A Mathematical Model of Force Generation by Flexible Kinetochore-Microtubule Attachments

Abstract

Important mechanical events during mitosis are facilitated by the generation of force by chromosomal kinetochore sites that attach to dynamic microtubule tips. Several theoretical models have been proposed for how these sites generate force, and molecular diffusion of kinetochore components has been proposed as a key component that facilitates kinetochore function. However, these models do not explicitly take into account the recently observed flexibility of kinetochore components and variations in microtubule shape under load. In this paper, we develop a mathematical model for kinetochore-microtubule connections that directly incorporates these two important components, namely, flexible kinetochore binder elements, and the effects of tension load on the shape of shortening microtubule tips. We compare our results with existing biased diffusion models and explore the role of protein flexibility inforce generation at the kinetochore-microtubule junctions. Our model results suggest that kinetochore component flexibility and microtubule shape variation under load significantly diminish the need for high diffusivity (or weak specific binding) of kinetochore components; optimal kinetochore binder stiffness regimes are predicted by our model. Based on our model results, we suggest that the underlying principles of biased diffusion paradigm need to be reinterpreted.

Introduction

The process of cell division involves a multitude of complex biochemical and mechanical events that lead to the equal partition of genetic material from the mother cell to the daughter cell. A fascinating and crucial process during division has to do with the generation and control of the movement of replicated chromosomes.

A chromosome must interact with microtubules (Mts), which are part of a dynamic network called the mitotic spindle (1–6). Connections between chromosomes and Mts are mediated by macromolecular structures called kinetochores (kts) (7–10). A variety of proteins that can associate with Mts directly localize at kts, however; Mts also undergo continuous growth and shortening while their ends are attached to the kt sites. A question of considerable interest in this context is how the kt site might function as a force-generating machine capable of moving chromosome several microns. A natural contender for this task would be molecular motor enzymes (11). However, molecular motor enzymes that localize at kts have been shown to be dispensable for kt motion in yeast (12,13). Kt nonmotor components should thus have the ability to generate movement by latching on to Mt tips that constantly lose or gain monomers; how such a task is achieved is not clear. Force generation at kts has consequently attracted considerable interest from both an experimental standpoint and quantitative modeling approaches (14–16).

Many components of kts have become known. Significant technical advances in high-resolution imaging have led to new insights regarding kt component spatial organization and copy numbers inside kts in a variety of organisms (17–19). A few proteins are emerging as important structural components of kts. The Ndc80 complex is an elongated dumbbell-like molecule with a high degree of flexibility because of a hinge site around its halfway point (20–22) that connects on one side to the kt structure and on the other to the Mts (23–25). KMN proteins are conserved kt components that form the primary kt-Mt interface (26); these proteins have affinity for the Mt, and importantly form a scaffold that acts to localize several kt kinases. The Dam1/DASH complex is essential in budding yeast and can form rings or spirals in the presence of Mt (27,28). Although there is no evidence that this complex can form rings in organisms other than yeast, it remains an important component at the kt-Mt interface. Further, it has been shown that Dam1 contain flexible elements for interaction with the kinetochore microtubule (kMt) (29). Finally, the Mis12 complex is another conserved kt component that can directly bind to the chromosomal chromatin (30,31).

Mts are polar hollow filaments composed of $\alpha \beta$ tubulin dimers that are arranged into linear chains called protofilaments. During mitosis, Mts undergo stochastic transitions between states of growth and shortening, known as “dynamic-instability” (32). Mts have a built-in polarity, with the plus-end experiencing faster growth/shortening than the minus-end. Tubulin adds to the Mt lattice in its Guanosine-triphosphate (GTP)-tubulin form; GTP is subsequently hydrolyzed into GDP-tubulin. The hydrolysis state of a tubulin dimer determines its preferred conformation: GTP dimers are thought to lie parallel to the Mt lattice, whereas Guanosine-diphosphate (GDP) tubulin prefers to bend away from the lattice (33). When a Mt disassembles, the tubulin at the Mt tips experience both loss of the GTP cap and lateral interactions that causes protofilaments to become relaxed and subsequently flare at the Mt ends (33,34). The plus-ends of Mts are embedded in the kt attachment site, and while attached, growth and shortening prevails. During this process, energy from GTP hydrolysis is released, and presumably this energy can be used by kt sites to generate motion (35–38).

