Effect of curvature and normal forces on motor regulation of cilia
Disclaimer. The following document is a preprint of the PhD dissertation of Pablo Sartori, defended in 2015 at the Technische Universität Dresden. There are only minor differences between this preprint and the final version (formatting changes, addition of acknowledgements, resizing of figures, and an additional appendix regarding the fitting procedure).
Abstract
Cilia are ubiquitous organelles involves in eukaryotic motility. They are long, slender, and motile protrusions from the cell body. They undergo active regular oscillatory beating patterns that can propel cells, such as the algae Chlamydomonas, through fluids. When many cilia beat in synchrony they can also propel fluid along the surfaces of cells, as is the case of nodal cilia.
The main structural elements inside the cilium are microtubules. There are also molecular motors of the dynein family that actively power the motion of the cilium. These motors transform chemical energy in the form of ATP into mechanical forces that produce sliding displacement between the microtubules. This sliding is converted to bending by constraints at the base and/or along the length of the cilium. Forces and displacements within the cilium can regulate dyneins and provide a feedback mechanism: the dyneins generate forces, deforming the cilium; the deformations, in turn, regulate the dyneins. This feedback is believed to be the origin of the coordination of dyneins in space and time which underlies the regularity of the beat pattern.
Goals and approach. While the mechanism by which dyneins bend the cilium is understood, the feedback mechanism is much less clear. The two key questions are: which forces and displacements are the most relevant in regulating the beat? and how exactly does this regulation occur?
In this thesis we develop a framework to describe the spatiotemporal patterns of a cilium with different mechanisms of motor regulation. Characterizing and comparing the predicted shapes and beat patterns of these different mechanisms to those observed in experiments provides us with further understanding on how dyneins are regulated.
Results in this thesis. Chapters 1 and 2 of this thesis are dedicated to introduce cilia, chapters 36 contain the results, and chapter 7 the conclusions.
In chapter 1 we introduce the structure of the cilium, and discuss different possible regulatory mechanisms which we will develop along the thesis. Chapter 2 contains a quantitative description of the ciliary beat observed in experiments involving Chlamydomonas.
In chapter 3 we develop a mechanical theory for planar ciliary beat in which the cilium can bend, slide and compress normal to the sliding direction. As a first application of this theory we analyze the role of sliding crosslinkers in static bending.
In chapter 4 we introduce a mesoscopic description of molecular motors, and show that regulation by sliding, curvature or normal forces can produce oscillatory behavior. We also show that motor regulation by normal forces and curvature bends cilia into circular arcs, which is in agreement with experimental data.
In chapter 5 we use analytical and numerical techniques to study linear and nonlinear symmetric beats. We show that there are fundamental differences between patterns regulated by sliding and by curvature: the first only allows wave propagation for long cilia with a basal compliance, while the second lacks these requirements. Normal forces can only regulate dynamic patterns in the presence of an asymmetry, and the resulting asymmetric patterns are studied in chapter 6.
In chapter 6 we study asymmetric beats, which allow for regulation by normal deformations of the cilium. We compare the asymmetric beat from Chlamydomonas wild type cilia and the beat of a symmetric mutant to the theoretically predicted ones. This comparison suggests that sliding forces cannot regulate the beat of these short cilia, normal forces can regulate them for the wild type cilia, and curvature can regulate them for wild type as well as for the symmetric mutant. This makes curvature control the most likely regulatory mechanism for the Chlamydomonas ciliary beat.
Contents
 1 Introduction
 2 Characterization of the Chlamydomonas beat
 3 Force balance of planar cilia
 4 Dynamics of collections of motors
 5 Nonlinear dynamics of the cilia beat
 6 Motor regulation in asymmetric beats
 7 Conclusions and future work
 A Calculus of variations and test of resistive force theory
 B Critical modes and numerical methods
Chapter 1 Introduction
Cilia are ubiquitous eukariotic organelles. Diverse in scale and function, they are involved in a number of different motile tasks, yet at the core of all cilia lies a common structure: the axoneme. Imaging of bending cilia, of their internal structure, and biochemical experiments, have given rise to models of the ciliary structure, as well as suggestions for their functioning. In this section we give an overview of the internal components of the cilium, its structure, and some of the suggested models for their dynamic regulation.
1.1 What cilia are and where to find them
Cilia are long thin organelles which protrude from eukaryotic cells. They are motile structures fundamental for the functioning of cells. Indeed, immotile cilia are related with problems in embryonal development, and lead to human diseases such as primary ciliary diskinisia [3]. Although with differences among species, the core structure of cilia is highly conserved, yet they are involved in a number of diverse functions.
Cilia can exhibit a variety of periodic ondulatory motions in the presence of chemical energy. These beat patters are involved in fluid flow and propulsion of microswimmers. Examples of cilia involved in fluid flow are the nodal cilia, responsible for the breaking of leftright symmetry in the development of vertebrates [62]. Another example are the cilia in the mammalian lungs, responsible for the flow of mucus [82] (see Fig. 1.1).
Many eukaryotic microswimmers use cilia to propel themselves. Such microorganisms can have as many as hundreds of cilia (as in the case of Paramecium, see Fig. 1.1), but in this thesis we will focus on microorganisms with one or two cilia. Examples of cells with one motile cilia are the sperm of sea urchin, bull or anopheles (see Fig. 1.1). An example of an organism with two cilia is the unicellular algae Chlamydomonas, the main model system studied in this thesis (see Fig. 1.1).
Organism  Length ()  Frequency ()  Wave number  Amplitude () 

Chlamydomonas  10  50  1  1 
Sea Urchin Sperm  60  30  2  5 
Trypanosome  30  20  3  0,6 
Bull Sperm  50  20  1  0,5 
Anopheles Sperm  80  ?  3  1,5 
There is great variability among the ciliary properties of the microswimmers in Fig. 1.1. Among them these cilia differ in size, beat frequency, wave number, and amplitudes. Estimates of these properties for the examples in Fig. 1.1 are in Table 1.1. But however different their beating properties are, underlaying all of them reside the same structural elements, which we review in the next section.
1.2 Inside a cilium: the 9+2 axoneme
In the core of cilia lies a cytoskeletal cylindrical structure called the axoneme. The axoneme is a cylindrical bundle of 9 parallel microtubules doublets. At its center there are 2 additional microtubules, the central pair (see Fig. 1.2 B). They extend from the basal end, attached to the cell body, to the distal end. While there are other possible structures [22, 64], the 9+2 axoneme is a highly evolutionary conserved structure [21]. Here we will focus on the Chlamydomonas axoneme which is an example of the 9+2 arrangement.
