alteration (C or D) requires thedevelopment of therapeutic strategies to rescue the mutation-mediated alteration of the system.


Page 18 of 26

the Ca2+ currents. The current via the channels of pas-sive Ca2+ transport is given by:

IpCa2+ = ipCa2+ · Np

Ca2+ , (2)

where NpCa2+ is the density of functionally active Ca2+

channels. ipCa2+ is the time averaged current via a singleCa2+ channel. Ca2+ channel activity is modulated exter-nally by a number of metabolic pathways. Therefore, weonly consider the pool of functionally active Ca2+ chan-nels. The time-averaged current via a single Ca2+ chan-nel is given by:

ipCa2+ =∑

i

giCa2+(Vm, Ca2+) · (Vm − ECa2+), (3)

where giCa2+(Vm, Ca2+) 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), ECa2+ =(

R · T2 · F

)· ln

(Ca2+

out

Ca2+in

)is

the Ca2+ potential in the equilibrium, Vm is the trans-membrane potential of the cilia membrane.The Ca2+ leakage current is given by:

IuCa2+ = gu

Ca2+ · (Vm − ECa2+ ) (4)

Assuming that the active Ca2+ transport systemextrudes one Ca2+ ion per cycle, the current generatedby the plasma membrane Ca2+ pump is given by:

IACa2+ = iA · NA

Ca2+ , (5)

where NACa2+ is the density of the plasma membrane

Ca2+ pump protein complexes bound to one Ca2+ ion,and iA is the time averaged Ca2+ current via a single Ca2+ channel. By introducing further assumptions that allthe channels of active Ca2+ transport are saturated byATP and that all bound Ca2+ molecules are releasedinto the extracellular space, this current would bedefined by the dynamics of the active transport channelsbound to a Ca2+ ion:

dNACa2+

dt= kp

A · [Ca2+r

] · (N00Ca2+ − NA

Ca2+

) −(km

A + kpA

)· NA

Ca2+ , (6)

where N00Ca2+ is the density of the active Ca2+ transport

channels, kpA and km

A are the association and dissociationconstants for the Ca2+ ion interaction with the activeCa2+ transport channels, respectively. kp

A is the constantthat defines the Ca2+ ion transition from the boundstate into the Ca2+ channel. By introducing new non-dimensional variables:

ω =NA

Ca2+

N00Ca2+

, η = nm · t, ka =kp

A · KCaM

nm, kb =

kmA + kp

A

nm, u =

Ca2+

KCaM,

(where nm is the dissociation constant of the proteinregulating the passive Ca2+ transport channels), equation(6) takes the following form:

dω

dη= ka · u · (1 − ω) − kb · ω, (7)

The steady state-current through these active trans-port channels is given by:

IACa2+ = τ · u

kA + u, (8)

where kA =km

A + kpA

kpA · KCaM

, τ = iA · N00Ca2+ , u =

Ca2+

KCaM.

The model in [91] represented the leakage currentfrom cilium into the cell body by the following expres-sion:

i = ϑ · (Ca2+r − Ca2+

t ). (9)

where ϑ is the effective diffusion constant, andCa2+

t and Ca2+t are the intraciliary and intracellular Ca2+

concentrations, respectively. The Nernst equations allowa more accurate modelling of the leakage current fromcilium into the cell body as follows:

ITCa2+ = gt(Vm) ·

(Vrt − R · T

2 · F· ln

[Ca2+r ]

[Ca2+t ]

), (10)

where gt(Vm) is the overall conductivity of the ciliumbase area, Vrt is the difference of the potential betweencell body and cilia, [Ca2+

r ]is the intraciliary Ca2+ concen-tration, and [Ca2+

t ] is the intracellular Ca2+ concentra-tion. Equation (10) can then be represented in thefollowing non dimensional form:

ITCa2+ = β1 ·

(ψrt − 0.5 · ln

uut

), (11)

where

β1 =gt(Vm) · R · T

F, ψrt =

Vrt · FR · T

, u =Ca2+

r

KCaM, ut =

Ca2+t

KCaM.

In the following sections we derive the models andanalyse the individual contributions of the differenttypes of Ca2+ currents to the intraciliary Ca2+

homeostasis.

