International Journal of
Molecular Sciences
Article
Decoding the Bell-Shaped Calcium Spikes in Phosphorylation
Cycles of Flagella
Miljko Satarić
Satarić 1,2,, Tomas
Miljko
Tomas Nemeš
Nemeš 1
1
2
3
*
and Jack Tuszynski 3, *
Faculty of Technical Sciences, University of Novi Sad, 21000 Novi Sad, Serbia;
sataric.miljko@gmail.com (M.S.); nemes.tomas@uns.ac.rs (T.N.)
Serbian Academy of Sciences and Arts, 11000 Belgrade, Serbia
Department of Physics, University of Alberta, Edmonton, AB T6G 2R3, Canada
Correspondence: jackt@ualberta.ca
Abstract: We investigate the messenger role of calcium ions implicated in the regulation of wave-like
bending dynamics of flagella. The emphasis is on microtubules of flagellar axoneme serving as
nonlinear transmission lines for bell-shaped spikes of calcium ions. The calcium sensitive proteins,
such as calmodulin, exhibit activation dependence on the spike train frequency and amplitude. Here,
we analyze a Ca2+ decoding module IDA-I1 whose activity is controlled by Ca2+ activated kinase. We
find that trains of Ca2+ spikes are advantageous compared to a constant rise in Ca2+ concentration as
being more efficient and much less prone to noisy fluctuations.
Keywords: flagellum; microtubule; axoneme; dynein motors; calcium spike signaling
Citation: Satarić, M.; Nemeš, T.;
Tuszynski,Satarić,
J. Decoding
the
M.; Nemeš,
T.;
Bell-Shaped Calcium Spikes in
Phosphorylation Cycles of Flagella.
Int. J. Mol. Sci. 2022, 23, 3760.
https://doi.org/10.3390/
ijms23073760
Academic Editor: Masatoshi Maki
Received: 17 January 2022
Accepted: 23 March 2022
Published: 29 March 2022
1. Introduction
Cilia and flagella are characterized by different patterns of movement but are identical
in structure and composition. In the following, for the sake of brevity, we will mostly
use the term flagella. Flagella are long thin appendages of many living cells, whose
oscillatory bending waves enable cells to be propelled through visco-elastic fluids, or to
drive fluid flows across the surface of the cell. Motile flagella are capable of complex,
subtly coordinated movements and can play versatile roles in fertilization and embryonic
developments [1]. Specific flagella perform the windshield wiper-like actions in trachea,
they clean
out of lungs.
flagellum
of a sperm
formscell
its forms
“tail” and
propels
it
where
theymucus
clean mucus
out ofAlungs.
A flagellum
of cell
a sperm
its “tail”
and
propels it in order to swim at speeds up to 3 mm/minute and attach to and fertilize an egg.
The hallmark structure of flagella is their central core, the axoneme.
The axoneme emanates from the flagellum cell body. The part inside the cell that
anchors axoneme is called basal body, Figure 1a.
Publisher’s Note: MDPI stays neutral
with
regard toNote:
jurisdictional claims in
Publisher’s
published maps and institutional affiliations.
Copyright: © 2022 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
Figure 1. (a) Cylindrical architecture of the flagellum with nine microtubule doublets forming the
axoneme; (b) cross section of axoneme. The figure shows the main ingredients; nine microtubule
doublets, the central microtubule pair, radial spokes, nexin linkers, and two sets of different dynein
motors (outer dynein arms and inner dynein arms). Some accessory proteins are also added [2].
Int. J. Mol. Sci. 2022, 23, 3760. https://doi.org/10.3390/ijms23073760
https://www.mdpi.com/journal/ijms
Int. J. Mol. Sci. 2022, 23, 3760
2 of 17
The microtubule (MT) cytoskeleton of flagella consists of triplet MTs in the basal body
and microtubule doublets (MTDs) in the axoneme. The axoneme contains 9 MTDs and the
central pair (CP) of single MTs, all in a parallel cylindrical arrangement. The MTDs and
CP are connected by the radial spokes (RSs), while neighboring MTDs are linked with sets
of nexin-linkers, see Figure 1b. The axonemal cytoskeleton is endowed with hundreds of
accessory protein structures, all encapsulated in the flagellum’s plasma membrane [1,3].
More than 600 different proteins were detected from the Chlamydomonas axoneme
by mass spectroscopy-based proteomics. Among these proteins just MTs and dynein
motors play the executive roles for flagellar bending motions. These motions are driven
by ATP-induced conformational changes of dynein motors. Other accessory proteins
constitute a complex toolkit for subtle support and aid in flagellar dynamics. It is known
that axonemes are reinforced by a network of tektin protofilaments that together with
nexin-linkers maintain the nine-fold integrity of axoneme and enhance its mechanical
robustness. Some accessory complexes are indicated in Figure 1b.
1.1. Microtubules
Microtubules are long hollow cylinders (Figure 2a) typically containing 13 parallel
protofilaments of α-β tubulin heterodimers (Figure 2b). There exist four types of nanopores in the MT wall located between neighboring tubulins connecting MT lumen with
an external part of a MT, (Figure 2c). These nanopores are very important for calcium
signaling along MTDs of axoneme as will be shown in the context of our model. The
outer diameter of a MT is 25 nm. Every MTD is composed of one A-MT and one B-MT.
The A-MT contains the usual number of 13 protofilaments and is tightly fused with B-MT
composed of 10 protofilaments, see Figure 2d. MTs can be generally highly dynamic in the
cytoskeleton, but in the case of the flagellar axoneme MTs are stable and have a constant
length. They are parallel to each other with MTD inter-spacing of about 30 nm, which is
closely comparable to the diameter of an MTD itself. Axonemal MTDs are implicated in very
versatile flagellar functions; starting from the construction of the axoneme’s architecture
and stability and playing the role of railways for intra-flagellar transport, then providing
a complex mechanism for flagellar bending. The transport along the axoneme is essential
for building and maintenance of eukaryotic flagella. Kinesin II motors carry cargoes in
an anterograde mode, from the proximal to the distal end of the axoneme, using B-MTs as
rails, while cytoplasmic dynein-1b (larger motors) carry cargoes in the opposite direction
using A-MTs since these are more spacious paths [4]. These functions are perhaps the main
reason why instead of single MTs the axoneme comprises MTDs. In most flagella the MTDs
numbered 5 and 6 in the standard numbering convention are permanently linked to one
another and cannot slide relative to each other. They do not have dynein motors attached
to them. This fact defines the position of the beating plane of the axoneme, (Figure 1b).
1.2. Dyneins
Axonemal dyneins are structured in two complexes of inner and outer dynein arms
(IDAs and ODAs—Figure 1b). There is a strikingly large difference in structure and
organization between ODAs and IDAs as well as with regard to their function. The ODAs
have two versions; one with a dimer (two dynein heavy chains-DHCs) and the other with
a trimer (three DHCs). They also possess light chains (LCs) and intermediate chains (ICs).
Species with dimers have protein calaxin as an associated Ca2+ sensor while trimers are
associated with calmodulin (CaM) LC4 as a Ca2+ sensor. The ODAs provide much of the
power required for flagellar movement. The ODAs repeat once every 24 nm and are very
homogenous in the structure of the axoneme.
Int. J. Mol. Sci. 2022, 23, 3760
3 of 17
Figure 2. (a) MT cylinder of 13 parallel protofilaments (one protofilament is depicted in blue, helical
symmetry in green; (b) the cross section of a MT with denoted dimensions, and tublin α-β heterodimer (secondary structure); (c) four types of nanopores in MT wall; (d) the microtubule doublet
consists of A-MT and B-MT, with 13 and 10 protofilaments, respectively [2].