Most of the existing theoretical models of Mt-kt coupling (35,39–42) are based on one of two postulated mechanisms for force generation. In the biased-diffusion model, initially proposed by Hill (35), the plus end of a kMt is assumed to be surrounded by a rigid coaxial “sleeve” the inner surface of which is composed of several binding elements that bind specific kMt sites. The one-dimensional Brownian motion of the sleeve along the axis of the kMt is biased to increase overlap, because a larger number of kMt-sleeve bindings lowers the total energy of the system. The interplay of this biased diffusion and the depolymerization of the kMt gives rise to the pull exerted by the coupler on the kt. The second proposed “power-stroke” coupling mechanism is based on the idea that the curling protofilament tips of a depolymerizing Mt exert a force on a rigid kt-connected sliding ring surrounding the Mt (40,41). These previous models capture several aspects of kt-Mt engagement; however, they ignore important mechanical features of the kt machine such as feedback between kMt protofilament shape and multiple flexible kt binder attachments under load. Specifically, kMt protofilaments at the curling ends of depolymerizing Mts can undergo shape changes when challenged by force. On the other hand, kMt protofilament shape can have significant effects on kMt depolymerization speeds, and on the ability of kt binders to engage with Mt tips. It is reasonable to expect that when modulation of Mt shape and kt binder attachment dynamics under load are combined together, mutual feedback might generate complex attachment responses. Novel modeling approaches are needed to account for these interactions.

Recent experimental results also highlight the need for a novel approach to kt-kMt modeling. There is evidence that kts engage kMts through multivalent attachments that move along Mts consistent with a biased diffusion mechanism (43). However, it also has been recently reported that all kt components are flexible, not rigid (44). Previous theoretical work that studied kt attachments in the biased-diffusion framework (including work from these authors (42)) assumes that kt multivalent attachments are rigid. Importantly, the role of kMt shape in depolymerization dynamics and kt attachments has been largely ignored in previous modeling work. Yet, recent data shows nonmonotonic sensitivity of kt-kMt attachment dynamics on the amount of tension load exerted on kts (45–47). These data collectively indicate that the kt/kMt juncture can respond to force in complex ways and thereby the kt machine may work in regimes where the two standard classes of models currently used do not apply.

The purpose of this paper is to develop a new model of force generation at the kt-Mt interface that incorporates kt-component flexibility, kMt protofilament shape mechanics, and kMt depolymerization kinetics. Using our model, we demonstrate how these features of kt junctures affect the ability of this attachment site to generate force for various parameter regimes. In so doing, we provide an alternate mechanism to rigid sleeve-type biased diffusion for kt force generation.

Methods and Materials

Description of the Model

We start by briefly describing the proposed location and geometrical arrangement of the components of the kt site. In Fig. 1 a we show a three-dimensional model derived from high-resolution data from a previous study (9), and in Fig. 1 b we show a diagram of the attachment site that we use for our model. The key assumption we make in this work is that several flexible kt proteins are bound to Mt protofilaments with variable deformation, generating force depending on the deformation of the flexible protein from its rest position.

Figure 1.

Diagram of model components for kt/kMt attachment. (a) three-dimensional model derived from high-resolution data of a vertebrate kt attached to a depolymerizing kMt, adapted from (9). Green ribbon representation for the kMt, and inner kt complexes are shown as spheres and rods, as described in (9). A nucleosome is shown as a ribbon model in dark purple, next to a simplified representation of chromatin (purple). (b) Diagram of our model for kt binders and kMt. The purple structure represents the flexible kMt binder elements, uniformly distributed on the kMt. To see this figure in color, go online.

Kinetochore binders

We assume that the kt components are uniformly distributed in the radial direction and that kMt protofilaments maintain rotational symmetry, so that the key dynamics of attachment are accurately represented by projecting and tracking the attachment site on a one-dimensional line, as shown in Fig. 1 b. We suppose that flexible and stretchable kt components (i.e., Ndc80 or KMN linkers etc.) are connected to a protein arm and bind the surface of the kMt. To describe the dynamics of the binder/kMt interactions in mathematical terms, we use two independent variables, y, the rest position of the binder, and z, the location of the binder attachment on the Mt, depicted in Fig. 1 b. The relative orientation of the tip of the Mt with the binder rest positions is determined by l. We distinguish three cases for the overlap, shown in Fig. 2 a:

• 1)For $l\le 0$ there is no overlap between the kMt and the binder rest positions; however, the kt is still engaged with the kMt, i.e., the binders are stretched outward from their rest positions.
• 2)For $0<l<L$ there is partial overlap between the rest positions of the binders and the kMt lattice.
• 3)For $l\ge L$ there is total overlap between all the binders and the kMt lattice.

Figure 2.

(a) Diagram of a flared mt protofilament and kMt overlap cases. A plot of the radial deformation of the kMt protofilaments u, under load f is also shown. (b) Diagram of the cross-section of a Mt with lateral attachments between protofilaments. The rest distance is given by $l_0$ and there are $n_f=13$ protofilaments per Mt. The radial displacement u measures distance from the center of the kMt. To see this figure in color, go online.