Model for intraciliary Ca2+-dependent Ca2+ channelconductivity inhibitionIn this model we assume that Ca2+ channels locatedwithin cilia have a Ca2+ binding site on the intracellularsite of the channel. According to such a model, Ca2+

ion binding to that site mediates the channel’s transitioninto the closed state with no conductivity. We furtherassume that the characteristic time for the transition


Page 19 of 26

from the conductive to the non conductive states ismuch smaller that the characteristic Ca2+ alterationtimes. In that case, the dynamics of the Ca2+ channelstransition into the closed state in response to Ca2+

increase is given by:

d [N]

dt= −np · [N] · [

Ca2+] + nm · ([N0] − [N]) , (12)

where N is the number of channels in the open state,N0 is the total number of channels, np and nm are theassociation and dissociation constants for the interactionof Ca2+ ions with the Ca2+ channels, respectively. Thesteady-state solution of equation (12) is given by:

[N] = [N0] · KC

KC + [Ca2+], (13)

where KC =nm

np. In the non dimensional form this

solution is given by:

n =kC

kC + u, (14)

where n =[N]

[N0], kC =

KC

KCaM, u =

Ca2+

KCaM.

When u = 0, n = 1 and for u = ∞ n = 0, the currentvia these channels equals:

ICa2+ = [N] · g (Vm) · (Vm − ECa2+) . (15)

The alteration of the overall conductivity of the Ca2+

channels in the Paramecium cilia in response to theshift of transmembrane potential from V0 to V1 underthe voltage clamped condition is given by:

g(V, t) = g(V1) − (g(V1) − g(V0)

) · exp(

tτp

), (16)

where τp is the characteristic time of the transmem-brane potential alteration from V0 to V1.The equation for the conductivity alterations (16) can

be represented as follows:

g(Vm, t) = g0 · (ν(V1) − (ν(V1) − ν(V0)) · exp(− tτP

), (17)

where ψ =V · FR · T

, v(ψ, t) is the Ca2+ channel conductiv-

ity dependence on the transmembrane potential and ontime.An early study investigated the Ca2+ channels’ con-

ductivity dependence on transmembrane potential ψ inParamecia [66] and approximated it by the followingequation:

ν(ψ) =exp(α · (ψ + d))

λ + exp(α · (ψ + d)), (18)

where a, d and l are the parameter values that allowthe best representation of the available experimentaldata. In this model, the steepness of the dependence ofthe conductivity on membrane potential is representedby the parameter a.For simplicity we assume that the elementary intracili-

ary volume with inward Ca2+ current does not containany Ca2+ binding proteins and Ca2+ is extruded into theextracellular space by the active Ca2+ transport only.The Ca2+ binding to the Ca2+ channels is assumed tooccur much faster than the characteristic times of Ca2+

channel inhibition. Under such assumptions, the intra-ciliary Ca2+ dynamics in response to the transmembranepotential shift under the voltage clamp is given by:

VR · d[Ca2+

]dt

=SR

z · F·(

− [N] · (g(Vm) · (Vm − ECa2+)) − β ·

[Ca2+

]KA +

[Ca2+

])

. (19)

In non dimensional form, equation (19) takes the fol-lowing form:

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

dudη

= s ·

⎛⎜⎜⎝

−b · n(η) ·(

ν(ψ1) − (ν(ψ1) − ν(ψ0)) · exp(

− η

τ0

))·

·(ψ1 − 0.5 · ln

(uout

u

))− u

kA + u

⎞⎟⎟⎠ ,

dndη

= −k · n · u + (1 − n),

(20)

where u =Ca2+

KCaM, uout =

Ca2+out

KCaM,

The initial conditions for the equations (20) when h =0 and ψ = ψ0 are equal to u0 (ψ0) and n(u0), which canbe obtained by numerical solution of these equations.The non dimensional Ca2+ current is equal to:

iCa2+ = −b · n (η) ·(

ν (ψ1) − (ν (ψ1) − ν (ψ0)) · exp(

− η

τ0

))·

·(ψ1 − 0.5 · ln

(uout

u

))− u

kA + u

(21)

Indirect Ca2+ channel conductivity regulationThe second model of intraciliary calcium regulationassumes the Ca2+ binding protein interacts with the Ca2+ channel and thereby causes the Ca2+ channels toclose. By considering that the Ca2+ ion-Ca2+ bindingprotein occurs on a faster time scale than the Ca2+ bind-ing protein-Ca2+ channels interaction, the steady-statesolution for the Ca2+ binding protein in complex with aCa2+ ion is given by:

[CaC] = [CaC0] · [Ca2+]KC + [Ca2+]

, (22)


Page 20 of 26

where [CaC0] is the total concentration of the Ca2+

binding protein and KC =km

kpis the equilibrium disso-

ciation constant.The Ca2+ binding protein interaction with the Ca2+

channels is given by:

d [C]

dt= −np · [C] · [CaC] + nm · ([C0] − [C]) , (23)

where C is the number of channels in the open con-ductive state, and C0 is the total number of channels.The steady-state solution of equation (23) is given by:

[C] = [C0] · KCC

KCC + [CaC], (24)

where KCC =nm

npis the equilibrium dissociation con-

stant for the Ca2+ binding protein-Ca2+ channels inter-action. By combing equations (22) and (24) one obtainsthe dependence of the open Ca2+ channels on the Ca2+

concentration:

c =kC + u

kC + (cac0 + 1) · u, (25)

where c =[C]

[C0], u =

Ca2+

KCaM, cac0 =

CaC0

KCC, kC =

KC

KCaM,

Equation (23) can then be represented in the followingnon dimensional form:

dcdη

= −c · cac0 · ukC + u

+ (1 − c), (26)

where

η = nm · t, c =[C]

[C0], kC =

KC

KCaM, u =

Ca2+

KCaM, cac0 =

[CaC0]

KCC,.

The solution of equation (26) that describes thedynamics of Ca2+ channels in the open conductive statein response to a Ca2+ shift from u0 to u1 Ca2+ level isgiven by:

c (t) = c∞ − (c∞ − c0) · exp(

−(

cac0 · u1

kC + u1+ 1

)· nm · t

), (27)

where

c∞ =kC + u1

kC + (cac0 + 1) · u1, c0 =

kC + u0

kC + (cac0 + 1) · u0,.

The solution (27) suggests that the number of openchannels would change exponentially in response to aCa2+ surge. The characteristic time for such an expo-nential change is given by:

τCa2+ =kC + u(

kC + (cac0 + 1) · u)) · nm

. (28)

For cases when cac0 > > 1 and u > > 1, the character-

istic time approximately equals τCa2+ ≈ 1[CaC0] · np

.

Under the assumption that the alteration of the trans-membrane potential difference influences the Ca2+ chan-nel conductivity, the channel conductivity as a functionof Ca2+ concentration can be represented as follows:

gCa2+ (t) = [C0] · g0 · ν(ψ) · c(t). (29)

When membrane potential changes from ψ0 to ψ1,and the alteration of the intraciliary Ca2+ concentrationis delayed, the conductivity changes from one value toanother exponentially with the characteristic time τV:

g (ψ , t) = g0 · (ν (ψ1) − (ν(ψ1) − ν(ψ0)

) · exp(−t/τV))

. (30)

Here we consider a simplified scenario when only pas-sive and active Ca2+ transport channels are present. Forsuch a model, the alteration of Ca2+ concentration in acilium is given by:

VR · d[Ca2+]dt

=SR

z · F· (IP

Ca2+ + IACa2+

). (31)

By substituting formulas for the passive and active Ca2+ currents into equation (31), one obtains:

VR · d[Ca2+

]dt

=SR

z · F·(

− [C0] · c (t) · g0 · ν (Vm, t) · (Vm − ECa2+) − β ·[Ca2+

]KA +

[Ca2+

])

. (32)

In this equation, we include the kinetics for the activeCa2+ channels due to the assumption that the dynamicsof currents via the active Ca2+ channels is much fasterthan the dynamics of currents through the passive Ca2+

transport.The non dimensional representation of Equations (32)

and (23) is given by:⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

dudη

= a ·

⎛⎜⎜⎝

−b · c(η, u) ·(

ν (ψ1) − (ν (ψ1) − ν (ψ0)) · exp(

− η

τ0

))·

·(ψ1 − 0.5 · ln

(uout

u

))− u

kA + u

⎞⎟⎟⎠ ,

dcdη

= −c · cac0 · ukC + u

+ (1 − c) ,

(33)

where u =Ca2+

KCaM, uout =

Ca2+out

KCaM,

By substituting the non dimensional representation ofthe Ca2+ current (21) into (33) the equation for theintraciliary Ca2+ dynamics (33) take the following form:

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

dudη

= a ·

⎛⎜⎜⎝

−b · c (η, u) ·(

ν (ψ1) − (ν (ψ1) − ν (ψ0)) · exp(

− η

τ0

))·

·(ψ1 − 0.5 · ln

(uout

u

))− u

kA + u+ ot (ψ1) ·

(ψst − 0.5 · ln

(uut

)),

dcdη

= −c · cac0 · ukC + u

+ (1 − c) ,

(34)

where u =Ca2+

KCaM, uout =

Ca2+out

KCaM, .


Page 21 of 26

Potassium currentHere we consider the mechanism of K+ current activa-tion by membrane potential. The full K+ current in acilium is given by:

IK+ = NK+ · g0K+ · νK+ (t, Vm) · (Vm − EK+) , (35)

where NK+ is the number of open K+ channels, g0K+ is

the maximal conductivity, and EK+ is the equilibrium K+

potential.We assume that the K+ channel conductance is

dependent on the membrane potential (the opposite ofthe case of Ca2+ channel conductance). The K+ channelconductance dependence on the membrane potential isgiven by:

νK+ (ψ) =exp (αK+ · (ψ + dK+))

λK+ + exp (αK+ · (ψ + dK+)). (36)

We further assume that in response to the fast mem-brane potential alteration from ψ0 to ψ1, the K+ conduc-tivity is changing exponentially with a characteristictime τK+. In that case the conductivity dynamics overtime are given by:

νK+ (t) = νK+ (ψ1) − (νK+ (ψ1) − νK+ (ψ0)) · exp( −t

τK+

).(37)

The parameters for the K+ channels’ conductivitydependence on the membrane potential can be esti-mated from experimental measurements of the fast K+

current (10 msec) as a function of membrane potential.In such a short time K+ current does not reach themaximum value and the Ca2+ current is almost equal tozero. Under such conditions, the equation for the K+

current is given by:

IK+ = i0 · νK+ (ψm) · (ψ − ψK+) . (38)

The full current measured under the voltage clampconditions is then given by:

iP = −b · c(η, u) ·(

ν (ψ1) − (ν (ψ1) − ν (ψ0)) · exp(

− η

τ0

))·(ψ1 − 0.5 · ln

(uout

u

))−

−b1 ·(

νK+ (ψ1) − (νK+ (ψ1) − νK+ (ψ0)) · exp(

− η

τK+

))· (ψ1 − ψK+) ,

(39)

where b1 = NK+ · g0K+

g0.

The transmembrane potential dynamicsAll the models described above were specifically devel-oped under the voltage clamp conditions. However, Ca2+ as well as K+ currents can alter the membrane poten-tial. In order to account for this effect, we consideredthe membrane potential dynamics in the absence of vol-tage clamp. Having established that the main contribu-tors to the registered currents are the Ca2+ and K+

currents, we analyse the membrane potential dynamicsas a function of Ca2+ and K+ current contributions:

Cm · dVm

dt= ICa2+ + IK+ . (40)

In the non dimensional form the equation for themembrane potential dynamics is given by:

dψ

dη= ρ ·

⎛⎝−b · (

c (η, u) · ν(η, ψ) + νstCa2+

) ·(ψ − 0.5 · ln

(uout

u

))−

−b1 · (νK+ (η, ψ) + νstK+

) · (ψ − ψK+)

⎞⎠ , (41)

where ψ =Vm · FR · T

, u =

[Ca2+

]KCaM

, νstCa2+, and νst

K+ are the

steady-state Ca2+ and K+ channel conductivities, respec-

tively, η = nm · t, ρ =g0

Cm · nm,.

