Effect of curvature and normal forces on motor regulation of cilia
EEffect of curvature and normal forces on motor regulation of cilia
Pablo SartoriMay 13, 2019 a r X i v : . [ q - b i o . S C ] M a y Disclaimer.
The following document is a preprint of the PhD dissertation of Pablo Sartori, defendedin 2015 at the Technische Universit¨at Dresden. There are only minor differences between this preprint andthe final version (formatting changes, addition of acknowledgements, resizing of figures, and an additionalappendix regarding the fitting procedure).
Abstract
Cilia are ubiquitous organelles involves in eukaryotic motility.
They are long, slender, andmotile protrusions from the cell body. They undergo active regular oscillatory beating patterns that canpropel cells, such as the algae
Chlamydomonas , through fluids. When many cilia beat in synchrony they canalso 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 thedynein family that actively power the motion of the cilium. These motors transform chemical energy in theform of ATP into mechanical forces that produce sliding displacement between the microtubules. This slidingis converted to bending by constraints at the base and/or along the length of the cilium. Forces and dis-placements within the cilium can regulate dyneins and provide a feedback mechanism: the dyneins generateforces, deforming the cilium; the deformations, in turn, regulate the dyneins. This feedback is believed to bethe 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, thefeedback mechanism is much less clear. The two key questions are: which forces and displacements are themost relevant in regulating the beat? and how exactly does this regulation occur?In this thesis we develop a framework to describe the spatio-temporal patterns of a cilium with differentmechanisms of motor regulation. Characterizing and comparing the predicted shapes and beat patterns ofthese different mechanisms to those observed in experiments provides us with further understanding on howdyneins are regulated.
Results in this thesis.
Chapters 1 and 2 of this thesis are dedicated to introduce cilia, chapters 3-6contain the results, and chapter 7 the conclusions.In chapter 1 we introduce the structure of the cilium, and discuss different possible regulatory mechanismswhich we will develop along the thesis. Chapter 2 contains a quantitative description of the ciliary beatobserved in experiments involving
Chlamydomonas .In chapter 3 we develop a mechanical theory for planar ciliary beat in which the cilium can bend, slideand compress normal to the sliding direction. As a first application of this theory we analyze the role ofsliding cross-linkers in static bending.In chapter 4 we introduce a mesoscopic description of molecular motors, and show that regulation bysliding, curvature or normal forces can produce oscillatory behavior. We also show that motor regulation bynormal 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 non-linear 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 theserequirements. Normal forces can only regulate dynamic patterns in the presence of an asymmetry, and theresulting 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 mutantto the theoretically predicted ones. This comparison suggests that sliding forces cannot regulate the beat ofthese short cilia, normal forces can regulate them for the wild type cilia, and curvature can regulate themfor wild type as well as for the symmetric mutant. This makes curvature control the most likely regulatorymechanism for the
Chlamydomonas ciliary beat. ontents
Chlamydomonas beat 17
Chlamydomonas cilia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Bending of disintegrated axonemes into circular arcs . . . . . . . . . . . . . . . . . . . . . . . 202.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Bull Sperm . . . . . . . . . . . 485
CONTENTS
A Calculus of variations and test of resistive force theory 65
A.1 Variations of mechanical and rayleigh functionals . . . . . . . . . . . . . . . . . . . . . . . . . 65A.2 Plane wave and hydrodynamic test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
B Critical modes and numerical methods 69
B.1 Characterization of critical modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69B.2 Non-linear dynamics of beating cilium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71B.3 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
References 74 hapter 1
Introduction
Cilia are ubiquitous eukariotic organelles. Diverse in scale and function, they are involved in a numberof different motile tasks, yet at the core of all cilia lies a common structure: the axoneme. Imaging ofbending cilia, of their internal structure, and biochemical experiments, have given rise to models of the ciliarystructure, as well as suggestions for their functioning. In this section we give an overview of the internalcomponents of the cilium, its structure, and some of the suggested models for their dynamic regulation.
Cilia are long thin organelles which protrude from eukaryotic cells. They are motile structures fundamentalfor the functioning of cells. Indeed, immotile cilia are related with problems in embryonal development, andlead to human diseases such as primary ciliary diskinisia [3]. Although with differences among species, thecore 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 beatpatters are involved in fluid flow and propulsion of micro-swimmers. Examples of cilia involved in fluid floware the nodal cilia, responsible for the breaking of left-right 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 micro-swimmers use cilia to propel themselves. Such microorganisms can have asmany as hundreds of cilia (as in the case of Paramecium, see Fig. 1.1), but in this thesis we will focus onmicroorganisms with one or two cilia. Examples of cells with one motile cilia are the sperm of sea urchin, bullor 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).There is great variability among the ciliary properties of the micro-swimmers in Fig. 1.1. Among themthese cilia differ in size, beat frequency, wave number, and amplitudes. Estimates of these properties for theexamples in Fig. 1.1 are in Table 1.1. But however different their beating properties are, underlaying all ofthem reside the same structural elements, which we review in the next section.Organism Length ( µ m) Frequency (s − ) Wave number Amplitude ( µ m) 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 ? > > Characteristics of the beat patterns of some flagellates.
There is great variability amongthe beat of flagellates. While
Chlamydomonas has a short and fast cilium,
Bull Sperm has a long slow one.7
CHAPTER 1. INTRODUCTION
Figure 1.1:
Examples of cilia creating flows and self-propelling micro-swimmers.
Nodal cilia [62]and cilia in bronchi [90] produce fluid flows. In the other examples cilia propel microorganisms. These micro-swimmers can have one cilium (as in the sperm examples shown), two (as in the case of
ChlamydomonasReinhardtii ), or many, as in the case of
Paramecium [1]. Bars correspond to 10 µ m. The figure is adaptedfrom [28]. In the core of cilia lies a cytoskeletal cylindrical structure called the axoneme. The axoneme is a cylindricalbundle of 9 parallel microtubules doublets. At its center there are 2 additional microtubules, the centralpair (see Fig. 1.2 B). They extend from the basal end, attached to the cell body, to the distal end. While .2. INSIDE A CILIUM: THE 9+2 AXONEME
Chlamydomonas axoneme which is an example of the 9+2 arrangement. central pairdoublets
A B inner dynein armsdoublets nexin links
Figure 1.2:
The 9+2 axoneme. A . Schematic cross-section of the 9+2 structure showing the doubletmicrotubules, the inner and outer dynein arms, the nexin links, the radial spokes and the central pair. B .Schematic of a bent and twisted axoneme showing the doublets (blue) and central pair (gray). Note therelative sliding between the doublets.The doublets and the central pair are connected by the radial spokes (see Fig. 1.2 A), responsible to keepthe radius of the diameter at ≈ ≈
30 nm. Each doublet is connected to its near-neighbors by cross-linkers suchas nexin, see section 1.2.3. These provide a resistance to the relative sliding of doublets, and to changesin the spacing between them. Doublets are also connected by dynein molecular motors 1.2.2, which createactive sliding forces when ATP is present. All these elements are present in a highly structured manner inthe axoneme, repeating along the long axis of the cylinder with a period of ≈
96 nm. In the following wedescribe in more detail the main characteristics of the axonemal elements.
The main structural elements of the axoneme are microtubules, present in the doublets as well as in thecentral pair. Microtubules are filamentous protein complexes which have scaffolding functions in the eukary-otic cytoskeleton. Structurally, they are hollow cylinders with a diameter of 24 nm composed of thirteenprotofilaments (see Fig. 1.3). The basic structural element of a protofilament is the αβ tubulin dimer, witha length of roughly 8nm. Because this dimer is polar the mircotubules are polar as well, which grow muchfaster from the + end than from their − end [92]. axonemal doubletmicrotubule Bprotofilamentαβ dimer microtubule microtubule A A B
Figure 1.3:
Microtubules and doublets. A.
Each microtubule is composed of thirteen protofilaments,shifted with respect to each other. A protofilament is a linear assembly of αβ dimers. B. Cross-section of adoublet. Each doublet consists of two microtubules, one complete (microtubule B, with thirteen protofila-ments) and one incomplete (microtubule A, with ten protofilaments).0
CHAPTER 1. INTRODUCTION powerstroke ATP binding+ hydrolisys DTP release+ binding stem ringstalk
Figure 1.4:
Mechanochemical cycle ofdynein.
A dynein motor is composed of aring with 6AAA domains, a stem (in yellow)and a stalk with a MT binding domain at itsend. ATP binding enhances the detachment ofdynein’s stalk from the microtubule, and its hy-drolysis locks it in a strained state in which thering is rotated. DTP release aids the dynein tobind to the microtubule. Finally, once bound, thepower stroke occurs, leaving the bound motor ina state with high ATP affinity.The elastic properties of microtubules can be accurately accounted for using a model of incompressiblesemi-flexible polymers with a bending stiffness of ∼
24 pN · µ m [37]. While recent experiments suggest thatcompressibility and shearing may play a role in determining this stiffness [88, 63], in this thesis we willconsider microtubules as inextensible and non shearing.In the axoneme microtubules appear in the central pair and in the doublets. The central pair consistsof two singlet microtubules (i.e., with all 13 protofilaments, see Fig. 1.3) longitudinally connected betweenthem. But while its basic component, namely the microtubules, have well known properties, those of thecentral pair are complex and elusive. For example, there is evidence suggesting that the central pair shows alarge twist when extruded from the axoneme [59]. The orientation of the central pair is used as a referenceto number the doublets, as in Fig. 1.2.The doublets are composed of one A-microtubule and one B-microtubule. The A-microtubule has all 13protofilaments, while the B-microtubule has only 10 and a half protofilaments (see Fig. 1.3). The stiffnessof microtubule doublets has not been directly measured, but simple mechanical considerations suggest thatthey are roughly three times that of a single microtubule, and so we estimate their bending rigidity as ∼
70 pN µ m [37].The doublets and the central pair are the main structural elements involved in determining the bendingstiffness of the axoneme. Using mechanical considerations one can estimate the stiffness of the axonemalbundle as being 30 times higher as that of a single microtubule, which yields ≈
600 pN · µ m for the stiffnessof 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 ≈ · pN · µ m has been measured, in good agreement tothe previous estimate. The ciliary beat is powered by axonemal dyneins, which are a family of molecular motors (see Fig. 1.4).They convert chemical energy into mechanical work, with one full mechano-chemical cycle correspondingto the hydrolysis of one ATP molecule. The
Chlamydomonas axoneme in particular contains 14 differenttypes of dyneins and has a total of ∼ of these motors [94] over its length of roughly 10 µ m. And whilethese dyneins can have diverse properties (such as different ATP affinities, gliding speeds, or capability ofgenerating torques [44, 84]), we focus here on their generic properties.Dyneins are periodically distributed along the nine microtubule doublets, with their stem (yellow inFig. 1.4) rigidly attached to the A-microtubule, and their stalks being briefly in contact with the adjacentB-microtubule during the power stroke process [37]. Stem and Stalk are connected by a 6AAA ring, and alltogether the weight of a motor domain is roughly ∼
300 kDa [78, 40]. Their size is roughly 30 nm, and thetheir structure is sketched in Fig. 1.4 [17, 75].It is known that during their power stroke, dyneins generate forces that tend to slide the axonemaldoublets with respect to each other. For instance, after a protease treatment of the axoneme the presenceof 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 .3. STRUCTURAL ASYMMETRIES ∼ k B T ≈
100 pN · nm, [37]) is invested in this motion, and thus we estimate a force of
100 pN · nm15 nm = 6 . ∼ . µ M − s − (for ATP concentrationsbelow 60 µ M) [68], which dramatically accelerates detachment of the dynein tail from the microtubule. Afterthis, ATP is hydrolyzed and a conformational change occurs which renders dynein to its pre-powerstrokeconfiguration. This slow step occurs at roughly ∼ . − , and is thus the rate-limiting step [47]. ADPincreases the affinity of dynein to microtubules (which may also be influenced by additional conformationalchanges [48]), leading to a binding state in which the powerstroke occurs, which completes the dyneincycle (see Fig. 1.4). Overall the duty ratio of dynein (fraction of cycle time which it spends bound theB-microtubule) has been estimated to be ∼
10% [37], and it is thus considered a non-processive motor.
Radial spokes and nexin cross-linkers are passive structural elements involved in maintaining the structuralintegrity of the axoneme. Coarsely, the radial spokes (see Fig. 1.2) help to sustain the radius of the axoneme,while the nexin cross-linkers prevent the doublets from sliding apart. We now review some of the moreintricate details of nexin and radial spokes.The nature of nexin linkers has long been debated. Indirect evidence suggests that a protein complexmust be involved in constraining the sliding of doublets, since protease treatment of axonemes results intelescoping of the doublets when ATP is present [86]. Furthermore, it is believed that such a constraint isfundamental in transforming the sliding dynein forces into axonemal bending, see section 1.4. But whileearly electro microscopy identified nexin as a separate protein complex [93, 32], recent work suggests thatit is part of the dynein regulatory complex [61, 34, 8]. This would imply that the cross-link nexin can beclosely 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 isclear that dynein provides passive resistance to sliding, as indeed it’s stiffness has been directly measured tobe ∼ . · nm − for 1 µ m of axoneme. This number increases five fold in the absence of ATP (when dyneinsare attached), thus further indicating that dynein and nexin are intimately related may be a highly regulatedpassive structure [58]. Finally, since it stretches about ten times its equilibrium length, its force-displacementbehavior has been suggested to be non-linear [55].In each 96 nm 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 anevolutionary vestige [5]. When the radius of the axoneme is reduced, radial spokes compress, which suggeststhat they are involved in sustaining the cross-section of the axoneme [102]. However, recent evidence hasshown that radial spokes are connected to the same regulatory complex identified as nexin, which suggeststhat 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 radialspokes [83].
In a coarse view the picture of the axoneme is highly symmetrical. It is periodical in its longitudinal direction,with a period of 96 nm. It also has ninefold rotational symmetry around its longitudinal axes, with motorsconnecting all the doublets. However, recent studies have shown that this picture is not accurate. There aremany important structural asymmetries in the axoneme, which can have an important role in regulating thebeating dynamics.2
CHAPTER 1. INTRODUCTION
Figure 1.5:
Chirality of
Chlamy-domonas . Representation of
Chlamy-domonas with two axonemes (polar, in-dicated by arrowheads) of the same chi-rality (note the doublets ordering) asthey are believed to be located in thecell [16].
The axoneme is a polar structure, and is thus not symmetric along its length. There are several polarasymmetries in the axoneme, the most obvious one is that the microtubule doublets themselves are made ofpolar proteins such as the doublets. Furthermore, along each repeat the distribution of dyneins and radialspokes 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 arc-length,due to inhomogeneous distribution of motors.The two ends of the axoneme are fundamentally different. First, the distal end (farthest from the cellbody) is the + end of the doublet microtubules and is thus constantly being polymerized. Second, at thebasal end of the axoneme there is a complex structure called the basal body. The basal body is the regionwhere the axoneme attaches to the cell body, and it’s composed of microtubule triplets which get transformedinto doublets as the basal body transforms to the axoneme (see Fig. 30 in [72], also [72]). Evidence suggeststhat 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 thebeat [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 thedistribution of dyneins [101, 16] as well as cross-linkers [66]. The 96 nm repeats are homogoneous only in thecentral part of the axoneme, with repeats changing as they approach the ends. For instance, certain typesof dyneins (like inner dynein arms) are missing towards the base of the axoneme [16]. Yet some other typesof dynein are present only near the base, the so-called minor-dyneins [101].
Since the axoneme is polar, and the dyneins are bound to the A tubule and exert their power stroke onthe B tubule, the axoneme is a chiral structure (compare in Fig. 1.5 the two axonemes, which have thesame chirality). Furthermore, to date only axonemes with one handedness have been found. One immediateconsequence for
Chlamydomonas is that it swims with two left arms . That is, unlike with human arms, itstwo flagella are not mirror-symmetric with respect to each other (see Fig. 1.5). But beyond this intrinsicchirality, detailed studies on the axonemal structure have revealed further chiral asymmetries which we nowdiscuss.Certain adjacent doublets are connected by various structures: the doublets 1 and 2 are connected by theso-called 1-2 bridge, believed to be important in setting the beating plane [16, 36]. Additional cross-linkershave recently been identified the doublets 9 and 1, 5 and 6; as well as 1 and 2 [66]. All these structureshave been suggested to constrain the sliding of the doublets they link, thus defining the beating plane asthat in which the non-sliding doublets lay (which is roughly that of doublets 1 and 5, see Fig. 1.2). Anotherasymmetry in the beating plane are the presence of beaks inside the B tubules, filamentous structures areall 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 axonemein 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 centralpair, whose interaction with the doublets remains unclear [46]. Furthermore, evidence in
Sea Urchin sperm has shown that the axoneme itself can take spiral-like shapes, which are intrinsically chiral [96]. .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 thisthesis. It is generally believed that the beat is generated by alternating episodes of activation of opposingsets of dynein [85]. Which dyneins are activated and which are deactivated is believed to be regulated bya mechano-chemical feedback. The beat is thus believed to be a self-organized process, with no a priori prescription of dynein activity: dyneins regulate the beat, and the beat regulates the dyneins. We now goover the key points of this process, reviewed in [98, 54, 15]. • Dyneins produce active sliding forces:
Since dyneins can slide doublets [84] and slide axonemesapart [86], it is believed that they produce sliding forces between the doublets, see section 1.2.2. Ina rotationally symmetric axoneme all these forces balance exactly and there is no net sliding force, asituation termed tug-of-war [38]. In this case fluctuations can create a small imbalance in force, whichis then amplified by the motors and finally produces a significant net sliding force. Alternatively, chiralasymmetries (see 1.3) in the axoneme can also result in a net sliding force. • Passive cross-linkers constrain sliding and convert it into bending:
The net sliding forceproduced when the dyneins on one side of the axoneme win the tug-of-war leads to relative motionbetween the doublets [11]. Thanks to cross-linkers along the axoneme [55, 8], and possibly also at thebase [4, 71, 13], this sliding is constrained and converted into bending. • A mechano-chemical feedback can amplify the bending and produce switching . In the tug-of-war scenario, when one set of dyneins wins and the axoneme bends, the other set of dyneins becomesgradually less active while the winning side becomes more active, thus the bending is increased [38].However, possibly due to a delay in the mechano-chemical cycle by the dyneins [57, 71, 38], eventuallythe switch occurs and the opposing dyneins become active. The bend is thus reversed, giving rise toan oscillation.Of the three processes described above, the one which is the least understood is that of the mechano-chemical feedback. It is not clear what mechanical cue is sensed by the dyneins, and how it is sensed. Wenow review some of the proposals.
In sliding control, the mechano-chemical feedback responsible for the switch occurs by the dynein motorssensing the sliding displacement between doublets. Motor models where response to sliding can induceoscillations of a group of motors have a long history [10, 31, 42, 43]. The way the feedback can appear variesfrom one model to another, typically for processive motors it is believed that sliding forces induce detachmentof motors [31], while for non-processive 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 per-pendicular to the bending plane. For example, if the beating plane is formed by the doublets 1 and 5, theopposing dyneins are those in doublets 3 and 8 (see Fig. ?? and [35]). In this case, the sliding experiencedby one set of motors is positive, in the sense that it favors their natural displacement to the + end of theaxoneme. The displacement of the opposing set of motors is negative. It is believed that this acts as aregulatory mechanism of the motors force, with the dyneins that slide to the positive direction creating anactive force and the others a resisting force.This mechanism has been shown to produce beating patterns [10, 19], that under certain special conditionsare very similar to the beat of Bull Sperm [71]. Furthermore, there is also direct observation of slidingoscillations in straight axonemes [45, 99].
In curvature control, the switching of molecular motors is regulated by the curvature of the flagellum. Thismechanism was initially proposed in [57], where the moment generated by the motors was suggested to becontrolled by a delayed reaction to curvature. Later works have used the motor moment density as thequantity to be affected by curvature [9, 14, 71].4
CHAPTER 1. INTRODUCTION sliding control normal force controlcurvature control detachdetach detachloadforce normalforce + + +
Figure 1.6:
Three mechanism of motor regulation.