The doublets and the central pair are connected by the radial spokes (see Fig. 1.2 A), responsible to keep the radius of the diameter at [61]. The doublets are parallel to each other and from the radius we estimate a spacing of . Each doublet is connected to its nearneighbors by crosslinkers such as nexin, see section 1.2.3. These provide a resistance to the relative sliding of doublets, and to changes in the spacing between them. Doublets are also connected by dynein molecular motors 1.2.2, which create active sliding forces when ATP is present. All these elements are present in a highly structured manner in the axoneme, repeating along the long axis of the cylinder with a period of . In the following we describe in more detail the main characteristics of the axonemal elements.
1.2.1 Scaffolding elements: doublet microtubules and the central pair
The main structural elements of the axoneme are microtubules, present in the doublets as well as in the central pair. Microtubules are filamentous protein complexes which have scaffolding functions in the eukaryotic cytoskeleton. Structurally, they are hollow cylinders with a diameter of composed of thirteen protofilaments (see Fig. 1.3). The basic structural element of a protofilament is the tubulin dimer, with a length of roughly . Because this dimer is polar the mircotubules are polar as well, which grow much faster from the end than from their end [92].
The elastic properties of microtubules can be accurately accounted for using a model of incompressible semiflexible polymers with a bending stiffness of [37]. While recent experiments suggest that compressibility and shearing may play a role in determining this stiffness [88, 63], in this thesis we will consider microtubules as inextensible and non shearing.
In the axoneme microtubules appear in the central pair and in the doublets. The central pair consists of two singlet microtubules (i.e., with all 13 protofilaments, see Fig. 1.3) longitudinally connected between them. But while its basic component, namely the microtubules, have well known properties, those of the central pair are complex and elusive. For example, there is evidence suggesting that the central pair shows a large twist when extruded from the axoneme [59]. The orientation of the central pair is used as a reference to number the doublets, as in Fig. 1.2.
The doublets are composed of one Amicrotubule and one Bmicrotubule. The Amicrotubule has all 13 protofilaments, while the Bmicrotubule has only 10 and a half protofilaments (see Fig. 1.3). The stiffness of microtubule doublets has not been directly measured, but simple mechanical considerations suggest that they are roughly three times that of a single microtubule, and so we estimate their bending rigidity as [37].
The doublets and the central pair are the main structural elements involved in determining the bending stiffness of the axoneme. Using mechanical considerations one can estimate the stiffness of the axonemal bundle as being 30 times higher as that of a single microtubule, which yields for the stiffness of the axoneme [37]. This number can be compared with several direct measurements of axonemal stiffness. While these vary with the experimental conditions (such as the presence of vanadate or ATP), for rat sperm [79] as well as sea urchin sperm [74] the value of has been measured, in good agreement to the previous estimate.
1.2.2 Active force generating elements: dynein motors
The ciliary beat is powered by axonemal dyneins, which are a family of molecular motors (see Fig. LABEL:fig:dynein). They convert chemical energy into mechanical work, with one full mechanochemical cycle corresponding to the hydrolysis of one ATP molecule. The Chlamydomonas axoneme in particular contains 14 different types of dyneins and has a total of of these motors [94] over its length of roughly . And while these dyneins can have diverse properties (such as different ATP affinities, gliding speeds, or capability of generating torques [44, 84]), we focus here on their generic properties.
Dyneins are periodically distributed along the nine microtubule doublets, with their stem (yellow in Fig. LABEL:fig:dynein) rigidly attached to the Amicrotubule, and their stalks being briefly in contact with the adjacent Bmicrotubule during the power stroke process [37]. Stem and Stalk are connected by a 6AAA ring, and all together the weight of a motor domain is roughly [78, 40]. Their size is roughly , and the their structure is sketched in Fig. LABEL:fig:dynein [17, 75].
figure\end@float
It is known that during their power stroke, dyneins generate forces that tend to slide the axonemal doublets with respect to each other. For instance, after a protease treatment of the axoneme the presence of ATP makes the doublets slide apart [86]. Finally it is worth noting that it has been shown in vitro that motors such as dynein and kinesin are capable of generating oscillatory behaviors reminiscent to beat patterns in the presence of ATP [81, 76]. All this evidence supports the idea that dyneins generate sliding forces which regulate the beat pattern of the axoneme.
Electron micrographs have shown that a motion of the tail of up to may occur [17]. As an upper limit to force generation we can consider that all the energy contained in one ATP molecule (which is , [37]) is invested in this motion, and thus we estimate a force of . On the other hand, indirect estimates have resulted in motor forces of up to [52], suggesting that axonemal dynein can generate as much force as cytoplasmic dynein [27] or kinesin [20, 87].
While the mechanochemical dynein cycle has not been fully deciphered, much is known about its main steps [39]. ATP binding is known to occur with an affinity of roughly (for ATP concentrations below ) [68], which dramatically accelerates detachment of the dynein tail from the microtubule. After this, ATP is hydrolyzed and a conformational change occurs which renders dynein to its prepowerstroke configuration. This slow step occurs at roughly , and is thus the ratelimiting step [47]. ADP increases the affinity of dynein to microtubules (which may also be influenced by additional conformational changes [48]), leading to a binding state in which the powerstroke occurs, which completes the dynein cycle (see Fig. LABEL:fig:dynein). Overall the duty ratio of dynein (fraction of cycle time which it spends bound the Bmicrotubule) has been estimated to be [37], and it is thus considered a nonprocessive motor.
1.2.3 Passive crosslinkers: nexin and the radial spokes
Radial spokes and nexin crosslinkers are passive structural elements involved in maintaining the structural integrity of the axoneme. Coarsely, the radial spokes (see Fig. 1.2) help to sustain the radius of the axoneme, while the nexin crosslinkers prevent the doublets from sliding apart. We now review some of the more intricate details of nexin and radial spokes.
The nature of nexin linkers has long been debated. Indirect evidence suggests that a protein complex must be involved in constraining the sliding of doublets, since protease treatment of axonemes results in telescoping of the doublets when ATP is present [86]. Furthermore, it is believed that such a constraint is fundamental in transforming the sliding dynein forces into axonemal bending, see section 1.4. But while early electro microscopy identified nexin as a separate protein complex [93, 32], recent work suggests that it is part of the dynein regulatory complex [61, 34, 8]. This would imply that the crosslink nexin can be closely involved in regulating the behavior of the dynein regulatory complex, and thus the flagellar beat. Furthermore, recently it has been suggested that dynein is linked to the radial spokes [49]. In any case, it is clear that dynein provides passive resistance to sliding, as indeed it’s stiffness has been directly measured to be for of axoneme. This number increases five fold in the absence of ATP (when dyneins are attached), thus further indicating that dynein and nexin are intimately related may be a highly regulated passive structure [58]. Finally, since it stretches about ten times its equilibrium length, its forcedisplacement behavior has been suggested to be nonlinear [55].