The IDAs are much more complex and contain at least 12 different DHCs [5,6]. These
DHCs are arranged and organized into seven different IDAs, labeled (a, b, c, d, e, I1, and
g) and are distinct in composition and respective location in a 96 nm repeating unit in
the axoneme. The IDA complex is responsible for control of the size and shape of the
forward and reverse flagellary bends. Special significance in this mechanism is assigned
to the major IDA motor I1. It is located in the proximal end of each 96 nm axonemal
repeat near the radial spoke (RS1) and it makes contacts with many other structures in
the axoneme including the nearest ODA and neighboring single-headed IDAs. The I1
comprises two DHCs (1α and 1β), three intermediate chains (IC97, IC140, IC138), and five
light chins. Importantly, the subcomplex IC138 is essential for the control of flagellary
waveform in terms of Ca2+ signals. The phosphorylation is mediated by the modifier of
inner arms (MIA) complex present in RS1. For example, MTD sliding in Chlamydomonas
flagella is regulated by phosphorylation/dephosphorylation of the IC138 complex of I1IDA. The phosphorylation and dephosphorylation of this protein inhibits and restores
wild-type MTD sliding, respectively.
It should be strongly stressed here that effective functioning of I1-IDA is provided
in terms of Ca2+ signaling. The I1 acts as a brake to slow and locally regulate MTDs
sliding driven by ODAs and other IDAs. Its 1β-DHC contributes to force production while
1α-DHC is the motor domain that resists or limits MTD’s sliding. Below we elaborate on
the decoding mechanism of calcium oscillations involved in the activation of I1 dyneins.
This represents the main focus of this article.
1.3. Radial Spokes
The RSs act as spacers to position MTDs in a circle around a central pair of MTs.
These RSs are large complexes comprising at least 23 different subunits including a diverse
group of structural and signaling proteins. They represent two important regulatory hubs
aimed at tuning of flagellar motility. The first spoke RS1 in association with I1 dynein is
positioned near the proximal end of the 96 nm axonemal repeat [7]. The second spoke RS2
and associated CaM, along with the spoke-associated complex (CSC) and nexin-dynein
regulatory complex (N-DRC) are located at the distal end of the 96 nm repeat. Together,
these two hubs coordinate dynein activity and regulate the waveform and beat frequency
of the flagellum. Below, we elucidate how calcium signaling pathways participate in
this regulation.
Int. J. Mol. Sci. 2022, 23, 3760
4 of 17
1.4. Bending of Axoneme
The MTDs are not contractile but, instead, slide relative to one another under the
action of forces and torques, produced by dynein motors. The variability of sliding velocity
is the basis of flagellar bend formation. Very notably the different IDAs and ODAs undergo
MTDs translocation at different speeds.
Bending of the axoneme originates from the imbalance of dynein motors on the
opposite sides of the bending plane. For instance, switching of dynein activity between
MTD7 and MTD3 (Figure 1b) is thought to be responsible for the periodical and planar
oscillations of sperm flagella.
During their power stroke dyneins produce force that tends to slide attached MTDs
with respect to each other, thus regulating the beat pattern of the axoneme. Accordingly,
the flagellar beat is enabled by alternating episodes of activation of sets of dynein. It is
concluded that these beats are highly organized processes such that Ca2+ signals indirectly
activate dynein sliding, initiating the beat, and conversely, the beat tunes the dyneins. When
activated, dyneins on one side of the axoneme win the tug-of-war, leading to relative motion
between MTDs. Passive nexin-linkers and RSs, along with the basal body of axoneme,
constrain sliding and convert it into bending [8].
2. Principles of Calcium Signaling
Calcium, being the most abundant metal on the earth and the fifth most abundant
element in the human body, was adopted as a cellular regulator at an early evolutionary
stage. Calcium acts as an intracellular second messenger in living processes ranging from
cell movement, important here for secretion and contraction, up to gene expression and
even cell apoptosis. The ability of a simple ion such as Ca2+ to play a pivotal role in cell
biology results from the ease with which cells have to shape Ca2+ signals in the dimensions
of space, time, amplitude, and frequency [9,10].
The advantage of calcium is its specific coordination chemistry. This is determined
by its valence, the ionic radius of 0.99 Å and its radius of hydration of 4.5 Å. A Ca2+ ion
can accumulate 8–12 oxygen atoms in the primary coordination sphere. The range of Ca-O
distances in complexes is 0.23–0.28 nm. The abilities of Ca2+ and negative phosphate ions
to trigger changes in the shape and charge of different proteins are two universal tools
of cellular signaling transduction. Thus, Ca2+ binds to thousands of different proteins
with over a million-fold range of affinities (nM to mM). These binding events can influence protein’s change in localization, association, and function. Interestingly, cells spend
a significant part of their totally consumed energy to maintain changes in Ca2+ concentration keeping the difference between their intracellular (0.1 µM free) and extracellular
(mM) concentration [11].
In extracellular fluids, the concentration of Ca2+ varies between 2.0 and 2.6 mM,
subdivided into three components: (a) ionized, (b) bound to small inorganic molecules,
and (c) complexed with organic molecules. The concentration of ionized Ca2+ forms is
of the order 1 mM. The total Ca2+ concentration in the cytosol is also on the order of
mM. But there, the concentration of free ionized fraction is about 104 -fold lower than the
bound and complexed components. To this end, cells are endowed with specific organelles
which contain sites that ligate Ca2+ with significant affinities. The first group of such
ligand peptides are sequestered inside mitochondria, the endoplasmic reticulum and the
sarcoplasmic reticulum (ER; SR), as well as in the Golgi apparatus. The second class of such
proteins involves cytoskeletal structures: microtubules, actin filaments, and intermediate
filaments. All these organelles buffer free Ca2+ within the nM range without modifying
its total content in the cell. A very important fact for this article is that ERs and SRs are
missing in mature flagella, including sperm cells. However, there are the ER remnants in
the membranous forms with similar Ca2+ buffering properties to those of the ER.
It is known that axonemes in mammalian sperm cells are reinforced by a network
of tectin-dense fiber axoneme complexes that together with nexin-linkers maintain the
nine-fold geometry and enhance the mechanical robustness of the axoneme. A significant
Int. J. Mol. Sci. 2022, 23, 3760
5 of 17
role in controlling Ca2+ concentration in sperm cells flagella is played by mitochondria.
Besides energy homeostasis, mitochondria play a vital role in maintaining ion homeostasis
in sperm cells [12].
Figure 3 illustrates the organization of the human sperm cell with its segmentation,
indicating the spatial dimensions. The mitochondria are restricted to the mid-piece of the
flagellum. They wrap helically around the outer dense fiber axoneme complex during spermatogenesis to form a cylinder-shaped mitochondrial sheet coaxial with the axoneme [13].
Figure 3. (a) Graphical illustration of the human sperm cell with its segmentation, indicating the
spatial dimensions; (b) the cell head and mid-piece, with some details; and (c) the cross-sections of
the mid-piece, principal-piece, and end-piece.
Within the cylinder-shaped sheet, adjacent mitochondria associate both end-to-end
and along their lateral surfaces. This arrangement sets a concentrated array of mitochondria
closely to the flagellum membrane from outer side, Figure 2c. This arrangement offers
an efficient way to provide most of the ATP energy supply required for flagellar motility.