In curvature control bending of the doubletsenhances their detachment (left panel). In sliding control sliding of the doublets to the minus end producesa load force with triggers detachment (center panel). In normal force control the increase in spacing of thedoublets produces a normal force that tends to detach dyneins.Notice that there is a crucial difference between curvature and sliding control. In sliding control oscilla-tions 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 operateby opposing motors being activated or inactivated when a critical value of the curvature is reached, togetherwith a delay [9, 13]. One reason why opposing motors may react differently to curvature is that active andinactive motors, perpendicular to the beating plane, experience different concavity of the adjacent doublet(see Fig. 1.6).This model is supported by the fact that some flagellar waves seem to show a traveling wave of constantcurvature [12]. Furthermore, while the study in [71] favored a sliding control model, curvature control alsoshowed good agreement with the
Bull Sperm beat. However, there is no accepted mechanism of curvaturesensing by dynein motors. Direct geometrical sensing is unlikely, given the small size of dynein comparedto curvatures in the axoneme [71]. Indeed, a radius of curvature of 2 µ m corresponds to the bending of an8 nm tubulin dimer through an angle of only 0.025 degrees which is two orders of magnitude smaller thanthe curved-to-straight conformation associated with the straightening of a free GDP-bound tubulin subunitneeded for its incorporation into the microtubule wall [70, 60]. An alternative explanation is provided bythe geometric clutch [53], reviewed in the next section. In the geometric clutch the key factor regulating the switch of dynein motors is the spacing between thedoublets, or equivalently the corresponding transverse force ( t − force) between them [50, 53]. The hypothesisis that there are two contributions to the t − force coming from curvature and sliding. The contribution ofthe curvature is termed global t − force, and tends to detach motors. This global t − force will be termednormal force in our description. The sliding contributes to the local t − force, which tends to attach motorsand is much smaller than the global t − force [52]. Thus the geometric clutch combines effects of sliding andcurvature control. There is an additional element required in the geometric clutch, which is a distributorof the t − force between the opposing sides of the axoneme. Radial spokes have been suggested to fulfill thisrole [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 spacingin bent regions, where the global t − force increases [56]. However, it is so far unclear how the t − force isdistributed in a cross-section of the axoneme [53]. This implies that it is not known how it gets distributedbetween the opposing motors. The same problem is present in curvature control, but not in sliding control:there opposing motors experience opposite sliding displacement. To gain insight into ciliary beat, in this thesis we combine tools of non-linear dynamics, elasticity, and fluidmechanics. This will allow us to provide a full description of the self-organized dynamics of a cilia propelledby molecular motors [57, 18]. We now give some brushstrokes on the main elements of the theory. .6. CONCLUSIONS G [ r ], where r ( s ) is the position of the arc-length point s of the cilium, which collects allthe 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 whichtends to slide the doublets apart. This dynamic equation will depend on the internal strains and stressesof 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 ξ n and ξ t (corresponding to normal andtangential motion). This simplification is possible because the Reynolds number of a cilium is small, whichallows us to neglect non-linear effects [69]; and because the cilium is a slender body, with a diameter a muchsmaller than its length L .Putting all these elements together, we will obtain a dynamic equation for the shape of the cilium. If weparametrize the cilium by its local tangent angle ψ ( s, t ) at arc-length s and time t , we will have to linearorder the following force balance equation ξ n ∂ t ψ = − κ∂ s ψ − a∂ s f . (1.1)This equation includes the effects of fluid friction, elasticity (with κ = EI , E the young modulus and I the second moment of inertia), and the motor force f . The full description of the system still requires anequation that couples the motor force dynamics to an internal strain in the cilium. For example, if the motorforce responds to sliding velocity (as has been suggested [10, 19]), we have to linear order f ( s, t ) = (cid:90) t χ ( t − t (cid:48) )∆( s, t (cid:48) )d t (cid:48) (1.2)where χ ( t − t (cid:48) ) is the linear response function, and ∆ is the local sliding between doublets. The two equationsabove 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 non-linearities). Alternatively, the motor force couldrespond to changes in the local curvature ∂ s ψ [57, 9] or the doublets spacing a [50]. In this thesis wecharacterize the beat patterns corresponding to these three different different regulatory mechanisms, andcompare the results to the experimentally observed ciliary patterns described in the next chapter. • The core element of the cilium is the axoneme: a cylindrical bundle of microtubules connected bydynein molecular motors. • The beat of cilia is a self-organized process powered through microtubule sliding forces that are pro-duced by dynein motors. • Self-organization 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.6
CHAPTER 1. INTRODUCTION hapter 2
Characterization of the
Chlamydomonas beat
Chlamydomonas cilia can be isolated from the cell and, in the presence of ATP, beat periodically. In thissection we mathematically describe the beat pattern of intact as well as disintegrated cilia. We show thatthe observed beat pattern of an intact cilium is well characterized by its static and fundamental harmonics.While for wild-type the zeroth harmonic is very important as the beat is asymmetric, this is not the casefor a symmetric mutant which we analyze. We also show that in disintegrated cilia pairs of doublets caninteract with each other, reaching a static equilibrium in which the shape is a circular arc. All of the dataappearing in this section was taken in the laboratory of Jonathon Howard: the data from the first sectionby Veikko Geyer, and that of the second by Vikram Mukundan.
Chlamydomonas cilia
The cilia of
Chlamydomonas can be isolated from the cell body [77]. When their membrane is removed andATP is added to the solution, they exhibit periodic beat patters [7]. This is evidence that the beat of theaxoneme is a self-organized property, as it occurs independent of the cell body and with ATP homogeneouslypresent around the axoneme.Our collaborators in the group of Jonathon Howard, in particular Veikko Geyer, have imaged the beatof isolated
Chlamydomonas axonemes using high speed phase contrast microscopy with high spatial andtemporal resolution. The pixel-size was of 139 ×
139 nm / pixel, and the frame-rate 10 s − . In comparison,the typical size of the axonemes was L ∼ µ m, and the characteristic beat frequency f ∼
50 s − . Nineframes 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 codepublished in [73]. As one can see, the beat of the axoneme is asymmetric, and because of this the axonemeswims in circles. While the typical beat frequency of the axoneme is f ∼
50 s − , the rotational frequency ofthe axoneme is ten times slower, on the order of f rot ∼ − .The tracked cilium is characterized at each time t by a set of two dimensional pointing vectors r ( s, t )from the fixed laboratory reference frame to each point along the axonemal arc-length s ∈ [0 , L ], with L the length of the cilium (see Fig. 2.2). To describe its shape decoupled from the swimming we use anangular representation. That is, for each point along the cilium we calculate the tangent angle Ψ( s, t ) withrespect to the horizontal axis x , see Fig. 2.2. Note that this angular representation contains less informationthan the vectorial representation, in particular the swimming trajectory is lost. However Ψ( s, t ) still retainsinformation about the rotation velocity of the axoneme f rot . To obtain a pure shape description we alsosubtract this rotation, that is ψ ( s, t ) = Ψ( s, t ) − f rot t , (2.1)where the tangent angle ψ ( s, t ) now only describes the shape of the axoneme.178 CHAPTER 2. CHARACTERIZATION OF THE
CHLAMYDOMONAS
BEAT
Figure 2.1:
Time series of a beating cilium. A.
An isolated
Chlamydomonas cilium has an asymmetricbeat pattern which makes it swim counter-clockwise. The tracking is noted with a green line, and thebase with a black circle (adapted from [28]). B. Beat pattern over one period of duration T obtained aftertracking, showing forward wave propagation (from base, indicated with a black circle, to tip) and thuscounter-clockwise swimming. As time progresses over one period the cilium changes color following thecircular legend counter-clockwiseSince the beat of Chlamydomonas is periodic in time, we analyze the shape information ψ ( s, t ) by trans-forming its temporal coordinate to frequency space. This can be done using a fast Fourier transform, andthe result for the mid-point of the axoneme is shown in Fig. 2.3 A. There are peaks around frequenciesmultiple of the fundamental frequency, and a background of noise. Neglecting the noise, the beat can thusbe approximated by the following Fourier decomposition ψ ( s, t ) ≈ n =+ ∞ (cid:88) n = −∞ ψ n ( s ) exp( i πnf t ) (2.2)Where f is the fundamental frequency of the beat, which in the example considered in Fig. 2.3 A is f =55 . − . The modes ψ n ( s ) are complex functions of s , and their amplitude decreases as the mode increases,see Fig. 2.3 A. Because the angle is a real quantity, its modes satisfy ψ − n = ψ ∗ n .Importantly, the above description includes the zeroth mode ψ ( s ), which corresponds to the averageshape of the cilium and defines its asymmetry. In Fig. 2.3 B this mode is shown for an example. As one r (s) y x tangentangle ψ (s) arc-length, s r (s) Figure 2.2:
Tracking the tan-gent angle.
The tangent an-gle ψ ( s ) at arc-length s measuredfrom the basal end, is definedwith respect to the lab frame xy using the pointing vector r ( s ). .1. THE BEAT OF ISOLATED CHLAMYDOMONAS
CILIA -6 -4 -2 Arc-length (μm)
Arc-length (μm)
Arc-length (μm)
A B C D f f f f -6 -4 -2 Arc-length (μm)
Arc-length (μm)
Arc-length (μm) f f f f E F G HMBO2 mutantWild Type
Figure 2.3:
Characterization of the
Chlamydomonas beat.
Top, characterization of Wild Type beat. A. The power spectrum of the angle at mid-length shows up to seven harmonics or modes. B. The zerothmode of the angle grows linearly over arc-length, resulting in a circular arc of constant curvature. C. Theamplitude of the first mode is roughly constant over arc-length. D. The phase of the first mode drops ∼ π over the length of the cilium. Bottom, characterization of mbo2 beat. The main difference resides in F,mbo2 shows a small zeroth mode since its beat is symmetric. Each line corresponds to an individual cilium,and the bolder one corresponds to the same cilium in panels B, C and D for WT, and in panels F, G and Hfor mbo2.can see, besides a flattening at the ends, the tangent angle decreases monotonically along the axoneme. Theamplitude and phase profiles of the first, second and third dynamic modes are shown in Fig 2.3 B and C.As one can see, the amplitude of the first mode is much larger than that of all higher modes, and roughlyconstant along the arc-length with a small dip in the middle. The phase profile of the first mode is shown inFig. 2.3 D, and is monotonically decreasing. Over the full length of the axoneme the phase decreases about2 π , which corresponds to a wave-length equal to the length of the axoneme.The curvature is the derivative of the tangent angle with respect to the arc-length. This means thatthe constant slope of the angle in the zeroth mode corresponds to a constant mean curvature of roughly ∼ .
25 rad · µ m − . Furthermore, since the amplitude of the first mode is approximately constant and its phasedecreases with a constant slope, we conclude that a good approximation of the Chlamydomonas ciliary beat,shown in Fig. 2.4 A, is the superposition of an average constant curvature, Fig. 2.4 B, and plane wave ofits angle, Fig. 2.4 C. Thus, the beat of
Chlamydomonas is fundamentally asymmetric, and its asymmetry iswell characterized by a constant mean curvature.It is important to note that this asymmetry is not present during the phototropic response of
Chlamy-domonas and in certain mutants that Move Backwards Only (MBO). The MBO 1-3 mutants beat symmetri-cally [80, 28]. In particular the mutant MBO 2 has a very small static component in the beat, see Fig. 2.3 A.Interestingly, the amplitude profile of the first mode is very similar to that of the wild type cilium, as one cansee by comparing Fig. 2.3 B and Fig. 2.3 C. The same is true of the phase profile, which appears in Fig. 2.3C. Thus, these mutants have a beat that, while lacking the asymmetry, are in the rest very similar to that ofthe wild-type cilium. Finally, it is worth noting that also wild-type
Chlamydomonas can exhibit symmetricbeat patterns in the presence of Calcium, although these are three-dimensional beats fundamentally differentfrom those of mbo2 [7, 28].0
CHAPTER 2. CHARACTERIZATION OF THE
CHLAMYDOMONAS
BEAT += B CA
Figure 2.4:
Schematic of asymmetric beat.
An asymmetric beat (A) can be described as a circular arc(constant mean curvature, B) plus a plane angular wave (C). Figure adapted from [28].
In the presence of a protease treatment, axonemes partially loose their cross-linkers. When this occurs,two doublets can interact via the dyneins of one of them. This provides a minimal system, which has beenreported to produce sliding and bending waves [4, 60]. In particular, at low concentrations of ATP, pairs offilaments associate, and propagate small bending waves towards the basal end as one filament slides alongthe other (Fig. 2.5, first row, arrows). Furthermore, in some occasions the two filaments re-associate alongtheir entire length and bend into a circular arc (Fig. 2.5, second row). The system then becomes unstableand the filaments separate again. μ m Figure 2.5:
Sliding of adjacent doublets in a split axoneme
One doublet slides along another and thendissociates (0.0 to 4.0 s). After reassociating (4.2 s), the two doublets remain in close apposition and bendinto a circular arc (4.5 s to 5.7 s) before dissociating again (not shown). Figure adapted from [60].To analyze the bending process in detail, the shapes of the filaments pairs are digitized, as shown inFig. 2.6 A. From this one can calculate the tangent angle as a function of arc-length in successive framesas the filaments become more and more bent, see Fig. 2.6 B. Importantly, the filament pair approaches asteady-state shape in which the tangent angle increases linearly with arc length, except at the very distal endwhere it flattens (Fig. 2.6, 5 . . Chlamydomonas cilia as was described in section 2.1.There is evidence that the two interacting filaments are two doublet microtubules and not two singletmicrotubules or one doublet microtubule interacting with the central pair. The two individual singlet mi-crotubules that comprise the central pair will each have a lower intensity than a doublet. However, theinteracting filaments (Fig. 2.7 A and B, red) have the same intensities as the non-interacting filaments(Fig. 2.7 A and B, blue). This implies that the interacting filaments are not two singlet microtubules (which .3. CONCLUSIONS
Tracking the doublets. A.
Theshape was characterized by tracking the centerlineof intensity along the filament contour marked bygreen crosses. B. The tangent angles are plottedas a function of arc-length. The tangent angle in-creased linearly with arc-length (times 5 . − . . Position (nm) μm -250 2500 S c a l ed i n t en s i t y BA Figure 2.7:
Intensity analysis of filaments. A.
Frame of a bending event with line-scans of fila-ments interacting (red), non-interacting (blue) andoverlapping (green). B. Corresponding intensityprofiles. The interacting and non-interacting fila-ments have similar intensity, while the intensity inthe overlap is 4 times higher than the non-overlapregions, as corresponds to dark-field microscopy.Figure adapted from [60].One of the advantages of this experiment is that the shapes reach a quasi-static limit, which can beanalyzed assuming mechanical equilibrium. The assumption of mechanical equilibrium means that thefrictional forces due to motion through the fluid can be ignored, and applies if the relaxation time of thebent beam in the fluid is much smaller than the typical time of the bending observed. The relaxation timeof a beam in a fluid is ∼ ξ n L /κ . Using κ = 120 pN · µ m for the doublets stiffness, ξ n = 0 . · s · µ m − for the fluid friction, and L = 5 µ m for the length, we obtain a relaxation time of ∼ .
01 s, which is muchsmaller than the typical duration of the process ∼ • The beat of the
Chlamydomonas cilium is well described by its static and fundamental mode. • In a wild-type cilium the static mode is large and has the shape of a circular arc, producing anasymmetry 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.2
CHAPTER 2. CHARACTERIZATION OF THE
CHLAMYDOMONAS
BEAT hapter 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 introducea two-dimensional representation of the axoneme as a pair of opposing inextensible filaments. We considerthe sliding as well as variable spacing between these filaments. By balancing mechanical and fluid forces, wederive the general non-linear dynamic equations of a cilium beating in a plane. As a first application of thistheory, we study the role of sliding cross-linkers in the static bending of a cilium.
We describe the axoneme by a pair of opposing filaments, which we label A and B (see Fig. 3.1). Eachfilament is parametrized by the arc-length s of the centerline, which ranges from 0 at the base to L at thetip. The filaments are separated by a distance a ( s ) which can depend on the arc-length, and they can slidewith respect to each other at every point.The geometry of the centerline is characterized by r ( s ), a two-dimensional pointing vector from thelaboratory frame. At any given point of the centerline we can define the local tangent vector t ( s ) and thelocal normal vector n ( s ), as shown in Fig. 3.1. The tangent vector is given by t = ∂ r ∂s = ˙ r , (3.1)where in the last expression we have introduced a notation in which upper dots denote arc-length derivatives.This notation will be kept throughout the rest of this thesis, with the number of dots denoting the orderof the arc-length derivative. The normal vector is defined simply as normal to t ( s ), with orientation suchthat t × n points out of the x y plane. Using the tangent vector we can also define the local tangent angle ψ ( s ) between the tangent vector t ( s ) and the x -axis. The relationship between the tangent angle and thepointing vector of the centerline is r ( s ) = r + (cid:90) s (cid:18) cos( ψ ( s (cid:48) ))sin( ψ ( s (cid:48) )) (cid:19) d s (cid:48) , (3.2)where r = r ( s = 0) is the position of the base of the centerline. The local curvature of the centerline C ( s )is given by the arc-length derivative of the tangent angle, C ( s ) = ˙ ψ ( s ). The geometry of the centerline isthus given by the set of equations ˙r = t ; ˙t = ˙ ψ n ; ˙n = − ˙ ψ t (3.3)which are the Frenet-Serret 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 a ( s ) / r A ( s ) = r ( s ) + a ( s )2 n ( s ) , r B ( s ) = r ( s ) − a ( s )2 n ( s ) . (3.4)234 CHAPTER 3. FORCE BALANCE OF PLANAR CILIA
Figure 3.1:
Planar geometry of a cilium.
