Curvature-sensitive kinesin binding can explain microtubule ring formation and reveals chaotic dynamics in a mathematical model
Simon P Pearce, Matthias Heil, Oliver E Jensen, Gareth W Jones, Andreas Prokop
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Curvature-sensitive kinesin binding can explain microtubule ringformation and reveals chaotic dynamics in a mathematical model
S. P. Pearce · M. Heil · O. E. Jensen · G. W. Jones · A.Prokop the date of receipt and acceptance should be inserted later
Abstract
Microtubules are filamentous tubular protein polymers which are essential for a rangeof cellular behaviour, and are generally straight over micron length scales. However, in some glid-ing assays, where microtubules move over a carpet of molecular motors, individual microtubulescan also form tight arcs or rings, even in the absence of crosslinking proteins. Understanding this phenomenon may provide important explanations for similar highly curved microtubules whichcan be found in nerve cells undergoing neurodegeneration. We propose a model for gliding as-says where the kinesins moving the microtubules over the surface induce ring formation throughdifferential binding, substantiated by recent findings that a mutant version of the motor proteinkinesin applied in solution is able to lock-in microtubule curvature. For certain parameter regimes,our model predicts that both straight and curved microtubules can exist simultaneously as stablesteady-states, as has been seen experimentally. Additionally, unsteady solutions are found, where awave of differential binding propagates down the microtubule as it glides across the surface, whichcan lead to chaotic motion. Whilst this model explains two-dimensional microtubule behaviour inan experimental gliding assay, it has the potential to be adapted to explain pathological curling innerve cells.
The skeleton of cells (cytoskeleton) is essential for cell structure, dynamics and function. It is formed by long filamentous protein polymers of three different classes: actin, intermediate filaments and
S. P. Pearce · M. Heil · O. E. Jensen · G. W. JonesSchool of Mathematics, University of Manchester, UKS. P. Pearce · A. ProkopFaculty of Biology, Medicine and Health, University of Manchester, UK a r X i v : . [ q - b i o . S C ] A ug S. P. Pearce et al. microtubules (MTs). Of these, MTs are the stiffest filaments with important roles in cellular pro-cesses, such as cell motility, division, organisation, adhesion, signalling and intracellular transport.MTs are composed of α - and β - tubulin heterodimers which are bonded in a polar head-to-tail fash-ion to form long chains known as protofilaments; these protofilaments are then assembled into ahelical tube. For a detailed description of how microtubules behave, see for example Hawkins et al.(2010) or Barsegov et al. (2017).Many different proteins bind to MTs, controlling MT behaviours, including their nucleation,(de)- polymerisation, stabilisation, severing, biochemical modification, and crosslinking to eachother or other cellular components (Lawson and Salas, 2013; Prokop, 2013). One particular classare MT-associated motor proteins, which use ATP as an energy source to walk along MTs, eitherto slide them against each other or to use MTs as intracellular highways to transport cargo aroundcells. Two fundamentally different classes of MT-associated motor proteins exist: the various mem-bers of the kinesin family of which most walk towards one end of the MT, and the dynein/dynactincomplex which moves towards the other (Prokop, 2013; Schliwa and Woehlke, 2003).Outside of cells, a powerful in vitro tool to study MT behaviour is a gliding (or motility) assay. In these experiments, motor proteins (typically kinesin-1) are adsorbed onto a solid surface in a dropof solution. When MTs are added, the surface-attached motor proteins attempt to walk along them,causing the MTs to glide over the surface. Typically in these assays, MTs stay relatively straight, aswould be expected from their large persistence length ( to (Howard, 2001)). However,in certain experimental conditions, MTs can form micron-sized rings; such conditions include highMT density or the presence of an air-medium interface (Weiss et al., 1991; Amos and Amos, 1991;Liu et al., 2011; Kawamura et al., 2008; Kabir et al., 2012). Strikingly, these MTs are able to trans-form from straight gliding to a curved circling motion and back again (Liu et al., 2011), showinga dynamic and reversible ability to change curvature, implying that this is not due to permanentdamage or irreversible damage/repair cycles (Schaedel et al., 2015) (see Figure 1).Studying the mechanisms that underlie MT curling has important applications. For example,systems based on MT-kinesin gliding assays have potential uses as lab-on-a-chip medical devices,utilizing the ability to bind only selected proteins to MTs through the choice of specific cargoadapters, leading to advective transport rather than mere diffusion (Bachand et al., 2014; Chaud-huri et al., 2017). These nano-devices need to be robust for potential clinical uses, but the presence of MT rings may disrupt their design.Furthermore, curved MTs as observed in gliding assays are similarly found in cells, particularlyin axons. Axons are the cable-like extensions of nerve cells; their structural backbone is formedby straight, parallel bundles of MTs. However, in the ageing brain or in nerves affected by certainneurodegenerative diseases (e.g. some forms of motor neuron disease), MTs are found to curl up urvature-sensitive kinesin binding can explain microtubule ring formation and reveals chaotic dynamics 3 with similar diameters as observed in gliding assays (Sanchez-Soriano et al., 2009; Voelzmann et al.,2016, 2017).To explain this phenomenon, the model of local axon homeostasis has been put forward (Voelz-mann et al., 2016). It proposes that MTs in axonal environments have a strong tendency to curl uplikely due to high abundance of MTs and MT-associated motor proteins, thus meeting the condi-tions known to cause rings in gliding assays. Various MT-regulating proteins are required to ‘tame’MTs into ordered bundles; functional loss of these regulators increases the risk of MT curling andcould explain neurodegeneration linked to them (Voelzmann et al., 2016). This model represents aparadigm shift for the explanation of certain forms of axon degeneration, by putting the emphasison MTs as the key drivers of axon decay.