Additionally, the vicinity of the axoneme and Ca2+ ionic channels enables mitochondria to
support the calcium induced calcium release (CICR) process which underlies the pulsatile
localized calcium waves along the flagella. The mammalian sperm mid-piece contains
50–57 mitochondria with diameters in the 0.7–3 µm range, which represents about 10% of
the cell volume. It was demonstrated in [14] that Ca2+ wave activities in Xenopus –
leaves
oocytes strengthened by oxidizable
substrates
that energize
mitochondria,
thusofincreasing
–3 μm range,
which
represents
about 10%
the
the Ca2+ wave amplitude and velocity. This clearly describes the basic role of mitochondria
in intracellular Ca2+ signaling. Therefore, mitochondria may be safely considered as the
guardians of the gate between life and death since they also play an important role in
cell apoptosis.
There is also clear evidence that other Ca2+ storage organelles are contained in mature
mammalian sperm cells due to the presence of inosital triphosphate receptors (IP3Rs) in
the acrosomal part of the cell head and over the sperm neck and mid-piece of the flagellum.
3. Calcium Signaling in Flagella
Flagella change their motility in response to Ca2+ concentration which is a crucial
regulator of the modulation of flagellar movement. These modulations include: (1) changes
in flagellar waveforms; (2) reversal of the direction of flagellar bending; (3) arrest of bending;
(4) changes in the beat frequency and amplitude [15].
For successful Ca2+ signaling there should exist four functional segments as follows:
Int. J. Mol. Sci. 2022, 23, 3760
6 of 17
1.
2.
3.
4.
In the case of the human sperm, the signaling is triggered by a stimulus (progesterone
or nitrogen monoxide -NO) that generates Ca2+ signal through Cat-sper channels;
It activates the ON mechanism that feeds Ca2+ into flagellum and sperm head. Since
flagella are thin cylinders with a very large surface-to-volume ratio [16], these Ca2+
fluxes are efficiently injected into the cytosol.
The pulses of Ca2+ ions function as messengers, which stimulate several axonemal
proteins, primarily CaM, calaxin and IC138 of IDA-I1, to perform the control of
dyneins and to modulate flagellary beats [17].
Eventually the OFF mechanism, composed of pumps and ionic exchangers, removes
Ca2+ from the cytoplasm to internal Ca2+ stores and buffers, as well as out of flagella,
in order to restore the resting state [18].
When activated, both Ca2+ entry and release channels introduce Ca2+ into flagellar
cytoplasm. However, since these channels (for example Cat Sper) have short open times,
they only introduce brief pulses of Ca2+ that form a small puff around the mouth of
each channel [19,20].
These elementary Ca2+ signals are the building-blocks by which recruitment of the
complex Ca2+ signals is constructed. This includes the localized “Ca2+ clouds” drifting
along MTDs within the scope of our model [2,16,21].
The advantages of such pulse-like oscillating Ca2+ waves instead of global increases of
2+
Ca concentration in controlling flagellary movement are manifold. These signals can have
a rapid action considered to have a much higher fidelity of information transfer than simple
tonic changes in Ca2+ concentration since they are much less prone to noisy fluctuations.
4. The Principal Sensors for Ca2+ Signals in Flagella
The main sensor for calcium signaling is a very ubiquitous Ca2+ binding protein CaM.
It bears structural Ca2+ -binding motifs known as “EF-hands”. The affinity of Ca2+ for
CaM is Kd ≈ 1 µM making it an ideal receiver for the rapid transient Ca2+ increase seen
with each incoming localized spike. One of the best-known enzymes that uses CaM to
help it “count” Ca2+ spikes is CaM dependent protein kinase C, which activates dynein
motors ODAs and IDAs via phosphorylation [22]. As a corroboration of the essential role
of CaM in calcium signaling, one can refer to the experiment in which a CaM antagonist
W-7 inhibits flagellum motility initiation [23]. In sea urchins, Chlamydomonas and teleost
fish sperm CaM affects the symmetry of flagellar beating.
Protein kinase C (PKC), which is activated via diacyglycerol in the presence of Ca2+ is
important for motility maintenance via phosphorylation of flagellar proteins involved in
dynein complexes [22,24].
The second important Ca2+ sensor is neuronal calaxin, which directly acts on an ODA
and regulates specific flagellar movement during sperm chemotaxis. Calaxin is essential
for generation and propagation of Ca2+ -induced asymmetric flagellar bending [17,25]. The
formation and propagation of asymmetric beating waves in sperm flagella depend on Ca2+
concentration, which regulates calaxin activity in catalyzing ODAs speeds. Immunoelectron
microscopy and biochemical analysis showed that calaxin interacts with the ODA heavy
chain in a Ca2+ -dependent manner. Knockout of calaxin caused reduced motility in sperm
flagella. In some cases, ODAs on the doublets 3, 5, and 8 were lost, suggesting that calaxin
might play a role in stabilizing the association of ODAs with MTDs within axoneme.
5. Mechanism for Propagation of Intracellular Calcium Waves—The Particular Aspect
of Flagellar Waves
Calcium waves were first seen at the calcium tsunami, which crosses a fertilizing
medaka fish egg [26]. These waves were subsequently inferred to cross a wide variety of
cm
cell parts, whole cells, and tissues with speeds ranging from 1 nm
s up to 3 s . Jaffe [27]
presented collated evidence about different classes of signaling calcium waves in the
context of various cellular processes. The long-standing dogma about the propagation of
calcium waves in living cells relies on the autocatalytic mechanism called CICR. This is
Int. J. Mol. Sci. 2022, 23, 3760
7 of 17
a regenerative cycle in which locally elevated intracellular calcium concentration induces
the release of new Ca2+ ions stored in ER or SR organelles. These additional Ca2+ ions
diffuse to nearby ER channels where they cause the release of yet more calcium. This
self-sustaining cycle is primarily modulated by the increased levels of inositol (1, 4, 5)
triphosphate (IP3 ) within the cytosol which forms calcium channels by the receptor proteins
IP3 Rs. These intracellular channels respond to Ca2+ concentration in a bell-shaped fashion;
that is IP3 R is inactive at low nM concentration of Ca2+ and active at mid-µM concentration,
then inactivated again by high concentrations that are in the mM range of Ca2+ . Thus, the
speed of CICR waves is determined by the diffusion of Ca2+ ions combined with diffusion
of IP3 Rs. The diffusion constant of Ca2+ ions is of the order of 20
µm2
s
µm2
s
while for IP3 Rs is
much greater, namely 300
[28].
If the density of IP3 Rs is higher, the speed of Ca2+ waves is further increased. We
already stressed that ER and SR are missing in mature flagella. However, the presence
of IP3 Rs in the sperm cell’s head, neck, and mid piece (Figure 3) indicates that this CICR
mechanism can be achieved in these compartments, especially in mid piece, where the
mitochondrial sheet around the axoneme is expressed.
The activation and inactivation of IP3 Rs within CICR mechanism occurs on a faster
time scale than the production and degradation of IP3 . The Ca2+ activation and inactivation
of IP3 Rs manifests itself on the time scale of milliseconds, while the production and
degradation of IP3 follows the time scale of several seconds.
The characteristic speeds of Ca2+ waves enabled by the CICR mechanism are of the
order of 3–35 µM
s [13,27]. For example, the speed of waves in Guinea pig myocytes is
µm
32 s [29]. However, of greater interest for our concept presented here is the contribution
by Jaffe [30] regarding fast calcium-induced calcium influx (CICI) waves. Much of that
model concerns calcium waves along functional flagella, including sperm cells and their
principal piece. These waves have the speeds ranging from 102 to 103 µm
s in a wide variety
of systems. Huang et al. [31] observed intracellular calcium waves in human fibrosarcoma
cells with a speed of 100 µm
s . These waves were blocked by the calcium channel blockers,
ions of gadolinium. Additionally, the speed of Ca2+ waves in human sperm was earlier
estimated to be of the order of 500 µm
s [32] and in the case of Hamster sperm (activated)
µm
was about 700 s [33].