Scheme of a cilium as a pair of opposing filaments (labeled Aand B) separated by a spacing a ( s ) with depends on the arc-length s (measured from the base, black circles).The relative sliding of the filaments ∆( s ) is shown, as well as the sliding at the base ∆ . Each position alongthe centerline (dashed line) is characterized by a position vector r ( s ) with origin at the laboratory frame xy . From the position vector we can obtain the tangent vector field t ( s ), as well as the normal n ( s ) fieldthrough Eq. 3.3. The tangent angle ψ ( s ) can be calculated at each point (see Eq. 3.2).Note that s does not parametrize the arc-length of either of the filaments, but that of the centerline. Thearc-length along each of the filaments is given by s A ( s ) = (cid:90) s | ˙r A ( s (cid:48) ) | d s (cid:48) and s B ( s ) = (cid:90) s | ˙r B ( s (cid:48) ) | d s (cid:48) . (3.5)Using the corresponding Frenet-Serret frame for each of the filaments, we obtain the curvature of the filaments(see Appendix A). To lowest order these curvatures are given by C A ( s ) ≈ ˙ ψ ( s ) + ¨ a ( s )2 , and C B ( s ) ≈ ˙ ψ ( s ) − ¨ a ( s )2 (3.6)where geometric non-linearities have been neglected.The sliding of one filament with respect to the other at centerline arc-length position s is given by themismatch in arc-length along one filament with respect to the other plus the reference sliding at the base.We thus have ∆ A ( s ) = ∆ + s B ( s ) − s A ( s ) and correspondingly ∆ B ( s ) = − ∆ A ( s ) ; (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 ∆( s ) = ∆ A ( s ). The explicit expression of the localsliding can be calculated using Eqs. 3.5 and is given to cubic order by∆( s ) ≈ ∆ + (cid:90) s a ( s (cid:48) ) ˙ ψ ( s (cid:48) )d s (cid:48) . (3.8)Note that Eqs. 3.6 and 3.8 take a particularly simple form in the limit of homogeneous spacing a → a , inwhich ∆( s ) = ∆ + a ( ψ ( s ) − ψ (0)), and C A = C B = ˙ ψ . As discussed in section 2, the axoneme is composed of passive and active mechanical elements. The passiveelastic elements such as the doublets and nexin cross-linkers provide structural integrity to the axoneme, andtend to restore it to a straight configuration without sliding. On the other hand the active elements createsliding forces between the doublets which ultimately bend it. .2. STATIC BALANCE OF FORCES G [∆ , a, r ] = (cid:90) L (cid:20) κ C + C ) + k − f m ( s )∆ + k ⊥ a − a ) + Λ2 ( ˙r − (cid:21) d s + k (3.9)where the explicit expressions of curvatures and sliding are given in Eqs. 3.6 and 3.8. The integral containsthe energy density of the bulk of the axoneme, and the last term is the energetic contribution of the base. Thefirst term in the integral is the bending energy characterized by the bending rigidity κ (in pN · µ m ) of eachfilament, which favors straight shapes. The second term is the energy of elastic linkers of stiffness density k (in pN · µ m − ) which are stretched by sliding (purple springs in Fig. 3.2). The stiffness (in pN · µ m − ) of basallinkers is denoted k (blue spring in Fig. 3.2 ). We have denoted by k ⊥ (in pN · µ m − ) the stiffness to normaldeformations relative to the reference spacing a (green springs in Fig. 3.2). The work performed by motorswhich generate relative force f m (in pN · µ m − ) between the two filaments is given by the contribution − f m ∆.We have also introduced a Lagrange multiplier Λ (in pN) to ensure the inextensibility of the centerline. k k k - F A filament A + F B filament B - F B + F A Figure 3.2:
Mechanics inside a cilium:
Op-posing motors in filaments A/B (green circles)produce forces F A / B , and reactions − F A / B .Normal compression is constrained by springs(green) of stiffness k ⊥ . Sliding is limited by bulksprings (purple, stiffness density k ) and a basalspring (blue, stiffness k ).To obtain the equations of static equilibrium of the axoneme under external forces we use the virtualwork principle. We consider as reference a straight configuration ( r (s)=s x ), with no basal sliding (∆ = 0)and homogeneous spacing ( a = a ). The virtual work principle establishes that the internal virtual work δW i = δG performed by a variation { δ r , δa, δ ∆ } , is equal to the external work δW e performed by externalforces against this variation. For the forces applied on the bulk of the axoneme we have δW i = δG = δW e , (cid:90) L d s (cid:18) δGδ r · δ r + δGδa δa (cid:19) + δGδ ∆ δ ∆ = (cid:90) d s f ext · δ r . (3.10)Where f ext are the bending force density applied externally in each point with a specified direction in theplane, and we are assuming that there are no external basal forces (conjugate of δ ∆ ) or compressive forces(conjugate of δa ). At the boundaries, the same principle applies, and we have F ext0 · δ r (0) = δGδ r (0) · δ r (0) , F ext L · δ r ( L ) = δGδ r ( L ) · δ r ( L ) ,T ext0 δψ ( L ) = δGδ ˙r ( L ) · δ ˙r (0) , T ext L δψ ( L ) = δGδ ˙r ( L ) · δ ˙r ( L ) , δGδa (0) δa (0) , δGδa ( L ) δa ( L ) , δGδ ˙ a (0) δ ˙ a (0) , δGδ ˙ a ( L ) δ ˙ a ( L ) . (3.11)Where we have now introduced the external forces F ext0 and F ext L at the boundaries. We have also allowedthe presence of external torques T ext0 and T ext L at the boundaries.To obtain the balance of forces in the axoneme the first step is to calculate the corresponding functional6 CHAPTER 3. FORCE BALANCE OF PLANAR CILIA derivatives. Doing standard variation calculus as detailed in Appendix A we obtain δGδ r = ∂ s (cid:104) ( κ ¨ ψ − ˙ aF + af ) n − τ t (cid:105) ,δGδa = k ⊥ ( a − a ) + κ .... a / − F ˙ ψ ,δGδ ∆ = − F (0) + F . (3.12)Where κ = 2 κ is the stiffness of both filaments, and we have introduced the static sliding force density f and the basal force F as f ( s ) = f m ( s ) − k ∆( s ) ,F = k ∆ ; (3.13)and the integrated force F ( s ) = (cid:82) Ls f ( s (cid:48) )d s (cid:48) . In the first equation we have introduced the Lagrange multiplier τ = Λ + κ ˙ ψ − F ˙ ψa to replace Λ. We prefer τ as it can be interpreted as the tension of the centerline. Thiscan be seen by the following relation τ ( s ) = τ (0) − t ( s ) · (cid:90) s δGδ r ( s (cid:48) )d s (cid:48) , (3.14)which can be derived directly from Eq. 3.12.The static equilibrium balance equations are thus given by f ext = ∂ s (cid:104) ( κ ¨ ψ − ˙ aF + af ) n − τ t (cid:105) ,k ⊥ ( a − a ) = F ˙ ψ − κ .... a / ,k ∆ = F (0) . (3.15)The forces and torques balances at the ends are obtained by using the boundary terms of the variations.This yields F ext0 = ( κ ¨ ψ − ˙ aF + af ) n − τ t , F ext L = ( κ ¨ ψ − ˙ aF + af ) n − τ t ,T ext0 = − κ ˙ ψ + ak ∆ , T ext L = κ ˙ ψ , a = ¨ a , a = ¨ a ; (3.16)where the equations to the left correspond to those at the basal end s = 0 and those to the right at thedistal end s = L . Note from Eq. 3.15 that the coupling between the spacing a and the angle ψ is non-linear,since F is of order ψ . This implies that for small bending the change in spacing is negligible. So far we have considered the static equilibrium of a cilium, where the mechanical passive and active elementsof the axoneme are balanced by time-independent external forces. In the case in which the cilium is immersedin a fluid, the external forces to which it is subject are fluid forces. This is the basis of the descriptionintroduced in 1.5, and can be summarized by f ext ( s ) = f fl ( s ) , (3.17)where f fl are the forces that the fluid exerts on the axoneme along its length.In general, obtaining the fluid forces acting on a moving object is not an easy task, as it involves solvingthe non-linear Navier-Stokes 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 f fl = − ( nn ξ n + tt ξ t ) · ∂ t r . (3.18) .3. DYNAMICS OF A CILIUM IN FLUID Sketch of Resistive Force Theory:
The motion of a fragment of axoneme is character-ized by the local velocity ∂ t r , which can be pro-jected in the tangential t and normal n directionswith respect to the arc-length. The local fluid dragforce opposes these components with correspond-ing drag coefficients ξ t and ξ n . Since ξ n > ξ t forslender rods, f fl and ∂ t r are not parallel, makingnet propulsion possible.This corresponds to forces opposing the moving filament with a drag coefficient ξ n in the normal direction,and a coefficient ξ t in the tangential direction (see Fig. 3.3).This description of the fluid-cilium interaction is known as Resistive Force Theory (RFT), and wasintroduced in a seminar paper by Hancock [33]. The effective friction coefficients can be related [33, 30] tothe fluid viscosity µ as ξ n = 4 πµ log(2 L/a ) + 1 / ξ t = 2 πµ log(2 L/a ) − / . (3.19)Crucially, the friction coefficients are different and satisfy ξ n > ξ t . This asymmetry allows that, withoutexerting any net force on the fluid, the cilium can propel itself. We note here that the validity of RFT forciliary beat has been verified in the literature for several swimming micro-organisms [41, 24], and is alsoverified 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 introducethe following Rayleigh dissipation functional R [ ∂ t ∆ , ∂ t a, ∂ t r ] = (cid:90) L (cid:20) ξ t t · ∂ t r ) + ξ n n · ∂ t r ) + ξ i ∂ t ∆) + ξ ⊥ ∂ t a ) (cid:21) d s + ξ ∂ t ∆ ) (3.20)The first two terms inside the integral refer to fluid friction. We have also included internal sliding friction ξ i and compressive friction ξ ⊥ , both measured in pN · s · µ m − . Finally, the base also contributes to the internalsliding friction with a coefficient ξ (in pN · s · µ m − ).Just as the mechanical forces are obtained through variations of the mechanical work functional G withrespect to the corresponding fields { ∆ , a, r } , the dissipative viscous forces are obtained through variationsof the Rayleigh functional with respect to their time derivatives. Thus, ignoring inertia, the force balance isnow established as δGδ r + δRδ∂ t r = 0 , δGδa + δRδ∂ t a = 0 , δGδ ∆ + δRδ∂ t ∆ = 0 . (3.21)From this we obtain the dynamics of the centerline r ( s, t ), the spacing a ( s, t ) and the basal sliding ∆ ( t ).Using the functional derivatives in Eq. 3.12, and computing those of the Rayleigh functional we obtain thefollowing set of dynamic equations( ξ n nn + ξ t tt ) · ∂ t r = − ∂ s (cid:104) ( κ ¨ ψ − ˙ aF + af ) n − τ t (cid:105) ,ξ ⊥ ∂ t a = F ˙ ψ − k ⊥ ( a − a ) − κ .... a / ,ξ ∂ t ∆ = F (0) − k ∆ ; (3.22)where we have omitted the dependences on time and arc-length, f denotes the sliding force density, and F = (cid:82) Ls f ( s (cid:48) )d s (cid:48) its integral. Analogously to f , we can define the normal force density f ⊥ and the basalforce F . We thus have f = f m − k ∆ − ξ i ∂ t ∆ ,f ⊥ = k ⊥ ( a − a ) + ξ ⊥ ∂ t a ,F = k ∆ + ξ ∂ t ∆ . (3.23)8 CHAPTER 3. FORCE BALANCE OF PLANAR CILIA
Note that while all three forces contain viscous and elastic components, only the sliding force f has acontribution of active motor forces.The equations above depend on the tension τ , which is a Lagrange multiplier. The standard procedure toobtain its value is to use the corresponding constraint equation, which in this case is ˙ r ( s, t ) = 1. Since thedynamic equations do not directly involve r ( s, t ), but instead involve its time derivative ∂ t r ( s, t ), we calculatethe time derivative of the constraint and obtain ∂ t ˙ r = 0. This equation can alternatively be written as ∂ t ˙ r = 2 t · ∂ t ˙ r = 0, from which we obtain the tension equation¨ τ − ξ t ξ n ˙ ψ τ = − ξ t ξ n ˙ ψ∂ s ( κ ¨ ψ − ˙ aF + af ) − ∂ s (cid:104) ˙ ψ ( κ ¨ ψ − ˙ aF + af ) (cid:105) . (3.24)Together with the boundary conditions given in Eq. 3.16, the set of Eqs. 3.22-3.24 constitute a complete setof integro-differential equations. Provided a prescription for the motor force f m and the external boundaryforces these equations can be solved to obtain dynamic shapes of the cilium. The dynamic equations of a cilium take a particularly simple form when it is imposed that the spacing a between the filaments is constant along the arc-length, that is a ( s ) → a . This requires that k ⊥ → ∞ , whilekeeping the normal force f ⊥ = k ⊥ ( a − a ) finite, such that it acts as a Lagrange multiplier. The resultingdynamic equations are ( ξ n nn + ξ t tt ) · ∂ t r = − ∂ s (cid:104) ( κ ¨ ψ + a f ) n − τ t (cid:105) , (3.25) ξ ∂ t ∆ = F (0) − k ∆ . (3.26)Note, that in this limit the sliding is determined by the shape through∆ = ∆ + a ( ψ ( s ) − ψ (0)) . (3.27)Finally, the equation for the Lagrange multipliers τ (tension) and f ⊥ (normal force) become ξ n ξ t ¨ τ − ˙ ψ τ = − ˙ ψ ( κ ... ψ + a ˙ f ) − ξ n ξ t ∂ s [ ˙ ψ ( κ ¨ ψ + a f )] , (3.28) f ⊥ = F ˙ ψ ; (3.29)which have to be solved at every time. Note that, while τ explicitly appear in the dynamic equation, f ⊥ doesnot. Another important fact is that the expression of f ⊥ is quadratic, which means that for small changes inthe shape the normal force is negligible. This is analogous to the non-linear coupling between spacing andbending seen before.The boundary equations in 3.16 simplify to the following four equations F ext0 = ( κ ¨ ψ + a f ) n − τ t , F ext L = ( κ ¨ ψ + a f ) n − τ t ,T ext0 = − κ ˙ ψ + a ( k ∆ + ξ ∂ t ∆ ) , T ext L = κ ˙ ψ . (3.30)With adequate choices of the external forces and torques at the basal and distal ends ( F ext0 /L and T ext0 /L ),these provide the six necessary boundary conditions. In this work, we will focus on three types of boundaryconditions summarized in Fig. 3.4. These are free ends (A), in which the cilium is not subject to externalforces or torques; pivoting base (B), in which the base of the cilium is held fixed, and it’s constrained fromrotating with a stiffness k p while the distal end is free; and clamped base (C), in which the base is held fixedand k p → ∞ enforces that ψ (0) = 0 with the distal end free.In section 2.1 we described the beat of Chlamydomonas cilia using an angular representation. Motivatedby this, we introduce an angular representation of the dynamic equation above by using ∂ t ˙ r = n ∂ t ψ . Thisdirectly gives ∂ t ψ = ξ − ( − κ .... ψ − a ¨ f + ˙ ψ ˙ τ + τ ¨ ψ ) + ξ − ˙ ψ ( κ ˙ ψ ¨ ψ + a f ˙ ψ + ˙ τ ) , (3.31) .4. REQUIREMENTS FOR STATIC BENDING pivoting clampedfree A B C
Figure 3.4:
Schematic of boundary conditions:
A free cilium (A) is not subject to any external forcesor torques. For a pivoting cilium (B) the force balance at the base is replaced by the conditions of no basalmotion, and there is an additional basal torque coming from the pivoting spring. In the clamped case (C)the base does not move, and the pivot is so stiff that the tangent angle at the base remains fixed.which, to linear order, is Eq. 1.1 of the introduction. Together with the tension equation and the boundaryconditions, this equation provides a mechanical description of the angles of all points along the cilium. Eq. 3.2relates angle and position, and shows that the angular representation does not contain the trajectory of thebasal point r ( t ). It can, however, be obtained by inserting the solution of the tangent angle in the righthand side of Eq. 3.25 and integrating r ( t ) over time. In order to understand the mechanism regulating the shape of the cilium, we begin by analyzing the staticlimit. In absence of external forces and torques, Eqs. 3.15 can be integrated. Constraining the spacing to behomogeneous (see 3.3.1), the force balances simply become κ ¨ ψ ( s ) = − a f ( s ) ,k ∆ = (cid:90) L f ( s (cid:48) )d s (cid:48) ; (3.32)with f = f m − k ∆ the sliding force density. The tension in this case is null (i.e., τ ( s ) = 0), and the normalforce is given by f ⊥ = κ ˙ ψ /a , (3.33)which as indicated before is a second order term. The normal force is always positive, indicating thatfilaments tend to split apart. Finally, there are two boundary conditions κ ˙ ψ (0) = a k ∆ and ψ (0) = 0 , (3.34)which allow to calculate static shapes ψ ( s ).In the static regime the force produced by the motors in either filament is the stall force, that is F A = F B = F st . Considering motor densities ρ A and ρ B , we have that f m = ( ρ A − ρ B ) F st . If both filaments havethe same densities of motors then ρ A = ρ B . In this case we have that the opposing forces balance each otherand f m = 0, thus the cilium doesn’t bend ( ψ ( s ) = 0) nor does it slide (∆ = 0). When ρ A (cid:54) = ρ B , thenEq. 3.32 gives non-trivial solutions.We can analytically solve Eq. 3.32 for the case of a constant motor force along the arc-length, and obtain ψ ( s ) = a(cid:96)k ∆ κ cosh[ L/(cid:96) ] − cosh [( L − s ) /(cid:96) ]sinh [ L/(cid:96) ] , (3.35)where we have used the two boundary conditions in Eq. 3.16, and defined the characteristic length (cid:96) = (cid:112) κ/ ( ka ) (3.36)0 CHAPTER 3. FORCE BALANCE OF PLANAR CILIA
C B C m a x ( μ m - ) k (pN · μ m -2 ) k (pN · μ m -1 ) D xy -1 -3 C m a x ( μ m - )
10 10 C u r v a t u r e , C ( μ m - ) μ m) A Figure 3.5:
Role of cross-linkers in bending. A.
For weak cross-linkers we have L (cid:29) (cid:96) , and the resultingshape has a linearly decreasing curvature. As cross-linkers get stronger (values for increasing shades of greenare k = 10 − pN /µ m , k = 10 pN /µ m and k = 10 pN /µ m ), L becomes comparable and eventually biggerthan (cid:96) , which flattens the resulting shape. B. Dependence of maximal curvature with cross-linker stiffnessfor three values of basal compliance. The softer the cross-linkers, the larger the curvature. Lighter shadesof red correspond to lower values of k ( k = 92352 pN /µ m, k = 923 . /µ m and k = 9 . /µ m), andthe dots correspond to the shapes in A. C. Dependence of maximal curvature with basal stiffness for threedecreasing cross-linker stiffness (decreasing shades of blue correspond to k = 2280 pN /µ m , k = 228 pN /µ m and k = 22 . /µ m ). Stiffer bases result in higher curvatures. D. Three examples of shapes correspondingto the dots in C. Other parameters are κ = 1730 pN · µ m ; L = 30 µ m; a = 0 . µ m; f m = 500 pN /µ m and,unless otherwise specified, k = 92352 pN /µ m and k = 228 pN /µ m , which results in (cid:96) ≈ is obtainedvia Eq. 3.27, and is ∆ = f m k + ( k /(cid:96) ) coth[ L/(cid:96) ] . (3.37)Note that due to the presence of static cross-linkers with stiffness k the net sliding force f = f m − k ∆ doesdepend on arc-length 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 cross-linkers is small. According to Eq. 3.36 in this case we have L (cid:28) (cid:96) . We thus expand in L/(cid:96) and to lowestorder obtain ψ ( s ) ≈ − a f st κ (cid:18) s − sL (cid:19) . (3.38)This parabolic solution implies that for filaments of length L (cid:28) (cid:96) the curvature ˙ ψ decreases linearly alongthe arc-length from its maximum value at the base to the minimum at the tip, as is shown in Fig. 3.5 A.As filaments get longer relative to (cid:96) the effect of cross-linkers becomes more prominent, which makes thecurvature decrease sub-linearly, and also reduces the maximum curvature at the base (see Fig. 3.5 A, darkershades of green).A key parameter that characterizes the bent cilium is thus its maximal curvature C max = ˙ ψ (0), which isshown in Fig. 3.5 B as a function of k for several values of k . The maximal curvature decreases monotonicallyas the cross-linkers become stiffer, and saturates for vanishing cross-linker stiffness. At the same time, themaximal curvature increases as the base of the cilium becomes stiffer ( k grows), and eventually saturatesin the limit of an incompressible base for which ∆ = 0 when k (cid:29) kL , see Fig. 3.5 C (three examples ofshapes appear in Fig. 3.5 D). Clearly in this limit the maximal curvature still depends on the bulk cross-linkerstiffness k . In particular, for k = 0 there is a sliding ∆ = f m /k , but the cilium remains straight, ψ = 0.Our analysis thus indicates that for the static bending of cilia a basal stiffness is necessary, and the stronger .5. CONCLUSIONS ε C m a x ( μ m - ) Figure 3.6:
Bending with a cross-linkergradient.
The maximal curvature growswith the cross-linker slope (cid:15) . Values are asin Fig. 3.5, with increasing shades of greencorresponding to ˜ k = 10 − pN /µ m , ˜ k =10 pN /µ m and ˜ k = 10 pN /µ m . (cid:96) is de-fined with k ( s = 0) = ˜ k .Until now we have shown that bending requires a lateral asymmetry ( ρ A (cid:54) = ρ B ) and a polar asymmetrywhich we have identified as the basal stiffness ( k (cid:54) = 0). An alternative way of introducing the polarasymmetry is to remove the basal constrain ( k = 0) but consider an inhomogeneous cross-linker stiffness.One possibility is a linear profile k ( s ) = ˜ k (1 − (cid:15)s/L ), with (cid:15) the slope of the linear gradient and ˜ k the cross-linker stiffness at the base. In this case, one recovers a flat solution when (cid:15) = 0. As the gradient slope grows(see Fig. 3.5 C), a bend develops and the maximal curvature increases (see Fig. 3.6). Since a linear gradientis a smaller inhomogeneity than a basal stiffness, the maximal curvature in the limit L (cid:29) (cid:96) is smaller thanin the case where a basal stiffness is present (compare Fig. 3.6 to Fig. 3.5 A). • We obtained the equations of motion for a pair of inextensible filaments with a variable spacing betweenthem that are immersed in a fluid and subject to active sliding forces. These equations show that thenormal force between the filaments is a non-linear effect. • The static bending of a filament pair requires: (i) the presence of a lateral asymmetry, such as a motordensity 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 (cid:96) defined by the cross-linkers.2
CHAPTER 3. FORCE BALANCE OF PLANAR CILIA hapter 4
Dynamics of collections of motors
The beat of cilia is powered by the action of molecular motors. In this chapter we introduce a mesoscopicdescription of the force generated by the dynein motors, and show how oscillatory instabilities emerge asa collective property of these. We also introduce a minimal stochastic biochemical motor model in whichmotors are regulated through their detachment rate. We finally demonstrate that if the detachment rate isregulated by normal forces or curvature, the static shape of the cilium corresponds to a circular arc. This isin agreement with experimental data obtained for the shapes of interacting doublet pairs.