The remaining assumptions for binder/kMt interactions are as follows. We assume that there are $n_T$ binders in total, uniformly distributed along the y axis and with rest position, y, where $0<y<L$. The kt binders can bind to the Mt at any position z with $z<l$. We track the density of bound binders, $n\left(z,y,t\right)$ using the partial differential equation, derived in the Supporting Material:

$$\frac{\partial n}{\partial t}=k_{on}\left(z,y\right)\left(\frac{n_T}{L}-{\int }_{-\infty }^{l}ndz\right)-k_{off}\left(z,y\right)n-\frac{\partial }{\partial z}\left(vn\right),$$ (1)

where v is the z-velocity of the binders bound to the kt. The rates $k_{on},k_{off}$ are position dependent attachment and detachment rates for binders respectively, derived in the Supporting Material. The numbers of binders in this model can change because of three factors: 1), a new binder bond is established, 2), an existing binder bond is broken, 3), a binder can change its $z-$ position because of movement of the Mt relative to the binder arm. At any given time, the total number of bound binders is ${\int }_0^L{\int }_{-\infty }^{l}n\left(z,y,t\right)dzdy$.

Kt binders can engage with the kMt and apply force in various directions; however, the structure of kts remains unresolved and we suppose that the primary movement of kt binders is along the horizontal axis of kMt symmetry, without significant changes in the vertical distance between the binder arm and the kMt lattice. Thus, we make the assumption that the kt binder force generated f, is horizontal, either pulling (to the right) or pushing (to the left), ignoring small forces along the vertical direction (Fig. 2 a). The force generated by an individual bound binder is assumed to be equal to $\kappa \left(y-z\right)$, where κ is the binder spring constant and $y-z$ is the displacement of the spring from its rest position, i.e., the protein springs are linear with zero rest length. A potential source of kt binder flexibility assumed here might be the observed flexible kink of the Ndc80 rod complexes (22); however, this does not preclude the existence of other compliant parts of kt elements, or potential contributions from pericentromeric chromatin stretching (48).

We next briefly summarize kt binder attachment/detachment kinetics. Unbound binders are free to diffuse but are constrained by the restoring force, hence their motion is described by an Ornstein-Uhlenbeck process. Consequently, they can bind to the Mt at a position z at a rate that is a Gaussian function of the force $\kappa \left(y-z\right)$. The unbinding rate is taken to be load-dependent in a way described by Bell’s law. The binding equilibrium constant is $\text{exp}\left(-a/k_{B}T\right)$, where a is the binding free energy. The binders have a strong affinity for the Mts, reflected by the fact that $a\gg k_{B}T$. The resulting model (Eq. 1) that tracks the number of bound and unbound kt binders is similar to Huxley’s model for muscle contraction (49), as detailed in the Supporting Material.

Shape of a depolymerizing kMt

Recent evidence indicates that kMt depolymerization is sensitive to opposing tension load on kt components (45,47). The depolymerizing Mt protofilaments are known to curl, or flare, at the plus ends. Umbreit et al. (50) found that incubating Ndc80 with Mts produced stabilized Mt tips with straighter protofilaments and slower depolymerization. Taken together, these findings indicate that kt components can have an effect on Mt growth/shortening kinetics, an effect that seems to be sensitive to load. We incorporate these effects into our model by assuming that the shape of kMts is modified by the tension from the binders.

Our model for the shape of a kMt is a generalization of a continuous model that represents the kMt as a tube made of uniform elastic material that can stretch and bend, Fig. 1 b. This approach removes the need for detailed descriptions on tubulin structure and interactions and is similar to the approach taken in Janosi et al. (51). Since the kMt here is assumed to be rotationally symmetric about the central axis, we focus our attention on a line, or “protofilament” with the understanding that the full kMt shape is obtained by rotating the filament about the central axis, as shown in Fig. 2 b.

Our assumptions for the shape equation model are as follows. We assume that a flaring protofilament has a constant preferred curvature ϕ and resists bending from its preferred curvature with bending rigidity α, with units of force per length-squared. Furthermore, we assume that flaring is constrained by elastic lateral forces between protofilaments, giving an unloaded protofilament the length constant ${\lambda }^{-1}$. Additionally, we assume that lateral bonds break in a load dependent fashion. A derivation of the shape equations for a loaded protofilament using energy minimization arguments is given in the Supporting Material. The resulting system of shape equations for our model generalizes the standard linear beam equations used to calculate equilibrium shapes for unloaded Mts considered in (51) and is given by the following equations:

$$\alpha \frac{d^{4}u}{d{z}^4}=\frac{d}{dz}\left(\rho \frac{du}{dz}\right)-k_{lat}u\text{exp}\left(-\frac{u}{u_f}\right),$$ (2)

$$\frac{d\rho }{dz}=f,\rho \left(l\right)=0,$$ (3)

where u is the radial deformation of the protofilament, f is the load force density on the filament, $k_{lat}=\alpha {\lambda }^4$ is the effective linear Hook’s constant for the lateral restoring force, and $\theta =du/dz$ is the angle of ascent of the kMt protofilament, see Fig. 2.