We earlier established [21,34] and further refined [2,16] the polyelectrolyte concept
of MTs aimed to explain how Ca2+ ions, albeit present in nM to µM concentrations in the
cytosol, use MTs as a buffers to condense close around their filaments and C-termini (CTTs).
Moreover, the accumulated Ca2+ ions form bell-shaped “Ca2+ clouds” that propagate along
MTs much faster than it could be achieved by simple three-dimensional diffusion. The
estimated speeds of Ca2+ waves within the scope of our model compared with some prior
experimental results are given in Table 1.
Table 1. A comparison between our estimated speeds and experimental values measured by
different authors.
Our Model
(Satarić et al.)
Estimated
Speeds [µm/s]
Experimental
Evidence
Experimental
Speed [µm/s]
2009 [34]
2010 [21]
2019 [16]
2020 [2]
6000-overestimated
160–240
530
620
/
Huang et al. [31]
Mortimer et al. [32]
Ishijima et al. [33]
/
100
500
700
The differences within our model are the consequences of different estimates of capacitance and resistance of elementary electric building blocks of MTD viewed as nonlinear
electric transmission line. These differences are caused by different post-translational modifications of CTTs which change the capacitance of these building blocks. Nonetheless, fairly
satisfactory agreement of our results with experimental data indicates that our concept
Int. J. Mol. Sci. 2022, 23, 3760
8 of 17
offers a very promising mechanism capable to explaining an efficient signaling pathway of
Ca2+ ions implicated in the modulation of flagellar bending.
6. Decoding of Axonemal “Ca2+ Clouds” by Phosphorylation Cycles
Prior experimental evidence has demonstrated that signaling by intracellular Ca2+
repetitive spikes regulates sensitive proteins in a way such that strengthening of appropriate
stimulus increases spikes frequency [35]. For example, the extracellular stimulus speract
induces changes of intra-flagellar Ca2+ concentration with oscillatory spatio-temporal
evolution from the flagellum to the sperm head. In the presence of 0.5 nM speract as
agonist, the frequency of Ca2+ pulses in Sea urchin sperm is of the order of 1 Hz, while
increased agonist concentration of 0.1 µM generates a frequency of about 5 Hz [36]. Signal
information can also be encoded in the amplitude of Ca2+ pulsatile train. This amplitude
usually can be changed with the external stimulus [37]. Experimental data additionally
indicate the capability of CaM and protein kinases to decode the amplitude of Ca2+ pulsatile
signal into an appropriate cellular response [38]. Furthermore, a very important catalyzer
of flagellar movement, CaM-dependent kinase C is sensitive to the frequency of Ca2+
signals [22,39]. It appears that such Ca2+ repetitive signals can be more efficient for protein
phosphorylation if compared to constant Ca2+ signals of equal average concentration.
Here, we consider the activation-inactivation cycle of an axonemal target motor protein, for example an IDA-I1 which plays the fundamental role in the regulation of MTDs
sliding. Its activation is triggered by phosphorylation through Ca2+ /CaM-dependent protein kinase II (PKII), and it is inactivated by dephosphorylation [40]. The stimulation of the
associated PKII is enabled by the binding of n Ca2+ ions to the E-F hands domains of the
associated CaM molecule. Thus, the Ca2+ /CaM-dependent PKII exerts the phosphorylation
of IC138, the vital ingredient of the IDA-I1 motor protein. This can be considered a rapid
“post-translational” modification accompanied by the conformation of target proteins. As
an example, the phosphorylation and dephosphorylation of IC138 inhibits and restores
wild-type MTDs sliding, respectively in Chlamydomonas flagella.
The fraction of activated PKII by Ca2+ /CaM, denoted by Y (t), and the phosphorylated
target protein IC138, denoted by X (t), obey the following time-dependent evolution equations:
dY
= aY [C (t)]n (1 − Y ) − bY Y
dt
(1)
dX
= a X YT Y (1 − X ) − bX X
(2)
dt
Here, YT represents the total available concentration of PKII in the axoneme, while the
time course of the Ca2+ concentration within a single spike is denoted by C (t). Symbols
aY and bY are the rate constants of Ca2+ binding and release in CaM, respectively, while
a X and bX represent the rate of phosphorylation and dephosphorylation of IC138 by PKII,
respectively. This is basically responsible for IDA-I1 motor actions in terms of the respective
two heavy chains (1α, 1β).
Next, we implement our concept of bell-shaped spikes, so called “Ca2+ clouds” propagating along microtubule doublets of the flagellar axoneme. The scenario of such trains
of spikes is elaborated in [16,21]. Similar models are developed in [41]. Interestingly our
concept is widely elaborated from a mathematical perspective [42–44].
The train of spikes arising from our model obtained in [42] is presented in Figure 4.
Int. J. Mol. Sci. 2022, 23, 3760
9 of 17
Figure 4. Our 3D representation of the train of spikes according to model described in Zahran [42].
𝑥
𝑣𝑡
𝜏 =vt .ℓ
Dimensionless space-time variables are: ==xℓ ℓ; τ =
ℓ
Equation (1) can be solved if the local Ca2+ concentration within a single spike is taken
2+
to be our bell-shaped
shaped“Ca
“Ca cloud”,
cloud”, [2,16]:
C0 𝐶0
2
C ( x, t𝐶(𝑥,
) = 𝑡) = (𝑐𝑜𝑠ℎ(
𝑥 𝑣𝑡
x ℓ−
vtℓ ))2
cosh ℓ − ℓ
(3)
𝑥
ℓ = 8 nm
𝑣
where x is the direction of an MT doublet, ℓ = 8 nm is the tubulin dimer length, v is the
drift velocity of “Ca cloud”,
𝐶
drift velocity of “Ca2+ cloud”, t stands for the time variable, and C00 is the amplitude of
Ca2+ concentration within each spike. We should fix the position of a specific dynein motor
with accompanying regulatory proteins (CaM, CaMKII, IC138) involved in the motor’s
with accompanying regulatory proteins (CaM, CaMKII, IC138) involved in the motor’s
𝑥=0
activations. Hence, we can take x = 0 for this site and the time evolution of the local Ca2+
concentration now reads
C0 𝐶
0
C (𝐶(𝑡)
t) = =
(4)
2
cosh vtℓ 𝑣𝑡 2
(𝑐𝑜𝑠ℎ ( ))
ℓ
Then, going over to the dimensionless time in Equation (1) by substituting
(5)
τ = bY t
𝜏 = 𝑏𝑌 𝑡
and inserting Equation (5) into Equation (1) we obtain
n
𝑛
𝑛 n
1 1
n
b
C00
C0 𝐶0
dY 𝑑𝑌
𝑏
𝐶
𝑌 𝑛Y
Y=
+ + [( 2 2)+ 1+1]
.
𝜅; κ==
(
𝑌=
(
)
2 )2
dτ
a
𝑎𝑌 Y
𝑑𝜏
vt
𝑣𝑡vt
𝑣𝑡
κ cosh𝜅(𝑐𝑜𝑠ℎ(
κ
cosh
𝜅(𝑐𝑜𝑠ℎ(
))
))
ℓb
ℓ𝑏ℓ b
ℓ𝑏
Y
(6)
𝑌Y
𝑌
The parameter κ 𝜅represents the half saturation concentration for Ca2+ /CaM activated
PKII. The bound number of Ca2+ ions in CaM (a so-called Hill coefficient) is n𝑛==4.4
We can solve Equation (6) numerically for an increasing set of parameter values ε𝜀
which is the ratio of the Ca2+ peak concentration and PKII half saturation
𝐶
𝜀C=0 0 = {1; 1.5; 2.0; ,5.0}
𝜅
ε=
=
(7)
{1; 1.5; 2.0; , 5.0}.