Consider that a cilium is dynamically bent. Its mechanical strains and stresses, such as the sliding ∆,curvature ˙ ψ , and normal force f ⊥ , will depend on time. In such a scenario, the motor force can be genericallycharacterized by its non-linear response to these strains and stresses. This reflects the idea that motors candynamically respond to the mechanics of the cilium. For example, in the case in which motors respond tothe sliding of the doublets, we will have f m ( s, t ) = F (0) + (cid:90) ∞−∞ F (1) ( t − t (cid:48) )∆( s, t (cid:48) )d t (cid:48) + (cid:90) ∞−∞ F (2) ( t − t (cid:48) , t − t (cid:48)(cid:48) )∆( s, t (cid:48) )∆( s, t (cid:48)(cid:48) )d t (cid:48) d t (cid:48)(cid:48) + (cid:90) ∞−∞ F (3) ( t − t (cid:48) , t − t (cid:48)(cid:48) , t − t (cid:48)(cid:48)(cid:48) )∆( s, t (cid:48) )∆( s, t (cid:48)(cid:48) )∆( s, t (cid:48)(cid:48)(cid:48) )d t (cid:48) d t (cid:48)(cid:48) d t (cid:48)(cid:48)(cid:48) + . . . , (4.1)where F ( h ) is the response kernel of order h , and terms of order higher than cubic have not been included.Note that the F ( h ) can in principle depend on arc-length, which would account for inhomogeneities of themotors along the cilium. In this thesis however we will consider them to be arc-length independent.For periodic dynamics with fundamental frequency ω , the response equation in Fourier space is f m ,j = F (0) + F (1) j ( ω )∆ j + F (2) k,j − k ( ω )∆ k ∆ j − k + F (3) k,l,j − k − l ( ω )∆ k ∆ l ∆ j − k − l + . . . , (4.2)with the nonlinear response coefficients F ( h ) i,j,k,... ( ω ) being the h − dimensional Fourier transform of the cor-responding h − order kernel. The coefficients F ( h ) i,j,k,... completely characterize the motor-filament interaction,and examples of them are given in sections 4.2 and 4.3. The linear response coefficient is of particularimportance, and we use a distinct notation for it: λ ( jω ) = F (1) j ( ω ) , (4.3)where the static response is λ (0) and the linear response to the fundamental mode is λ ( ω ). The lineardynamic response then becomes f m , = λ ( ω )∆ . (4.4)334 CHAPTER 4. DYNAMICS OF COLLECTIONS OF MOTORS
Analogous relations can be written for regulation via normal forces and curvature, and for each case wewill use a different greek letter for the linear response coefficient. Thus for curvature response we have f m , = β ( ω ) ˙ ψ , (4.5)where in this case β ( ω ) = F (1)1 ( ω ), with F (1) j obtained from a relation analogous to Eq. 4.2 involvingcurvature. Finally, for normal force response we use f m , = γ ( ω ) f ⊥ , , (4.6)with γ ( ω ) = F (1)1 ( ω ). c o ll e c t i v e m o t o r f o r c e , f m collective motor velocity, � t ∆ v Figure 4.1:
Force-velocity curve ofa collection of motors.
For valuesof the control parameter Ω in the sub-critical region Ω < Ω c the force veloc-ity curve is monotonic. In this regime,in the absence of any force, motorsmove at the collective velocity v > c an insta-bility occurs, and in the region whereΩ > Ω c the system is unstable andthere are two possible collective ve-locities. Coupling the motors with anelastic element will then give rise tooscillations [42]. For Eq. 4.7 we haveΩ = α , and v = 0. 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 forceresponds to the local sliding velocity of filaments. In particular, we consider the following model ∂ t f m = − τ ( f m − α ∂ t ∆ + α ( ∂ t ∆) ) , (4.7)where the model parameters are the relaxation time τ , the response parameter α , and the saturationstrength α >
0. These parameters do not depend on the arc-length s , corresponding to a homogeneousdistribution of motors along the cilium. In this description of the motor force there are no quadratic terms,which is a consequence of time-reversal symmetry. The response coefficients of this model can be easilycalculated, and are F (1) j ( ω ) = α iωj iωjτ and F (3) j,k,l ( ω ) = α iω jkl iωτ ( j + k + l ) , (4.8)with ω = 2 π/T the fundamental angular frequency and T the period of the oscillation.At the steady state in which the sliding velocity ∂ t ∆ is stationary, this model is characterized by anon-linear force velocity relation schematized in Fig. 4.1, in which Ω = α is the control parameter. Whilefor values α ≤ α > κ → ∞ . In this limitbending and fluid forces are not relevant, and setting the basal stiffness and viscosity to zero, the filamentwill remain straight (thus ˙ ψ = 0). The internal sliding forces are then balanced, and we have f m − k ∆ − ξ i ∂ t ∆ = 0 . (4.9) .3. A STOCHASTIC BIOCHEMICAL MODEL OF CILIARY MOTORS s li d i ng time2 4 6 80-0.050.00+0.05+0.10 A B C time2 4 6 80time2 4 6 80-0.10
Figure 4.2:
Oscillatory instability of motor force. A . For values of the control parameter α < α c thesystem is stable and shows damped oscillations. B . At the critical point α = α c sinusoidal oscillations occur. C . Far from the critical point α > α c non-linear oscillations occur, with higher harmonics and a differentfrequency to that of critical sinusoidal oscillations. In this example, triangular waves. The parameters usedfor this numerical integration of Eqs. 4.7 and 4.9 are k = 1812 pN /µ m , ξ i = 9 .
76 pNs /µ m , τ = 0 . α = { , . , } pN s /µ m and α = 4 · − pN s /µ m . The same initial sliding was used in all cases,and the axes of all three panels are equal.Together with the motor model in Eq. 4.7, this equation defines a dynamical system. To see how this systemcan exhibit an instability, consider as an ansatz small perturbations of force and sliding which are ∝ exp( σt )[42, 43]. In general σ = τ − + iω is complex, with τ the characteristic relaxation time of the perturbation,and ω the frequency of the perturbation. Using this ansatz in the equations one obtains for σ the followingsecond order characteristic equation σ = ( k + ξ i σ )(1 + τ σ ) /α , (4.10)which we now discuss.Using the coefficient α as control parameter, we study the stability of this dynamical system. Inparticular, note that for α = 0 the system is clearly stable and non-oscillatory, and so τ < ω = 0.As α increases the system is still stable (i.e. τ < ω (cid:54) = 0 it will exhibit damped oscillations asshown in Fig 4.3 A. Eventually, the control parameter may reach a critical value α c for which one of the twosolutions to Eq. 4.10 becomes critical, that is τ → ∞ . In this case small amplitude sinusoidal oscillationsappear with frequency ω , as can be seen in Fig. 4.3 B. Finally, for values larger than the critical one (theregion α > α c ) non-linear oscillations occur. Their amplitude is set by the nonlinear saturation term α ,see Fig. 4.3 C.Similar models can be worked out for curvature and normal force control. In such cases, however, thedynamics of the motor model cannot be studied independently from those of the filaments. It is indeed thiscoupling 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 changesin curvature is ∂ t f m = − τ ( f m − α ˙ ψ + α ˙ ψ ) . (4.11)Here τ corresponds to a delay, α to the linear response, and α to the saturation which controls theamplitude. The response coefficients in this case are given by F (1) j ( ω ) = α j iωjτ and F (3) j,k,l ( ω ) = α ω jkl iωτ ( j + k + l ) . (4.12) So far we have considered an effective description of the motor force, we now introduce a minimal stochasticbiochemical model capable of giving active response to sliding, curvature, and normal force. To do so, we first6
CHAPTER 4. DYNAMICS OF COLLECTIONS OF MOTORS note that in the description of the axoneme as a pair of opposing filaments, motors are present on filament Aand filament B (see Fig. 3.2). The motors rigidly attached through their stem to filament B exert with theirstalks a force per motor F B on filament A during their power stroke, and a reaction force F B in filament B.The analogous holds for motors with their stem attached to A. Since motors can stochastically attach anddetach from the filaments, there are probabilities p A and p B of a motor being attached to either filament.Thus, given line densities of motors ρ A and ρ B along the two filaments, the total motor force density exertedon filament A is f m = ρ B p B F B − ρ A p A F A , (4.13)and analogously filament B is subject to a force − f m .In steady state where motors are stalled the motor force is F st = F A = F B , where F st is the stall force.When the motors are moving at a certain velocity, they are characterized by a force-velocity relationship,which we consider for simplicity to be linear with slope α . Thus for the motors in filament A we have F A = F st − α∂ t ∆ A , (4.14)where we note that α ≥ F B = F st + α∂ t ∆ B . Since ∆ A = ∆ and ∆ B = − ∆ (see 3.1), we can write for the net motor densitythe following relation f m = ( ρ A p A − ρ B p B ) F st + ( ρ A p A + ρ B p B ) α∂ t ∆ . (4.15)Considering the densities of motors to be homogeneous along the filaments, the description of the motorforce is completed with the binding kinetics of the motors.Since motors can either be bound and exert force or unbound, we have for the motors in filament A thefollowing equation d p A d t = − k off , A p A + k on (1 − p A ) , (4.16)and an analogous equation for the binding probabilities of motors in filament B. It is known that thedetachment rate of a molecular motor can be enhanced by the forces which they sustain, and the dependenceis exponential according to Bell’s law [6]. If we generalize this to also allow curvature dependent detachment,we have k off , A = k exp (cid:20) F A F c + f ⊥ f c ⊥ + C A C c (cid:21) , (4.17)where F c is the characteristic sliding detachment force, f ⊥ is the normal force as defined in Eq. 3.23, f c ⊥ thecharacteristic normal spacing detachment force, and C c the characteristic detachment curvature. TogetherEqs. 4.15 to 4.17 form a minimal biochemical description of the motor force. This description, alternativeto the generic approach used in the previous section, also gives rise to oscillatory instabilities as can be seenin Fig. 4.3. The nonlinear response coefficients of the generic approach corresponding to this biochemicalmodel can be obtained by expanding these equations in powers of the strains and stresses.In the static regime, the motors are stalled, and the motor force density takes the value f m = ( ρ A − ρ B ) p F st , (4.18)where p is the stall attachment probability which in the limit C A = C B = ˙ ψ of constant spacing (see Eq. 3.6)is the same for motors on both filaments. This static attachment probability is given by p = (cid:32) k k on exp (cid:34) F st F c + f ⊥ , f c ⊥ + ˙ ψ C c (cid:35)(cid:33) − , (4.19)and is independent of the static sliding, but depends on the static curvature and normal force making staticregulation of bent shapes possible. Importantly, the static motor force vanishes for the case ρ A = ρ B . Thus,in this model static active force can only be generated when lateral symmetry is broken. .4. MOTOR REGULATION PRODUCES CIRCULAR BENDS b i nd i ng p r obab ili t y f o r c e B time s li d i ng -0.30-0.100.00+0.20+0.30 C time0 4 8 12 16+0.10-0.200.00.20.3 A time4 8 12 160.1 p A p B Figure 4.3:
Sliding oscillations of antagonistic biochemical motors. A . The probabilities of themotors in the rigid filaments A and B to be bound oscillate with a phase shift. B . The mismatch in bindingprobability generates a net active motor force f m , which is also oscillatory. C . This finally results in slidingoscillations. The parameters used for this numerical integration of Eqs. 4.15 and 4.17 are k = 1812 pN /µ m , ξ i = 9 .
76 pNs /µ m , k on = 14 s − , k = 2 . − , α = 0 . ρ A = ρ B = 10 motors /µ m, F c = 1 pN, and F st = 4 pN.Following the path outlined in the previous section, the response coefficients of the motor force to sliding,curvature, and normal forces can be calculated. In particular, the linear coefficients are λ ( ω ) = − ( ρ A + ρ B ) p (1 − p ) iω + ω p /k on ωp /k on ) F st F c α + ( ρ A + ρ B ) p αiω ,β ( ω ) = ( ρ A − ρ B ) p (1 − p ) 1 − iωp /k on ωp /k on ) F st C c ,γ ( ω ) = ( ρ A − ρ B ) p (1 − p ) 1 − iωp /k on ωp /k on ) F st f c ⊥ . (4.20)Importantly, for the case ρ A = ρ B in which lateral symmetry is preserved, the response coefficients tocurvature and normal force vanish ( β ( ω ) = 0 and γ ( ω ) = 0), but the one to sliding does not (this is only truein the limit a → a , otherwise curvature control would produce a response to spacing). On the other hand,in the case that ρ A (cid:54) = ρ B this model exhibits static regulation if it is sensitive to curvature and normal forces,since according to Eq. 4.19 p will be a function of arc-length. In this case the linear response coefficientsabove will not be homogeneous along the arc-length. Finally, we note that in the sliding response coefficientonly the first term is active, with the second one corresponding to protein friction. In section 3.4 we saw that the shapes resulting from a constant motor force in the cilium are spirals, whichcorrespond to a linear decrease in curvature. The presence of cross-linkers makes the decay of curvature evenstronger (see Fig. 3.5). However, we described in section 2.2 experiments where the shapes of statically bentdoublets were circular arcs. That is, they were characterized by a linearly growing tangent angle and thus aconstant curvature (see Fig. 2.6). This suggests that motor regulation as introduced in this chapter may beresponsible for producing a non-homogeneous distribution of bound motors leading to the observed shapesof bent cilia.We can infer the distribution of active motors from the shape using the moment balance equation(Eq. 3.32), which after one integration is F ( s ) = κC ( s ) /a . This balance equation implies that the mo-tor activity is concentrated at the distal end of the filaments pair, s = L . This follows because a circulararc (i.e., a shape with constant curvature) requires a constant total force, which in turn requires the slidingforce density f ( s ) to be zero except at the distal end. Indeed, in Fig. 4.4 A we show three examples of motorforce distributions, with the corresponding shapes in Fig. 4.4 B obtained from numerically solving Eq. 3.32.As the motor force accumulates more at the tip of the cilium, the shape becomes closer to a circular arc.8 CHAPTER 4. DYNAMICS OF COLLECTIONS OF MOTORS arc-length, s L f o r c e den s i t y L/20f max f max /20 s=0s=L xy A B
Figure 4.4:
Formation of arcs under force regulation.
A uniform force distribution (panel A, red curve)produces a spiral shape (panel B, red curve). Tip-concentrated force distributions (panel A, yellow and bluecurves) produce nearly circular arcs (panel B, yellow and blue curves).To obtain such circular shapes we consider the biochemical model from the previous section in the staticlimit. In this case there is no sensitivity to sliding, as ∂ t ∆ = 0, but motors can sense curvature and normalforces. The moment balance Eq. 3.32 together with Eq. 4.19 for the motor binding probability results in κ ¨ ψ ( s ) = − ρ A A F st u exp (cid:34)(cid:18) ˙ ψ ( s ) √ a F c ⊥ /κ (cid:19) + ˙ ψ ( s ) C c (cid:35) . (4.21)where we have used that the normal force is given by f ⊥ = ˙ ψ /a and that ρ B = 0, since in the doubletexperiment only the motors of one filament are exerting active forces. We have also defined the parameter u = k exp[ F st /F c ] /k on , which is the fraction of time that the motor spends unbound. This equation leadsto force concentration at the distal end, as can be seen by the following argument.Motor sliding forces cause bending, which results in a normal force that tends to separate the filamentpair. When the normal force f ⊥ ( s ) exceeds the characteristic normal force density F c , the motors detach,resulting in a decrease in sliding force. Only near the distal end, at which the curvature decreases to zero(according to the boundary condition in Eq. 3.34), will the normal force fall below its critical value and themotors will remain attached. Thus, this motor regulation mechanism results in feedback: as the doubletstarts to bend, the higher curvature at the base (Fig. 4.4) causes basal motors to detach, and as the benddevelops there will be a wave of detachment that only stops at the distal end. Note that this argument isequally valid for normal force and curvature regulation. To compare quantitatively the predictions of this model with the experimental data, we numerically inte-grated Eq. 4.21 using the boundary conditions in Eq. 3.34. For regulation via normal force, which is shownin yellow in Fig. 4.5, we use as parameters a stall motor force F st = 5 pN, a motor density ρ A = 200 µ m − , abending stiffness κ = 120 pN µ m , a critical-motor force f c ⊥ = 200 pN /µ m and no curvature control C c → ∞ .For the solution under curvature control which is shown in blue, we use the same parameters but a criticalcurvature of C c = 0 . µ m − and no normal force regulation by setting f c ⊥ → ∞ . In both cases we consideredthe fraction of unbound time to be u = 0 . .4. MOTOR REGULATION PRODUCES CIRCULAR BENDS C u r v a t u r e ( μ m ) T angen t ang l e A B - π π /20 0.51 Arc-length (μm)
Arc-length (μm)
Figure 4.5:
Comparison of model and tracked data. A.
Tangent angle vs arc-length for several controlmechanisms. Red line corresponds to no control (or sliding control), yellow to normal force control, and blueto curvature control. Black circles correspond to tracked data, note that no control departs significantlyfrom the data. B. Curvature vs arc-length for three control mechanisms. The data, which is noisier than inA, shows a roughly constant curvature that decreases towards the end. Normal force control (yellow) andcurvature control (blue) show the same behavior, but no control (in red) does not.linearly decreasing curvature (see Figs. 4.4 and 4.5, red lines), which is not consistent with the experimentaldata. The sliding control model, where detachment is proportional to the sliding force experienced by themotors (given by Eq. 4.14), also leads to a constant force density because, at steady state, the shear forceexperienced by all the motors is the same, and corresponds to the stall force. Thus, the sliding control modelalso has a constant shear force per unit length, and, like the unregulated case, leads to a non-circular shape,inconsistent with the observations. Figure 4.6:
Dependence of average curvatureon filament length.