In each axonemal repeat there are two radial spokes per doublet in the Chlamydomonas axoneme [65]. Recently, it has been shown that there is also an incomplete third radial spoke, which is believed to be an evolutionary vestige [5]. When the radius of the axoneme is reduced, radial spokes compress, which suggests that they are involved in sustaining the crosssection of the axoneme [102]. However, recent evidence has shown that radial spokes are connected to the same regulatory complex identified as nexin, which suggests that they may also act as regulators of the beat [5]. While axonemes can beat in the absence of radial spokes [100], it has been observed that the gliding speed of dynein increases substantially in the presence of radial spokes [83].
1.3 Structural asymmetries
In a coarse view the picture of the axoneme is highly symmetrical. It is periodical in its longitudinal direction, with a period of . It also has ninefold rotational symmetry around its longitudinal axes, with motors connecting all the doublets. However, recent studies have shown that this picture is not accurate. There are many important structural asymmetries in the axoneme, which can have an important role in regulating the beating dynamics.
1.3.1 Axoneme polarity
The axoneme is a polar structure, and is thus not symmetric along its length. There are several polar asymmetries in the axoneme, the most obvious one is that the microtubule doublets themselves are made of polar proteins such as the doublets. Furthermore, along each repeat the distribution of dyneins and radial spokes is not homogeneous. More importantly, there are two polar asymmetries which occur on a larger scale: the asymmetry between the ends, due mainly to the basal body; and the asymmetry along its arclength, due to inhomogeneous distribution of motors.
The two ends of the axoneme are fundamentally different. First, the distal end (farthest from the cell body) is the + end of the doublet microtubules and is thus constantly being polymerized. Second, at the basal end of the axoneme there is a complex structure called the basal body. The basal body is the region where the axoneme attaches to the cell body, and it’s composed of microtubule triplets which get transformed into doublets as the basal body transforms to the axoneme (see Fig. 30 in [72], also [72]). Evidence suggests that the basal body is elastic, and can allow for doublets sliding at the base of the axoneme [91, 95, 97]. This has led to the proposal that a basal constraint is an important regulatory mechanism for regulating the beat [71, 76], with some evidence that it may be necessary for beating [26, 29].
But besides of differences at the ends of the axoneme, along its length there are also asymmetries in the distribution of dyneins [101, 16] as well as crosslinkers [66]. The repeats are homogoneous only in the central part of the axoneme, with repeats changing as they approach the ends. For instance, certain types of dyneins (like inner dynein arms) are missing towards the base of the axoneme [16]. Yet some other types of dynein are present only near the base, the socalled minordyneins [101].
1.3.2 Chirality
Since the axoneme is polar, and the dyneins are bound to the A tubule and exert their power stroke on the B tubule, the axoneme is a chiral structure (compare in Fig. LABEL:fig:chirality the two axonemes, which have the same chirality). Furthermore, to date only axonemes with one handedness have been found. One immediate consequence for Chlamydomonas is that it swims with two left arms. That is, unlike with human arms, its two flagella are not mirrorsymmetric with respect to each other (see Fig. LABEL:fig:chirality). But beyond this intrinsic chirality, detailed studies on the axonemal structure have revealed further chiral asymmetries which we now discuss.
figure\end@float
Certain adjacent doublets are connected by various structures: the doublets 1 and 2 are connected by the socalled 12 bridge, believed to be important in setting the beating plane [16, 36]. Additional crosslinkers have recently been identified the doublets 9 and 1, 5 and 6; as well as 1 and 2 [66]. All these structures have been suggested to constrain the sliding of the doublets they link, thus defining the beating plane as that in which the nonsliding doublets lay (which is roughly that of doublets 1 and 5, see Fig. 1.2). Another asymmetry in the beating plane are the presence of beaks inside the B tubules, filamentous structures are all along doublets 1, 5 and 6 [36, 66].
Several species of inner dyneins are also missing in certain doublets, for example in the doublet 1 [16]. Interestingly, the same dynein arms that are missing in the basal region are also missing all along the axoneme in the doublet 9, thus coupling the polar asymmetry with the chiral asymmetry.
Further contribution to this chiral asymmetry can come from the inherent twist observed in the central pair, whose interaction with the doublets remains unclear [46]. Furthermore, evidence in Sea Urchin sperm has shown that the axoneme itself can take spirallike shapes, which are intrinsically chiral [96].
1.4 The mechanism of ciliary beat
How the beating patterns of cilia occur is a subject of intense research, and the main topic we address in this thesis. It is generally believed that the beat is generated by alternating episodes of activation of opposing sets of dynein [85]. Which dyneins are activated and which are deactivated is believed to be regulated by a mechanochemical feedback. The beat is thus believed to be a selforganized process, with no a priori prescription of dynein activity: dyneins regulate the beat, and the beat regulates the dyneins. We now go over the key points of this process, reviewed in [98, 54, 15].

Dyneins produce active sliding forces: Since dyneins can slide doublets [84] and slide axonemes apart [86], it is believed that they produce sliding forces between the doublets, see section 1.2.2. In a rotationally symmetric axoneme all these forces balance exactly and there is no net sliding force, a situation termed tugofwar [38]. In this case fluctuations can create a small imbalance in force, which is then amplified by the motors and finally produces a significant net sliding force. Alternatively, chiral asymmetries (see 1.3) in the axoneme can also result in a net sliding force.

Passive crosslinkers constrain sliding and convert it into bending: The net sliding force produced when the dyneins on one side of the axoneme win the tugofwar leads to relative motion between the doublets [11]. Thanks to crosslinkers along the axoneme [55, 8], and possibly also at the base [4, 71, 13], this sliding is constrained and converted into bending.

A mechanochemical feedback can amplify the bending and produce switching. In the tugofwar scenario, when one set of dyneins wins and the axoneme bends, the other set of dyneins becomes gradually less active while the winning side becomes more active, thus the bending is increased [38]. However, possibly due to a delay in the mechanochemical cycle by the dyneins [57, 71, 38], eventually the switch occurs and the opposing dyneins become active. The bend is thus reversed, giving rise to an oscillation.
Of the three processes described above, the one which is the least understood is that of the mechanochemical feedback. It is not clear what mechanical cue is sensed by the dyneins, and how it is sensed. We now review some of the proposals.