κ
𝑌=0
𝑡=0
𝑌(𝜏̅)
The initial condition is: Y = 0 for t = 0. The shapes of resultant functions of Y (τ ) are
depicted in Figure 5.
Int. J. Mol. Sci. 2022, 23, 3760
10 of 17
Figure 5. Calcium/calmodulin activation and inactivation increases with increasing amplitude of
“Ca2+ cloud
1.5green, ε =
𝜀=
2
5
“Ca
cloud spikes”
spikes” (𝜀
(ε =
= 11 red, ε𝜀==1.5
2 blue,
ε =𝜀 5=orange).
The effective dimensionless
𝑣𝑡
vt
time is now τ = ℓ𝜏b̅ Y=. ℓ𝑏
𝑌
In flagella, it is quite reasonable to assume that Ca2+ binding to CaM is much faster
than it’s
≫ 𝑏bY)) [45]. It implies that the ratio 𝜀ε is several times greater than
’ unbinding (a𝑎Y𝑌 ≫
𝑌
unity for spikes with amplitudes high enough. This is reflected in the above curve with
ε𝜀 =
= 55 (see Figure 5, orange line), which is practically flat, giving
dY𝑑𝑌 = 0
= 0,
dτ𝑑𝜏̅
(8)
equilibrium for activated PKIIs.
This implies the following solution of Equation (1):
𝑏𝑌
1
≪1
𝑌=
𝑛
𝑎𝑌
1 𝑣𝑡 2
bY
𝑏𝑌
1+
[(𝑐𝑜𝑠ℎ(
))
]
Y=
(9)
𝑎𝑌
ℓ𝑏𝑌 2 n ; a ≪ 1
Y
1 + baY cosh ℓvt
bY
𝑌Y
𝑡
𝑡→0
This expression shows that Y does depend on time and for large t tends to be zero.
𝑌 has a “constant” value approximately equal to unity as
But for t → 0 , such that the second term in the denominator of Equation (9) is much less
𝜀=5
than unity, this indicates that Y has a “constant” value approximately equal to unity as was
to a “constant” 𝑌
in Figure 5, for ε = 5. If a Ca2+ pulse has a greater speed, the segment corresponding to
𝑌(𝑡)
a “constant” Y is shorter in time, which is plausible.
𝑋
Thus, by inserting Y (t) from Equation (1) into Equation (2), under the condition of
𝑑𝑋
Equation (8), yields the following form
the
for X:
[𝐴𝑋 (𝐶) + 𝑏equation
(𝐶)
− evolution
= 𝐴𝑋of
𝑋 ]𝑋
𝑑𝑡
dX
= A X (C ) − [ A X (C ) + bX ] X.
dt
(10)
The effective phosphorylation rate of IC138 is1the function of Ca2+ concentration
]
= 𝑎𝑋 𝑌𝑇 as
[ follows:
𝑋 (𝐶)
𝜅 𝑛
within the spike progressing along𝐴the
axoneme
1+(
)
𝐶(𝑡)
shaped “Ca cloud”,
1
(11)
A X (C ) = a X YT
n .
κ
1 + C (t)
𝑑𝑋
𝑎 𝑌
𝑎 𝑌
𝑋 𝑇
𝑋 𝑇
=[
]−[
+ 𝑏𝑋 ] 𝑋
2 𝑛
2 𝑛
𝑑𝑡 the spike
𝜅
𝑣𝑡 bell-shaped𝜅 “Ca2+𝑣𝑡cloud”, Equation (3), and insert
Again, we use here
as
the
1+( (𝑐𝑜𝑠ℎ( )) )
1+( (𝑐𝑜𝑠ℎ( )) )
𝐶0
ℓ
𝐶0
ℓ
it into Equations (10) and (11), yielding
a X YT 𝜏 = 𝑎 𝑌 𝑡 𝑑𝑡 = 𝑑𝜏a X YT
dX
=
𝑎𝑋𝑌𝑇
2 𝑋n 𝑇 −
dt
1 + Cκ0 cosh vtℓ
1 + Cκ0 cosh
vt
ℓ
2 n + bX X.
(12)
Int. J. Mol. Sci. 2022, 23, 3760
11 of 17
Introducing the new scaled dimensionless time into Equation (12) by substitution
τ = a X YT t ; dt =
one finds that
dX
+
dτ
dτ
,
a X YT
(13)
1
1
X (τ ) =
n
𝑑𝑋
1 n + gX
1
, (14)
2
2
+ 𝑔𝑋 𝑋(𝜏) =
vτ
vτ
κ+
κ
2 𝑛
2 𝑛
𝑑𝜏
1 + C0 cosh 𝜅ℓa Y
1 + C0 𝜅 cosh ℓ a𝑣𝜏Y
𝑣𝜏
[
1+(
𝐶0
X T
(𝑐𝑜𝑠ℎ(
)) )
ℓ𝑎𝑋 𝑌𝑇
]
[
1+(
where the dimensionless activation rate of the IC138 target is
b 𝑏
gX =
𝑔𝑋 =X .𝑋
a X Y𝑎T𝑋 𝑌𝑇
𝐶0
(𝑐𝑜𝑠ℎ(ℓ𝑎 X 𝑌T )) )
𝑋 𝑇
]
(15)
In order to obtain numerical solutions of Equation (14) we should choose a set of
appropriate parameters taking into account that the activation rate of the phosphorylation
of IC138𝑎𝑋a X is greater than its dephosphorylation rates bX , as
𝑏𝑋follows
1; 0.5;
gX𝑔=
0.2;0.2;
0.010.01
{1;{0.5;
}. }
𝑋 =
(16)
We also adopt the reasonable physiological values of concentrations given as: 𝜅 =
(0.3
− 0.4)μM
𝑛 = is4 n = 4 again. Finally, the
κ=
C0 = 1.5 µM. The Hill coefficient
(0.3 − 0.4) µM𝐶and
0 = 1.5μM
= 0is: X =𝜏 0=for
0 τ = 0. The corresponding
initial condition for the solution of Equation𝑋(14)
graphical solutions for target activation
X (τ ) are represented in Figure 6.
𝑋(𝜏)
Figure 6. The target activation as a function of time. Increase of the spike amplitude along
bX leads to an elevation of the activation curve for the
with decrease of dephosphorylation 𝑏rate
𝑋
𝑔𝑋 —gX (1—red;
—
target protein. Solutions of Equation (14) for: (a) different values of the parameter
𝜅
0.5—green; 0.2—blue;
0.01—orange) and fixed values
—
—
𝜅 of the𝐶κ0 and C0 ; (b) different values of the ratio
𝐶0
κ1 1
11 —green; 1 —blue) and fixed value of the dimensionless activation rate gX = 0.2. The
5
1.5 —red; 3—
C0 (—
𝑔𝑋 = 0.2
3
5
.
effective dimensionless
time here𝑣𝜏
is τ̂ = ℓ avτ
X YT
1
1.5
—
𝜏̂ = ℓ𝑎 𝑌
𝑋 𝑇
Generally,
activation of the target protein LC138 has a rapidly rising phase
reached by a peak spike of Ca2+ concentrations of the order of 1 µM before slowly decaying
when the spike drops to the basal Ca2+ concentration level. We stress
that elevated global
1 μM
cytoplasmic Ca2+ concentrations can be very toxic so, instead of a sustained manner, Ca2+
signals in flagellum are commonly generated in terms of pulsatile forms. Since the spikes
of bell-shaped “Ca2+ clouds” (Equation (3)) drifting along the axoneme are fast compared
to the rate of phosphorylation of the target IC138 protein, it is reasonable to conclude
shaped effect
“Ca of
clouds”
that the cumulative
several(spikes is responsible for an efficient activation of the
I1-IDA motor.