The curvatures of the fivetracked split axonemes (insets) show a weak depen-dence on their length (green symbols). Curvaturecontrol (blue line) and normal-force control (yellowline) both predict the observed weak length depen-dence, without introducing additional parameters.In the absence of regulation, the predicted linearscaling vastly deviates from the data (red line).To determine how curvature depends on filament length, we analyzed five pairs of microtubule doubletsthat showed the arcing behavior whose lengths ranged from 5 to 9 µ m. A total of 24 arcing events wereobserved, with up to 8 events for a single pair of filaments. All bent into nearly circular arcs (Fig. 4.6, upperimages). The average curvature (excluding the last 1 µ m from the distal tip) increased only weakly withthe length of the doublets (Fig. 4.6, filled points). The same parameters used in Fig. 4.5 were then usedto fit the curvature vs. length data for all five doublet pairs, without any additional parameters. Both thecurvature control and the normal-force control models were in very good agreement with the data (Fig. 4.6,blue and yellow curves respectively). The models predict a weak dependence of the curvature on filamentlength because it is only the most-distal motors that generate the bending forces. By contrast, if the densityof active motors were constant along the doublets, as in the sliding control model, then the average curvaturewould be proportional to length (integrating Eq. 3), which is inconsistent with the data (Fig. 4.6, red line).0 CHAPTER 4. DYNAMICS OF COLLECTIONS OF MOTORS • We developed an effective description for the non-linear dynamic response of a collection of motorsto changes in curvature, sliding, and normal forces; and showed that it can give rise to oscillatoryinstabilities. • Regulation of motor detachment by normal forces or curvature gives rise to tip accumulated forcesin the static limit, thus bending pairs of filaments into circular arcs. A sliding control mechanismproduces homogeneous force distribution and thus spiral shapes. • Circular arcs similar to those obtained by the theory are observed for pairs of doublets, which supportsregulation by normal forces or curvature instead of sliding. hapter 5
Non-linear dynamics of the cilia beat
Cilia exhibit a great variety of beating patterns. In this chapter we analyze how different mechanisms ofmotor regulation give rise to non-linear beat patterns which allow ciliary propulsion. The critical beats areanalyzed analytically to linear order. Finite amplitude beats, which are non-linear, are supercritical. Weinvestigate them by numerically integrating the time-domain equations of motion.We first consider sliding regulation, and show that wave propagation is only possible for long cilia and inthe presence of a boundary asymmetry. This is the case of single-flagellates such as
Bull Sperm , where thecilium is long and the head provides the required asymmetry. We further show that under sliding controlthe boundaries play a fundamental role in determining the direction of wave propagation.We then move on to curvature control, where we show that the direction of wave propagation is determinedby the motor parameters instead of the boundaries. Furthermore, under curvature control wave propagationis possible for short cilia such as those of
Chlamydomonas . It was shown in section 4.2 that motors regulated by sliding of the filament pair can become dynamicallyunstable. Consider a motor model characterized by its response coefficients λ ( h ) i,j,... to sliding. Further, weassume that the motor does not couple to curvature or normal forces. In such a case, we say that motorsare controlled by sliding.Consider now that at time t = 0 a small complex sliding perturbation ∆( s, t ) = ∆( s ) exp( σt ) is turnedon, where in general σ = τ − + iω with τ the relaxation time of the perturbation and ω its frequency. Theeffect of the sliding perturbation on the force can then be described by f ( s, t ) = − χ ( σ )∆( s ) exp( σt ) , (5.1)where χ ( σ ) is the sliding response coefficient. This coefficient combines passive and active elements, and isgiven by χ ( σ ) = − λ ( σ ) + k + σξ i , (5.2)with k the sliding stiffness, ξ i the internal viscosity, and λ ( σ ) the response coefficient of the motor force. Theresponse to small perturbations λ ( σ ) can be calculated by inserting the perturbation ansatz in the non-linearresponse Eq. 4.1. Keeping the linear order results in λ ( σ ) = (cid:82) ∞ λ ( t − t (cid:48) )e − σ ( t − t (cid:48) ) d t (cid:48) , which is the analyticalcontinuation of the Fourier coefficient λ ( ω ) introduced before.The tangent angle likewise evolves according to ψ ( s, t ) = ψ ( s )e σt . We can obtain an equation for thecomplex amplitude ψ ( s ) of the angle by linearizing Eq. 3.31, which results in σξ n ψ ( s ) = − κ .... ψ + a χ ( σ ) ¨ ψ . (5.3)Provided the dependence on σ of the response χ ( σ ) and a choice of boundary conditions, this boundary valueproblem can be solved. Using the boundary conditions a discrete set of solutions or modes can be obtained,each characterized by a relaxation time τ and an angular frequency ω .412 CHAPTER 5. NON-LINEAR DYNAMICS OF THE CILIA BEAT subcritical critical supercriticalfrequency, ω (rad/s)time, ( s ) - - - - - - ω (rad/s)0 200 400 600 800 1000frequency, ω (rad/s)0 200 400 600 800 1000 0.0 0.01 0.02 0.03 0.040.0 0.01 0.02 0.03 0.04 0.0 0.01 0.02 0.03 0.04 ψ ( L ) ( - r ad s ) ψ ( L ) ( - r ad s ) ψ ( L ) ( - r ad s )r e l a x a t i on r a t e , ( s - ) A B CD E F time, ( s ) time, ( s ) r e l a x a t i on r a t e , ( s - ) r e l a x a t i on r a t e , ( s - ) Figure 5.1:
Onset of modes instability in cilia. A, B, C.
The value of | Γ | − (with Γ the boundary valuedeterminant of a free cilium) is plotted as a function of the relaxation rate σ (cid:48) and frequency σ (cid:48)(cid:48) using the motormodel in Eq. 4.7 (red dots correspond to divergencies, where modes lie). In the subcritical regime all modesare stable, with negative relaxation rate. In the critical regime the mode denoted by 1 becomes unstable witha zero relaxation rate. In the supercritical regime mode 1 is unstable. D, E, F.
Time traces for the tip angleafter solving the non-linear dynamic equations for the values used in A, B and C correspondingly. Thereis a clear transition from damped, to critical, and supercritical oscillations. Parameters used as in Fig. 5.2,and for the motor model k = 228 pN /µ m , ξ i = 0 . · s /µ m , α = 2 · − pN · s /µ m , τ = 0 . α = 0 . · s /µ m (for A and D), α = 1 .
24 pN · s /µ m (for B and E) and α = 1 .
52 pN · s /µ m (for C andF). In Fig. 5.1 we show the stability of the different modes of the motor model in 4.2, for which λ ( σ ) = α σ/ (1 + στ ) with τ the relaxation time of the motors, and α the control parameter. For values α < α c all modes (which appear as red dots in Fig. 5.1 A) have a negative relaxation rate τ < ω (see Fig. 5.1 D). At the critical value α = α c one mode, which we noted 1 in Fig. 5.1 B, becomes critical: it’s characterized by τ → ∞ . In thiscase the system shows small amplitude oscillations (see 5.1 E). Finally, in the supercritical regime α > α c ,the system is stabilized by non-linearities and shows large amplitude oscillations (see Fig. 5.1 F, where thefull non-linear problem was solved). For a cilium that beats at its critical poin, a perturbation does not decay nor does it increase, as τ → ∞ .Critical perturbations thus evolve as exp[ σt ] = exp[ iω c t ], producing sustained small amplitude oscillationsof critical frequency ω c . Since at the critical point only one harmonic is present, we have ψ ( s, t ) = ψ ( s )e iω c t + ψ ∗ ( s )e − iω c t , (5.4)where we have used that the angle is a real quantity and thus ψ − n = ψ ∗ n , and the sub-index 1 denotes thefundamental mode. For simplicity, we are considering that the oscillation has no static mode. Equationsanalogous to 5.4 apply to the basal sliding and the motor force. Further, the relationship between the sliding .1. BENDING WAVES UNDER SLIDING CONTROL f ( s ) = − χ c ∆ ( s ) , (5.5)where χ c = χ ( σ = iω c ) with the control parameter at the critical point, Ω = Ω c . From now on we drop thesub-index 1, since it applies to all geometrical and mechanical variables of the system. The relation betweenbasal sliding ∆ and basal force F is F = χ ∆ , (5.6)with the critical value of the passive basal compliance χ = k + iω c ξ .Consistent with the presence of just one harmonic, at the critical point the amplitude of the oscillationsis small. Critical oscillations are thus described by the linear equation 5.3. Using σ = iω c and χ = χ c resultsin i ¯ ω c ψ = − .... ψ + ¯ χ c ¨ ψ , (5.7)where bars denote dimensionless quantities, and we have used the following transformations¯ s = sL , ¯ t = tω , ¯ τ = L κ τ , ¯∆ = ∆ a , ¯ f = a L fκ , ¯ k = a Lκ k , ¯ ξ = a Lωκ ξ i , ¯ k = a L κ k , ¯ ξ i = a L ωκ , ¯ λ = a L κ λ , ¯ ξ = ξ n ξ t , ¯ ω = L ωξ n κ , ¯ χ = ¯ k + i ¯ ξ , ¯ χ c = ¯ k + i ¯ ξ i − ¯ λ . (5.8) -40-60-80-20-10000-50-100-150-200-250-300 response, χ’ r e s pon s e , χ ’’ χ c = χ’+i χ’’
12 3456 mode 4mode 2mode 1color code
AB C t/T
Figure 5.2:
Critical beats of free cilia under sliding control. A.
Space of critical solutions for free ends.The value of | Γ | − is plotted as a function of χ , where Γ is the determinant to the boundary value problem.Yellow to blue corresponds to increasing values, and divergences (red dots) correspond to critical modes.These are numbered by | χ n | < | χ n +1 | (one divergence corresponds to a trivial solution, and is not marked). B. Color code used throughout this thesis, as time progresses over one period the shade of the cilium changescolor counter-clockwise. C. Three critical beats for free cilia. While mode 1 shows a forward (base-to-tip)traveling wave which results in swimming towards the left, mode 2 has a backward traveling wave, andmode 4 shows no wave propagation, being a standing mode. Arrows denote direction of wave propagation.The parameters used were the typical ones of
Bull Sperm : κ = 1730 pN · µ m , ξ n = 0 . · s /µ m , L = 58 . µ m, ω c = 2 π · · s − , a = 0 . µ m and k = ∞ (which ensures no basal sliding).Note that, since Eqs. 3.29 and 3.28 are non-linear, tension and normal force vanish to first order, τ = 0and f ⊥ = 0. We will see in the next chapter that this does not apply to asymmetric ciliary beats. Given the4 CHAPTER 5. NON-LINEAR DYNAMICS OF THE CILIA BEAT critical frequency ω c , the linear ordinary differential equation for ψ ( s ) is an eigenvalue problem, which hasa discrete set of solutions for the eigenvalues χ c = χ n with n = 1 , , . . . (5.9)The set of eigenvalues χ n is determined by the condition that the determinant Γ( ¯ χ c ; ¯ ω c , ¯ χ ) of the homoge-neous system of equations imposed by the boundary conditions is zero (see Appendix B for details). Theyare numbered according to the rule | χ n | < | χ n +1 | .In Fig. 5.2 we show the value of | Γ( χ (cid:48) + iχ (cid:48)(cid:48) ) | − over the space of negative values of χ (cid:48) and χ (cid:48)(cid:48) , which iswhere the active motor response λ dominates over the passive response of k + iω c ξ i . The value of ¯ ω c ∼ is a typical one for Bull Sperm , and the basal compliance has been taken very stiff | χ | ∼ . Red dotscorrespond to divergences where a mode exists. A time-trace of some of the modes is also shown in Fig. 5.2.The first mode for a free axoneme is forward swimming. The wave propagates forward from base to tip,thus allowing for propulsion towards the left. This mode is also the one that was numbered 1 in Fig. 5.1.This is the first to become unstable with the motor model of 4.2. The second mode is backwards swimming.Finally, there is an infinite number of standing modes, of which an example is shown (see Appendix B forphase and amplitude profile of modes 1 and 2).At the critical point the characteristics of the modes, such as direction of wave propagation or amplitudeprofile, depend on the boundary conditions, on the basal compliance ¯ χ , and on the dimensionless frequency¯ ω . But besides this, they are generic and independent of the motor model dynamics. That is, they areindependent of the functional dependence of ¯ χ on ¯ ω , as long as χ ( ω c ) = χ n for a certain n . For now werestrict ourselves to the case of free cilia, and study the effect of ¯ χ and ¯ ω c on the modes. In particular, thedimensionless frequency ¯ ω can be understood as a ratio of two length scales¯ ω = (cid:18) L(cid:96) (cid:19) , with (cid:96) = (cid:18) κξ n ω (cid:19) / . (5.10)The characteristic length (cid:96) is the typical wavelength of a cilium when driven from a boundary. It is alsothe length beyond which hydrodynamic effects become negligible [57]. The length L is simply the length ofthe flagellum. -10 -2 -10 -10 w a v e s peed , length,length, Bull SpermChlamydomonas -1012 m o t o r r e s pon s e , mode 1mode 2mode 1mode 2 BA Figure 5.3:
Length scaling of modes. A.
Scaling of the real ( χ (cid:48) , solid lines) and imaginary ( χ (cid:48)(cid:48) , dashedlines) parts of the eigenvalues for the first and second modes in Fig. 5.2. The real part vanishes (linearly)for short lengths. B. Scaling of the wave velocity (see Eq. 5.11) with the length of the cilium, arrows marktypical values for
Chlamydomonas and
Bull Sperm . Short cilia show no wave propagation, as all solutionsbecome standing waves. Long cilia allow forward (in mode 1) and backward (in mode 2) wave propagation.The basal stiffness is k → ∞ , which prevents basal sliding.For short cilia L (cid:28) (cid:96) , we have ¯ ω c (cid:28)
1. In the limit of ¯ ω c → χ (cid:48)(cid:48) →
0, as can be seen in Fig. 5.3 A [18]. In this regime viscous components (of the surroundingfluid and the sliding force) vanish and the phase profile of ψ ( s ) becomes flat for all modes. They becomestanding modes, which do not allow for wave propagation. This effect is better described using the wavepropagation velocity v w = − (cid:90) L | ψ ( s ) | ∂ s arg( ψ ( s ))d s , (5.11) .1. BENDING WAVES UNDER SLIDING CONTROL v w has been plot as a function of L/(cid:96) , as cilia get shorter their wave velocitydecreases, vanishing asymptotically as L/(cid:96) →
0. It is important to note that, while in
Bull Sperm
L/(cid:96) issufficiently large to allow for wave propagation, this is not the case for Chlamydomonas . This suggests that
Chlamydomonas cilia are not regulated by sliding control.The basal compliance χ has a fundamental role in regulating the flagellar beat, particularly when motorsare controlled by sliding. The reason is that Eq. 5.7 is symmetric around the midpoint s = 1 /
2, thus notreflecting the polar asymmetry of the axoneme (see section 1.3.1). The breaking of polar symmetry has tocome from the boundary conditions or, for a free axoneme, from the presence of a basal compliance χ (cid:54) = 0.To better understand this we expand the eigenvalues in k : χ n = χ (0) n + k χ (1) n + k χ (2) n + . . . . Introducingthis in the equation of the determinant and matching terms we can determine the coefficients χ ( i ) n . Wedistinguish between standing modes (such as mode 2), for which χ (0) n (cid:54) = 0; and traveling modes (such as 1),for which χ (0) n = 0 (see Fig. 5.4 A). -1.0 10 -0.50.00.51.0-10 -2 -10 -10 normalized basal stiffness, w a v e v e l o c i t y , m o t o r r e s pon s e , mode 1mode 2 mode 1mode 2 A B normalized basal stiffness,
Figure 5.4:
Scaling of modes with normalized basal stiffness. A.
Since mode 2 is a standing modeit does not vanish for small values of ¯ k = a Lk /κ . Mode 1 however is a directional mode, and vanishesin the absence of basal compliance (as k → χ (cid:48) ∝ k and χ (cid:48)(cid:48) ∝ k ). B. Mode 1, which is adirectional mode, does not substantially alter its directionality as the basal stiffness changes. Mode 2 loosesits directionality when the base decreases its stiffness. Parameters as those in Fig. 5.2, which correspond to
L/(cid:96) = 7 . k = 0, in this limit they are standing waves.So while static bending is impossible in the absence of a basal constrain, as shown in section 3.4, dynamicsolutions do exist. These standing modes gain directionality as the basal compliance grows, resulting in anincrease of wave velocity (see Fig. 5.4 B, mode 2). On the other hand, traveling modes vanish when k = 0,but for an arbitrarily small basal compliance allow for strong wave propagation (see mode 1 in Fig. 5.4 A). Although Fig. 5.1 B shows that mode 1 is the first to become unstable, in order to obtain finite amplitudebeat patterns it is necessary to numerically integrate the set of Eqs. 3.26-3.31 supplemented with Eq. 4.7 forthe motor force in the supercritical regime. This system of equations is a set of non-linear coupled partialdifferential equations of fourth order with an integral term for the basal sliding. The oscillatory transitioncorresponds to a Hopf bifurcation, and as the control parameter α moves away from the critical point α c the amplitude of the oscillation grows as ( α − α c ) / . Furthermore, the saturation term α of the motormodel also controls the amplitude, in this case following the inverse scaling α − / (see Appendix B, whereboth these scalings where verified). Thus by sufficiently decreasing the saturation term and increasing thecontrol parameter, finite amplitude beats will occur. We note here that, since sliding control only produceswave propagation for long cilia, we consider parameters such that L/(cid:96) ≈
7, as is the case in
Bull Sperm .The numerical integration of this dynamical system is far from trivial, and the custom-made numericalalgorithm is described in Appendix B. In Fig. 5.5 A we can see the time trajectory of a typical beat pattern inthe supercritical regime. This beat corresponds to mode 1 in the non-linear regime, and it shows a forwardtraveling wave which makes the free cilia swim. To show that this beat is non-linear we plot the power6
CHAPTER 5. NON-LINEAR DYNAMICS OF THE CILIA BEAT Frequency, f (Hz) B wave propagationswimming direction f f f f f
300 400 A t/T non-linear theory (numerical) Figure 5.5:
Non-linear beat pattern of a free cilium. A.
Time trace of the swimming cilium withmode 1 dominating (see Fig. 5.2). The base is to the left, and the wave propagates right towards the tip,making the cilium move towards the left. Parameters as those in Fig. 5.1, with α = 5 . · s /µ m and α = 2 · − pN · s /µ m . B. The power spectrum of the angle at the mid-point of the cilium (obtainedthrough a Discrete Fourier Transform of the numerical integration) shows peaks in the odd harmonics, dueto the antisymmetry of the problem to the transformation t → − t .spectrum of the mid-angle, which exhibits higher odd harmonics. The lack of even harmonics is due to theodd parity of the equations of motion and the motor model.Figure 5.6: Tangent angle of linear and non-linear solution. A.
Kymograph of the tangent angle ofthe analytical critical solution over three periods. As one can see the waves propagate from base to tip. It isalso clear that the amplitude decreases towards the tip. Height normalized to match that of the non-linearsolution. B. Same as A but for the non-linear solution obtained numerically. We can see that frequency,wave profile and amplitude are in good agreement with the analytical linear solution. Parameters as inFig. 5.5.To better compare this non-linear beat pattern with the critical (thus, linear) beat of mode 1, we showin Fig. 5.6 a kymograph of the angle ψ ( s, t ) by adjusting the amplitude of the linear analytical solution tothat of the non-linear numerical solution. As one can readily see, the beat patterns are very similar. Thefrequency is roughly the same, and in both cases the wave clearly propagates towards the tip (forward) witha slowing down of the propagation towards the end. Furthermore, in both cases the amplitude of the beatdecays over arc-length, with the decay being stronger in the non-linear theory than in the linear one. Boundary conditions have an important effect on the patterns of beating cilia [19, 18]. This can be readilyseen in Fig. 5.7, where we directly compare the non-linear beat patterns for freely pivoting and clamped .1. BENDING WAVES UNDER SLIDING CONTROL B Freely pivoting base (non-linear) Clamped base (non-linear) A low basal stiffness, frequency 10 s -1 high basal stiffness, frequency 13.4 s -1 low basal stiffness, frequency 10.3 s -1 high basal stiffness, frequency 16.2 s -1 t/T Figure 5.7:
Beat patterns with different boundary conditions. A.
Beat pattern for a freely pivotingbase, the wave travels and the basal stiffness has little relevance. B. Beat patterns for a clamped base show abackwards traveling wave. Changing the basal stiffness produces a significant change in the beat pattern. InA top ξ = 26 . · s/µ m, k = 10 pN /µ m, α = 16 . · s /µ m , α = 1 . · − pN · s /µ m ; bottom samebesides k = 10 pN /µ m and α = 4 · − pN · s /µ m . In B α = 18 pN · s /µ m , α = 4 · − pN · s /µ m , ξ = 268 pN · s/µ m, with top k = 10 pN /µ m and bottom k = 5 · pN /µ m. Other parameters as inFig. 5.1, with ξ i = 9 . · s/µ m and k = 228 pN /µ m .This change in direction of wave propagation is not affected by changes in the basal stiffness. In fact, adecrease in basal stiffness has little effect in the beating patterns for a pivoting cilium (other than reducingthe amplitude, which we compensated by also reducing α , see Fig. 5.7 B). In the case of a clamped basea change in basal stiffness changes the beating mode (this is detailed in Appendix B). Yet, this change ofmode does not affect the direction of wave propagation (see Fig. 5.7 A). pivoting stiffness, k p ( L/ κ ) mode 1 (linear) s peed , v w -1 -1.0-0.50.00.51.0 Figure 5.8:
Wave velocity vs pivoting stiff-ness.