1.4.1 Sliding control
In sliding control, the mechanochemical feedback responsible for the switch occurs by the dynein motors sensing the sliding displacement between doublets. Motor models where response to sliding can induce oscillations of a group of motors have a long history [10, 31, 42, 43]. The way the feedback can appear varies from one model to another, typically for processive motors it is believed that sliding forces induce detachment of motors [31], while for nonprocessive motors like dynein ratchet mechanisms have been suggested [42, 43]
An axoneme bends because of the sliding forces generated by the opposing dyneins in the plane perpendicular to the bending plane. For example, if the beating plane is formed by the doublets 1 and 5, the opposing dyneins are those in doublets 3 and 8 (see Fig. LABEL:fig:raxoneme and [35]). In this case, the sliding experienced by one set of motors is positive, in the sense that it favors their natural displacement to the + end of the axoneme. The displacement of the opposing set of motors is negative. It is believed that this acts as a regulatory mechanism of the motors force, with the dyneins that slide to the positive direction creating an active force and the others a resisting force.
1.4.2 Curvature control
In curvature control, the switching of molecular motors is regulated by the curvature of the flagellum. This mechanism was initially proposed in [57], where the moment generated by the motors was suggested to be controlled by a delayed reaction to curvature. Later works have used the motor moment density as the quantity to be affected by curvature [9, 14, 71].
Notice that there is a crucial difference between curvature and sliding control. In sliding control oscillations can occur for straight filaments (as, for example, occurs in actin fibers in the presence of myosin [67]). This is impossible in curvature control. Models of the axoneme where curvature regulates the beat operate by opposing motors being activated or inactivated when a critical value of the curvature is reached, together with a delay [9, 13]. One reason why opposing motors may react differently to curvature is that active and inactive motors, perpendicular to the beating plane, experience different concavity of the adjacent doublet (see Fig. 1.4).
This model is supported by the fact that some flagellar waves seem to show a traveling wave of constant curvature [12]. Furthermore, while the study in [71] favored a sliding control model, curvature control also showed good agreement with the Bull Sperm beat. However, there is no accepted mechanism of curvature sensing by dynein motors. Direct geometrical sensing is unlikely, given the small size of dynein compared to curvatures in the axoneme [71]. Indeed, a radius of curvature of corresponds to the bending of an tubulin dimer through an angle of only 0.025 degrees which is two orders of magnitude smaller than the curvedtostraight conformation associated with the straightening of a free GDPbound tubulin subunit needed for its incorporation into the microtubule wall [70, 60]. An alternative explanation is provided by the geometric clutch [53], reviewed in the next section.
1.4.3 Geometric clutch
In the geometric clutch the key factor regulating the switch of dynein motors is the spacing between the doublets, or equivalently the corresponding transverse force (force) between them [50, 53]. The hypothesis is that there are two contributions to the force coming from curvature and sliding. The contribution of the curvature is termed global force, and tends to detach motors. This global force will be termed normal force in our description. The sliding contributes to the local force, which tends to attach motors and is much smaller than the global force [52]. Thus the geometric clutch combines effects of sliding and curvature control. There is an additional element required in the geometric clutch, which is a distributor of the force between the opposing sides of the axoneme. Radial spokes have been suggested to fulfill this role [52].
Computer models using the geometric clutch have successfully replicated the beat of Chlamydomonas [51]. Furthermore, there is direct experimental evidence suggesting that bent axonemes show higher spacing in bent regions, where the global force increases [56]. However, it is so far unclear how the force is distributed in a crosssection of the axoneme [53]. This implies that it is not known how it gets distributed between the opposing motors. The same problem is present in curvature control, but not in sliding control: there opposing motors experience opposite sliding displacement.
1.5 Physical description of axonemal beat
To gain insight into ciliary beat, in this thesis we combine tools of nonlinear dynamics, elasticity, and fluid mechanics. This will allow us to provide a full description of the selforganized dynamics of a cilia propelled by molecular motors [57, 18]. We now give some brushstrokes on the main elements of the theory.
First, to derive the mechanical forces we will use a variational approach. That is, we will construct an energy functional , where is the position of the arclength point of the cilium, which collects all the elastic properties of the cilium. By performing variations, we will obtain the mechanical forces. Second, to describe the effect of molecular motors, we will write down a dynamic equation for the motor force which tends to slide the doublets apart. This dynamic equation will depend on the internal strains and stresses of the cilium, thus coupling the system. Third, to model the fluid we will use resistive force theory [30], in which the fluid force is characterized by two friction coefficients and (corresponding to normal and tangential motion). This simplification is possible because the Reynolds number of a cilium is small, which allows us to neglect nonlinear effects [69]; and because the cilium is a slender body, with a diameter much smaller than its length .
Putting all these elements together, we will obtain a dynamic equation for the shape of the cilium. If we parametrize the cilium by its local tangent angle at arclength and time , we will have to linear order the following force balance equation
(1.1) 
This equation includes the effects of fluid friction, elasticity (with , the young modulus and the second moment of inertia), and the motor force . The full description of the system still requires an equation that couples the motor force dynamics to an internal strain in the cilium. For example, if the motor force responds to sliding velocity (as has been suggested [10, 19]), we have to linear order
(1.2) 
where is the linear response function, and is the local sliding between doublets. The two equations above define a linearized dynamical system which can undergo oscillatory instabilities and go to a limit cycle (which, to be described, requires to take into account nonlinearities). Alternatively, the motor force could respond to changes in the local curvature [57, 9] or the doublets spacing [50]. In this thesis we characterize the beat patterns corresponding to these three different different regulatory mechanisms, and compare the results to the experimentally observed ciliary patterns described in the next chapter.
1.6 Conclusions

The core element of the cilium is the axoneme: a cylindrical bundle of microtubules connected by dynein molecular motors.

The beat of cilia is a selforganized process powered through microtubule sliding forces that are produced by dynein motors.

Selforganization arises through motor regulation via one (or several) of the following mechanisms: regulation by microtubules sliding, by their curvature, or by the normal force arising between them.
Chapter 2 Characterization of the Chlamydomonas beat
Chlamydomonas cilia can be isolated from the cell and, in the presence of ATP, beat periodically. In this section we mathematically describe the beat pattern of intact as well as disintegrated cilia. We show that the observed beat pattern of an intact cilium is well characterized by its static and fundamental harmonics. While for wildtype the zeroth harmonic is very important as the beat is asymmetric, this is not the case for a symmetric mutant which we analyze. We also show that in disintegrated cilia pairs of doublets can interact with each other, reaching a static equilibrium in which the shape is a circular arc. All of the data appearing in this section was taken in the laboratory of Jonathon Howard: the data from the first section by Veikko Geyer, and that of the second by Vikram Mukundan.