In this equation, we take into consideration the contri-bution of the independent parts of the Ca2+ and K+ cur-rents, conductivities νst

Ca2+ and νstK+, respectively, that do

not depend on membrane potential or Ca2+ concentra-tion. We also use the fact that Ca2+ and K+ channelconductivities change on a much faster time scale incomparison with membrane potential or Ca2+ concen-tration. Such an approximation allows us to employ thesteady-state solutions for the channel conductivities as afunction of membrane potential and Ca2+ concentration.The described assumptions and considerations lead tothe following system of two nonlinear coupled equationsfor intraciliary Ca2+ concentration and membranepotential:

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

dudη

= s ·(

−b · (c (u) · ν (ψ) + νst

Ca2+

) ·(ψ − 0.5 · ln

(uout

u

))− u

kA + u

),

dψ

dη= −ρ ·

⎛⎝−b · (c (u) · ν (ψ) + νst

Ca2+

) ·(ψ − 0.5 · ln

(uout

u

))−

−b1 · (νK+(ψ) + νst

K+

) · (ψ − ψK+) + I0

⎞⎠ ,

(42)

where I0 is the non dimensional inward current.

Currents activated by the membrane hyperpolarisationThe membrane hyperpolarisation current is given by:

IhtCa2+ = Nht

Ca2+ · ghtCa2+ (ψ , t) · (Vm − ECa2+

t

)(43)

where NhtCa2+ is the number of open, membrane hyper-

polarisation-activated Ca2+ channels located on the cellbody, ght

Ca2+is the conductivity of the channel, and ECa2+t is

the equilibrium potential for Ca2+ ions.

ghtCa2+ (ψ , t) = g0 · νh (ψ , t) , (44)

where

g0 = max(ght

Ca2+ (ψ , t))

, νh (ψ) =exp (αh · (ψ + dh))

λh + exp (αh · (ψ + dh))

By combining the contribution from different types ofcurrents, one can derive the dynamics of the membrane


Page 22 of 26

potential alteration as follows:

Cm · dVm

dt= nr ·

((−N0

Ca2+ · (w1 · g1Ca2+ + w3 · g3

Ca2+

)+ Nht

Ca2+ · ghtCa2+

)·(

Vm − R · T2 · F

· ln(

uout

ut

)))− iA · N00

Ca2+ · ω+

+(N01

K+ · g1K+(t, Vm) + N02

K+ · g2K+(t, Vm) + N03

K+ · g11K+(Vm) · ν1 + N04

K+ · g22K+(Vm) · ν2

) · (Vm − EK+) − IAK+ .

(45)

The coupled system of differential equations for theCa2+ and membrane potential dynamics can be furtherformulated as:

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

dudη

= s ·(

−b · (c (u) · νCa2+ (ψ) + νstCa2+

) ·(ψ − 0.5 · ln

(uout

u

))− u

kA + u

),

dψ

dη= −ρ · ((−b · (c (u) · ν (ψ) + νst

Ca2+

) − νCah · νh (ψ)) ·

(ψ − 0.5 · ln

(uout

ut

))−b1 · (

ν1K+ (ψ) + νKca · ν2

K+ (ψ , u) + νstK+

) · (ψ − ψK+) + I0,

− (46)

where vh(ψ) is the Ca2+ current contribution, activatedby membrane depolarization, and νK+ (ψ , u) is the Ca2+-dependent K+ current contribution.

Cilia-to body Ca2+ currentThe incorporation of the Ca2+ current from cilia to cellinto the equations for the intraciliary Ca2+ and mem-brane potential system leads to an additional term beingadded to the equations (46):

du

dη= s ·

⎛⎜⎜⎝

−b · (c (u) · ν (ψ) + νst

Ca2+

) ·(ψ − 0.5 · ln

(uout

u

))−

− ukA + u

+ νCat · νt (ψ) ·(

ψtr − 0.5 · ln(

uut

))⎞⎟⎟⎠ ,

dψ

dη= −ρ · (−b · (

c (u) · ν (ψ) + νstCa2+

) − νCah · νh (ψ)) ·

(ψ − 0.5 · ln

(uout

ut

))−

−b1 · (νK+ (ψ) + νKca · ν1

K+ (u, ψ) + νstK+

) · (ψ − ψK+) + I0,

(47)

In the general case the conductivity of the proteincomplexes governing the Ca2+ current from cilia to thecell body depends on the membrane potential:

Table 1 Parameter values employed in the systems model for the ciliary excitation

Parameter Value (dimensionless unlessotherwise stated)

