Force transduction by the microtubule-bound Dam1 ring

Several models have been proposed to explain how the Dam1 ring can couple the kinetochore to a depolymerizing MT so as to produce a force. Over two decades ago Hill proposed the first quantitative model describing how a depolymerizing microtubule could be harnessed for the production of force (6). In this model a hypothetical sleeve surrounds the MT and provides the attachment to the kinetochore. An attraction between the sleeve and MT provides an energy barrier preventing detachment, but this sleeve may still be able to slide along the MT without paying the energy of detachment. More recent computational models are detailed, mechanistic and micromechanical. One such model has taken into account the energy predicted to be available due to the curling of PFs (17, 18, 19) and, following the discovery of the Dam1 ring, was extended to reflect current structural knowledge and incorporate the hypothesis that Dam1 forms rigid transient links to the MT (20). Another independent model postulated an electrostatic attraction maintaining the rings position at the tip of the MT (21), combined with a powerstroke. All recent models include a combination of the following features: 1) the intrinsic diffusion of the Dam1 ring; 2) an effective powerstroke due to curling PFs; 3) an attractive potential between Dam1 and the MT. However, whilst these models include many of the relevant physical features of the system and produce satisfactory simulations of a reliable force transduction system, the problem cannot be considered “solved” because many variants on these models are possible and they have not been quantitatively compared to data. Furthermore, the lack of discriminatory experimental data precludes validation. In light of this we feel that much can be gained from rigorously analyzing the contribution of the various features in order to determine their possible role.

In what follows we describe two distinct “minimal” models, both of which describe a functional Dam1-mediated force transduction system. In the protofilament model the splaying PFs at the depolymerizing end physically prevent the ring from sliding off. In binding models an attraction between the ring and MT provides an energy barrier preventing detachment. The two models are not mutually exclusive – a hybrid model, incorporating both contributions, may also apply although one of the constituent mechanisms will typically dominate. While it is straightforward to modify our analysis to include such hybrid models we neglect them here for clarity, since our purpose is to differentiate the contributions. In common with previous models we also neglect other molecular components, e.g., microtuble-associated proteins (MAPs) and kinases (22, 23), that certainly play important additional roles in vivo.

Some previous studies have incorporated a powerstroke, arising from the motion of PFs, in driving the motion of the Dam1 ring (17, 20, 21). Indeed, it has been demonstrated that PFs can push a bead attached to the side of a MT (24), with a force of about 5 pN per 1–2 PFs. However, it is also known that models in the “burnt bridges” (25) class require no powerstroke per se to generate motion (26). Rather, purely diffusive Brownian motion can be rectified if “bridges” (here segments of MT) are lost (depolymerize) after they have been crossed. No instantaneous physical force is required, although the resultant rectified Brownian motion does give rise to a force in the thermodynamic sense. Such models are, in turn, members of a larger class of models known as “Brownian ratchets” (27). These models exhibit velocities that depend on applied force, and stall for sufficiently high forces, as also seen in more complex models (21). It is not clear a priori to what extent the powerstroke plays an important role. In the present work we also seek to answer this question.

We seek to analyze a general model that includes both a diffusive burnt bridge mechanism and a powerstroke, in order to determine their relative contribution. Here the powerstroke involves a depolymerization event which “unzips” PFs and moves the position of the last unbroken section of MT; a new section takes on this identity when the previous one unzips (contains separated, splayed protofilaments). As a highly energetic powerstroke this is assumed to occur even when the ring is very close to the MT end, with a rate that gives rise to a depolymerization velocity $v_{\text{ps}}$. The sequence of microscopic PF unzippering events gives rise to a well defined velocity for the last fully intact MT section, irrespective of the sequence in which the neighboring PFs unzip, and the precise MT helicity. Critically, we also assume that polymerized MT can be lost with a second rate, giving a depolymerization velocity $v_{\text{bb}}$, whenever the ring has diffused a distance $\delta$ from the end. This can be thought of as the depolymerization velocity of a bare MT because, in this case, there is no Dam1 ring anywhere on the MT. We make no prior assumptions as to which contribution dominates, rather we determine this by fitting the parameters $v_{\text{bb}}$, $v_{\text{ps}}$ and $\delta$ to data for the variation of the Dam1 velocity with load (12).