0s 15s 30s35s 60s 75s80s 85s 90s
Fig. 1
Time-series of an initially straight MT forming a loop at
35 s , rotating until
75 s before re-straightening, extractedfrom Supplementary Movie 1 of Liu et al. (2011). Another adjacent MT stays in a loop through the whole video. Eachframe is µ m square. Used with permission of Prof. J. Ross. To lend credibility to this model, it is pivotal to identify and validate the mechanisms that canexplain the phenomenon of MT curling. So far, Ziebert et al. (2015) introduced a model to explain
S. P. Pearce et al. the formation of MT rings which suggests that, in the presence of the MT-stabilising drug taxol,each tubulin dimer may exist in two distinct conformations, one slightly shorter than the other. Intheir model, protofilaments are able to switch between these two states; when only some of theprotofilaments are switched this leads to a longitudinally curved MT as an energetically favouredcondition, providing a mechanism to create rings via an internal change to the MT.Here, we explore the complementary possibility that differential binding of external factors canactively contribute to MT curling. Peet et al. (2018) show that MTs which are being bent in a flowchamber normally straighten after the flow is removed, but stay curved in the presence of a non-motile version of kinesin-1. They propose that this non-motile kinesin has a tendency to bind prefer-entially to the convex side of curved MTs and, by doing so, stabilise them in bent confirmation (seeFigure 2); at higher concentrations this behaviour disappears, presumably because oversaturationoccurs so the kinesin binds in equal amounts on all sides of the MT.This behaviour is consistent with findings for other MT-associated proteins, in particular tau(Samsonov et al., 2004) and doublecortin (Bechstedt et al., 2014; Ettinger et al., 2016), which binddifferentially between straight and curved MTs due to conformational changes that happen on the structural scale of the individual tubulin dimer: at a curvature of µ m − the tubulin dimer spacingat the outside of the MT is 2.5% larger compared to that of the inside. Fig. 2
Sketch of the proposed model. A) When the MT is straight, the surface-bound kinesin is equally likely to bindto each side, with no effect on the overall curvature. B) If a MT becomes curved, the likelihood of kinesin bindingfrom each side becomes asymmetric; this asymmetry in the amount of bound kinesin to each side induces curvatureby acting as a lateral reinforcement. As the MT then continues to glide across the array, the newly encountered kinesinwill also preferentially bind to the same side.
Here we present a model based on the hypothesis that curvature-selective binding can occur inMT-kinesin gliding assays; the flexible neck linker of the surface-attached kinesins can extend up to
45 nm from the surface (Palacci et al., 2016), and is therefore long enough to reach the curved sides ofthe MT which are typically held at around
17 nm above the surface (Kerssemakers et al., 2006)). Ourmodel reproduces key behaviours of MTs observed in gliding assays, with a bistable regime wherestraight MTs and MT rings can coexist, and predicts how they can be controlled. Furthermore, we urvature-sensitive kinesin binding can explain microtubule ring formation and reveals chaotic dynamics 5 find unsteady propagating wave solutions and chaotic dynamics within the system, which havenot been previously reported for filaments and may reflect true MT behaviours that have escapedthe attention of experimenters so far. x ( s, t ) , parametrized by itsarclength s measured at time t , which lies within a two-dimensional plane. This is a reasonable ap-proximation for gliding assays, where the MTs remain close to the surface throughout their move-ment. The treatment shown here follows the standard case where the filament has no referencecurvature (see, for example, Audoly, 2015; Ziebert et al., 2015; De Canio et al., 2017).As the individual tubulin dimers are of fixed length, the total arclength is constant and weimpose the inextensibility constraint, x (cid:48) · x (cid:48) = 1 , where a prime denotes differentiation with respect to s . Relative to fixed Cartesian coordinate axes e x , e y , we define the tangent vector as t = x (cid:48) = cos θ e x + sin θ e y , (1)where θ ( s ) measures the angle between t and e x . The normal vector n is then given by the relation t (cid:48) = κ n , which defines the curvature κ = θ (cid:48) .We will assume that the filament has a variable reference curvature (to be specified later), ˜ κ ( s, t ) ,and that the mechanical energy of the MT is a function of the squared deviation of the curvaturefrom the reference curvature, E = 12 (cid:90) L (cid:104) B ( κ − ˜ κ ) + λ (cid:0) x (cid:48) · x (cid:48) − (cid:1)(cid:105) ds. (2)Here λ is a Lagrange multiplier enforcing the inextensibility constraint and B is the bending (flex-ural) modulus. Taking the variational derivative of (2), utilizing δκ = n · δ x (cid:48)(cid:48) , we find δE = (cid:90) L B ( κ − ˜ κ ) δκ + λ ( x (cid:48) · δ x (cid:48) ) ds = (cid:2) B ( κ − ˜ κ ) n · δ x (cid:48) (cid:3) L + [( λ x (cid:48) − B (( κ − ˜ κ ) n ) (cid:48) ) · δ x ] L + (cid:90) L (( B ( κ − ˜ κ ) n ) (cid:48) − ( λ x (cid:48) )) (cid:48) · δ x ds (3)where δ represents the variation of a quantity and n (cid:48) = − κ t . The elastic force density, f , acting onan element of the MT is therefore given by f = − δEδ x = − B (( κ − ˜ κ ) n ) (cid:48)(cid:48) + ( λ x (cid:48) ) (cid:48) , (4)subject to x (cid:48) · x (cid:48) = 1 . Here f has units of force per unit length, and may be considered as thecircumferentially averaged surface stress (Lindner and Shelley, 2015). The MT is immersed within S. P. Pearce et al. a viscous medium at very low Reynolds number, and so we use resistive force theory, a Stokes-flowapproximation which takes advantage of the aspect ratio being small, (cid:15) = h/L (cid:28) , where h isthe MT diameter (