2.1 The beat of isolated Chlamydomonas cilia
The cilia of Chlamydomonas can be isolated from the cell body [77]. When their membrane is removed and ATP is added to the solution, they exhibit periodic beat patters [7]. This is evidence that the beat of the axoneme is a selforganized property, as it occurs independent of the cell body and with ATP homogeneously present around the axoneme.
Our collaborators in the group of Jonathon Howard, in particular Veikko Geyer, have imaged the beat of isolated Chlamydomonas axonemes using high speed phase contrast microscopy with high spatial and temporal resolution. The pixelsize was of , and the framerate . In comparison, the typical size of the axonemes was , and the characteristic beat frequency . Nine frames from a sample beat pattern appear in Fig. 2.1, the tracking of the axoneme appears as a green line, the basal end of the axoneme is marked by a black circle. The tracking was performed using the fiesta code published in [73]. As one can see, the beat of the axoneme is asymmetric, and because of this the axoneme swims in circles. While the typical beat frequency of the axoneme is , the rotational frequency of the axoneme is ten times slower, on the order of .
The tracked cilium is characterized at each time by a set of two dimensional pointing vectors from the fixed laboratory reference frame to each point along the axonemal arclength , with the length of the cilium (see Fig. LABEL:fig:tangent). To describe its shape decoupled from the swimming we use an angular representation. That is, for each point along the cilium we calculate the tangent angle with respect to the horizontal axis , see Fig. LABEL:fig:tangent. Note that this angular representation contains less information than the vectorial representation, in particular the swimming trajectory is lost. However still retains information about the rotation velocity of the axoneme . To obtain a pure shape description we also subtract this rotation, that is
(2.1) 
where the tangent angle now only describes the shape of the axoneme.
figure\end@float
Since the beat of Chlamydomonas is periodic in time, we analyze the shape information by transforming its temporal coordinate to frequency space. This can be done using a fast Fourier transform, and the result for the midpoint of the axoneme is shown in Fig. 2.2 A. There are peaks around frequencies multiple of the fundamental frequency, and a background of noise. Neglecting the noise, the beat can thus be approximated by the following Fourier decomposition
(2.2) 
Where is the fundamental frequency of the beat, which in the example considered in Fig. 2.2 A is . The modes are complex functions of , and their amplitude decreases as the mode increases, see Fig. 2.2 A. Because the angle is a real quantity, its modes satisfy .
Importantly, the above description includes the zeroth mode , which corresponds to the average shape of the cilium and defines its asymmetry. In Fig. 2.2 B this mode is shown for an example. As one can see, besides a flattening at the ends, the tangent angle decreases monotonically along the axoneme. The amplitude and phase profiles of the first, second and third dynamic modes are shown in Fig 2.2 B and C. As one can see, the amplitude of the first mode is much larger than that of all higher modes, and roughly constant along the arclength with a small dip in the middle. The phase profile of the first mode is shown in Fig. 2.2 D, and is monotonically decreasing. Over the full length of the axoneme the phase decreases about , which corresponds to a wavelength equal to the length of the axoneme.
The curvature is the derivative of the tangent angle with respect to the arclength. This means that the constant slope of the angle in the zeroth mode corresponds to a constant mean curvature of roughly . Furthermore, since the amplitude of the first mode is approximately constant and its phase decreases with a constant slope, we conclude that a good approximation of the Chlamydomonas ciliary beat, shown in Fig. 2.3 A, is the superposition of an average constant curvature, Fig. 2.3 B, and plane wave of its angle, Fig. 2.3 C. Thus, the beat of Chlamydomonas is fundamentally asymmetric, and its asymmetry is well characterized by a constant mean curvature.
It is important to note that this asymmetry is not present during the phototropic response of Chlamydomonas and in certain mutants that Move Backwards Only (MBO). The MBO 13 mutants beat symmetrically [80, 28]. In particular the mutant MBO 2 has a very small static component in the beat, see Fig. 2.2 A. Interestingly, the amplitude profile of the first mode is very similar to that of the wild type cilium, as one can see by comparing Fig. 2.2 B and Fig. 2.2 C. The same is true of the phase profile, which appears in Fig. 2.2 C. Thus, these mutants have a beat that, while lacking the asymmetry, are in the rest very similar to that of the wildtype cilium. Finally, it is worth noting that also wildtype Chlamydomonas can exhibit symmetric beat patterns in the presence of Calcium, although these are threedimensional beats fundamentally different from those of mbo2 [7, 28].
2.2 Bending of disintegrated axonemes into circular arcs
In the presence of a protease treatment, axonemes partially loose their crosslinkers. When this occurs, two doublets can interact via the dyneins of one of them. This provides a minimal system, which has been reported to produce sliding and bending waves [4, 60]. In particular, at low concentrations of ATP, pairs of filaments associate, and propagate small bending waves towards the basal end as one filament slides along the other (Fig. 2.4, first row, arrows). Furthermore, in some occasions the two filaments reassociate along their entire length and bend into a circular arc (Fig. 2.4, second row). The system then becomes unstable and the filaments separate again.
To analyze the bending process in detail, the shapes of the filaments pairs are digitized, as shown in Fig. LABEL:fig:bendtrack A. From this one can calculate the tangent angle as a function of arclength in successive frames as the filaments become more and more bent, see Fig. LABEL:fig:bendtrack B. Importantly, the filament pair approaches a steadystate shape in which the tangent angle increases linearly with arc length, except at the very distal end where it flattens (Fig. LABEL:fig:bendtrack, to ). Such a linearly increasing tangent angle implies that the steadystate shape is approximately a circular arc. This static constant curvature is analogous to that observed in the wildtype Chlamydomonas cilia as was described in section 2.1.
figure\end@float
There is evidence that the two interacting filaments are two doublet microtubules and not two singlet microtubules or one doublet microtubule interacting with the central pair. The two individual singlet microtubules that comprise the central pair will each have a lower intensity than a doublet. However, the interacting filaments (Fig. LABEL:fig:intensities A and B, red) have the same intensities as the noninteracting filaments (Fig. LABEL:fig:intensities A and B, blue). This implies that the interacting filaments are not two singlet microtubules (which would both be much dimmer than the noninteracting filaments) or a singlet and a doublet (one of the two interacting filaments would be much dimmer than the other).
figure\end@float
One of the advantages of this experiment is that the shapes reach a quasistatic limit, which can be analyzed assuming mechanical equilibrium. The assumption of mechanical equilibrium means that the frictional forces due to motion through the fluid can be ignored, and applies if the relaxation time of the bent beam in the fluid is much smaller than the typical time of the bending observed. The relaxation time of a beam in a fluid is . Using for the doublets stiffness, for the fluid friction, and for the length, we obtain a relaxation time of , which is much smaller than the typical duration of the process .