Our model involves a clear distinction between two mechanisms (only) and represents the simplest possible model capable of explaining this data. It can be biophysically motivated on the grounds that the Dam1 ring interacts with neighboring tubulin and so the rate of PF unzippering at the MT end should depend on how close the Dam1 ring is to the end (a concept already introduced in (21)). In the section below we discuss this mechanism in terms of a putative energy landscape for the depolymerization (unzipping) reaction. We believe that it would be unjustified to postulate the existence of any features on this energy landscape beyond the minimum required to explain the data. This amounts to a model involving two (distinct) depolymerization mechanisms. Our results suggest that both mechanisms play important roles at moderate loads. In particular powerstroke-only models are clearly inconsistent with the data showing a velocity that is dependent on force (11, 12), assuming a strong power-stroke as previously measured on the order of $30-65$ pN (24). This is because, with a powerstroke-only model, we would not expect the velocity of Dam1 ring to be significantly slowed under a force as low as $2$ pN; the data shows a significant slowing. We find that the length scale $\delta$ controlling burnt-bridge reactions, a free fit parameter, is close to the axial length of a tubulin dimer. We speculate that this may provide indication of “crack”-like splitting of the MT, as discussed below.

The powerstroke and burnt-bridge reactions can be thought of as arising from transitions over an energy barrier of the form shown in Fig. 2, where the free energy $F$ of PF curling is shown as varying with protofilament angle $\theta$ and the distance of the Dam1 ring from the MT tip $x$. The figure shows only a putative schematic of the free energy of PF curling reaction, and should not be confused with the potential $V(x)$ in which the Dam1 ring diffuses.

PFs may produce a power stroke that pushes the ring with force $f_{\text{pf}}$, estimated from experimental evidence to be $30-65$ pN (24). This is the slope down the descending valley, diagonally right to left, in Fig. 2. Provided that the load force $f\ll f_{\text{pf}}$ the powerstroke will give rise to a depolymerization velocity $v_{\text{ps}}$ that is the rate at which the last intact dimer on the MT crosses the highest part of the ridge-like energy barrier in Fig. 2 ($x<\delta$). Since the estimate for $f_{\text{pf}}$ is so much larger than any force considered here, it is reasonable to make the limited assumption that $v_{\text{ps}}$ is constant for all experimentally measurable load forces of a few pN or less.

In addition the MT can also depolymerize when the Dam1 ring is further than a critical distance $\delta$ from the end of the MT. In this case the burnt-bridge reaction gives rise to a depolymerization velocity $v_{\text{bb}}$ that is the rate at which the last intact dimer on the MT crosses the lower part of the ridge-like energy barrier in Fig. 2 ($x>\delta$). That the rate of MT unzippering is retarded when the Dam1 ring is near the MT end is a result of the fact that the velocity decreases as the load force is increased and the ring is more often closer to the MT end. Although it is not necessary to interpret our model in terms of the Dam1 ring physically occluding the unzippering of the tubulin dimers, this interpretation may not be unreasonable, particularly in view of the fact that we find $\delta$ to be comparable with the axial length of the last intact ring of tubulin dimers.

The resultant velocity due to both mechanisms is the sum of the probability that the ring is close to the MT end $x<\delta$, multiplied by the powerstroke velocity, and the probability that it is far $x>\delta$, multiplied by the burnt-bridge velocity,

The velocity follows from Eqs. 3 and 2

The variation of this velocity with load is shown in Fig. 3 for $v_{\text{bb}}=580$ nm/s (29), and the values $v_{\text{ps}}=55$ nm/s and $\delta=14$ nm that correspond to the best fit to data (12). Since a “burnt-bridges”-only model fails to fit the data sufficiently (i.e. $v_{\text{ps}}>0$) it suggests that a powerstroke plays a role in forced Dam1 motion. It should be noted that, although in this model $v\rightarrow v_{\text{ps}}$ as $f\rightarrow\infty$, we do not suggest this is a physical feature of the system. Rather it is the consequence of the assumption that protofilaments are perfectly rigid and the powerstroke reaction is asymptotically strong. Our model would need modification for forces approaching $f_{\text{pf}}$. As discussed later, PFs are estimated to require tens of pN to bend.