Finally, the shape of the kMt can be determined once we compute the load imparted on the kMt by the kt binders. This binder load is computed using the following:

$$f\left(z\right)=\frac{\kappa }{n_f}{\int }_0^L\left(z-y\right)n\left(z,y,t\right)dy,$$ (4)

where the parameter κ is the spring coefficient for kt binder springs and $n_f=13$ is the number of protofilaments per Mt.

Depolymerization rate

A key component that allows for feedback between kMt loading and kMt dynamics is a force-dependent model for kMt depolymerization. We next outline our assumptions. Consistent with the findings of Umbreit et al. (50), we assume that the flaring protofilament can break (i.e., depolymerize) at any position along its length; however, highly curved segments are more likely to break than less curved segments. Specifically, we assume that the depolymerization rate at position z is exponential in curvature as follows:

$$k_{break}\left(z\right)=\frac{{\beta }_{max}}{n_{f}S}\text{exp}\left({\kappa }_b\left(\dot{\theta }-\varphi \right)\right)\text{exp}\left(\frac{-\rho }{{\rho }_0}\right),$$ (5)

where $\dot{\theta }$ is the local curvature, and ρ is the tension in the protofilament. Here, ${\kappa }_b$ and S are positive parameters, both having units of length, chosen so that with no load from binders, the expected depolymerization length is that of a tubulin dimer, ${\delta }_t$, and the expected rate of tubulin removal from a protofilament is ${\beta }_{max}/n_f$, where the rate of removal of tubulin from Mts is ${\beta }_{max}$. Both ${\kappa }_b$ and S are determined numerically. The multiplicative factor $\text{exp}\left(-\rho /{\rho }_0\right)$ is motivated by Bell’s law. The motivation for this term is to allow the protofilament breakage rate to increase if the protofilament tension increases. (Note that according to (3), if f, the load, is positive, then ρ is negative.) This rate relationship with tension load reflects a potential catch-bond behavior created by protofilament shape modification under tension. A catch-bond rate assumption has been used in other models of kt/kMt interactions (45,46). We highlight that this last load assumption is not necessary for our model; however, its inclusion provides a natural extension of the depolymerization rate model and allows us to explore the complex feedback between kMt load force and kMt dynamics. We also distinguish our approach from previous works, because our depolymerization rate, Eq. 5, allows for the explicit incorporation of the effects of both the shape of the kMt, and the load forces because of kt binders on the kMt depolymerization dynamics; these two components have not been previously studied together in other catch-bond type models.

The model is closed when we specify that the velocity of kMt depolymerization using the position dependent depolymerization rate in Eq. 5 as follows:

$$v=v_d\equiv {\int }_{-\infty }^l\left(l-z\right)k_{break}\left(z\right)dz.$$ (6)

We study the dynamic properties of the kt-kMt connection in our model as follows. For a fixed velocity v and overlap length l, we calculate the steady-state binder distribution function, $n\left(z,y\right)$, and corresponding load and Mt shape. For a kMt shape, we then calculate the corresponding rate of kMt depolymerization $v_d$, using Eq. 6. Then, we adjust the overlap variable, l, until the kt velocity v and depolymerization rate $v_d$ agree. Details of our calculation are shown in the Supporting Material. Model results were obtained by numerically computing equilibrium protofilament shapes and resulting depolymerization velocities. All the parameter values used along with corresponding references are listed in Table 1.

Table 1.

Parameter values

Parameter	Description	Value	This study
L	Binder domain length	30–100 nm (43, 52, 53)	90 nm
$n_T$	Maximal number of binders	8–50 (18, 46, 54)	50
$n_f$	Protofilaments per microtubule	13	13
κ	Ndc80 spring constant	5–500 pNμm−1 (46)	1–100 pNμm−1
a	Free energy of binding	2.5–13 kBT	5 kBT
${\kappa }_{off}$	Basal binder off rate	0.01–0.5 s−1 (16, 46)	0.25 s−1
${\kappa }_{on}$	Basal binder on rate	${\kappa }_{off}\text{exp}\left(a/k_{B}T\right)$ (46)	37.1 s−1
η	Bell’s law coefficient for basal binder off rate	Estimated	100 μm−1
ϕ	Mt preferred curvature	50 μm−1 (55)	50 μm−1
α	Mt protofilament flexural rigidity	$1.5-13\times {10}^{-3}$ pNμm2 (16, 40, 56, 57)	40 pNnm2
${\lambda }^{-1}$	Protofilament length constant	20 nm (56, 57)	10 nm
${\beta }_{max}$	Maximum rate of tubulin removal	100–300 s−1 (39, 42, 46, 58)	300 s−1
$u_f$	Bell’s law coefficient for lateral connections	Estimated	3.5 nm
${\kappa }_b$	Depolymerization length constant	Computed	1.87μm
S	Depolymerization rate scale factor	Computed	15 nm
${\delta }_t$	Tubulin subunit length	8 nm	8 nm

Results

Protofilament shapes under load

We first examine the protofilament shape and corresponding depolymerization rates. Fig. 3 shows the shapes and depolymerization rates for several loading protocols. The shapes of the Mts result from the interplay of the stresses and energies stored in the elastic kMt lattice. Away from the Mt tip, the shape is the one that minimizes the energy of the system, where the energy is the sum of the energy in lateral connections and the energy associated with bending the protofilament away from its preferred curved state. At the tip of the Mt, the preferred configuration represents a balance between all the forces at play, including kt binder forces. This leads to longitudinal bending accompanied by an increase in total Mt circumference (i.e., protofilament flaring) near the tip.