25 nm ). This is the simplest approximation of slender-body theory, and gives thelocal dynamic relation (Lindner and Shelley, 2015), v = c πµ P · ( f + f ext ) (5)where v = ∂ x ∂t = ˙ x is the velocity of material points, f ext is the external force per unit lengthacting along the MT, µ is the fluid viscosity, and the tensor P ≡ ( I + ( ξ − tt ) = nn + ξ tt reflectsthe anisotropic drag on the filament due to its shape. The constant c = ln(2 (cid:15) − ) is a free-spaceslender body ratio (Becker and Shelley, 2001), and we also use the free-space approximation ξ = 2 for an idealised slender filament. Both of these neglect the effect of the nearby surface; a morerefined approach is likely to predict higher values of overall drag. There is nothing to prevent self-intersection of the filament within this model; if self-intersection does occur the filament is thereforeassumed to go out of plane, crossing over or under itself. The equations of motion are thereforegiven by ˙ x = c πµ P · (cid:0) − B (( κ − ˜ κ ) n ) (cid:48)(cid:48) + ( λ x (cid:48) ) (cid:48) + f ext (cid:1) , x (cid:48) · x (cid:48) = 1 , ≤ s ≤ L. (6)Due to the constraint, this is a ninth-order (in s ) system of differential algebraic equations withindex 3, so six spatial boundary conditions are required to fully specify the system. It is easier towork in terms of intrinsic coordinates which move with the filament, so we take the derivative of(6) with respect to s , and use ˙ x (cid:48) = ˙ θ n to give two equations in the normal and tangential directionsrespectively, ˙ θ = c πµ (cid:0) − ( BK (cid:48)(cid:48) − τ κ ) (cid:48) + ξ ( τ (cid:48) + BκK (cid:48) + f m ) κ (cid:1) (7a) c πµ (cid:0) ( BK (cid:48)(cid:48) − τ κ ) κ + ξ ( τ (cid:48) + BκK (cid:48) + f m ) (cid:48) (cid:1) (7b)where K = κ − ˜ κ is the excess curvature and τ = λ + BκK is a generalised tension. Here, as inZiebert et al. (2015), we have assumed that the external force comes solely from the action of thekinesin motors, which force the microtubule along its tangent and so f ext = f m t , where f m is aconstant. Equation (6) nevertheless allows the MT to move normal to its centreline. The position x may then be found from the filament angle θ by integrating (1). The natural boundary conditionsfor a free end of the MT come from the variational principle (3), and are given by K = 0 , K (cid:48) = 0 , τ = 0 . (8)For a fixed end we have x = x , and therefore we set ˙ x = 0 in (6) which leads to the force conditions, τ κ − BK (cid:48)(cid:48) = 0 , τ (cid:48) + BκK (cid:48) + f m = 0 , (9)which are supplemented with either K = 0 for a freely rotating pinned end or θ = θ for a clampedend. urvature-sensitive kinesin binding can explain microtubule ring formation and reveals chaotic dynamics 7 For a straight filament with κ = ˜ κ = 0 , (6) gives v = | ˙ x | = ξ cf m πµ , (10)which allows us to estimate an appropriate tangential force being applied from measurements ofthe MT velocity. Typical velocities for gliding assays are . µ m s − to µ m s − , with differencesdue to factors such as viscosity, ATP concentration, temperature and salt concentrations. While itwould be reasonable to expect that the speed of the MT would be proportional to the amount ofkinesin attached to the MT (and hence the surface kinesin density), as equation (10) implies, this isnot seen in experiments, which show that the velocity is constant for kinesin densities of µ m − to
10 000 µ m − (Howard et al., 1989). We therefore will use v to set an appropriate choice of f m .2.2 Kinesin BindingWe now turn to the binding of the surface-bound kinesin to the MT, which we will model as acontinuous field, assuming that the concentration is sufficiently high for this to be valid. Here we focus on the ‘sides’ of the filament, arbitrarily denoting them with + and − with associated boundconcentrations c + and c − ; we do not model the protofilaments which are directly above the surfaceas we assume that they will not affect the reference curvature. The proteins bind and unbind to thefilament according to standard protein binding kinetics, ˙ c ± + v s c ±(cid:48) = a ± ( κ ) (cid:18) − c ± c max (cid:19) − δc ± + Dc ±(cid:48)(cid:48) , (11)where a ± ( κ ) is a curvature-dependent association rate, δ is a disassociation rate, c max is the maxi-mum number of binding sites per unit length and D is a diffusion constant (measured as . µ m s − for kinesin-1 (Lu et al., 2009)). The left hand side of (11) is a material derivative, incorporating thefact that we are working in intrinsic coordinates while the kinesin is fixed to the surface, where v s is the instantaneous MT velocity (6) projected in the tangential direction, v s ≡ ˙ x · t = ξ ( τ (cid:48) + BκK (cid:48) + f m ) . (12)Dividing (11) by c max , we use the bound ratios φ ± = c ± /c max as dependent variables, giving ˙ φ ± = a ± ( κ ) c max (1 − φ ± ) − δφ ± + Dφ ±(cid:48)(cid:48) − v s φ ±(cid:48) . (13) At the ends of the MT, we allow no diffusion-based flux of the protein (although it will ‘fall off’the trailing end with the velocity v s ) and hence impose ∂φ ± ∂s = 0 at both ends. Although we aremodelling the MT as a one-dimensional rod, in reality it has a complex protein structure, with eachtubulin monomer consisting of approximately 450 amino acids folded into a 3D arrangement withcharged residues protruding from the surface. Bending the entire filament moves these residuesin relation to each other, expanding those on one side and contracting those on the other; such S. P. Pearce et al. changes can be expected to change the binding kinetics of associated proteins, as these are alsocomplex charged structures. The precise nature of this relationship is unknown, but here we willassume a sigmoidal relationship of the protein association rate a on the local curvature κ , a ± ( κ ) = α (1 ± tanh( βκ )) , (14)where β acts as a scaling factor (with dimensions of length) to determine the degree of preferentialbinding of the kinesin to curved MTs. This implies that when the MT becomes curved the bindingrates to each side of the MT will locally change, and increasing the value of β will mean that thedifferential binding is more sensitive to small curvatures.The choice of this sigmoidal relationship ensures that the association rate both saturates at highcurvature and that a ± ( κ ) is always positive; we have checked that other functional forms can beused to similar effect.We assume that the average on-rate α is proportional to both the number of kinesin moleculesavailable in the vicinity of the MT and their ability to reach one side of the MT, α = dΓ ω on , (15)where Γ is the surface kinesin density, assumed to be sufficiently large for depletion not to be aconcern, d is the maximum distance kinesin can extend (
45 nm (Palacci et al., 2016)), and ω on isan attachment rate per kinesin molecule within range, given as
20 s − (Chaudhuri and Chaudhuri,2016). We note that this is a high estimate, because we neglect the binding to protofilaments directlyabove the surface.Our final model assumption is that the local concentration of bound protein influences the in-trinsic curvature of the filament, by acting as a brace on the side of the filament or some otherconformational change, as suggested by Peet et al. (2018). As the protofilaments are bonded to eachother via lateral bonds, it is assumed that this is able to affect the entire MT. If only one side ofthe MT has a high concentration this will prevent the filament from straightening, altering the MTreference curvature, while if both sides have bound protein then there will be no net effect on thecurvature. Again, we assume a sigmoidal dependence of ˜ κ on the difference between the two ‘sides’of the MT, ˜ κ = κ c tanh (cid:16) γ ( φ + − φ − ) (cid:17) , (16) where γ > is a scaling factor that controls the steepness of the MT response to differential bindingand κ c > is the maximum characteristic curvature. These unknown constants will depend on theprecise nature of the bracing effect, but the measured lattice expansion of . in Peet et al. (2018)suggests κ c = 0 . µ m − . The MT will therefore curve towards the side with less bound protein,as shown in Figure 2. urvature-sensitive kinesin binding can explain microtubule ring formation and reveals chaotic dynamics 9 L and the unbindingtime δ − , resulting in the domain of integration being s ∈ (0 , , yielding χ − ˙ θ = − ( K (cid:48)(cid:48) − τ κ ) (cid:48) + ξ ( τ (cid:48) + κK (cid:48) + F ) κ (17a) K (cid:48)(cid:48) − τ κ ) κ + ξ ( τ (cid:48) + κK (cid:48) + F ) (cid:48) (17b) ˙ φ ± = α (1 ± tanh( βκ ))(1 − φ ± ) − φ ± − v s φ ±(cid:48) + Dφ ±(cid:48)(cid:48) , (17c) ˜ κ = κ c tanh (cid:16) γ ( φ + − φ − ) (cid:17) , (17d) v s = ξ ( τ (cid:48) + κK (cid:48) + F ) , (17e)where the three dimensionless numbers, F = f m L B , χ = cB πµL δ , α = α c max δ (18)are the ratio of forcing to bending rigidity, called the flexure number in Isele-Holder et al. (2015),the ratio of elastic to viscous forces (inversely related to the Sperm number (Lowe, 2003)) and the updated base on-rate respectively. The boundary conditions at a free end are K = K (cid:48) = τ = φ + (cid:48) = φ −(cid:48) = 0 , (19)while at a pinned end we have, τ κ − K (cid:48)(cid:48) = τ (cid:48) + κK (cid:48) + F = K = φ + (cid:48) = φ −(cid:48) = 0 . (20)To connect back to the spatial positions, (6) may be written as, ˙ x = χ ( − sin θ ( K (cid:48)(cid:48) + τ κ ) + ξ cos θ ( τ (cid:48) + K (cid:48) κ + F )) (21a) ˙ y = χ ( cos θ ( K (cid:48)(cid:48) + τ κ ) + ξ sin θ ( τ (cid:48) + K (cid:48) κ + F )) , (21b)which we can calculate after solving for κ . We can also use (1) to get the shape, supplementing with(21) to find the position at a single point.For the examples shown here, we will set D = 0 . µ m s − (Lu et al., 2009), v = 0 . µ m s − , κ c = 1 µ m − (comparable to the . µ m − suggested in Peet et al. (2018); the exact value does notaffect the primary conclusions here), δ = 1 s − (Chaudhuri and Chaudhuri, 2016). For calculating χ , there is a wide range of measured values of the bending modulus B , depending on the measuringtechnique and conditions, with a noticeable length dependence (Pampaloni et al., 2006). Similarly,the viscosity µ is not clear, as the medium of the gliding assay is more viscous than pure water, andso we shall set χ = χ /L , where χ = 1 µ m or µ m in the examples below.As we are considering only the protofilaments on the sides of the MT, we have c max = 250 µ m − if we only consider two protofilaments as being available for binding on each side. Combined withthe estimates of the other values above, this gives α = 3 . when Γ = 1000 µ m − . κ = ˜ κ = κ , τ , φ + and φ − are all constant. Evaluating (17) leads to values for the steady-state protein concentrations, φ ± = α (1 ± tanh( βκ )) α (1 ± tanh( βκ )) + 1 (22)and the following transcendental equation for κ : κ − κ c tanh (cid:18) αγ tanh( βκ )( α + 1) − α tanh( βκ ) (cid:19) = 0 . (23)The straight MT with no differential binding is always a solution to (23), while non-zero roots of κ correspond to curved states with radius of curvature κ − . Note that (23) depends only on theparameters involved in the protein binding, not those connected to the mechanical response. - - - PitchforkSaddle - nodeUnstable Hopf A B