Figure No Equation

a 4 2A 18

d 0.4 2A 18

l 0.5 2A 18

b 2 2B, 3, 5, 6, 7, 9, 10 20, 21, 34, 39

kA 1 2B, 3, 5, 6, 7, 11 20, 34, 42

uout 1000 2B, 3, 5, 6, 7 20, 21, 34

ψ0 -1.2 2B, 3, 5, 6, 7, 8 20, 21, 34, 39

V0 30 mV 2B, 3, 5, 6, 7, 8 20, 21, 34, 39

ψ1 -1, -0.8, -0.5, -0.2, 0, 0.2, 0.5 2B, 3, 5, 6, 7, 8 20, 21, 34, 39

V1 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

τ0 0.02 2B, 3, 5, 6, 7, 9, 10 20, 34, 39

k 2 2B, 3, 5, 6, 7 20

KC 1 4, 5, 6, 7 25

CaC0 5, 10, 50, 100 4, 5, 6, 7 25, 34

a 4 7, 11 34, 42

αK+ 0.5 8, 10 36

λK+ 0.005 8, 10 36

dK+ 0.5 8, 10 36

b1 1 9 39

b 8 11, 12, 13, 14 42, 46

cac0 20 11, 12, 13, 14 42

νstCa2+ 0.01 11, 12, 13, 14 42, 46

νstK+ 0.01 11, 12, 13, 14 42, 46

r 10 11, 12, 13, 14 42, 46

s 0.5 11, 12, 13, 14 42, 46

vCah 0.9 12, 13, 14 46

lh 5 12, 13, 14 44

dh 1 12, 13, 14 44

ah 4 12, 13, 14 44


Page 23 of 26

νstK+ (48)

All parameter values used in the above equations aregiven in Table 1. The relationship between dimensionaland non-dimensional quantities for Ca2+ concentrationand membrane potential are given in Table 2.

AcknowledgementsThis work was supported by the Strategic Research Development Awardfrom Faculty of Sciences, University of Kent (NVV) and the Russian Fund forBasic Research (NVK).

Author details1Centre for Molecular Processing, School of Biosciences, University of Kent,Canterbury, Kent CT2 7NJ, UK. 2Biophysics & Bionics Lab, Department ofPhysics, Kazan State University, Kazan 420008, Russia. 3Centre for Systems,Dynamics and Control, College of Engineering, Mathematics and PhysicalSciences, University of Exeter, Harrison Building, North Park Road, Exeter EX44QF, UK. 4Inter-regional Diagnostic Centre, Karbisheva-12A, Kazan 420101,Russia. 5Department of Mechanical Engineering, The University of HongKong, Pokfulam Road, Hong Kong. 6School of Automation, NanjingUniversity of Science and Technology, 200 Xiao Ling Wei, Nanjing 210094, P.R. China. 7Centre for Bioinformatics, Department of Computer Science,School of Physical Sciences & Engineering, King’s College London, Strand,London WC2R 2LS, UK. 8St. John’s Institute of Dermatology, King’s CollegeLondon, 9th Floor Tower Wing, Guy’s Hospital, Great Maze Pond, SE1 9RTLondon, UK. 9Laboratoire de Biochimie, CNRS UMR7654, Department ofBiology, Ecole Polytechnique, 91128 Palaiseau, France. 10School ofEngineering and Digital Arts, University of Kent, Canterbury, Kent CT2 7NT,UK.

Authors’ contributionsNVK, ANG, BG, MZQC, ID, CH and AA developed and implemented theproject under the supervision of DGB, RNK, YU and NVV. XY, SKS and CMScontributed to the analysis of the model. All authors contributed to thewriting of the final manuscript. All authors read and approved the finalmanuscript.

Received: 13 June 2011 Accepted: 15 September 2011Published: 15 September 2011

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Table 2 The relationship between dimensional and non-dimensional quantities for Ca2+ concentration and membranepotential

Variables Dimensional variables Non-dimensional variables Coefficient value

Calcium Ca2+ (M/L)u =

Ca2+

KCaM

KCaM = 4 μM

Transmembranepotential

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F · Vm

R · T

RT

F= −0.025 V


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doi:10.1186/1752-0509-5-143Cite this article as: Kotov et al.: Computational modelling elucidates themechanism of ciliary regulation in health and disease. BMC SystemsBiology 2011 5:143.

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Computational modelling elucidates the mechanism of ciliary regulation in health and disease

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