As the pivoting stiffness increases, the wavevelocity of mode 1 decreases. At a certain value thewave reverses and the wave propagates backwards.This is in agreement with Fig. 5.7, which showsbackwards waves for clamped ( k p → ∞ ) and for-ward traveling for pivoting ( k p → k = 8 . · pN /µ m.To better understand the change in direction of wave propagation as the base goes from pivoting toclamped, we analytically studied how the first mode is influenced by the pivoting stiffness k p . In Fig. 5.8 weplot the wave velocity of mode 1 as a function of the pivoting stiffness. For k p = 0 the cilium is freely pivoting,and for k p → ∞ we have that ψ (0) = 0 and the cilium is clamped. As one can readily see, stiffening thebasal pivot produces a reversal of wave direction, in agreement with the non-linear beat patterns in Fig. 5.7.This dependence on the direction of wave propagation on the boundary conditions provides a simple andelegant mechanism of change in swimming direction for a micro-organism. Simply stiffening the connectionbetween the head and the cilium can produce wave reversal. It can however also be seen as a limitation ofsliding control, not allowing forward wave propagation for clamped cilia. In the next section we show a wayto circumvent this issue.8 CHAPTER 5. NON-LINEAR DYNAMICS OF THE CILIA BEAT
Bull Sperm
The choice of motor model determines which mode gets selected, as shown in Fig. 5.1. Since in Fig. 5.2 we sawthat different modes have different direction of wave propagation the relevance of the motor model is clear.In the case of a clamped cilium, we saw that the first mode is a backwards traveling one (see Fig. 5.8), butthere is a higher mode which is forward traveling (see Fig. 5.9), in agreement with experimental observationsfor the beat of
Bull Sperm [71]. -40-60-80-20-10000-50-100-150-200-250-300 response, χ’ r e s pon s e , χ ’’ χ = χ’+i χ’’ mode 3mode 2mode 1 AB Figure 5.9:
Linear beats of clamped cilia under sliding control. A.
Space of solutions for a clampedcilium, with red dots in the critical modes (again we see a trivial mode). B. Three critical beats. The firstmode shows backwards wave propagation, as already discussed. The second mode is a standing mode. Mode3 however shows forward propagation, and the beat patter is similar to that of
Bull Sperm . Parameters asin Fig. 5.2, but for a frequency of 14 . − .To excite this forward mode without activating the backwards one a more complex motor model thanthat of Eq. 4.7 is necessary. A simple modification to that model is to include a higher order time-derivativefor the motor force, that is α ∂ t f m + τ ∂ t f m + f m = α ∂ t ∆ − α ( ∂ t ∆) , (5.12)where α accounts for effective inertial effects (these need not come from mechanical inertia, they may beof chemical origin). Indeed, for an adequate choice of parameters this model permits to select the forwardtraveling mode. The resulting beat pattern of a numerical integration of the equations is given in Fig. 5.10.As one can see, the wave propagation is very clearly backwards and the amplitude grows along the arc-length. wave propagation Figure 5.10:
Forward wave in clamped cilium.
Using the motor model of Eq. 5.12 allows to excite thesecond mode at a frequency of 14 . − . This beat is similar to that of Bull Sperm , although with loweramplitude. The parameters used are those in Fig. 5.2, with a basal stiffness k = 92352 pN /µ m, a basalcompliance ξ = 268 pN · s /µ m. The motor parameters are τ = 10 − s, m = 3 . · − s , α = 110 pN · s /µ m ,and α = 0 .
93 pN · s /µ m .The implication of this is that forward modes also exist for clamped cilia under sliding control, howeverthey are modes of order higher than one. To be excited, a complex motor model which includes inertial .2. BENDING WAVES UNDER CURVATURE CONTROL Bull Sperm is well explained by a sliding controlmechanism, the microscopic details of such a mechanism must include some kind of chemical inertia.
So far we have studied the beat patterns produced when motors are controlled by sliding of the doublets.However motors can instead be controlled by changes in the curvature of the cilium. In this case the responseof the sliding force to a sliding perturbation ∆( s, t ) = ∆( s ) exp( σt ) is passive, characterized by the slidingstiffness k and internal friction ξ i . However its response to curvature ˙ ψ = ˙∆ /a can be active. We thus have f ( s ) = − χ ( σ )∆( s ) − β ( σ ) ˙ ψ ( s ) , (5.13)with χ ( σ ) = k + ξ i σ the passive viscoelastic response, and β ( σ ) the linear response coefficient of the motorsto curvature (see Eq. 4.5). β’χ=k>0 mode 2mode 1-10-15-20-250.0 0.1 0.3 0.5 r e s pon s e , β ’’ β c =β’+iβ’’ forwardmodesbackwardmodes (-) χ=k>0β c =β’+iβ’’ -0.5 -0.3 -0.1 0.0response, β’ r e s pon s e , β ’’ AC mode 2 - mode 1 - - - - BD t/T Figure 5.11:
Critical beats of free cilia under curvature control without basal compliance. A, C.
Space of critical solutions for free ends. The value of | Γ | − is plotted as a function of β for a fixed value of χ . Divergences (red dots) correspond to critical modes, which are numbered by | β n | < | β n +1 | . The region oforward modes is in A, and that of backward modes in C (these regions are symmetric to each other). B , D.
Time traces of the first two modes, which in A show forward wave propagation (thus swimming to the left),and backwards traveling waves in D. Arrows denote direction of wave propagation.The parameters used werethe typical ones of
Chlamydomonas cilium: κ = 400 pN · µ m , k = 15 · pN /µ m , ξ n = 0 . · s /µ m , L = 12 µ m, ω c = 2 π · · s − , a = 0 . µ m and k = 0. At the critical point Ω = Ω c , the beat patterns exhibit only one harmonic with critical frequency ω c . Inthis case the small amplitude beats are characterized by linearization of Eq. 3.31 together with the linearresponse of Eq. 5.13, which using σ = iω c results in i ¯ ωψ = − .... ψ + ¯ χ ¨ ψ + ¯ β c ... ψ , (5.14)0 CHAPTER 5. NON-LINEAR DYNAMICS OF THE CILIA BEAT where we have defined the dimensionless curvature response at the critical point as ¯ β c = a Lκ β c , and barsas before denote dimensionless quantities. Provided boundary conditions, obtaining the critical modes is aneigenvalue problem. In this case the discrete set of eigenvalues are given by the condition β c = β n with n = 1 , , . . . (5.15)We now use the condition | β n | < | β n +1 | to label the modes, and drop bars as all quantities are dimensionless.Note that there is a fundamental difference between Eq. 5.7, characteristic of sliding control; and Eq. 5.14,characteristic of curvature control. The former is symmetric with respect to the change s → − s , whereas thelatter is not. Thus, motor regulation by curvature explicitly takes into account the polarity of the motor-doublet interaction. Importantly, Eq. 5.14 does remain unchanged under a change of s → − s and β → − β .This means that there will be pairs of modes with opposite directions of wave propagation corresponding tovalues of β c with opposite signs. In Fig. 5.11 A and C we show two regions of the space of critical modes ofa freely swimming cilium regulated by curvature, and see that this is indeed the case. The first two modescorresponding to forward traveling waves appear in Fig. 5.11 B, while those corresponding to backwardwaves (which we note 1 − and 2 − ) are shown in Fig. 5.11 D. These modes are identical aside from a differentdirection of wave propagation. w a v e v e l o c i t y , A B m o t o r r e s pon s e , β mode 1mode 1 - β’’β’ -1 -10-50510 10 -1 -4-2024 mode 1mode 1 - normalized basal stiffness, normalized basal stiffness, Figure 5.12:
Depdence of modes with normalized basal stiffness. A.
The eigenvalues of first forwardand backward modes show little dependence on the basal compliance ¯ k = a Lk /κ . In particular, they areboth directional. B. The wave velocity decreases as the basal stiffness decreases, but always maintaining itssign unaffected. Parameters as those in Fig. 5.11, which correspond to normalized length
L/(cid:96) ≈ . k ≈ χ = 0), which means that under curvature control there are no standing modes: all modesare traveling. In Fig. 5.12 we see how the forward and backward first modes (1 and 1 − ) depend on the basalcompliance. The eigenvalues show little change. Furthermore, also the wave velocity remains little affectedas the modes exhibit only a small change (see Appendix B). This is in contrast with the role of the basalcompliance under sliding control.In section 3.4 we saw that not only a basal compliance was necessary to produce wave propagation infreely swimming cilia under sliding control, but also a minimum length (see Fig. 5.3). In Fig. 5.13 we showthat this is not the case under curvature control. While the eigenvalues of the modes depend strongly on thelength of the cilium (see Fig. 5.13 A), wave propagation can occur for short and long cilia, see Fig. 5.13 B. Integrating in time and space the dynamical system posed by Eqs. 3.26-3.31 together with Eq. 4.11 forthe motor force we obtain non-linear beat patterns under curvature control. The numerical scheme usedwas the same as for sliding control. The control parameter α was adjusted to the supercritical regime,and α was regulated so that the amplitude of the beats was large. Finally, since (unlike sliding control)curvature control can produce wave propagation for short cilia, we restricted ourselves to the regime relevantfor Chlamydomonas in which
L/(cid:96) ≈ − , and thus show wave propagation in the .2. BENDING WAVES UNDER CURVATURE CONTROL w a v e s peed , length,length, Bull Sperm m o t o r r e s pon s e , β mode 1mode 1 - A β’’β’ mode 1mode 1 - B Chlamydomonas
Figure 5.13:
Scaling of modes with cilium length. A.
Scaling of the real ( β (cid:48) , solid lines) and imaginary( β (cid:48)(cid:48) , dashed lines) parts of the eigenvalues for the first and second modes in Fig. 5.11. The real part vanishesfor short lengths. B. Wave velocity vs length of the cilium, arrows note typical values for
Chlamydomonas and
Bull Sperm . Unlike in sliding control, short and long cilia allow for strong wave propagation. Parametersas in Fig. 5.11. Frequency, f (Hz) C wave propagationswimming direction f f f A wave propagationswimming direction B t/T non-linear theory (numerical) non-linear theory (numerical) Figure 5.14:
Non-linear beat pattern of a free cilium regulated by curvature. A, B.
Time tracesof swimming cilia with mode 1 (in A) and mode 1 − in B dominating (see Fig. 5.2). In A the wave travelsforward and the cilium to the left, vice-versa in B. C. The power spectrum of the angle at the mid-point ofthe cilium for the trajectory in A has a fundamental frequency of f = 29 . − . Parameters in A as those inFig. 5.11, with α = 700 nN, α = 5000 nN · µ m and τ = 0 . α → − α and α → − α .forward (for A) and backward (for B) directions. The difference in the motor model to activate one or theother is the following change of sign: α → − α and α → − α . In each case there is strong propulsion inone beat, unlike for the case in Fig. 5.5. This is because the large amplitude of the beat is roughly constantalong the arc-length.To compare this non-linear beat patterns with the critical modes we again use kymographs. Since mode1 − is the mirror symmetric of mode 1, we just focus on the latter. In Fig. 5.15 the kymographs of the angle ψ ( s, t ) show that the waves propagate from base to tip steadily. The amplitude is homogeneous along thelength of the cilium, with a small dip around the midpoint in agreement with what was shown in Fig. 2.3for the beat of Chlamydomonas . There is good agreement between the numerical solution in Fig. 5.15 Bcorresponding to the non-linear beat patterns with the critical solution in Fig. 5.15 A obtained analyticallyin frequency domain.
We have seen that boundary conditions play a crucial role in the beat patterns produced by sliding control,being capable of reverting the direction of wave propagation. Under curvature control the direction of wavepropagation is set by the motors, and preserved with different boundary conditions.In the top panels of Fig. 5.16 A and B we have the non-linear beat patterns of a freely pivoting andclamped cilium with motors regulated by curvature. In both cases the wave propagates backwards, and theamplitude increases along the arc-length. The difference between the patterns is minor, thus unlike undersliding control changing the pivoting stiffness has a small effect on wave propagation. The same is true forthe basal stiffness, which only has a minor effect on the beat patterns under curvature control (see Fig. B.2).These beat patterns are remarkably similar to those of
Bull Sperm , however they correspond to short cilia:2
CHAPTER 5. NON-LINEAR DYNAMICS OF THE CILIA BEAT while for the top panels of A and B cilia of L = 12 . µ m were used, the typical length of the Bull Sperm cilium is L = 52 . µ m. We thus studied the beat patterns of long cilia under curvature control for clampedand freely pivoting boundary conditions, see bottom panels of Fig. 5.16 A and B. As in the case of shortcilia, the wave propagation is forward for both boundary conditions. The wave number for these long ciliais around three, which is larger than that observed for Bull Sperm .To conclude, the effect of boundary conditions on the beat pattern on a curvature control mechanism aresmaller than on a sliding control mechanism. To obtain wave-reversal under curvature control a change inthe microscopic details of the motors is necessary, changes at the boundaries do not suffice. Loosely speaking,the reason is that the fundamental modes of curvature control are directional, while those of sliding controlare standing and only become directional when a boundary asymmetry is included. • A cilium in which motors are controlled by sliding has the following requirements for producing wavepropagation: a basal asymmetry, such as a basal compliance; and a length larger than the characteristicdynamic length (cid:96) . In the absence of these conditions the cilium shows standing waves. • The direction of wave propagation of the first sliding control mode is determined by the boundaryconditions. These conclusions are also valid for supercritical beats, analyzed numerically in the timedomain. • Contrary to sliding control, when motors are regulated by curvature wave propagation exists in theabsence of a basal asymmetry and for lengths of the cilium short compared to (cid:96) . They are thus notsubjected to the requirements of sliding control. • The direction of wave propagation in curvature control is determined by the motor response, and themain effect of changing the length of the cilium is changing the wave-number of the beat. Theseconclusions are valid for supercritical beats, analyzed numerically in the time domain.Figure 5.15:
Tangent angle of linear and non-linear solution. A.
Kymograph of the tangent angle ofthe analytical critical solution over three periods for mode 1. Waves propagate forward (base to tip) and theamplitude is roughly constant along arc-length with a dip around the mid-point. Height normalized to matchthat of the non-linear solution. B. Same as A but for the non-linear solution which was obtained numerically.Frequency, wave profile and amplitude are in good agreement with the analytical linear. Parameters as inFig. 5.14. .3. CONCLUSIONS B Clamped base (non-linear) A long cilium, short cilium,long cilium, t/T Freely pivoting base (non-linear)short cilium,
Figure 5.16:
Beat patterns with different boundary conditions. A.
Beat patterns of the forward andbackward modes for a freely pivoting base. B. Forward and backward beat patterns for a clamped base.Parameters as in Fig. 5.11, besides: in A and B top α = 500 nN, α = 2000 nN · µ m and τ = 0 .
03 s; in aand B bottom α = 180 nN, α = 20 nN · µ m , and τ = 0 .
03 s. The emerging beat frequencies in A and Btop to bottom are: f = 199 . − , f = 248 . − , f = 132 . − and f = 132 . − .4 CHAPTER 5. NON-LINEAR DYNAMICS OF THE CILIA BEAT hapter 6
Motor regulation in asymmetric beats
The beat of
Chlamydomonas cilia is intrinsically asymmetric. In this chapter we explore the consequencesof this asymmetry in motor regulation. We develop a theory of asymmetric beats, and show that due tothe asymmetry normal forces can regulate the beat. We discuss the asymmetric beat patterns activated bysliding, curvature and normal force control. We end the chapter by comparing the beats patterns obtainedfrom the asymmetric theory with those tracked of wild-type and mbo2
Chlamydomonas cilia.
So far we have discussed beat patterns that are symmetric, as the one depicted in Fig. 6.1 A. Symmetricbeats have a flat mean shape (green line). The beat of
Chlamydomonas cilia however is asymmetric and thushas an average curvature, as seen in Fig. 6.1 B.If the beat is planar, the swimming path of an isolated asymmetric cilium is circular. This is indeedthe case for the
Chlamydomonas cilium swimming near a surface, as seen in Fig. 2.1. Furthermore, thesame is also true for single flagellates such as
Bull Sperm [23]. For non-planar asymmetric beats the pathis a spiral [25]. Thus the asymmetry is of fundamental importance in determining the swimming path of amicroswimmer. We now provide a physical description for asymmetric ciliary beats.Figure 6.1:
Symmetric and asymmetric planar beats. A.