2.3 Conclusions

The beat of the Chlamydomonas cilium is well described by its static and fundamental mode.

In a wildtype cilium the static mode is large and has the shape of a circular arc, producing an asymmetry in the beat. In an mbo2 cilium the static mode is small, and the resulting beat symmetric.

Pairs of doublets are statically bent into circular arcs due to motor regulation.
Chapter 3 Force balance of planar cilia
The planar beat of a cilium can be described as a pair of opposing filaments. In this section we introduce a twodimensional representation of the axoneme as a pair of opposing inextensible filaments. We consider the sliding as well as variable spacing between these filaments. By balancing mechanical and fluid forces, we derive the general nonlinear dynamic equations of a cilium beating in a plane. As a first application of this theory, we study the role of sliding crosslinkers in the static bending of a cilium.
3.1 Planar geometry of the axoneme
We describe the axoneme by a pair of opposing filaments, which we label A and B (see Fig. 3.1). Each filament is parametrized by the arclength of the centerline, which ranges from 0 at the base to at the tip. The filaments are separated by a distance which can depend on the arclength, and they can slide with respect to each other at every point.
The geometry of the centerline is characterized by , a twodimensional pointing vector from the laboratory frame. At any given point of the centerline we can define the local tangent vector and the local normal vector , as shown in Fig. 3.1. The tangent vector is given by
(3.1) 
where in the last expression we have introduced a notation in which upper dots denote arclength derivatives. This notation will be kept throughout the rest of this thesis, with the number of dots denoting the order of the arclength derivative. The normal vector is defined simply as normal to , with orientation such that points out of the plane. Using the tangent vector we can also define the local tangent angle between the tangent vector and the axis. The relationship between the tangent angle and the pointing vector of the centerline is
(3.2) 
where is the position of the base of the centerline. The local curvature of the centerline is given by the arclength derivative of the tangent angle, . The geometry of the centerline is thus given by the set of equations
(3.3) 
which are the FrenetSerret formulas for the special case of a planar geometry in the absence of torsion.
Having introduced the geometry of the centerline we relate it to that of the pair of opposing filaments. Since each of the filaments is at a distance from the centerline, we can write
(3.4) 
Note that does not parametrize the arclength of either of the filaments, but that of the centerline. The arclength along each of the filaments is given by
(3.5) 
Using the corresponding FrenetSerret frame for each of the filaments, we obtain the curvature of the filaments (see Appendix A). To lowest order these curvatures are given by
(3.6) 
where geometric nonlinearities have been neglected.
The sliding of one filament with respect to the other at centerline arclength position is given by the mismatch in arclength along one filament with respect to the other plus the reference sliding at the base. We thus have
(3.7) 
where is the basal sliding of filament A with respect to B. From now on we take as reference filament B, and define the local sliding as that of A, we thus have . The explicit expression of the local sliding can be calculated using Eqs. 3.5 and is given to cubic order by
(3.8) 
Note that Eqs. 3.6 and 3.8 take a particularly simple form in the limit of homogeneous spacing , in which , and .
3.2 Static balance of forces
As discussed in section 2, the axoneme is composed of passive and active mechanical elements. The passive elastic elements such as the doublets and nexin crosslinkers provide structural integrity to the axoneme, and tend to restore it to a straight configuration without sliding. On the other hand the active elements create sliding forces between the doublets which ultimately bend it.
All the key elements of our ciliary description are provided in Fig. LABEL:fig:mech. To characterize their mechanical properties, we introduce the following work functional:
(3.9) 
where the explicit expressions of curvatures and sliding are given in Eqs. 3.6 and 3.8. The integral contains the energy density of the bulk of the axoneme, and the last term is the energetic contribution of the base. The first term in the integral is the bending energy characterized by the bending rigidity (in ) of each filament, which favors straight shapes. The second term is the energy of elastic linkers of stiffness density (in ) which are stretched by sliding (purple springs in Fig. LABEL:fig:mech). The stiffness (in ) of basal linkers is denoted (blue spring in Fig. LABEL:fig:mech ). We have denoted by (in ) the stiffness to normal deformations relative to the reference spacing (green springs in Fig. LABEL:fig:mech). The work performed by motors which generate relative force (in ) between the two filaments is given by the contribution . We have also introduced a Lagrange multiplier (in ) to ensure the inextensibility of the centerline.
figure\end@float
To obtain the equations of static equilibrium of the axoneme under external forces we use the virtual work principle. We consider as reference a straight configuration (r(s)=sx), with no basal sliding () and homogeneous spacing (). The virtual work principle establishes that the internal virtual work performed by a variation , is equal to the external work performed by external forces against this variation. For the forces applied on the bulk of the axoneme we have
(3.10) 
Where are the bending force density applied externally in each point with a specified direction in the plane, and we are assuming that there are no external basal forces (conjugate of ) or compressive forces (conjugate of ). At the boundaries, the same principle applies, and we have
(3.11) 
Where we have now introduced the external forces and at the boundaries. We have also allowed the presence of external torques and at the boundaries.
To obtain the balance of forces in the axoneme the first step is to calculate the corresponding functional derivatives. Doing standard variation calculus as detailed in Appendix A we obtain
(3.12) 
Where is the stiffness of both filaments, and we have introduced the static sliding force density and the basal force as
(3.13) 
and the integrated force . In the first equation we have introduced the Lagrange multiplier to replace . We prefer as it can be interpreted as the tension of the centerline. This can be seen by the following relation
(3.14) 
which can be derived directly from Eq. 3.12.
The static equilibrium balance equations are thus given by
(3.15) 
The forces and torques balances at the ends are obtained by using the boundary terms of the variations. This yields
(3.16) 
where the equations to the left correspond to those at the basal end and those to the right at the distal end . Note from Eq. 3.15 that the coupling between the spacing and the angle is nonlinear, since is of order . This implies that for small bending the change in spacing is negligible.