Ca2+
In order to analyze the cumulative activation outcome of a series of “Ca clouds”,
tively by approximating the “Ca
Int. J. Mol. Sci. 2022, 23, 3760
12 of 17
In order to analyze the cumulative activation outcome of a series of “Ca2+ clouds”,
Equation (14) needs to be solved analytically but superposition of these spikes is analytically
intractable. Therefore, we consider it semi-quantitatively by approximating the “Ca2+
clouds” with a piece-wise constant function having the equal profile area
Z +∞
−∞
C0
(coshτ )2
e = 2C0 ,
dτ = Ce f f ∆
(17)
e is the dimensionless time width of a rectangular constant function in Figure 7.
where ∆
e = 2 one gets Ce f f = C0 , or for ∆
e = 1 it follows that Ce f f = 2C0 . In real time ∆
Taking ∆
is the time width and T is the spike train period (see Figure 7). The expression for that
𝑇 given by:
piece-wise function is
Ce f𝐶f 𝑒𝑓𝑓
, mT
≤ ≤t <
++
∆∆
, 𝑚𝑇
𝑡 <mt𝑚𝑡
m = 1, 2, 3 . . .
C (t) =
𝐶(𝑡) =
0, {mT
+
∆
<
t
<
m
+
1
T 𝑚 = 1,2,3 …
(
(𝑚
0, 𝑚𝑇 + ∆< 𝑡 <
+ )1)𝑇
(18)
Figure 7. The approximation of a bell-shaped spike by a piece-wise constant function.
The corresponding duty ratio is
∆
𝛾=
∆
γ = .𝑇
T
(19)
We assume
that of
the“Ca
target
proteinIn
IC138,
in orderthe
to be
phosphorylated,
integrates
the
the incoming
spikes
clouds”.
that respect,
average
IC138 activity
for a single
2+
incoming
spikes of “Ca clouds”. In that respect, the average IC138 activity for a single
𝑚
m-th spike has the form
Z ∆
1 1∆
h Xm⟨𝑋
i𝑚
=⟩ = 𝑇 ∫ X𝑋(𝑡)𝑑𝑡
(20)
(t)dt. .
T 00
= 1s
1𝑠 and γ𝛾==1/2,
1⁄2 then Equation (20) in terms of
If we consider a spike train with T𝑇 =
=
0.2
Equation (14) for g𝑔
=
0.2
yields
x𝑥
1
1 ⁄2
Z
⟨𝑋𝑚 ⟩ =
1 ∫1/2 𝑋(𝑡)𝑑𝑡 = 0.086
h Xm i = 𝑇 0 X (t)dt = 0.086.
T 0
(21)
We now use the analytical result for cumulative activation of the target protein for the
spike train containing the greater number of piecewise functions of Equation (18) evaluated
by [45] as:
𝛾−1
−𝛾(1+𝜎)
𝜔𝜎 [1−𝑒𝑥𝑝 ( 𝜔 )][1−𝑒𝑥𝑝 ( 𝜔 )]
𝜎
⟨𝑋⟩ =
⟨∑∞
{𝛾 + h( )
ih
} i
1+𝛾𝜎
𝑚=1 𝑋𝑚 ⟩ =
[1−𝑒𝑥𝑝
−γ
(1+σ)(− 𝜔 )]
γ −1
𝜎+1 1𝜎+1
−
exp
1
−
exp
ω
ω
ωσ
σ
∞
i
h
h ∑ m =1 Xm i = h X i =
γ+
, (22)
σ + 1
σ+1
1 − exp − 1+γσ
ω
where in our case the corresponding parameters, the effective activation, and duty ratio are
∆
1
1
𝜎 = ( ) [
given by
𝑛] 𝛾 =
𝑇
𝑔𝑥
𝜅
1+(
)
𝐶
1 𝑒𝑓𝑓
1
∆
(23)
σ=
; γ = .
gx 𝜔 1 + κ n
T
1
1
Ce f f
= 𝑏𝑥
𝑇
𝛾=
𝜔=
1
2
𝑇𝑑𝑝ℎ
1
𝑇𝑏𝑥
1
Int. J. Mol. Sci. 2022, 23, 3760
13 of 17
The dimensionless frequency ω is the ratio of the frequency of Ca2+ spike repetition
and the frequency of the dephosphorylation cycle ( T 1 = bx ):
1
T
dph
ω=
Now, we fix the duty ratio to be γ =
1
2
1
.
Tbx
(24)
and consider the function from Equation (22)
1
h X i = f γ = ; σ, ω ,
2
(25)
representing it in the 3D form as shown in Figure 8.
Figure 8. The 3D plateau-like function of cumulative activation of the target protein.
As a special example, we can apply this concept to the flagellar Ca2+ spike trains from
experimental assays with sea urchin sperm [46–48].
–
–
The estimated parameter values in this case are as follows:
∆∆
1
⁄2 T
𝑇=
0.2𝑠−−1
10 σ 𝜎==5 5
= 11/2;
= 0.5𝑠
0.5s; b𝑏x𝑥==0.2s
; ω𝜔==10;
.
(26)
γ𝛾== 𝑇 =
∆
T
𝛾 = = 1⁄2 𝑇 = 0.5𝑠 𝑏𝑥 = 0.2𝑠 −1 𝜔 = 10 𝜎 = 5
𝑇 = 10
𝜎 obtain from Equation (22) the function
If we fix the frequency ω𝜔=
10 and change σ we
𝜔 = 10
h X⟨𝑋⟩
i ==f 1𝑓(σ;
),
(𝜎;ω𝜔==10
10)
1
𝜎
(27)
which is shown in Figure 9. It is clear that it saturates
increasing
⟨𝑋⟩ = 𝑓1with
(𝜎; 𝜔
= 10) concentration of PKII
and with increasing the phosphorylation rate a x . 𝑎𝑥
𝑎𝑥
Figure 9. The cumulative activation as the function of the effective activation rate σ.
𝜎
𝜎=5
⟨𝑋⟩ = 𝑓2 (𝜔; 𝜎 = 5)
𝜎
Int. J. Mol. Sci. 2022, 23, 3760
14 of 17
Otherwise, setting σ = 5, the cumulative activation as the function of frequency is
h X i = f 2 (ω; σ = 5)
(28)
and is presented in Figure 10.
Figure 10. The cumulative activation as a function of the relative spike frequency ω.
𝜔
The comparison of Figures 9 and 10 shows that the frequency of spikes causes a steeper
increase in the activation of the target protein, saturating at a value of ω ≥ 10.
𝜔 ≥ 10
Eventually, on the basis of Equation (22) the cumulative activation of the LC138
complex in sea urchin sperm reads
1
hXi = f γ = ; σ =
(29)
1 5; ω = 10 = 0.714
2 = ; 𝜎 = 5; 𝜔 = 10) = 0.714
⟨𝑋⟩ = 𝑓 (𝛾
2
On the basis of Equation (21) for the activation by a single bell-shaped spike, and using
result of Equation (29) we estimate that approximately eight spikes are needed to achieve
cumulative activation of the LC138 complex, a vital ingredient of the I1 inner dynein arm.