Figure 3.

Model results of kMt shapes and depolymerization. (a) Protofilament displacement u, and (b) normalized depolymerization rate, $k_{break}\left(z\right)/k_{break}\left(0\right)$ for an end-loaded protofilament with three load amounts: $G=0$ unloaded (solid line), $G=1$ (dashed line), and $G=10$ (short dashed line). For this depolymerization rate, ${\kappa }_b=1.87\mu$ m and $S=15$ nm. (c) Depolymerization rate $\left(v_d\right)$ for a protofilament with end-load G, relative to that for the unloaded mt, for two different values of the Bell’s law coefficient $L^2{\varphi }_0/\alpha =2.5$ (upper, dashed) and $\infty$ (lower, dashed). (d) and (e) Mt shape shown as the rotationally symmetric “cylinder” derived from the protofilament shape in (a), for the loads $G=0$, unloaded shown in (d), and $G=10$ in (e), respectively. To see this figure in color, go online.

The exact solution of the shape equations can be found in the case that the Mt is end-loaded and $u_f=\infty$ (i.e., Eq. 2 is linearized). By end-loaded, we mean that the total load, F is applied at the tip of the protofilament at $z=l$, so that in Eq. 2, $\rho =-F$ for $z<l$. For an unloaded Mt with $u_f=\infty$, one finds the same shape as found in a previous study (51), with ∼ 1 nm indentation and 5 nm total radial flare from the nominal radius of 12.5 nm. Since data on flaring kt mts show 30 to 40 nm of flaring (59), this linear model of Mt shape is not appropriate here. Instead, with $u_f=3.5$ nm, the equations can no longer be solved with analytical methods, but the amount of flaring is closer to that seen in data.

Examples of the protofilament shape and scaled depolymerization rates for Mts with $u_f=3.5$ nm are shown in Fig. 3 a and b. Fig. 3 a shows shapes corresponding to three different Mts with end-loading G = 0, 1, and 10, where $G=F/\alpha {\lambda }^2$ is the nondimensional load corresponding to total load F. The solid curve shows the shape with no load G = 0, and the solid curve in Fig. 3 b shows the scaled depolymerization rate for this Mt, $k_{break}\left(z\right)/k_{break}\left(0\right)$. For this Mt, the expected depolymerization distance is ${\delta }_t$ and the expected depolymerization velocity for a mt is ${\beta }_{max}{\delta }_t/n_f$. The remaining two dashed curves in Fig. 3 show the protofilament shape and depolymerization rate for protofilaments that are end-loaded with a total force F, for two different values of nondimensional load $G=F/\alpha {\lambda }^2$, (G = 1, dashed, and G = 10, short dashed). Clearly, for larger values of G, the protofilament is straighter, leading to a slower depolymerization rate. Fig. 3 c shows the relative depolymerization rate (relative to that of the unloaded protofilament with G = 0) as a function of G, the nondimensional load, for two different values of the Bell’s law coefficient ${\hat{\rho }}_0=L^2{\rho }_0/\alpha =2.5$ (dashed curve) and $\infty$ (solid curve). Note that ${\rho }_0=\infty$ corresponds to a tension-independent depolymerization rate, where the rate of monomer removal is related to the local curvature. The upper curve (corresponding to $L^2{\rho }_0/\alpha =2.5$) shows nonmonotone behavior typical of catch-bonds. This results from competition between decreased curvature, which slows depolymerization, and tension, which increases depolymerization. The parameter ${\rho }_0$ scales the effect of tension on depolymerization, with smaller values of ${\rho }_0$ leading to a more rapid increase in depolymerization as a function of increasing load.