3.3 Dynamics of a cilium in fluid
So far we have considered the static equilibrium of a cilium, where the mechanical passive and active elements of the axoneme are balanced by timeindependent external forces. In the case in which the cilium is immersed in a fluid, the external forces to which it is subject are fluid forces. This is the basis of the description introduced in 1.5, and can be summarized by
(3.17) 
where are the forces that the fluid exerts on the axoneme along its length.
figure\end@float
In general, obtaining the fluid forces acting on a moving object is not an easy task, as it involves solving the nonlinear NavierStokes equation. However at low Reynolds number and in the limit of slender filaments (which applies to freely swimming cilia, see discussion in section 1.5), the fluid forces take the simple form
(3.18) 
This corresponds to forces opposing the moving filament with a drag coefficient in the normal direction, and a coefficient in the tangential direction (see Fig. LABEL:fig:rft).
This description of the fluidcilium interaction is known as Resistive Force Theory (RFT), and was introduced in a seminar paper by Hancock [33]. The effective friction coefficients can be related [33, 30] to the fluid viscosity as
(3.19) 
Crucially, the friction coefficients are different and satisfy . This asymmetry allows that, without exerting any net force on the fluid, the cilium can propel itself. We note here that the validity of RFT for ciliary beat has been verified in the literature for several swimming microorganisms [41, 24], and is also verified in Appendix A for some examples of freely swimming Chlamydomonas cilia.
To collect the effect of the fluid viscosity and other viscous components inside the axoneme, we introduce the following Rayleigh dissipation functional
(3.20) 
The first two terms inside the integral refer to fluid friction. We have also included internal sliding friction and compressive friction , both measured in . Finally, the base also contributes to the internal sliding friction with a coefficient (in ).
Just as the mechanical forces are obtained through variations of the mechanical work functional with respect to the corresponding fields , the dissipative viscous forces are obtained through variations of the Rayleigh functional with respect to their time derivatives. Thus, ignoring inertia, the force balance is now established as
(3.21) 
From this we obtain the dynamics of the centerline , the spacing and the basal sliding . Using the functional derivatives in Eq. 3.12, and computing those of the Rayleigh functional we obtain the following set of dynamic equations
(3.22) 
where we have omitted the dependences on time and arclength, denotes the sliding force density, and its integral. Analogously to , we can define the normal force density and the basal force . We thus have
(3.23) 
Note that while all three forces contain viscous and elastic components, only the sliding force has a contribution of active motor forces.
The equations above depend on the tension , which is a Lagrange multiplier. The standard procedure to obtain its value is to use the corresponding constraint equation, which in this case is . Since the dynamic equations do not directly involve , but instead involve its time derivative , we calculate the time derivative of the constraint and obtain . This equation can alternatively be written as , from which we obtain the tension equation
(3.24) 
Together with the boundary conditions given in Eq. 3.16, the set of Eqs. 3.223.24 constitute a complete set of integrodifferential equations. Provided a prescription for the motor force and the external boundary forces these equations can be solved to obtain dynamic shapes of the cilium.
3.3.1 Constrain of constant filament spacing
The dynamic equations of a cilium take a particularly simple form when it is imposed that the spacing between the filaments is constant along the arclength, that is . This requires that , while keeping the normal force finite, such that it acts as a Lagrange multiplier. The resulting dynamic equations are
(3.25)  
(3.26) 
Note, that in this limit the sliding is determined by the shape through
(3.27) 
Finally, the equation for the Lagrange multipliers (tension) and (normal force) become
(3.28)  
(3.29) 
which have to be solved at every time. Note that, while explicitly appear in the dynamic equation, does not. Another important fact is that the expression of is quadratic, which means that for small changes in the shape the normal force is negligible. This is analogous to the nonlinear coupling between spacing and bending seen before.
The boundary equations in 3.16 simplify to the following four equations
(3.30) 
With adequate choices of the external forces and torques at the basal and distal ends ( and ), these provide the six necessary boundary conditions. In this work, we will focus on three types of boundary conditions summarized in Fig. 3.2. These are free ends (A), in which the cilium is not subject to external forces or torques; pivoting base (B), in which the base of the cilium is held fixed, and it’s constrained from rotating with a stiffness while the distal end is free; and clamped base (C), in which the base is held fixed and enforces that with the distal end free.
In section 2.1 we described the beat of Chlamydomonas cilia using an angular representation. Motivated by this, we introduce an angular representation of the dynamic equation above by using . This directly gives
(3.31) 
which, to linear order, is Eq. 1.1 of the introduction. Together with the tension equation and the boundary conditions, this equation provides a mechanical description of the angles of all points along the cilium. Eq. 3.2 relates angle and position, and shows that the angular representation does not contain the trajectory of the basal point . It can, however, be obtained by inserting the solution of the tangent angle in the right hand side of Eq. 3.25 and integrating over time.
3.4 Requirements for static bending
In order to understand the mechanism regulating the shape of the cilium, we begin by analyzing the static limit. In absence of external forces and torques, Eqs. 3.15 can be integrated. Constraining the spacing to be homogeneous (see 3.3.1), the force balances simply become
(3.32) 
with the sliding force density. The tension in this case is null (i.e., ), and the normal force is given by
(3.33) 
which as indicated before is a second order term. The normal force is always positive, indicating that filaments tend to split apart. Finally, there are two boundary conditions
(3.34) 
which allow to calculate static shapes .
In the static regime the force produced by the motors in either filament is the stall force, that is . Considering motor densities and , we have that . If both filaments have the same densities of motors then . In this case we have that the opposing forces balance each other and , thus the cilium doesn’t bend () nor does it slide (). When , then Eq. 3.32 gives nontrivial solutions.
We can analytically solve Eq. 3.32 for the case of a constant motor force along the arclength, and obtain
(3.35) 
where we have used the two boundary conditions in Eq. 3.16, and defined the characteristic length
(3.36) 
beyond which the sliding compliance of the crosslinkers becomes significant. The basal sliding is obtained via Eq. 3.27, and is
(3.37) 
Note that due to the presence of static crosslinkers with stiffness the net sliding force does depend on arclength even if the motor force does not.
To characterize the static shapes given by Eq. 3.35 we first consider the limit in which the role of crosslinkers is small. According to Eq. 3.36 in this case we have . We thus expand in and to lowest order obtain
(3.38) 
This parabolic solution implies that for filaments of length the curvature decreases linearly along the arclength from its maximum value at the base to the minimum at the tip, as is shown in Fig. 3.3 A. As filaments get longer relative to the effect of crosslinkers becomes more prominent, which makes the curvature decrease sublinearly, and also reduces the maximum curvature at the base (see Fig. 3.3 A, darker shades of green).