7. Conclusions
In this paper we provided a description of the structure and bending dynamics
of flagella. Special emphasis was placed on the microtubules within the axoneme and
the associated dynein motor proteins. We revisited the accumulated knowledge about
15 different axonemal dyneins that are distinct in both their composition and location within
the axoneme. These dyneins are implicated in complex mechanisms of flagellar beating.
Specific attention has been dedicated to the unique signaling role of Ca2+ ions in the
regulation of flagellar dynamics. For example, in sperm cells the intracellular Ca2+ level,
pH, and ATP are the key regulatory elements of bending motility. The first two factors
are controlled by ion channels. The principal Ca2+ channel is CatSper, which is activated
by progesterone and flagellar alkalinity. The most important Ca2+ sensors in flagella are
EF-hand proteins, primarily calmodulin and neuronal calaxin whose roles in Ca2+ signaling
are very crucial.
We have paid significant attention to the concept of the bell-shaped “Ca2+ cloud”.
Calcium ions injected into the flagellum by ionic channels (CatSper, for example), through
a so-called calcium induced calcium influx, and further elevated by CICR release mechanism, accumulate closely around microtubules in the axoneme. These “Ca2+ clouds” are
shaped “Ca
not static but instead, move with the speeds of the order of hundreds of micrometers
per cloud”.
second, see Table 1. They form trains of narrow spikes, see Figure 4. Increased intracellular
Ca2+ influx elevates their amplitude and speed. Such oscillations of concentration of free
intracellular Ca2+ are a basic control
for beating
of flagella. These “Ca clouds” are
ndmechanism
microtubules
in the axoneme.
The main part of this article, which is our original and novel contribution, elucidates
the decoding of these spike trains in the context of the phosphorylation cycle mediated
Int. J. Mol. Sci. 2022, 23, 3760
15 of 17
by calmodulin and protein kinase. It appears that the decoding process is dependent on
the amplitude and the frequency of these spikes within the train. We investigated the case
where these spikes activate the target protein IC138 more efficiently if the frequency is
higher. In that respect the oscillation frequency should be greater than the inactivation
(dephosphorylation) rate of the IC138 target. As an illustrative example, we applied the
concept of cumulative activation of IC138 phosphorylation to the case of Ca2+ signaling in
sea urchin sperm. The obtained numerical estimation of dynein I1 activation has an important physiological relevance as demonstrated in Equation (29). Our analysis indicates that
trains of Ca2+ spikes are advantageous compared to a constant rise in Ca2+ concentration as
being more efficient and much less prone to noisy fluctuations. We expect that our results
shed more light for deeper understanding of subtle calcium control mechanisms implicated
in flagellar dynamics.
Author Contributions: Conceptualization, M.S.; methodology, M.S.; software, T.N.; validation, J.T.;
investigation, M.S.; resources, M.S.; writing—original draft preparation, M.S.; writing—review and
editing, T.N. and J.T.; visualization, T.N. All authors have read and agreed to the published version of
the manuscript.
Funding: This research was funded by Department of Fundamental Sciences at Faculty of Technical
Sciences at University of Novi Sad within the project “Application of Fundamental Disciplines in
Technical and Information Sciences”.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design
of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or
in the decision to publish the results.
Abbreviations
MT
MTD
CP
RS
ATP
IDA
ODA
DHC
LC
IC
CaM
MIA
CSC
N-DRC
ER
SR
CICR
PKC
IP3
IP3 R
CICI
CTT
microtubule
microtubule doublet
central pair
radial spoke
adenosine triphosphate
inner dynein arm
outer dynein arm
dynein heavy chain
light chain
intermediate chain
calmodulin
modifier of IDA
spoke-associated complex
nexin-dynein regulatory complex
endoplasmic reticulum
sarcoplasmic reticulum
calcium induced calcium release
protein kinase C
inositol triphosphate
inositol triphosphate receptor
calcium induced calcium influx
carboxyl terminal tail
Int. J. Mol. Sci. 2022, 23, 3760
16 of 17
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
Satir, P.; Christiansen, S.T. Overview of structure and function of mammalian cilia. Annu. Rev. Physiol. 2007, 69, 377–400.
[CrossRef] [PubMed]
Satarić, M.V.; Zdravković, S.; Nemeš, T.; Satarić, B.M. Calcium signaling modulates the dynaimcs of cilia and flagella. Eur. Biophys.
J. 2020, 49, 401–406. [CrossRef] [PubMed]
Ishikawa, T. Axoneme structure from motile cilia. Cold Spring Harb. Perspect. Biol. 2017, 9, a028076. [CrossRef] [PubMed]
Stepanek, L.; Pigino, G. Microtubule doublets are double-track railways for intraflagellar transport trains. Science 2016, 352, 721–724.
[CrossRef] [PubMed]
Kamiya, R.; Yagi, T. Functional diversity of axonemal dyneins as assessed by in vitro and in vivo motility assays of Chlamydomonas mutants. Zool. Sci. 2014, 31, 633–644. [CrossRef] [PubMed]
Viswanadha, R.; Sale, W.S.; Porter, M.E. Ciliar Motility: Regulation of axonemal dynein motors. Cold Spring Harb. Perspect. Biol.
2017, 9, a018325. [CrossRef]
Lindemann, C.B.; Leich, K.A. Flagellar and ciliary beating: The proven and the possible. J. Cell Sci. 2010, 123, 519–528. [CrossRef]
King, S.M.; Sale, W.S. Fifty years of microtubule sliding in cilia. Mol. Biol. Cell 2018, 29, 698–701. [CrossRef]
Bootman, M.D.; Collins, T.J.; Peppiatt, C.M.; Prothero, L.S.; MacKenzie, L.; De Smet, P.; Travers, M.; Tovey, S.C.; Seo, J.T.;
Berridge, M.J.; et al. Calcium signaling—An overview. Semin. Cell. Dev. Biol. 2001, 12, 3–10. [CrossRef]
Berridge, M.J.; Lipp, P.; Bootman, M.D. The versality and universality of calcium signalling. Net. Rev. Mol. Cell Biol. 2000, 1, 11–21.
[CrossRef]
Burgoyne, R.D. Neuronal calcium sensor proteins: Generating diversity in neuronal Ca2+ signaling. Nat. Rev. Neurosci. 2007,
8, 182–193.
Srivastava, N.; Pande, M. Mitochondrion: Features, functions and comparative analysis of specific probes in detecting sperm cell
damages. Asian. Pac. J. Rep. 2016, 5, 445–452. [CrossRef]
Zhao, W.; Li, Z.; Ping, P.; Wang, G.; Yuan, X.; Sun, F. Outer dense fibers stabilize the axoneme to maintain sperm motility. J. Cell
Mol. Med. 2018, 22, 1755–1768. [CrossRef]
Jouaville, L.S.; Ichas, F.; Holmuhamedov, E.L.; Camacho, P.; Lechleiter, J.D. Synchronization of calcium waves by mitochondrial
substrates in Xenopus laevis oocytes. Nature 1995, 377, 438–441. [CrossRef] [PubMed]
Inaba, K. Calcium sensors of ciliary outer arm dynein; functions and phylogenic considerations for eukaryotic evolution. Cilia
2015, 4, 6. [CrossRef] [PubMed]
Satarić, M.V.; Nemeš, T.; Sekulić, D.; Tuszynski, J.A. How signals of calcium ions initiate the beats of cilia and flagella. BioSystems
2019, 182, 42–51. [CrossRef] [PubMed]
Mizuno, K.; Shiba, K.; Okai, M.; Takahashi, Y.; Shitaka, Y.; Oiwa, K.; Tanokura, M.; Inaba, K. Calaxin drives sperm chemotaxis by
Ca2+ medicated direct modulation of a dynein motor. Proc. Natl. Acad. Sci. USA 2012, 109, 20497–20502. [CrossRef] [PubMed]
Lishko, P.V.; Botchkina, I.L.; Kirichok, Y. Progesterone activates the principal Ca2+ channel of human sperm. Nature 2011,
471, 387–391. [CrossRef]
Publicover, S.; Harper, C.V.; Barratt, C. Ca2+ signaling in sperm–making the most of what you’ve got. Nat. Cell Biol. 2007,
9, 235–242. [CrossRef]
Darszon, A.; Nishikagi, T.; Beltran, C.; Treviño, C.L. Calcium channels in the development, maturation and function of spermatozoa. Physiol. Rev. 2011, 91, 1305–1355.