We now proceed to calculate the mean time the Dam1 ring will remain on a MT and transduce force¹¹1Recently it has been discovered that the Dam1 oligomers track the tip of depolymerizing MTs without forming a ring (30, 31). It seems unlikely that a protofilament model could operate without a full ring, however, it is not known to what extent, if at all, small oligomers contribute to force production. Furthermore, it has been shown that 16-20 Dam1 complexes are present at the kinetochore during metaphase (32), enough to form the ring. We await the result of experiments where tension is applied to putative Dam1 oligomers. . This time is controlled by different physics in the protofilament and the binding  models, see Figs. 1 and 4. However, in both cases, the velocity of the ring is governed by the model described above, see Eq. 4.

We now consider an oscillating applied force of the form

Provided the period is sufficiently long $\omega^{-1}\gg\lambda^{2}/D$ our quasi-static approximation for the ring position should give an accurate estimate for its probability density $\phi(x,t)$.

The depolymerization velocity will be retarded according to Eq. 4, relating $v$ to $f(t)$.

The protofilament model exhibits the most sensitivity to the intrinsic (bare) MT depolymerization velocity $v_{\text{bb}}$, as is shown in Fig. 6 A. For the protofilament model, $\tau$ is strongly dependent on ${\langle{t_{\text{unzip}}}\rangle}$ and consequently $v_{\text{bb}}$. The binding model, on the other hand, is only weakly dependent on $v$ (see Supporting Material), and on this range of $v_{\text{bb}}$ we can assume that depolymerization is quasistatically slow with respect to ring diffusion. The result can be understood physically by realising that as $v_{\text{bb}}$ increases, the rate of PF unzipping $k_{\text{unzip}}$ also increases, while $k_{\text{break}}$ remains constant making it less likely that the PFs will break off sufficiently quickly to release the ring.

An experimental test that might be able to distinguish which model operates could be achieved, e.g., by addition of a depolymerization inducing agent, such as $\text{Ca}^{2+}$ or XMCAK1.

The diffusion constant $D$ of the ring is determined by the ring’s dimensions and the roughness of the binding energy landscape along the MT, rather than the magnitude of the binding energy itself. A more rough landscape reduces the mobility of the ring. Fig. 6 B shows the effect of the diffusion constant on the runtime for both models. Only the binding model is sensitive to change in $D$, having reduced runtime with faster diffusion. This is because the increased mobility of the ring increases the chance it is able to scale the potential barrier constraining it to the MT.

Although it may be possible to alter $D$ biochemically, for example by phosphorylation (30), it is difficult to do so independently of $\Delta G_{\text{ Dam1}}$. Decreasing $D$ may be better accomplished by attaching a long inert polymer to the complex to increase viscous drag.

The runtime in the binding model is sensitive only to the instantaneous force provided $2\pi/\omega\gg\lambda^{2}/D$, see the low frequency portion of Fig. 6 C. If $f_{1}=1$ pN then $2\pi/\omega_{\text{max}}$ is on the order of 1 kHz. The runtime in the protofilament model is sensitive to the time over which changes in $v$ persist. If the force is oscillating with a long period then the rate of detachment will be greater in the high force part of the cycle than if the period is short. This is because the Dam1 ring takes some time to detach if it needs to first wait for the PF curls to break, see Fig. 6 C. Sigmoidal increase of $\tau$ would be a signature of a system that depends on a second time (1/$k_{\text{break}}$), like the protofilament model; insensitivity of $\tau$ to frequency would imply a binding-style coupling.