Tension-independent depolymerization

Fig. 4 shows a typical example of the load velocity curve, in this case for $\kappa =10$ pN μm⁻¹, $\alpha =40$ pN nm², ${\rho }_0=\infty$. Recall that ${\rho }_0=\infty$ corresponds to the case where tension in the protofilament has no direct effect on depolymerization. An interesting feature of this load-velocity curve is that there are actually two curves, one (shown dashed) for which $\delta =l/L$, the percentage overlap, is negative (we refer to this as no overlap, and binders are attached but stretched), and one for which δ is positive. For this load-velocity curve, the maximum load of ∼ 7 pN occurs close to the minimal velocity of ∼ 0.009 μm s⁻¹. There is no steady-state solution of the equation for larger loads because the couplers detach at larger loads. The dashed curve intersects the velocity axis (load $=0$) at the free depolymerization velocity 0.175 μms⁻¹. The solid curve has the interesting feature that it can sustain velocities that are larger than the free depolymerization velocity, which arises when a proportion of kt binders help to increase the local curvature at the kMt tip ends. This last feature highlights potentially counterintuitive feedbacks that can arise in this coupler, since kt binders can be locked to increase kMt depolymerization by bending the protofilaments, and subsequently increasing kt translocation velocities in the face of resistive load. The force-velocity curves obtained here also distinguish this model from the Hill-type biased diffusion force-velocity relations (42), because in the classical biased diffusion model velocities cannot exceed free depolymerization rates, and coupler velocity is independent of the specific amount of overlap.

Figure 4.

(a) Load as a function of velocity and (b) velocity as a function of δ, the percentage overlap, for Ndc80 spring constant $\kappa =10$ pNμm⁻¹, and protofilament bending rigidity $\alpha =40$ pN nm²$\left(\gamma =700\right)$. The solid curve represents a binder/Mt arrangement for which the overlap is positive, whereas the red dashed curve represents a binder/mt arrangement for which the overlap is negative. To see this figure in color, go online.

To give an idea of how the binders work to generate load, Fig. 5 shows a series of figures detailing the forces and loads for three different values of overlap $\delta =-0.3402$ (a), 0.6748 (b), and 0.4017(c). Fig. 5 a shows the force density generated by the Ndc80 complex of proteins, plotted as a function of $x=y/L$. The dotted portion of the curves b and c are the regions where Ndc80 overlaps with protofilaments. For curve a, even though there is no overlap with the kt binder arm $\left(\delta <0\right)$, the binders are engaged because they are extended (stretched), hence generating positive force. For the other two curves, some Ndc80 proteins are extended (with force $>0$) whereas others are compressed (with force $<0$). Fig. 5 b shows the load on the protofilaments plotted as a function of $z<l$; for curve a, there is no overlap with Ndc80, so $z<0$. For curves a and b, the load is strictly positive, so that the protofilaments are straightened (with protofilament shape shown in Fig. 5 c), which leads to a slowing of depolymerization. On the other hand, for Fig. 5 c, the load on protofilaments is of both signs, so that the protofilaments are more curved than the unloaded protofilament, and therefore, according to our model (Eq. 5), depolymerization is enhanced. It is for this reason that depolymerization can be faster than for an unloaded protofilament, even though there is a net positive load. The dashed curve in Fig. 5 c is the shape of an unloaded protofilament, as shown in Fig. 3 a. These results imply that flexible kt binders can play a complex role in modulating kMt shortening dynamics, with the potential for enhancing kMt depolymerization rates in some overlap regimes.

Figure 5.

Ndc80 binder force density (a), protofilament load (b), and protofilament displacement (c) shown for the three values of overlap $\delta =-0.3402$ for curve a, 0.6748 for curve b, and 0.4017 for curve c. The dashed curve in (c) shows the shape of an unloaded protofilament, from Fig. 3 a. The dotted portion of the curves b and c are the regions of overlap of Ndc80 with protofilaments. To see this figure in color, go online.

Although the shape of the load-velocity curve is qualitatively the same for a large range of parameter values, the quantitative details are modified in significant ways. The nondimensional parameter $\gamma =n_T{L}^3\kappa /n_f\alpha$ determines the relative stiffness of the protofilaments compared with the force applied by Ndc80 binders and, therefore, determines the ability of the binders to bend the protofilament from their preferred unloaded configuration. Hence, this parameter determines the variability of the velocity of the load-velocity curve. If γ is large, there is large variation in the velocity, whereas if γ is small, the variation in velocity is also smaller. Fig. 6 shows the load-velocity curve and velocity-overlap curve in the two cases that $\gamma =140$ and $\gamma =700$. The main impact of this change is that the range of velocities is smaller ($v>0.26\mu$ m s⁻¹ for $\gamma =140$ compared with $v>0.09\mu$ m s⁻¹ for $\gamma =700$.) These results indicate that overall velocity and maximal loads supported by kts are lowered when kt binders are “weakened” because they cannot affect kMt depolymerization dynamics. In a previous study (50) it was reported that phosphorylation of kt components by Aurora B kinase not only weakens kt attachment, but also nearly abolishes the ability of the Ndc80 complex to influence kMt dynamics. Our results support the possibility that posttranslational phosphorylation of kt binders affects attachment dynamics by altering the stiffness parameter γ.

Figure 6.

(a) Load as a function of velocity and (b) velocity vs. δ, the percentage overlap, for Ndc80 spring constant $\kappa =10$ pNμm⁻¹, and protofilament bending rigidity $\gamma =140$ and $\gamma =700$. The solid curve represents a binder/mt arrangement for which the overlap is positive, whereas the red dashed curve represents a binder/mt arrangement for which the overlap is negative. To see this figure in color, go online.