Fig. 3 (A) Bifurcation diagram for b = 3 , γ = 1 , plotting normalised curvature z against dimensionless binding rate α . Unstable branches are marked by dashes, dots represent the location of the Hopf bifurcations shown in B. Verticallines I-IV show positions with 1, 3, 5 and 1 values of κ respectively. (B) Parameter space map (over binding rate α and preferential binding affinity b ) showing the position of the three types of local bifurcations seen in the system, for γ = 1 , χ = 3 µ m , L = 10 µ m . The dotted line corresponds to the bifurcation diagram shown in (A). Defining z = κ /κ c , b = βκ c , steady-states are associated with the roots of the following three-parameter equation, j ( z ) = z − tanh (cid:18) αγ tanh( bz )( α + 1) − α tanh( bz ) (cid:19) ≡ z − ˜ j ( z ) = 0 , (24)defining the function ˜ j ( z ) . Non-zero roots of j ( z ) will therefore correspond to steady-states with anon-zero uniform curvature, and so we wish to understand how the parameters affect the existence urvature-sensitive kinesin binding can explain microtubule ring formation and reveals chaotic dynamics 11 of these roots. If j (cid:48) (0) < , then a positive root must exist as lim z →±∞ j ( z ) = ±∞ , which leads to abifurcation condition from where non-zero roots intersect z = 0 , j (cid:48) (0) = 2 αbγ ( α + 1) − . (25)As shown in Figure 3A, (25) defines two values of the binding rate α , ( α , α ), at which pitchforkbifurcations occur; we focus on α as it is the parameter most available for experimental control.A linear stability analysis (see Appendix) shows that the uniform solution becomes unstablewith a single real positive eigenvalue at α = α , until it re-stabilizes at α = α . The first bifurcationat α = α is supercritical, producing stable non-uniform solutions, but the solutions arising from α = α can be either stable or unstable, depending on the values of b and γ , with a saddle-nodebifurcation occurring further along the branch to re-stabilize solutions when they are initially un-stable (Figure 3B). This saddle-node bifurcation occurs for a wide range of parameters and leads tomultiple non-zero roots as shown in Figure 3A. These additional roots appear when ˜ j ( z ) is initiallysmaller than z but grows sufficiently fast to exceed z , giving a second non-zero root.Provided that α and α are far enough apart, we also find that a nested series of Hopf bi- furcations occur along both the unstable branches as the periodically spaced complex eigenvaluesmove across the real axis, generating unstable limit cycles (due to the positive eigenvalue still be-ing present), with a corresponding reverse bifurcation when they pass back over the real axis. Thepositions of these Hopf bifurcations are indicated as points in Figure 3A and as curves on Figure3B, which shows the parameter space as α and b are varied for a fixed γ = 1 . A similar picture isfound as the curvature-binding parameter γ is changed, but with a negative relationship betweenthe two feedback parameters b (or β ) and γ ; when one is small the other needs to be large in orderfor the feedback strength to be large enough to create non-zero roots, as can be seen in (25).We have therefore shown there exists a range of parameter values for which multiple stablesteady-states exist, allowing for the simultaneous existence of straight and curved MTs at the sameparameter values and for a single MT to be transferred between the two when suitably perturbed, aswas shown experimentally by Liu et al. (2011) (Figure 1). As can be seen in Figure 3A, the non-zeroconstant value of κ is generally below the prescribed characteristic curvature κ c (i.e. z is less than1); this value is approached for some parameter values but can be significantly less, allowing forvariation in the ring sizes with a fixed κ c . Furthermore, the Hopf bifurcations point to the potential existence of oscillating states, which we will explore below.3.2 Numerical SolutionsWe now solve the full set of partial differential equations, (17)-(21), to see how the curvature of theMT evolves in time. To do this, we use the method of lines, discretising in s with a fourth-order central-difference formula to give a set of coupled nonlinear ordinary differential equations in t ,which are solved using Mathematica.As the straight solution is always a steady-state, we need to introduce an initial curvature per-turbation to the MT to be able to see other behaviours. One option is to temporarily pin the leadingtip of a gliding MT (as is observed in gliding assays when the MT encounters defective motors(Bourdieu et al., 1995)). For a large enough F , this will cause the MT to buckle and rotate aroundthe pinned point (Sekimoto et al., 1995; Chelakkot et al., 2014; De Canio et al., 2017). We then allowthe MT to unpin after some curvature (and therefore also differential binding) has been generated,and the MT will then either reach a curved configuration or re-straighten. This is shown in Figure4 and Supplementary Movie S1, where two MTs that are unpinned after slightly different amountsof time settle into the two different steady-states. Fig. 4