Symmetric beat patterns such as the one of
Bull Sperm have a flat mean shape (green line). B. Asymmetric beats such as that of the
Chlamydomonas cilium show a curved mean shape (green line).We study the physics of asymmetrically beating cilia by describing the small amplitude dynamics of thecilium around a mean shape. To do this we perform an expansion of the dynamic equations of a beatingcilium around a static shape. That is, consider the dynamics of the beating cilium described by ψ ( s, t ) = ψ ( s ) + ψ ( s )e iωt + ψ ∗ ( s )e − iωt , (6.1)where ψ is the static mode and ψ the fundamental mode. Analogous expressions hold for other mechanicalquantities (tension, sliding force, etc.). By introducing this into the dynamic equations of the beating ciliumand expanding in powers of ψ we obtain a hierarchy of equations. Using the static component of this556 CHAPTER 6. MOTOR REGULATION IN ASYMMETRIC BEATS hierarchy allows us to calculate the mean shape. With the static mode at hand, we can then solve theequation for the fundamental mode ψ .The equations of the zeroth mode correspond to a static force balance, as derived in section 3.4. Theseequations show that the average curvature of the cilium ˙ ψ is determined by the static contribution of thesliding force f . In the limit of small amplitude ψ we can write the dynamics of the fundamental Fouriermode if the average shape is given: iωξ n ψ = − .... ψ − a ¨ f + ˙ ψ ˙ τ + ¨ ψ τ + ξ n ξ t ˙ ψ ( κ ˙ ψ ¨ ψ + a ˙ ψ f + ˙ τ ) ,ξ n ξ t ¨ τ − ˙ ψ τ = − ˙ ψ ( κ ... ψ + a ˙ f ) − ξ n ξ t ∂ s [ ˙ ψ ( κ ¨ ψ + a f )] . (6.2)This pair of equations is the generalization of the equations for the symmetric beat. In them, the fundamentalmode is coupled to the static mode. Since the force balance of the static mode has been used, the couplingonly appears through the mean curvature ˙ ψ and its derivatives. In the case in which the static modevanishes ( ˙ ψ = 0) we recover Eq. 1.1, and the tension also vanishes. However, for asymmetric beats allterms may be equally relevant. Solutions to the system above are obtained using the sliding force balance χ ∆ , = (cid:82) s f ( s (cid:48) )d s (cid:48) and the boundary conditions of Eq. 3.30. The static mode of the
Chlamydomonas beat corresponds within a good approximation to a circular arc (seeFigs. 2.3 A and 2.4). This motivates studying asymmetric beats in which the average curvature is constant.The balance of static forces states that κ ˙ ψ = a F (see section 3.4). Thus to obtain a circular arc withconstant curvature ˙ ψ = C , the static force must be constant along the length of the cilium with magnitude F = κC /a . This implies that the sliding force density is given by f = δ ( s − L ) κC /a , (6.3)which corresponds to an accumulation of the static sliding force at the distal end of the cilium, as alreadydescribed in section 2.2. The static mode of the basal sliding corresponding to this shape with constant staticcurvature is ∆ , = κC / ( k a ), and the sliding displacement along the cilium is ∆ ( s ) = κC / ( k a )+ aC s .For beats in which the static shape corresponds to a circular arc, the linear dynamic equations become. iωξ n ψ = − κ .... ψ − a ¨ f + (cid:18) ξ n ξ t (cid:19) C ˙ τ + ξC ( κ ¨ ψ + a f ) ,ξ n ξ t ¨ τ − C τ = − (cid:18) ξ n ξ t (cid:19) C ( κ ... ψ + a ˙ f ) . (6.4)These equations have constant coefficients, and as such can be analytically solved if boundary conditionsand an additional equation for the motor force are provided. Just like in the symmetric case, in asymmetric beats the motor force responds to the strains and stresseswithin the cilium (see chapter 4). There is, however, a fundamental difference. Due to the asymmetry, thenormal force f ⊥ has a non-vanishing fundamental mode, which is given by f ⊥ , = ˙ ψ ( F + κ ˙ ψ /a ) . (6.5)Note that in the symmetric case ˙ ψ = 0 this mode vanishes, and the normal force only has modes of ordertwo and higher.This fundamental mode of the normal force in asymmetric beats allows that motors are dynamicallyregulated to linear order by normal stress. The most general form of the linear motor force response is thus f m , = λ ( ω )∆ + β ( ω ) ˙ ψ + γ ( ω ) f ⊥ , , (6.6) .3. UNSTABLE MODES OF A CIRCULAR CILIUM f = ( λ (0) − k )∆ + β (0) ˙ ψ + γ (0) f ⊥ , + δ ( s − L ) κC /a , (6.7)which also incorporates the role of cross-linkers of stiffness k , and where the last term corresponds to thestatic force at the tip responsible for creating the asymmetry. So far this motor description is completelygeneral. We now discuss the conditions under which the static curvature is constant. We have seen that when the static mode has a constant curvature, the static sliding force accumulates atthe tip. In this scenario all terms besides the last one in Eq. 6.7 vanish. Since in sliding control motorsare regulated by sliding velocity, the linear response λ ( ω ) vanishes at zero frequency, that is λ (0) = 0.The same is true for curvature control. Thus in mechanisms in which the motor force is regulated by thetime-derivatives of the curvature, we will have β (0) = 0. Note that such a mechanism is different fromwhat is usually referred to as curvature control, which is a delayed response of the motor force to curvature[57, 14, 71]. The same applies to normal force control: when the motor force depends on time-derivatives ofthe normal force, we have that γ (0) = 0. Under these conditions the static force has a term correspondingto the cross-linkers, and one which accumulates at the tip.The cross-linkers with stiffness k will deviate the static mode away from a circular arc. To estimate howlarge their effect is, we compare the tip accumulated force F = κC a with the total static sliding force (cid:82) L k ∆ . Dividing these two terms we obtain (cid:82) L k ∆ ( s )d sF = kLk + L (cid:96) , (6.8)where (cid:96) is the characteristic length associated to cross-linkers (see Eq. 3.36). In the case in which (cid:96) (cid:29) L andfor large values of the basal compliance k (cid:29) kL we have that the effect of cross-linkers can be neglected. Inconclusion, for motor models in which the motor force depends on the time derivatives of sliding, curvature,or normal forces, and where the role of cross-linkers is small, we can have beats around a circular arc. Before studying the critical modes of an asymmetric cilium, it is instructive to study the stability of per-turbations with a fixed wave-length λ = 1 /q . To do so we interpret the cilium as an infinite elastic mediumwhich can become unstable due to the action of molecular motors. A perturbation of fixed wavelength willevolve as ψ ( s, t ) = e σt − iqs , (6.9)where σ = τ − + iω with τ − the growth rate and ω the frequency at which the perturbation evolves.Analogous equations hold for the tension, normal force and basal sliding. Introducing this perturbation inthe dynamic equations provides a dispersion relation σ ( q ), which determines the stability of the differentwave-lengths 1 /q .We now focus on the case of constant static curvature C with the general motor model introduced inthe previous section. After replacing the perturbation in the dynamic equations, we obtain the followingdispersion relation in implicit form: a χ ( σ ) − ia qβ ( σ ) − iκqC γ ( σ ) = − κq − ξ n (cid:18) − i C γ ( σ ) q (cid:19) Aσ , (6.10)8
CHAPTER 6. MOTOR REGULATION IN ASYMMETRIC BEATS which has units of a force balance. Here χ ( σ ) = k + σξ i − λ ( σ ) is the net sliding compliance, and A is thefunction A ( q, K ) = q + ξ t C /ξ n ( q − C ) . (6.11)Provided the dependence of the motor response on σ we can study the dispersion relation σ ( q ).As an example, consider the case in which motors are regulated by curvature and so γ = 0 and λ = 0.For simplicity we study the symmetric case C = 0 with the motor model introduced in Eq. 4.11, whichcorresponds to β ( σ ) = α / (1 + τ σ ). In Fig. 6.2 A we plot the dispersion relation for positive values of thecontrol parameter Ω = α . As one can see at the critical value α c > q c > ω c >
0, see also Fig. 6.2 B. Simultaneously, a mode − q c becomes unstable at a negative frequency − ω c . Both these modes correspond to forward traveling waves.For negative values of the control parameter α , modes can also become unstable. Indeed, at α = − α c the mode q c > − ω c and the mode q c < ω c . Both modes correspond in this case to backwards traveling waves. We can conclude that theinstabilities result in traveling waves whose direction of propagation is determined by the value of α , thusby motor model details.This is different from the behavior for a sliding control mechanism, in which β = 0 and γ = 0. Insliding control Eq. 6.10 is symmetric with respect to q , which indicates that for positive and negative q thefrequencies will have the same sign. Thus at the critical point waves traveling in opposite directions appear,which result in a standing mode. These conclusions are in agreement with the conclusions of the previouschapter for the critical beats of a cilium of finite length with free ends as boundary conditions. wave vector q wave vector qq c -q c q c -q c r e l a x a t i on r a t e / τ f r equen cy ω ω c -ω c A B
Figure 6.2:
Dispersion relation for curvature control. A.
Relaxation rate of a fixed wave-lengthperturbation for three values of the control parameter. At the critical point two modes become unstable. B. Characteristic frequency of the modes. Since q c has frequency ω c and − q c has frequency − ω c , both modeshave the same direction of propagation. The values of the control parameter used are { α c / , α c , α c / } (inred, green and blue respectively), and time is in units of τ .To analyze the dispersion relation the dependence of the motor model on σ is necessary. However atthe critical point Ω = Ω c the behavior of the system is generic. When Ω = Ω c the relaxation time ofa critical wave-length diverges, and thus σ = iω c with ω c the critical frequency. Using this condition inEq. 6.10 we obtain that the critical values of the response coefficients satisfy certain conditions for eachmotor regulation mechanism. In particular, for sliding control where β = 0 and γ = 0, we obtain that thereal and imaginary part of χ have to be negative. For curvature control we have that the real part of β is positive and the imaginary part negative for forward traveling waves, the opposite is true for backwardtraveling waves. For normal force control the imaginary part of γ has to be positive for forward and negativefor backward traveling waves. The sign of the real part depends on motor parameters. Importantly, whilefor sliding and curvature control these signs are independent of the sign of the static curvature C , undernormal force control the signs flip with the sign of C . This reveals that for the same set of motor parametersunder normal force regulation the sign of the curvature regulates the direction of wave propagation. Theseresults are summarized in table 6.1. .3. UNSTABLE MODES OF A CIRCULAR CILIUM χ (cid:48) , χ (cid:48)(cid:48) β (cid:48) , β (cid:48)(cid:48) γ (cid:48) , γ (cid:48)(cid:48) sliding control ↔ − , − , , → + , + + , − , → ( C <
0) + , + 0 , ∓ , +Table 6.1: Response coefficients for different regulatory mechanisms.
The signs of the real part(single prime) and imaginary part (double prime) of the response coefficients to sliding ( χ ), curvature ( β )and normal force ( γ ) is given for three regulatory mechanisms: sliding control (with either direction of wavepropagation), curvature control (for forward waves) and normal force control (for forward waves with anegative mean curvature). To obtain critical beat patterns we need to consider a cilium of finite length, and provide boundary conditions.The procedure is the same as the one outlined in Appendix B, but using this time equations 6.4 for thedynamics and equation 6.6 for motor regulation. In Fig. 6.3 we provide some examples of critical asymmetricbeats of cilia with a clamped and pivoting base, with a free tip (free base beats are considered in the nextsection). Typical values of
Chlamydomonas were used for the ciliary parameters (see caption).Under sliding control there is no wave propagation, and no net swimming of the cell is possible. Asalready shown in the previous chapter, this is because wave propagation under sliding control only occurs forcilia long compared to the characteristic length (cid:96) (see section 5.1.1), but the Chlamydomonas cilium satisfies L ∼ (cid:96) . Curvature control produces strong wave propagation both for clamped and pivoting cilia. Finally,normal force control also produces strong wave propagation, revealing this new mechanism as a candidatefor regulation of the beat of asymmetric cilia.Figure 6.3: Examples of critical asymmetric beat patterns. A.
Beat patterns of a clamped cilium undersliding, curvature, and normal force control. Sliding control produces standing waves, curvature and normalforce control produce forward traveling waves. B. Same as A but for a pivoting base boundary conditions,as corresponds to the intact
Chlamydomonas cilium. Arrows denote direction of wave propagation. Theparameters used for the cilium were those typical of
Chlamydomonas used in the previous chapter (as inFig. 5.11) with a static curvature C = − . µ m − . The resulting response coefficients preserved the signconvention of table 6.1, with γ (cid:48) < CHAPTER 6. MOTOR REGULATION IN ASYMMETRIC BEATS
Which motor regulatory mechanism is responsible for producing the bending waves observed in the beat of
Chlamydomonas cilia? To answer this question we explored the three regions of parameter space in table 6.1for free end boundary conditions (no external torques or forces, see Fig. 3.4). We defined the mean squaredisplacement distance R ( ψ the , ψ exp ) between the theoretically obtained critical mode and the experimentaldata, and found its minimum value in the corresponding region of parameter space. The result of a typicalfit is shown in Fig. 6.4. Sliding control Curvature control Normal force control0.0 0.2 0.4 0.6 0.8 1.0-1.0-0.50.00.51.0 arc-length, s/L R e (cid:31) Ψ (cid:29) & I m (cid:31) Ψ (cid:29) R (cid:28) (cid:27) R (cid:28) (cid:27) R (cid:28) (cid:27) A B s/L R e (cid:31) Ψ (cid:29) & I m (cid:31) Ψ (cid:29) s/L R e (cid:31) Ψ (cid:29) & I m (cid:31) Ψ (cid:29) C Sliding control Curvature control Normal force control
E F G
Experiment D t/T Figure 6.4:
Fits of
Chlamydomonas wild-type beat.
Best fits of the real (orange) and imaginary (blue)parts of the first Fourier mode of the tangent angle under sliding control (A), curvature control (B) and normalforce control (C). Below, the corresponding position space tracked data for the first mode (D) followed bythe fits (E to G). Arrows denote direction of wave propagation. The data corresponds to a cilium of length L = 10 . µ m, frequency f = 73 .
05 s − , and mean curvature C = − . µ m − As we can see, both normal force control and curvature control provide good fits, while sliding controldoes not. Sliding control does not produce good fits since the length of the cilium is comparable to thecharacteristic dynamic length (cid:96) , and thus does not allow for wave propagation (see section 5.1.1). Curvaturecontrol was already shown to produce traveling waves of constant amplitude for symmetric beats, this is nowshown to also be true for asymmetric beats and to agree quantitatively with the Chlamydomonas beat .Normal force control produces similar fits to curvature control since, according to Eq. 6.5, normal force isa probe for curvature sensing. Table 6.2 collects average parameters resulting from the fits of 10 differentcilia. The resulting basal stiffness is very large, since taking χ (cid:48) to be due to cross-linkers, we have k (cid:29) χ (cid:48) L .However, the characteristic length defined by χ (cid:48) is short, (cid:96) ≈ µ m. With such strong cross-linkers it wouldbe impossible to bend the Chlamydomonas cilium as much as observed, which suggests the origin of thesliding compliance may be motor response and not cross-linkers.Normal force control and curvature control can both produce beats very similar to that of wild-type,and a priori there is no argument favoring one rather than the other in this asymmetric beat. However,for symmetric beats where C = 0, the dynamic component of the normal force vanishes and normal forcecontrol as described in our planar model cannot be a regulatory mechanism. We thus fitted the symmetricbeating mutant mbo2, where the static curvature is reduced by one order magnitude as compared to wild-type (see section 2.1). In this case the results obtained were similar to the case of wild-type (see Fig. 6.5):sliding control cannot produce wave propagation, while curvature and normal force control are in good .4. BEAT OF FREELY SWIMMING CILIA: EXPERIMENTAL COMPARISON R (%) 46 ± ± ± χ (cid:48) , χ (cid:48)(cid:48) (nN · µ m − ) − ± , − . ± . ± , ± ± , ± β (cid:48) , β (cid:48)(cid:48) (pN) 0 5 ± , − ±
768 0 γ (cid:48) , γ (cid:48)(cid:48) . ± . , . ± . χ (cid:48) , χ (cid:48)(cid:48) (mN · µ m − ) 0 . ± , . ± . ± . , · − ± − . ± . , · − ± · − ∆ , (nm) − ± − ± − ± | ∆ , | (nm) 16 . ± . ±
10 8 ± Average parameters extracted from fits of wild-type cilia.
Curvature controla and normalforce control provide very good fits, unlike sliding control which is not capable of producing wave propagationdue to the shortness of the axoneme (see Fig. 6.4).agreement with the data. There is however an important difference between the fits of mbo2 and wild-typeunder curvature and normal force control. Under curvature control the value of the response coefficient β iswell preserved from wild-type to mbo2, but this is not the case for γ under normal force control (see table6.3). The reason is that in mbo2 the small static curvature results in a small normal force, which requires acorrespondingly larger response coefficient γ to be sensed. While not impossible, such a change in sensitivityis hard to justify. Finally, we note that the inferred basal compliance for mbo2 is several orders of magnitudebelow the value for wild-type.To conclude: a length scale argument suggests that the beat of Chlamydomonas cilia can not be regulatedby sliding, and the existence of wave propagation in symmetric beats suggests that it cannot be regulatedby normal forces. This suggests that curvature is the likely mechanism through which the beat is regulated,in agreement with the good fits obtained.
Sliding control Curvature control Normal force control R (cid:31) (cid:30) R (cid:31) (cid:30) R (cid:31) (cid:30) A B C s/L R e (cid:29) Ψ (cid:27) & I m (cid:29) Ψ (cid:27) s/L R e (cid:29) Ψ (cid:27) & I m (cid:29) Ψ (cid:27) s/L R e (cid:29) Ψ (cid:27) & I m (cid:29) Ψ (cid:27) E F G
Experiment D t/T Figure 6.5:
Fits of
Chlamydomonas mbo2 beat.
Best fits of the real (orange) and imaginary (blue) partsof the first Fourier mode of the tangent angle under sliding control (A), curvature control (B) and normalforce control (C). Below, the corresponding position space tracked data for the first mode (D) followed bythe fits (E to G). The data corresponds to a cilium of length L = 9 . µ m, frequency f = 38 .
00 s − , andmean curvature C = − . µ m − .2 CHAPTER 6. MOTOR REGULATION IN ASYMMETRIC BEATS
Sliding control Curvature control Normal Force control R (%) 70 ± ± ± χ (cid:48) , χ (cid:48)(cid:48) (nN · µ m − ) − ± , − . ± . ± , ± ± , ± β (cid:48) , β (cid:48)(cid:48) (pN) 0 400 ± , − ±
600 0 γ (cid:48) , γ (cid:48)(cid:48) . ± . , ± χ (cid:48) , χ (cid:48)(cid:48) (nN · µ m − ) 40 ± , . ± .
03 14 ± , ± ± , ± , (nm) − ± − ± − ± | ∆ , | (nm) 40 ±
10 40 ±
20 50 ± Average parameters extracted from fits of mbo2 cilia.
Curvature and normal force controlprovide very good fits, but the values of γ in normal force control are very spread and different from thosein the wild-type fits. • We developed a theory for the asymmetric beat of cilia. In contrast to symmetric beats, the normalforce has a linear dynamic component, which allows for normal force motor regulation. • For cilia short compared to the characteristic dynamic length (cid:96) , curvature control and normal forcecontrol produce wave propagation for clamped, pivoting and free end boundary conditions. Due to theshort length, in all these cases sliding control produces standing waves. • Curvature and normal force control produce asymmetric beat patterns as those observed in isolatedwild-type cilia. With similar motor parameters, curvature control also produces beats as those of thesymmetric mbo2 mutant. hapter 7
Conclusions and future work
Cilia are highly conserved structures involved in many cellular processes. Much is known about their struc-ture, and their main constitutents are well identified. Yet, how these components self-organize in order toproduce an orchestrated beat has remained a challenging question for the past fifty years. Today, it is widelybelieved that the response of dynein motors to the stresses and strains within the cilium are the key tounravel its beat pattern. Yet what the differences are between alternative regulatory mechanisms is poorlyunderstood. In this thesis we characterized the different beat patterns produced by three motor regulatorymechanisms, sliding, curvature, and normal force control; and compared them to experimental data.
In this thesis we have used numerical and analytical tools in order to analyze different static and dynamicpatterns of cilia, and compared them to experiments performed by our collaborators. Below we summarizethe main findings. • In chapter 3 we developed a planar theory for beating cilia by balancing hydrodynamic and mechanicalforces. Importantly, we incorporated sliding and normal cross-linkers at the bulk of the cilium, as wellas at the base. We then showed that basal cross-linkers are necessary to produce bending, and thatthose at the bulk define a fundamental length-scale beyond which the cilium is straight. • In chapter 4 we introduced the collective behavior of molecular motors, and showed that they arecapable of producing dynamic instabilities such as oscillations. We then proposed a model by whichmotors can sense sliding, curvature and normal forces. In the static regime sensitivity to curvatureand normal forces gives raise to circular arcs, in agreement with experimental evidence in disintegratedcilia. • In chapter 5 we characterized the symmetric beat patterns corresponding to motor regulation by slidingand curvature. We showed that sliding control requires an asymmetry at the boundaries as well as aminimal length to produce wave propagation. This is in contrast to curvature control, which is capableof producing wave propagation in both directions irrespective of boundary conditions and length. Wealso characterized the critical beats under several conditions, and obtained non-linear beat patterns bynumerically solving the dynamic equations. • In chapter 6 we showed that asymmetrically beating cilia, such as those in
Chlamydomonas , can beregulated via normal forces. We indeed showed that the resulting beat patterns obtained by curvatureand normal force control are in good agreement with those measured for wild-type cilia. Further-more, with similar parameters curvature control also produced beat patterns analogous to those of thesymmetric mbo2
Chlamydomonas mutant. Because these cilia are very short, sliding control does notproduce wave propagation.To conclude, our work shows that curvature, sliding, and normal forces can all regulate beat patterns.However, important differences exist on the conditions under which any of these mechanisms can operate:634
CHAPTER 7. CONCLUSIONS AND FUTURE WORK sliding requires long cilia and a constraint at one end, and normal force requires an asymmetry. Furthermore,the resulting beat patterns can also be very different. In the case of the wild-type
Chlamydomonas cilia inparticular, we have good evidence that curvature or normal forces can regulate its beat, while sliding cannot.
The results in this thesis suggest new lines of research to gain further insight into the regulatory mechanismsbehind ciliary beat. Perhaps the most prominent unanswered question is the origin of the static asymmetryof the
Chlamydomonas axoneme. While in chapter 4 we suggested a mechanism capable of producing circulararcs for disintegrated axonemes, the intact cilium is a complicated structure whose full understanding requiresa three-dimensional treatment. Given the new light shed here about ciliary beat by the incorporation ofnormal forces and asymmetries, it is a necessary next step to develop a three-dimensional model of the ciliumwhich includes radial and transverse elastic elements which may regulate the static bend.Such a model could also point towards mechanisms by which motors can sense curvature or normal forces.For instance, radial stresses or the three-dimensional extension of the normal force here studied could be themeans by which curvature regulation arises. Furthermore, new imaging techniques could allow to indirectlyobserve the strains conjugate to such stresses. Together with a technique to observe the waves of dyneinactivity, this could provide direct evidence on the mechanism of motor regulation.In this work we have used a coarse grained description of the motor force, which proved rich enough toallow for distinctive selection of beating modes, as well as amplitude and frequency selection. If differentbeating modes are experimentally characterized and then compared to the models here introduce, this couldprovide new information about the motor model. New patterns can be accessed changing experimentalconditions as temperature, ATP, and viscosity; or characterizing mutants other than mbo2. Mutants withlong flagella are particularly interesting, given the important role of the length in determining the beatpattern. ppendix A
Calculus of variations and test ofresistive force theory
In this appendix we complete the calculus of variations outlined in 3.2 and 3.3. We also show how ResistiveForce Theory is a valid model for axoneme-fluid interactions by comparing theoretical predictions of RFTto experimental measurements.