An important parameter for kt couplers is the maximal number of binders in each attachment site. In our model, both parameters γ and the load F are proportional to the number of the total binders in the kt complex, $n_T$. Specifically, both γ and F are decreased if the numbers of binders, $n_T$, is decreased. As shown by our calculations, the range of loads supported by the kt, as well as the variations in the velocity responses are diminished for small γ, as seen in Fig. 6. These results indicate that the total number, or size, of the kt complex has direct effects on the range of velocities with which the kt responds, with more complex responses in velocity arising as the size of the kt, or $n_T$, is increased. We expect that varying the number of kt binders (such as Ncd80 copy numbers) will lead to transitions between the different load-velocity regimes seen in our model when γ is varied. This prediction may be testable, since the stoichiometry of kt couplers has been studied recently (52,60).

The second important determinant of the load velocity curve is the parameter $\kappa L^2/k_{B}T$. This parameter is important because it determines how readily the Ndc80 spring can reach to bind with the protofilaments. If κ is large, the reach of the binders is small and so they cannot generate much force; whereas if κ is small, the binders have extended reach but also may not develop much force. Consequently, we expect to get a “sweet spot” for κ for which the maximal load is optimized. Fig. 7 shows three load-velocity curves (Fig. 7 a) and velocity-overlap curves (Fig. 7 b) for the three values of $\kappa =1,6,15$ pNμm⁻¹. For these parameter values, the maximal load occurs for the intermediate value $\kappa =6$ pNμm⁻¹. The nonmonotonic response in attachment strengths is more clearly displayed in Fig. 8, where we show the maximum kt load supported as a function of the binder stiffness κ. It would be interesting to test these results experimentally by systematically studying the effect of binder “ease-of-reach” on kt loads, perhaps by modifying the Ndc80 kink region and measuring the maximal load supported by the attachment. Our model predicts a nonlinear relation between maximal load and kt binder stiffness.

Figure 7.

(a) Load as a function of velocity and (b) velocity as a function of δ, for three values of κ (in units of pNμm⁻¹), and $\alpha =40$ pN nm². The solid curve represents a binder/mt arrangement for which the overlap is positive, whereas the red dashed curve represents a binder/mt arrangement for which the overlap is negative. To see this figure in color, go online.

Figure 8.

Maximum kt load as a function of binder spring coefficient κ. Kt attachments show an optimal stiffness regime of flexible kt binders. To see this figure in color, go online.

Tension-dependent depolymerization

A third important determinant of the load-velocity relationship is the Bell’s law coefficient ${\rho }_0$ in the protofilament breakage rate. For all the previous plots, ${\rho }_0=\infty$ for which there is no effect of tension on depolymerization. However, catch-bond rate assumptions have been recently used to interpret attachment kinetics, based on a recent in vitro study (45). We can study the effects of this nonlinear tension dependence on the load-velocity relation of our model. Fig. 9, shows examples of the load velocity relationship and velocity-δ relationship for three values of the Bell’s law coefficient ${\rho }_0=\infty ,10$, and 5 pN. As one would expect, the effect of tension-dependent depolymerization is to generally decrease the supported load and increase the velocity of kts. This occurs for ${\rho }_0\ne \infty$, because the depolymerization rate increases for high tension loads, see Fig. 3 c (dashed curve). In our depolymerization model, the nondimensional parameter ${\rho }_0/\alpha {\lambda }^2$ controls the balance between two competing effects: 1), the tension forces that increase protofilament breaking rates, and 2), load induced protofilament straightening that slows depolymerization. A so-called “catch-bond” regime emerges here when kMt tension load can increase depolymerization rates and in so doing eliminate curvature effects for large loads. Thus, small ${\rho }_0/\alpha {\lambda }^2$ signals that the filaments are less resistive to load and can easily break under tension, whereas large ${\rho }_0/\alpha {\lambda }^2$ indicates that protofilaments are more resilient to load and become less curved without breaking, thus causing slower depolymerization. Most kMt protofilaments become elongated and less curled while attached to kts in anaphase, where depolymerization of kMt prevails (59), and measurements of attachments times for reconstructed kts in (45) indicate that large forces are required to break kt-kMt attachments (more than 10 pN of tension was required to break most attachments in (45)). Based on these observations, it is reasonable to expect that kt-kMt attachments operate in a high ${\rho }_0$ regime, where the inclusion of an explicit catch-bond response to tension may not be necessary. In this regime, tension forces primarily serve to slow down depolymerization by straightening protofilaments until large tension loads detach the kt binders.

Figure 9.

(a) Load as a function of velocity and (b) velocity as a function of δ, for three values of ${\rho }_0$ (in units of pN), and $\alpha =40$ pN nm².The solid curve represents a binder/Mt arrangement for which the overlap is positive, whereas the red dashed curve represents a binder/mt arrangement for which the overlap is negative. To see this figure in color, go online.