A key parameter that characterizes the bent cilium is thus its maximal curvature , which is shown in Fig. 3.3 B as a function of for several values of . The maximal curvature decreases monotonically as the crosslinkers become stiffer, and saturates for vanishing crosslinker stiffness. At the same time, the maximal curvature increases as the base of the cilium becomes stiffer ( grows), and eventually saturates in the limit of an incompressible base for which when , see Fig. 3.3 C (three examples of shapes appear in Fig. 3.3 D). Clearly in this limit the maximal curvature still depends on the bulk crosslinker stiffness . In particular, for there is a sliding , but the cilium remains straight, . Our analysis thus indicates that for the static bending of cilia a basal stiffness is necessary, and the stronger it is the higher the bend. Biologically, this basal stiffness can arise from the distinct properties of the basal body. Thus severing the basal body and other basal constrains is expected to straighten actively bent cilia.
figure\end@float
Until now we have shown that bending requires a lateral asymmetry () and a polar asymmetry which we have identified as the basal stiffness (). An alternative way of introducing the polar asymmetry is to remove the basal constrain () but consider an inhomogeneous crosslinker stiffness. One possibility is a linear profile , with the slope of the linear gradient and the crosslinker stiffness at the base. In this case, one recovers a flat solution when . As the gradient slope grows (see Fig. 3.3 C), a bend develops and the maximal curvature increases (see Fig. LABEL:fig:grad). Since a linear gradient is a smaller inhomogeneity than a basal stiffness, the maximal curvature in the limit is smaller than in the case where a basal stiffness is present (compare Fig. LABEL:fig:grad to Fig. 3.3 A).
3.5 Conclusions

We obtained the equations of motion for a pair of inextensible filaments with a variable spacing between them that are immersed in a fluid and subject to active sliding forces. These equations show that the normal force between the filaments is a nonlinear effect.

The static bending of a filament pair requires: (i) the presence of a lateral asymmetry, such as a motor density higher in one filament than in the other; (ii) a polar asymmetry, such as basal constraint; (iii) a filament length larger than the characteristic length defined by the crosslinkers.
Chapter 4 Dynamics of collections of motors
The beat of cilia is powered by the action of molecular motors. In this chapter we introduce a mesoscopic description of the force generated by the dynein motors, and show how oscillatory instabilities emerge as a collective property of these. We also introduce a minimal stochastic biochemical motor model in which motors are regulated through their detachment rate. We finally demonstrate that if the detachment rate is regulated by normal forces or curvature, the static shape of the cilium corresponds to a circular arc. This is in agreement with experimental data obtained for the shapes of interacting doublet pairs.
4.1 Nonlinear response of the motor force
Consider that a cilium is dynamically bent. Its mechanical strains and stresses, such as the sliding , curvature , and normal force , will depend on time. In such a scenario, the motor force can be generically characterized by its nonlinear response to these strains and stresses. This reflects the idea that motors can dynamically respond to the mechanics of the cilium. For example, in the case in which motors respond to the sliding of the doublets, we will have
(4.1) 
where is the response kernel of order , and terms of order higher than cubic have not been included. Note that the can in principle depend on arclength, which would account for inhomogeneities of the motors along the cilium. In this thesis however we will consider them to be arclength independent.
For periodic dynamics with fundamental frequency , the response equation in Fourier space is
(4.2) 
with the nonlinear response coefficients being the dimensional Fourier transform of the corresponding order kernel. The coefficients completely characterize the motorfilament interaction, and examples of them are given in sections 4.2 and 4.3. The linear response coefficient is of particular importance, and we use a distinct notation for it:
(4.3) 
where the static response is and the linear response to the fundamental mode is . The linear dynamic response then becomes
(4.4) 
Analogous relations can be written for regulation via normal forces and curvature, and for each case we will use a different greek letter for the linear response coefficient. Thus for curvature response we have
(4.5) 
where in this case , with obtained from a relation analogous to Eq. 4.2 involving curvature. Finally, for normal force response we use
(4.6) 
with .
figure\end@float
4.2 Oscillatory instability of sliding filaments with motors
A collection of molecular motors can cooperate to produce collective behavior such as spontaneous motion [42] or oscillatory instabilities [43]. To show this, we consider a simple motor model in which the motor force responds to the local sliding velocity of filaments. In particular, we consider the following model
(4.7) 
where the model parameters are the relaxation time , the response parameter , and the saturation strength . These parameters do not depend on the arclength , corresponding to a homogeneous distribution of motors along the cilium. In this description of the motor force there are no quadratic terms, which is a consequence of timereversal symmetry. The response coefficients of this model can be easily calculated, and are
(4.8) 
with the fundamental angular frequency and the period of the oscillation.
At the steady state in which the sliding velocity is stationary, this model is characterized by a nonlinear force velocity relation schematized in Fig. LABEL:fig:forvel, in which is the control parameter. While for values the forcevelocity is stable, for there is a region of instability of the motor force (see Fig. LABEL:fig:forvel). This instability may give rise to beat patterns of the cilium, which we will study in the next chapters of this thesis. To illustrate this it is helpful to consider a very stiff cilium, with . In this limit bending and fluid forces are not relevant, and setting the basal stiffness and viscosity to zero, the filament will remain straight (thus ). The internal sliding forces are then balanced, and we have
(4.9) 
Together with the motor model in Eq. 4.7, this equation defines a dynamical system. To see how this system can exhibit an instability, consider as an ansatz small perturbations of force and sliding which are [42, 43]. In general is complex, with the characteristic relaxation time of the perturbation, and the frequency of the perturbation. Using this ansatz in the equations one obtains for the following second order characteristic equation
(4.10) 
which we now discuss.
Using the coefficient as control parameter, we study the stability of this dynamical system. In particular, note that for the system is clearly stable and nonoscillatory, and so while . As increases the system is still stable (i.e. ), but since it will exhibit damped oscillations as shown in Fig 4.2 A. Eventually, the control parameter may reach a critical value for which one of the two solutions to Eq. 4.10 becomes critical, that is . In this case small amplitude sinusoidal oscillations appear with frequency , as can be seen in Fig. 4.2 B. Finally, for values larger than the critical one (the region ) nonlinear oscillations occur. Their amplitude is set by the nonlinear saturation term , see Fig. 4.2 C.
Similar models can be worked out for curvature and normal force control. In such cases, however, the dynamics of the motor model cannot be studied independently from those of the filaments. It is indeed this coupling between the motor force and filaments, together with the dynamic instability of the motor force, which underlies the ciliary beat. An example of a mechanism in which the motor force responds to changes in curvature is
(4.11) 
Here corresponds to a delay, to the linear response, and to the saturation whic