A.1 Variations of mechanical and rayleigh functionals
We begin by giving the full expressions for the curvature of filament A. First note that˙ r A = (cid:16) − a ψ (cid:17) t + ˙ a n ¨ r A = − (cid:16) ˙ a ˙ ψ + a ψ (cid:17) t + (cid:18) (1 − a ψ ) ˙ ψ + ¨ a (cid:19) n . (A.1)The tangent vector of filament A is defined as t A = ∂ r A ∂s A = ˙ r A | ˙ r A | , (A.2)which can be readily calculated using Eqs. A.1. It is then straight-forward to calculate the normal vector n A which is normalized and perpendicular to t A . Once normal and tangent vectors are known, the curvature issimply C A = n A · ∂ t A ∂s A = n A · ˙ t A | ˙ r A | , (A.3)which again can be calculated using Eqs. A.1. Replacing a → − a we obtain the expression of C B , andexpanding to leading (second) order we have Eq. 3.6.To derive the variations of the energy functional, we begin by writing Eq. 3.9 in the alternative form G = (cid:90) L (cid:34) κ ψ + κ (cid:18) ¨ a (cid:19) + k − f m ( s )∆ + k ⊥ a − a ) + Λ2 ( ˙r − (cid:35) d s + k (A.4)for which we have simply used Eq. 3.6. We begin by calculating the variation with respect to the spacing a ,656 APPENDIX A. CALCULUS OF VARIATIONS AND TEST OF RESISTIVE FORCE THEORY which is: (cid:90) L d s δGδa δa = (cid:90) L (cid:26) κ ¨ a δ ¨ a + k ⊥ ( a − a ) δa − f (cid:90) s ˙ ψδa d s (cid:48) (cid:27) d s = (cid:20) κ ¨ a δ ˙ a (cid:21) L − (cid:90) L f d s (cid:90) L ˙ ψδa d s + (cid:90) L (cid:26) − κ ... a δ ˙ a + k ⊥ ( a − a ) δa + ˙ ψδa (cid:90) s f d s (cid:48) (cid:27) d s == (cid:20) κ ¨ a δ ˙ a (cid:21) L − (cid:20) κ ˙ a δa (cid:21) L + (cid:90) L δa (cid:26) κ .... a k ⊥ ( a − a ) − ˙ ψF (cid:27) d s (A.5)Where we have defined the net static sliding force density f = f m − k ∆, the integrated force F = (cid:82) Ls f d s (cid:48) ,we have used Eq. 3.8 for relating sliding to spacing, and in the last equality we have introduced a boundaryterm inside the integral.We now calculate the variation with respect to the position of the center-line r . Before proceeding , wenote that ˙ ψ = n · ˙ t , so that δ ˙ ψ = n · δ ˙ t = n · δ ¨ r . In the first equality we have used that ˙ t · δ n = 0, which comesfrom imposing | n | = 1. After this preamble, we have (cid:90) L d s δGδ r δ r = (cid:90) L (cid:26) κ ˙ ψ n · δ ¨ r − f (cid:90) s a n · δ ¨ r d s (cid:48) + Λ t · δ ˙ r (cid:27) d s = (cid:90) L (cid:110) ( κ ˙ ψ − aF ) n · δ ¨ r + Λ t · δ ˙ r (cid:111) d s = (cid:104) ( κ ˙ ψ − aF ) n · δ ˙ r (cid:105) L + (cid:104)(cid:16) − ( κ ¨ ψ − ˙ aF + af ) n + τ t (cid:17) · δ r (cid:105) L + (cid:90) L ∂ s (cid:110) ( κ ¨ ψ − ˙ aF + af ) n − τ t (cid:111) · δ r d s (A.6)Where we have integrated by parts twice, and defined the tension τ = Λ + κ ˙ ψ − aF ˙ ψ following Eq. 3.14.The variation with respect to the basal sliding ∆ is easy to calculate: δGδ ∆ δ ∆ = ( k ∆ − (cid:90) L f d s ) δ ∆ . (A.7)The variations of the Rayleigh dissipation function given in Eq. 3.20 are calculated in much the sameway, with all terms balancing those of the equilibrium variations. For the basal sliding velocity variationswe have δRδ∂ t ∆ δ∂ t ∆ = ( ξ ∂ t ∆ − F i (0)) δ∂ t ∆ , (A.8)where we have defined the internal friction force F i ( s ) = (cid:82) Ls f i d s (cid:48) , with f i = − ξ i ∂ t ∆ being the internalfriction force density. For the spacing velocity the variation is (cid:90) L d s δRδ∂ t a δ∂ t a = (cid:90) L (cid:110) ξ ⊥ ∂ t a − ˙ ψF i (cid:111) δ ( ∂ t a )d s . (A.9)Finally, for the velocity of the filament the variation gives (cid:90) L d s δRδ∂ t r δ∂ t r = [ − aF i n · δ∂ t ˙ r ] L + [( − ( − ˙ aF i + af i ) n + τ i t ) · δ∂ t r ] L + (cid:90) L (( ξ n nn + ξ t tt ) · ∂ t r + ∂ s { ( − ˙ aF i + af i ) n − τ i t } ) · δ∂ t r d s (A.10)where we have defined τ i = − aF i ˙ ψ as the dissipative component of the tension. For completeness, we also note that δψ = n · δ t .2. PLANE WAVE AND HYDRODYNAMIC TEST A.2 Plane wave and hydrodynamic test
In section 2.1 it was stated that the axonemal beat of the
Chlamydomonas axoneme can be understood asan angular plane wave (see Fig. 2.4), that is ψ ( s, t ) = f rot t + C s + A cos (2 π ( νt − s/λ )) (A.11)where C is the mean curvature, f rot the global rotational frequency , λ is the wavelength, A is the amplitudeand ν the frequency. We now use this simple description to verify the model of axoneme-fluid interactionintroduced in 3.3 (Resistive Force Theory). In RFT the fluid force is given by f fl = − ( nn ξ n + tt ξ t ) · ∂ t r , (A.12)with r ( s, t ) the position of the axoneme over time which relates to the shape ψ ( s, t ) via r ( s, t ) = r ( t ) + R ( s, t ) = r ( t ) + (cid:90) s (cid:18) cos( ψ ( s (cid:48) , t ))sin( ψ ( s (cid:48) , t )) (cid:19) d s (cid:48) , (A.13)with r ( t ) being the trajectory of the basal end, and R ( s, t ) the shape of the axoneme. Note that since˙ r = ˙ R , the shape alone defines the tangent and normal vector fields, independent of the position of the base r . Since in RFT the inertia of the fluid is neglected, the sum of all forces exerted on it must vanish, and wethus have (cid:90) L f fl ( s, t )d s = − (cid:90) L ( nn ξ n + tt ξ t ) · ( ∂ t r ( t ) + ∂ t R ( s, t ))d s = 0 . (A.14)Taking the tangent angle ψ ( s, t ) given by Eq. A.11 the only unknowns on this equation are the componentsof the basal velocity ∂ t r ( t ). We can thus numerically solve this equation in order to obtain the velocity ofthe base at each time-step, and reconstruct the axonemal trajectory. μ m μ m A B C
Figure A.1:
Reconstructed vs tracked trajectories. . Tracked trajectories (in yellow) and reconstructedtrajectories (in black) using the plane wave approximation of the beat (in Eq. A.11) and a RFT modelof the fluid (Eq. A.12). The values used in the panels { A, B, C } are: C = { . , . , . } µ m − , λ = { . , . , . } µ m, L = { , , } µ m, f rot = { . , . , . } s − , ν = { . , . , . } s − , and A = { . , . , . } . Figure adapted from [28].In Fig. A.1 three comparisons of the reconstructed trajectories (in red) to the actual trajectories (inblack) are given. Notice the very good agreement, given that the shape information used corresponds to aplane wave approximation of the beat. This neglects amplitude profile, higher harmonics, and fluctuations.Fig. A.1 thus confirms not only that RFT applies to Chlamydomonas axonemes, but that the plane waveapproximation on the axonemal beat discussed in 2.1 is accurate.8
APPENDIX A. CALCULUS OF VARIATIONS AND TEST OF RESISTIVE FORCE THEORY ppendix B
Critical modes and numericalmethods
In this appendix detail different beating modes referred to in chapter 5. We show the beat patterns, amplitudeand phase profiles of several critical beats, as well as non-linear beats obtained numerically. We also describethe numerical method used to solve the dynamical system, and characterize the dynamical transition.
B.1 Characterization of critical modes
The equation which characterizes the critical beats of symetric cilia regulated via sliding or curvature is iωψ = − .... ψ + χ c ¨ ψ + ¯ β c ... ψ , (B.1)where χ c and β c are the rlinear response coefficients to sliding and curvature respectively, and we are usingthe normalization rules in Eqs. 5.8 to make all quantities dimensionless. The general solution to this linearequation is ˜ ψ ( s ) = (cid:88) α =1 A α exp( k α s ) (B.2)where { A α } are the amplitudes determined by the boundary conditions, and { k α } are the solutions to thecharacteristic equation, which is: iη = − k α + χ c k α + β c k α . (B.3)To obtain the coefficients { A α } we insert Eq. B.2 into the boundary conditions, which gives a linearset of equations in { A α } . This linear system will depend on ω , χ c and β c through the solutions of thecharacteristic polynomial k α ( ω, χ c , β c ). It will also depend on the basal sliding ∆ , which makes the systemappear homogeneous. However, using the sliding force balance χ ∆ = (cid:90) (cid:104) χ c ∆( s ) + β ˙ ψ ( s ) (cid:105) d s (B.4)together with ∆( s ) = ∆ + ψ ( s ) − ψ (0) we can eliminate the basal sliding dependence, so that we havea homogeneous system of four equations. The determinant of the coefficient matrix of this system Γ is afunction of the set of seven parameters { ω, χ (cid:48) , χ (cid:48)(cid:48) , β (cid:48) , β (cid:48)(cid:48) , χ (cid:48) , χ (cid:48)(cid:48) } (where primes and double primes denote realand imaginary part), and it has to be null to obtain non-trivial solutions. Given five of these parameters,the condition Γ = 0 is a complex equation that gives a discrete spectrum of solutions for the other two.For example, in sliding control we typically fix all parameters besides χ (cid:48) and χ (cid:48)(cid:48) , which we determine fromthe condition Γ = 0. In this case we order the modes according to | χ n | < | χ n +1 | . In curvature control it690 APPENDIX B. CRITICAL MODES AND NUMERICAL METHODS is β (cid:48) and β (cid:48)(cid:48) which are determined from the determinant condition, and the ordering of the modes follows | β n | < | β n +1 | .In Fig. B.1 we show the amplitude and phase profile of the first two modes of a freely swimming ciliumwith a small and a high basal compliance. As one can see mode 1 is a directional mode, which shows strongwave propagation from base to tip for high and low basal compliance alike (yellow curves, panes A and Cfor small basal stiffness and B and D for large). This is contrast with mode 2 (red curves), which shows adramatic change as the basal compliance becomes smaller: while for a large value of the basal compliancemode 2 show wave propagation from tip to base (panels B and D), when the basal compliance is small itbecomes a standing mode with no wave propagation (panels A and C). The length in all cases is L/(cid:96) ≈ Bull Sperm ), guaranteeing that wave propagation is possible.
Arc-length, s A m p li t ude | p s | Arc-length, s A m p li t ude | p s | k = 10 -2 k = 10 s P ha s e a r g s P ha s e a r g k = 10 -2 k = 10 mode 1mode 2mode 1mode 2 mode 1mode 2mode 1mode 2 Figure B.1:
Modes 1 and 2 of sliding control for free ends.
As one can readily see, for a largebasal compliance (panels to the right) both modes are directional and have opposing directions of wavepropagation. As the basal compliance vanishes mode 2 looses its directionality and becomes a standingmode, while mode 1 remains directional. Parameters as in Fig. 5.2.The basal compliance has a less dramatic effect in beats controlled by curvature. In Fig. B.2 we comparethe amplitude and phase profiles of the first unstable mode under curvature control for a high (in red) andlow (in blue) basal compliance. As one can see, while the increase in basal compliance makes the amplitudedecrease near the base, the effect is much smaller than for sliding controlled beat patterns. s/L s/L A m p li t ude P ha s e A B
Figure B.2:
Mode 1 of curvature control for free ends.
A large change in the basal stiffness hasa minimal effect on the beat pattern of this mode. Parameters as in Fig. 5.11, with low and high basalcompliances corresponding to k = 0 and k = 10 pN · µ m − .As stated in the main text, all sliding control modes loose directionality when the length of the cilium L becomes comparable to the characteristic length (cid:96) . In Fig. B.3 A, B and C we have a time trace as well asamplitude and phase profiles of mode 1 of a freely beating cilium under sliding control. Although the basalcompliance is high such that all modes can be directional, due to the small length of the cilium L/(cid:96) ≈ .2. NON-LINEAR DYNAMICS OF BEATING CILIUM A m p li t ude A m p li t ude P ha s e P ha s e s/L s/L s/L s/L A B CD E F t/T
Figure B.3:
Beat patterns of long and short cilia. A, B & C.
Beat pattern of a short (
L/(cid:96) ≈ D, E & F.
Beat pattern of a long (
L/(cid:96) ≈
15) freely swimming ciliumregulated by curvature. Wave propagation is strong, just as for short cilia (see Fig. 5.13)
B.2 Non-linear dynamics of beating cilium
The onset of the oscillatory regime in our description of cilia occurs via a Hopf bifurcation. That is, as thecontrol parameter Ω crosses some critical value Ω c , the system starts oscillating with small amplitude at acritical frequency ω c . We characterized the properties of this Hopf bifurcation by numerically integratingthe dynamical system for a range of values of the control parameter Ω = α . As shown in Fig. B.4 A theoscillatory behavior can be understood as a dynamic phase transition. There are two associated scaling lawsnear the critical point: the amplitude of the beat increases as ( α − α c ) / , in Fig. B.4 B; and it decreasesas α − / , in panel Fig. B.4 C beatingphase control parameter, staticphase 10 -5 -4 -3 -2 -1 control parameter, -2.5 -1.5 -0.5 amplitude parameter, -6 -4 -2 B CA
Figure B.4:
Numerical characterization of bifurcation. A.
For values of the control parameter smallerthan the critical one, the system is stable and there are no oscillations (green area). For values higher thanthe critical, finite amplitude oscillations appear. B. The amplitude increases as the critical parameter growsas a power law. C. The saturation parameter also scales with the amplitude as a power law. Results obtainedby performing a series of simulations for a pivoting cilium regulated via sliding.It was seen in chapter X that modifying the basal stiffness could produce a significant change in thebeat pattern of a clamped cilium. In Fig. B.5 A we analyze how the motor response of the first two modeschanges as a function of the basal stiffness k . The crucial point is that at some critical value of k theirorder flips. Thus the mode which is second at a low basal stiffness, becomes first at a high basal stiffness.As a consequence the mode gets excited. By numerically computing a series of simulations of the non-lineartheory we indeed observed a change in the beat patterns of the cilium, which we represent in Fig. B.5 B andC by the amplitude and frequency of the beat respectively. Interestingly, this change is not abrupt as should2 APPENDIX B. CRITICAL MODES AND NUMERICAL METHODS be expected. The reason is that near the critical point both modes become very similar, although not equal,thus the discrete jump to be expected in all quantities is very much smoothed out.
A B basal stiffness, mode 1mode 2 r e s pon s e , basal stiffness,
345 10 -1 basal stiffness, f r equen cy , H z C mode 1 mode 2 mode 1 -1 mode 2 Figure B.5:
Mode swapping as a function of the basal stiffness. A.
As the basal stiffness changesthe two modes swap their order: for high basal compliance | χ | > | χ | , for low | χ | < | χ | . B. As one or theother modes are gets activated there is a jump in the amplitude of the beat. While this jump is discrete,the two modes are very similar near the cross-over point so the jump appears continuous. C. As the modeschange, so does their characteristic frequency.
B.3 Numerical methods
To solve the dynamical system defined by the ciliary mechanics and the motor dynamics in the time domain,we used a custom made numerical algorithm. It consists of an adaptation of the IMEX finite differencesalgorithm in [89], but including a predictor-corrector iteration loop [2]. We now outline the key steps of thealgorithm.For concreteness, we write down the equations that were solved for the case of a clamped cilium wheremotors were regulated via sliding velocity. The dynamic equations are in this case: ∂ t ψ = ξ − ( − κ .... ψ − a ¨ f + ˙ ψ ˙ τ + τ ¨ ψ ) + ξ − ˙ ψ ( κ ˙ ψ ¨ ψ + a f ˙ ψ + ˙ τ ) (B.5) ξ n ξ t ¨ τ − ˙ ψ τ = − ˙ ψ ( κ ... ψ + a ˙ f ) − ξ n ξ t ∂ s [ ˙ ψ ( κ ¨ ψ + a f )] (B.6) f = f m − k ∆ − ξ i ∂ t ∆ (B.7) ∂ t f m = − τ ( f m − α ∂ t ∆ + α ( ∂ t ∆) ) (B.8) ξ ∂ t ∆ = (cid:90) L f ( s )d s − k ∆ (B.9)with ∆( s ) = ∆ + a ( ψ ( s ) − ψ (0)), and boundary conditions κ ... ψ (0) + a ˙ f (0) − ˙ ψ (0) τ (0) = 0 ; κ ¨ ψ ( L ) + af ( L ) = 0 ψ (0) = 0 ; ˙ ψ ( L ) = 0 κ ˙ ψ (0) ¨ ψ (0) + a ˙ ψ (0) f (0) + ˙ τ (0) = 0 ; τ ( L ) = 0 . (B.10)The meaning of the color scheme will be explained below. The main challenges to numerically integrate theseequations are the following: they have a high order in space, they are non-linear, and they have coupledtime-derivatives.To proceed, we first discretize the angle, tension, and motor force along the arc-length. We also discretizedall the differential operators using a second order finite differences scheme, with side derivatives at theboundaries [89]. To deal with the fourth order spatial derivative we use an IMEX scheme. That is, wetreat the fourth order term implicitly (that is, in the new time step, noted by green) and all the othersexplicitly (in the old time step, noted by blue). Finally, in order to evolve the system in time, we discretizethe time-derivatives with a first order Euler scheme, thus ∂ t ψ = ( ψ − ψ ) / d t with d t the discretization time.There is a draw-back with the previous iteration scheme. In the boundary conditions and tension equa-tions all terms must be evaluated in the new time step. The boundary conditions, which are required to .3. NUMERICAL METHODS ψ , involve the updated tension τ . But according to Eq. B.6 to obtain τ we need ψ .To solve this issue we perform a loop within each time step, such that initially we use the old tension τ inthe boundary conditions, and with it obtain an estimate for the updated angle. We then solve the tensionequation, and obtain an estimate for the updated tension. We then use this estimated tension in the bound-ary conditions to re-estimate the angle, and also re-estimate the tension. This process is iterated until theboundary conditions are satisfied with enough accuracy, and then we move to the next time step.The accuracy of the method was ensured by calculating the residual sliding forces (using Eq. B.9, allquantities evaluated in the same time step) and the differential equation residue (using Eq. B.5, alsoquantities in the same time step). In all cases the relative errors were kept below 10 − , with often muchsmaller errors. The stability of the algorithm changed depending on the region of the parameter space, mainlydetermined by the length of the cilium which strongly affects the sperm number ( L/(cid:96) ) . This parameterwas changed up to three orders of magnitude. The different types of boundary conditions, and the ratioof time-scales defined by the different viscosities was also a limiting factor in the numerical stability of thealgorithm. While several simple calibration methods were used for debugging the code, the most reliable isthe direct comparison of the obtained numerical solutions with the analytical ones at the bifurcation point(see for example Figs. 5.5 and 5.14).4 APPENDIX B. CRITICAL MODES AND NUMERICAL METHODS ibliography [1] Cnrs phototeque. http://phototheque.cnrs.fr/ . Accessed: 2015-01-05.[2]
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