Shape evolution of two MTs which are temporarily pinned before being released. All the parameters are thesame except the time of release, which are t = 13 , . respectively, with α = 5 , b = 3 , γ = 1 , χ = 3 µ m , L = 5 µ m .The resulting end-states correspond to the two stable steady-states seen at line III in Figure 3A. Instead of pinning an initially straight MT, we can also generate an initial perturbation by bend-ing the MT into a non-straight configuration and then allowing it to relax, mimicking an interactionwith other MTs. In both of these cases, the exact basins of attraction of the two steady-states dependson both the binding and the mechanical parameters; a relatively large perturbation from straightgliding, which persists for long enough to produce binding differences, is required to move the MTinto the curved configuration. urvature-sensitive kinesin binding can explain microtubule ring formation and reveals chaotic dynamics 13
20 40 60 80 100 - - - - AB C
Fig. 5
A) Shape evolution of a MT showing oscillatory behaviour as it moves across the surface. At these parametervalues propagating waves in ˜ κ cause the MT to periodically undulate as it moves across the surface, settling intoa steady rhythm. One period of these shapes are shown superimposed in (B). For these parameters, the trailing tippreferred curvature, ˜ κ ( L, t ) , plotted in (C), has a regular gap between zeroes of T = 3 . . Here α = 8 , b = 3 , γ =1 , χ = 1 µ m , L = 5 µ m . curvature being translated along the MT as it glides. The curvature of the MT trailing tip, ˜ κ ( L ) ,shows the regular periodicity where the waveform reaches the tip (Figure 5B), and we denote thetime between zeroes of ˜ κ ( L ) by T .As the parameters are changed, the period and amplitude between these oscillations changessmoothly. Additionally, as the diffusion D is increased from D i = 0 . µ m s − (Lu et al., 2009),we also see period-doubling bifurcations occur as shown in Figure 6, where T breaks symmetry,leading to a non-symmetric behaviour in the waveform (Figure 6C). As D is increased further aperiod-doubling cascade continues, leading to chaotic behaviour where the MT never settles into aperiodic regime, as shown in Figure 6A. Upon further increasing of D , the wave solution disappearsand the system settles into the uniform curvature solutions described in Section 3.1. While it isunlikely that D could be used as a control parameter in an experiment, these results demonstratethe range of possible outcomes in this system, and their sensitivity to parameter values.After the initial period-doubling, the overall motion of the MT is biased, leading to the MT mov- ing in a large circle whilst undulating, as shown in the insets of Figure 6A and in SupplementaryMovies S3-S7. The radius of this large circle, R c , decreases monotonically with D , as the waveformbecomes increasingly asymmetric, shown in Figure 6D.These oscillatory solutions, and the period doubling cascade to chaos, appear to exist in regionsof parameter space around the second pitchfork bifurcation, α , shown in Figure 3, provided thatthe ‘wings’ of the bifurcation diagram are large enough to include the Hopf bifurcations. They also
460 470 480 490 500 - - AB C D
Fig. 6 (A) Time T between the zeroes of the trailing tip curvature ˜ κ ( L, t ) (after initial transient behaviour) as thediffusion D is increased, showing a period doubling cascade to chaos. Inset above are plots (all on the same scale)showing the motion of the trailing tip of the MT after the initial transient behaviour. For D/D i = 3 . , . , . , theMT settles into a steady orbit with decreasing radius, whereas for D/D i = 3 . the MT moves around the planein a chaotic manner. (B) Superimposed shapes for D/D i = 3 . over a sub-interval of the orbit, showing a largeramplitude than in Figure 5. (C) Plot of the reference curvature at the end of the MT, ˜ κ ( L ) , showing the two values of T which can be seen in (A). (D) Plot showing how the radius of the large orbit traced out by the oscillating MT decreasesmonotonically as D/D i is increased. All other parameters are the same as in Figure 5.urvature-sensitive kinesin binding can explain microtubule ring formation and reveals chaotic dynamics 15 exist when different sigmoidal functions ˜ κ = κ c G ( γ ( φ + − φ − )) are used instead of (16), for instance G ( x ) = 2 π arctan (cid:16) πx (cid:17) , erf (cid:18) √ πx (cid:19) , x √ x , (26)where erf is the error function (results not shown). We have presented here a model for the occurrence of rings and arcs seen in MT gliding assays,based on the extrinsic effect of differential binding of the surface-attached kinesins which move theMTs forward. Our model goes beyond that of Ziebert et al. (2015), who assume that the formationof these rings is the result of an intrinsic property of the MTs. Our model incorporates the presenceof the kinesins as external factors that drive a feedback loop where MT bending is recognised andstabilised, thus recruiting further kinesins; intrinsic properties of MTs as considered by Ziebertet al. (2015) may contribute to these effects and can be considered in our model by modifying theequation for the reference curvature (16).