Perhaps not expected, however, is that there is hysteretic behavior when ${\rho }_0/\alpha {\lambda }^2$ is sufficiently small. In particular, with ${\rho }_0/\alpha {\lambda }^2$ sufficiently small, there is a range of velocities for which there are three possible loads, with hysteretic transitions into and out of this range. The actual appearance of these hysteric regimes in kt function depends on the value of ${\rho }_0/\alpha {\lambda }^2$; but if present, it might have interesting mechanical consequences for chromosome motion.

Discussion

In this study we derived and analyzed a model of force generation at the kt-mt juncture. The primary novelty of our model is that we include two important features that have not been previously considered together at this attachment site. These are the effect of flexibility of key kt components that interact with kMt fiber, and the effects of tension load on the shape and corresponding depolymerization velocity of attached kMts. Further, the modeling framework we propose in this paper allows for an analytically tractable treatment of kMt protofilament shapes under load that arise while a depolymerizing kMt is attached to a moving kt. Our model extends and generalizes previous work on continuous models of Mt shape (51). The model we have constructed shares some common features with Huxley’s model for muscle contraction (49). Indeed, we propose that the Huxley muscle model is a useful framework for kt attachment models, especially in light of the reported kt binder flexibility.

The inclusion of a flexible array of binders distinguishes our model from previous biased diffusion models (35,39,42). For these previous kt-kMt models, the assumption is made that kt components are rigid. This rigidity requirement then allows an attached kts’ binder components to change position relative to the kMt either via thermal diffusion, or kMt depolymerization/polymerization. However, for this biased-diffusion process to work, it is also required that the binding of kts to mts be sufficiently weak so that there can be significant thermal diffusion on the kMt lattice.

Our model achieves biased diffusion by a different mechanism, one that is more physically reasonable. Because the binders are flexible, when they are unbound from the Mt, they undergo diffusion, constrained by the restoring force of stretching. So, although the diffusion of the binders is not biased, their binding is biased by the location of binding sites relative to the rest location of the binder. In our model, once the binder is attached to the mt, it is no longer free to diffuse along the mt; this is in direct contrast to previous models of biased diffusion. In fact, for our proposed mechanism to work, the binders must be flexible, as inflexible binders do not diffuse.

Another important component of our model is the incorporation of a model of kMt shape and explicit connection between protofilament shape and the rate of kMt shortening. Our model also directly connects mechanical features of the kMt with its depolymerization rate by using an exponential rate function. This rate is reminiscent of catch-bonds that can gain strength under tensile load, and as seen in Fig. 3 c, this gain of strength under load slows depolymerization. Of course, under high enough loads bonds break more readily, and this feature is reflected by the parameter ${\rho }_0$. Smaller values of ${\rho }_0$ lead to more rapid breakage because of loading, and can even change the character of the load-velocity curve (see Fig. 9,).

Our model results indicate that kt binder flexibility can have a significant effect on the tracking ability of a kt coupler with a depolymerizing kMt. We highlight two important results here. First, the velocity range that the coupler can sustain under load is sensitive to the ratio of the stiffness of the kMt protofilaments with the force applied by the Ndc80 linkers. This makes sense since the generation of velocity in this model is directly dependent on the ability of the protofilament to bend, thus a more rigid protofilament can easily overwhelm a flexible kt component and prevent movement, because of slowed depolymerization. Second, the range of velocities is also sensitive to the flexibility of Ndc80. This feature is related to the ability of Ndc80 to capture a binding site and generate force. If too soft, the kt component can reach multiple binding sites; however, the connection cannot support much load. On the other hand, if the kt component is too stiff, then it cannot explore space and access binding sites as easily. Thus, stiffer binders are not able to support kt coupling. In fact, we observe an optimal stiffness range for these couplers.

The optimal stiffness can be directly related to some intrinsic properties of sleeve-type biased diffusion couplers. We have previously shown (42) that in the Hill sleeve-type biased diffusion model, thermal diffusion, as well as binder spacing, can significantly affect the force velocity response of the coupler. In this study we show that flexibility and diffusion of unbound kt binders eliminate the need for diffusion of the binders on the kMt lattice. More specifically, highly flexible (soft) kt components generate an effective thermal motion of the coupler on the kMt lattice binding sites allowing for swift adjustment of the kt juncture on the depolymerizing filament; this leads to successful coupling. On the other hand, stiff couplers can support more force when engaged with a binder; however, they cannot reach enough binding sites, effectively operating as a sleeve-type biased diffusion coupler with strong specific binding and/or small diffusion coefficient (we have previously referred to these couplers as sticky couplers).

We conclude that diffusion of kt couplers is crucial to their functionality; however, the nature of that diffusion, whether on the kMt lattice, or because of the flexibility of unbound binders, is yet to be definitively established. Based on our model results and the observed kt flexibility, we expect the latter option to play a dominant role in kt motion.