Satarić, M.V.; Sekulić, D.; Živanov, M. Solitonic ionic currents along microtubules. J. Comput. Theor. Nanosci. 2010, 7, 1–10.
[CrossRef]
White, D.; de Lamirande, E.; Gagnon, C. Protein kinase C is an important signaling mediator associated with motility of intact sea
urchin spermatozoa. J. Exp. Biol. 2007, 210, 4053–4064. [CrossRef] [PubMed]
Wargo, M.J.; Dymek, E.E.; Smith, E.F. Calmodulin and PF6 are components of a complex that localizes to the C1 microtubule of
the flagellar central apparatus. J. Cell Sci. 2005, 118, 4655–4665. [CrossRef] [PubMed]
Morita, M.; Igushi, A.; Takemura, A. Roles of calmodulin and calcium/calmodulin-dependent protein kinase in flagellar motility
regulation in the coral Acropora digitifera. Mar. Biotechnol. 2009, 11, 118–123. [CrossRef] [PubMed]
Shiba, K.; Baba, S.A.; Fujiwara, E.; Inaba, K. Calaxin is essential for the transmission of Ca2+ dependent asymmetric waves in
sperm flagella. bioRxiv 2020, 109, 50. [CrossRef]
Gilkey, J.C.; Jaffe, L.F.; Ridgway, E.B.; Reynolds, G.T. A free calcium wave traverses the activating egg of the medaka, Oryzias
latipes. J. Cell Biol. 1978, 76, 448–466. [CrossRef]
Jaffe, L.F. Fast calcium waves. Cell Calcium 2010, 48, 102–113. [CrossRef]
Atri, A.; Amundson, J.; Clapham, D.; Sneyd, J. A single-pool model for intracellular calcium oscillations and waves in the
Xenopus laevis oocyte. Biophys. J. 1993, 65, 1727–1739. [CrossRef]
Keller, M.; Kao, J.P.Y.; Egger, M.; Niggli, E. Calcium waves driven by “sensitization” wave-fronts. Cardiovasc. Res. 2007, 74, 39–45.
[CrossRef]
Jaffe, L.F. Stretch-activated calcium channels relay fast calcium waves propagated by calcium-induced calcium influx. Biol. Cell
2007, 99, 175–184. [CrossRef]
Huang, J.B.; Kindzelskii, A.L.; Clark, A.J.; Petty, H.R. Identification of calcium channels promoting calcium spikes and waves in
HT1080 tumor cells: Their apparent roles in cell motility and invasion. Cancer Res. 2004, 64, 2482–2489. [CrossRef] [PubMed]
Int. J. Mol. Sci. 2022, 23, 3760
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
17 of 17
Mortimer, S.T.; Schëväert, D.; Swan, M.A.; Mortimer, D. Quantitative observations of flagellar motility of capacitating human
spermatozoa. Hum. Reprod. 1997, 12, 1006–1012. [CrossRef] [PubMed]
Ishijima, S.; Baba, S.A.; Mohri, H.; Suarez, S.S. Quantitative analysis of flagellar movement in hyperactivated and acrosome-reacted
golden hamster spermatozoa. Mol. Reprod. Dev. 2002, 61, 376–384. [CrossRef] [PubMed]
Satarić, M.V.; Ilić, D.I.; Ralević, N.; Tuszynski, J.A. A nonlinear model of ionic wave propagation along microtubules. Eur. Biophys.
J. 2009, 38, 637–647. [CrossRef]
Berridge, M.J.; Bootman, M.D.; Roderick, H.L. Calcium signalling: Dynamics, homeostasis and remodelling. Nat. Rev. Mol. Cell
Biol. 2003, 4, 517–529. [CrossRef]
Wood, C.D.; Darszon, A.; Whitaker, M. Speract induces calcium oscillations in the sperm tail. J. Cell Biol. 2003, 161, 89–101.
[CrossRef]
Dolmetsch, R.E.; Lewis, R.S.; Goodnow, C.C.; Healy, J.I. Differential activation of transcription factors induced by Ca2+ response
amplitude and duration. Nature 1997, 386, 855–858. [CrossRef]
Reither, G.; Schaefer, M.; Lipp, P. PKCα: A versatile key for decoding the cellular calcium toolkit. J. Cell Biol. 2006, 174, 521–533.
[CrossRef]
Dupont, G.; Houart, G.; De Konick, P. Sensitivity of CaM kinase II to the frequency of Ca2+ oscillations: A simple model.
Cell Calcium 2003, 34, 485–497. [CrossRef]
Schingmann, K.; Michaut, M.A.; McElwee, J.L.; Wolf, C.; Travis, A.J.; Turner, R.M. Calmodulin and CaMKII in the sperm Principal
Piece: Evidence for a Motility-Related Calcium/Calmodulin Pathway. J. Androl. 2007, 28, 706–716. [CrossRef]
Hunley, C.; Marucho, M. Electrical propagation of condensed and diffuse ions along actin filaments. bioRxiv 2021, 50, 91–107.
[CrossRef] [PubMed]
Zahran, E.H.M. Exact travelling wave solutions of nano-ionic solitons and nano-ionic current of microtubules using the
exp(− ϕ(ξ ))-expansion method. Adv. Nanoparticles 2015, 4, 25–36. [CrossRef]
Sirisubtawee, S.; Koonprasert, S. Exact traveling wave solutions of certain nonlinear partial differential equations using the
( G ′ /G2 )-expansion method, Hindawi. Adv. Theor. Math. Phys. 2018, 2018. [CrossRef]
Kayum, M.A.; Seadawy, A.R.; Akbar, A.M.; Sugati, T.G. Stable solutions to the nonlinear RLC transmission line equation and the
Sinh-Poisson equation arising in Mathematical Physics. Open Phys. 2020, 18, 710–725. [CrossRef]
Koldenkova, V.P.; Nagai, T. Genetically encoded Ca2+ indicators; Properties and evaluation. Biochem. Biophys. Acta 2013,
1833, 1787–1797. [CrossRef]
Salazar, C.; Politi, A.Z.; Höfer, T. Decoding of calcium oscillations by phosphorylation cycles: Analytic results. Biophys. J. 2008,
94, 1203–1215. [CrossRef]
Böhmer, M.; Van, Q.; Weyand, I.; Hagen, V.; Beyermann, M.; Matsumoto, M.; Hoshi, M.; Hildebrand, E.; Kaupp, U.B. Ca2+ spikes
in the flagellum control chemotactic behavior of sperm. EMBO J. 2005, 24, 2741–2752. [CrossRef]
Priego-Espinosa, D.A.; Darszon, A.; Guerrero, A.; González-Cota, A.L.; Nishikagi, T.; Martinez-Mekler, G.; Carneiro, J. Modular
analysis of the control of flagellar Ca2+ -spike trains produced by CatSper and CaV channels in sea urchin sperm. PLoS Comput.
Biol. 2020. [CrossRef]