Both models are able to reproduce the key aspect of the experiments, with regimes of bistabilitywhere both curved and straight MTs exist simultaneously for the same parameters, differentiatedby the local forcing on the MT. In order for rings to be formed, we find that the system needs to bewithin a parameter regime ( α, β, γ ) which permits multiple steady-states, and the MT must undergohigh induced curvature. Going beyond the model of Ziebert et al. (2015), our approach leads to twopredictions which can be experimentally tested:1. Our model predicts that varying the effective binding rate α should affect whether or not MTrings can form, as well as their size. This on-rate includes both the binding distance d , whichcan be experimentally altered by truncations of the kinesin-1 tail region as in the experimentsperformed in VanDelinder et al. (2016), as well as the surface-bound kinesin density Γ whichmay be directly varied by altering the kinesin concentration.2. Our model predicts that high MT density will encourage the formation of rings, consistent withthe experiments of Liu et al. (2011). MT density contributes in two ways: it encourages the bend-ing of MTs through MT-MT interactions, and it promotes the pushing down of MTs towards thesurface of the assay during cross-over events, significantly enhancing the access to the convex MT side.The non-kinesin parameters used in our model are already well-known or easily measurable,but the parameters β , γ and κ c which connect the protein binding to the preferred curvature areentirely unknown. However, they could be characterised via fluid-bending experiments of the kindperformed in Peet et al. (2018), and then fitting an appropriate variation of the model described here. The measurements of the curvature sensitivity for the protein doublecortin in taxol-stabilisedMTs (Figure 3G of Ettinger et al. (2016)) suggest a value of β of approximately µ m .Additionally, the exact values for the parameters involved in the combined kinesin on-rate α arenot very well characterised. For instance, the surface kinesin density Γ is not routinely measured ingliding assays, but can be obtained via landing-rate experiments (Katira et al., 2007). Furthermore,other motor proteins than kinesin-1 (e.g. other members of the kinesin superfamily or the minus-end directed dynein/dynactin motor complex), are able to translocate MTs in gliding assays. Ifparameter changes in these assays facilitate the occurrence of rings, this would provide additionaldata sets that could be compared and help to refine parameter determination.The unsteady solutions where the MT oscillates while moving across the surface, includingthe period-doubling cascade to chaos, are particularly interesting mathematically as they are notimmediately obvious from the governing equations. An oscillating regime for a clamped or pinnedfilament (without the preferred curvature) driven by the tangential motors, as considered here,was found by Sekimoto et al. (1995), with a Hopf bifurcation occurring above a critical forcing F ;De Canio et al. (2017) also show this behaviour for the case of a filament with a follower force acting on the free end, and expand upon its origin. The behaviour seen here is similar, with a self-sustainedoscillatory waveform, but these authors did not report chaotic dynamics.These oscillations may be biologically relevant; the wavy MTs generated by our model looksimilar to those shown in an experimental figure of Gosselin et al. (2016), as well as a MT seen inSupplementary Movie 1 of Scharrel et al. (2014), and similar regularly undulating curved MTs areseen in cells (Brangwynne et al., 2007, 2006); although these are assumed to be caused by mechanicalbuckling, it is possible that this kind of curvature-dependent binding may enhance the effect. Theremay also be a connection to the MT phenomenon described as ‘fishtailing’, where MTs oscillatelaterally whilst their head is stuck, as shown in Applewhite et al. (2010) and Weiss et al. (1991) forexample. We therefore encourage experimentalists to look out for this kind of behaviour. Our current model is only a starting point, and a number of further aspects can or should be incor-porated.
1. The arrangement of protofilaments into the MT is via a helical arrangement, with a skew an-gle inducing a global supertwist for the MTs where moving forward along one protofilamentinvolves rotating around the MT; the exception is for MTs with 13 protofilaments which arealmost straight, and this is therefore the type considered in our model so far. MTs with moreor fewer protofilaments are shown to rotate as the kinesin moves along them (Ray et al., 1993), urvature-sensitive kinesin binding can explain microtubule ring formation and reveals chaotic dynamics 17 inducing a torque that could be considered in future versions of the model, particularly as ax-onal MTs are not always the 13 protofilament type (Chaaban and Brouhard, 2017). Furthermore,Kawamura et al. (2008) find that more rings occur in gliding assays when they use freshly pre-pared MTs, which have fewer 13-protofilament MTs than their 24 hour aged MTs, suggestingthat this MT rotation might enhance the ring-formation by making it easier for the kinesin tobe bound to the outer side of the MTs; this effect may be explained by kinesin stepping fromone protofilament to another as it reaches its maximum extension, as it does to move aroundobstacles (Schneider et al., 2015).2. The inclusion of the crosslinking protein streptavidin into gliding assays induces the formationof bundles of curved MTs, known as spools, where multiple MTs are attached together and theentire structure is bent into an arc (VanDelinder et al., 2016). Luria et al. (2011) found more small-circumference spools than predicted by their simulations; these spools have a similar diameterto that of the non-crosslinked MT loops (Liu et al., 2011; Kawamura et al., 2008), and we suggestthat the mechanism we propose may be responsible. In particular, Lam et al. (2014) found thatthe size and number of crosslinked spools depends upon the kinesin density, with the presence of more kinesin leading to fewer small-diameter spools, which is consistent with our results.Incorporation of MT-MT interactions via crosslinking proteins would therefore be a naturalextension of our model.3. The core of this model is suitable for other situations where differential protein binding mayinfluence filament curvature, for instance where the protein is freely diffusing in solution asin Peet et al. (2018). The model assumes that the MT stays in the same vertical plane, as it isattached to the surface by the kinesin, and the extension to three dimensions may be requiredto properly model the situation in cells, incorporating both the twisting as well as the bendingof the MT, as well as modelling all the protofilaments individually.4. As mentioned in the introduction, other MT-binding proteins occurring in nerve cells, such astau and doublecortin, have been shown to be curvature-sensitive, and we therefore recommendthat these are tested to see if they can also reinforce the curvature. Additionally, an extension tothe model to incorporate competition between proteins for the same binding sites may explainhow certain proteins have a protective function.
Acknowledgements
SPP thanks the Leverhulme Trust for the award of an Early Career Fellowship. AP acknowl-edges the support of the Biotechnology and Biological Sciences Research Council [grant numbers BB/L026724/1,BB/M007553/1, BB/L000717/1].8 S. P. Pearce et al.
Appendix: Linear Stability Analysis
To investigate the stability of the steady-states, we expand every variable about the constant solu-tion as, θ = Ωt + κ s + (cid:15)θ ( s ) e ωt , τ = (cid:15)τ ( s ) e ωt ,φ ± = φ ± + (cid:15)φ ± ( s ) e ωt , ˜ κ = κ + (cid:15) ˜ κ ( s ) e ωt , where the steady-state values are given by (22) and (23), and the angular velocity Ω is given by Ω = ξχ F κ = vκ , connecting the rotation speed of the arc solutions with the free gliding speed.Linearising with respect to (cid:15) and letting K = θ (cid:48) − ˜ κ gives at first order, ωθ = χ (2 F θ (cid:48) + 3 κ τ (cid:48) + 2 κ K (cid:48) − K (cid:48)(cid:48)(cid:48) )0 = κ τ − κ K (cid:48)(cid:48) ωφ ± = − (1 + α ± α tanh( βκ )) φ ± ± αβ sech ( βκ )(1 − φ ± ) κ − F χξφ ±(cid:48) + Dφ ±(cid:48)(cid:48) ˜ κ = γ ( κ c + κ tanh( βκ )) . Writing as a first order matrix system, y (cid:48) = Ay where y = ( θ , K , K (cid:48) , K (cid:48)(cid:48) , τ , τ (cid:48) , φ +1 , φ + (cid:48) , φ − , φ −(cid:48) ) ,the matrix A is given by A = a − a
00 0 1 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 − ωχ − ξ F ξκ ξ ) κ ξ F a − ξ F a
00 0 0 0 0 1 0 0 0 00 0 0 − (1 + ξ ) κ ξ − κ ξ − − a +2 a +3 + ωD − − F ξχD − γa a +2
00 0 0 0 0 0 0 0 0 10 a − a a − a − + ωD − − F ξχD − where a = γ (cid:18) κ c + κ tanh (cid:18) αγ tanh( βκ ) α tanh( βκ ) − (1 + α ) (cid:19)(cid:19) a ± = αβ sech ( βκ ) D (1 + α ± α tanh( βκ )) a ± = 1 + α (2 − a βγ ) + α ± α (1 + α ) tanh( βκ ) + α ( α + a βγ ) tanh( βκ ) D (1 + α ± α tanh( βκ )) . urvature-sensitive kinesin binding can explain microtubule ring formation and reveals chaotic dynamics 19 We find the eigenvalues of this linear boundary-value problem using the Compound Matrix methodto calculate the Evans function (Afendikov and Bridges, 2001) in Mathematica (a package imple-mentation of which is available from github.com/SPPearce/CompoundMatrixMethod) and cheb-fun (Driscoll et al., 2014) in Matlab. The spectrum of eigenvalues here is discrete, with regularlyspaced pairs of complex conjugates.
Appendix: Supplementary Video Captions
Supplementary Video 1: Shape evolution of two MTs which are temporarily pinned before beingreleased. All the parameters are the same except the time of release, which are t = 13 , . respec-tively, with α = 5 , b = 3 , γ = 1 , χ = 3 µ m , L = 5 µ m . Corresponds to the still images in Figure 4.Supplementary Video 2: Shape evolution of a MT showing oscillatory behaviour as it moves acrossthe surface. Here α = 8 , b = 3 , γ = 1 , χ = 1 µ m , L = 5 µ m , and corresponds to the extractedframes in Figure 5.Supplementary Video 3: Shape evolution of a MT showing oscillatory behaviour, before the period doubling bifurcation, with D = 3 . D i . Here α = 8 , b = 3 , γ = 1 , χ = 1 µ m , L = 5 µ m .Supplementary Video 4: Shape evolution of a MT showing oscillatory behaviour, past the perioddoubling bifurcation, with D = 3 . D i . Here α = 8 , b = 3 , γ = 1 , χ = 1 µ m , L = 5 µ m . Corre-sponds to the first vertical dotted line in Figure 6.Supplementary Video 5: Shape evolution of a MT showing oscillatory behaviour, past the perioddoubling bifurcation, with D = 3 . D i . Here α = 8 , b = 3 , γ = 1 , χ = 1 µ m , L = 5 µ m . Corre-sponds to the second vertical dotted line in Figure 6.Supplementary Video 6: Shape evolution of a MT showing oscillatory behaviour, past the perioddoubling bifurcation, with D = 3 . D i . Here α = 8 , b = 3 , γ = 1 , χ = 1 µ m , L = 5 µ m . Corre-sponds to the third vertical dotted line in Figure 6.Supplementary Video 7: Shape evolution of a MT showing oscillatory behaviour, past the perioddoubling bifurcation, with D = 3 . D i . Here α = 8 , b = 3 , γ = 1 , χ = 1 µ m , L = 5 µ m . Corre-sponds to the fourth vertical dotted line in Figure 6. References
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