Mechanics and kinetics of dynamic instability
Abstract
During dynamic instability, selfassembling microtubules (MTs) stochastically alternate between phases of growth and shrinkage. This process is driven by the presence of two distinct states of MT subunits, GTP and GDPbound tubulin dimers, that have different structural properties. Here, we use a combination of analysis and computer simulations to study the mechanical and kinetic regulation of dynamic instability in threedimensional (3D) selfassembling MTs. Our model quantifies how the 3D structure and kinetics of the distinct states of tubulin dimers determine the mechanical stability of MTs. We further show that dynamic instability is influenced by the presence of quenched disorder in the state of the tubulin subunit as reflected in the fraction of nonhydrolysed tubulin. Our results connect the 3D geometry, kinetics and statistical mechanics of these tubular assemblies within a single framework, and may be applicable to other selfassembled systems where these same processes are at play.
Introduction
Microtubules (MTs) are polar tubular polymers formed by the selfassembly of the protein tubulin. MTs are ubiquitous in eukaryotic cells, where they are a major component of the cellular cytoskeleton, and participate in a number of essential cellular functions, such as cell migration, morphogenesis, transport within cells and cell division (Desai, 1997; Gupta et al., 2015; Huber et al., 2013; Fletcher and Mullins, 2010). They are also involved in regulating the shape and dynamics of axons, cilia and flagella (MimoriKiyosue, 2011).
The basic building blocks of MTs are tubulin heterodimers. These are formed by αtubulin and βtubulin, two structurally similar globular proteins with mass of about 55 kDa. αβtubulin dimers are arranged longitudinally into flexible tubulin filaments called protofilaments (PFs). A number (between 9 and 16, typically 13) of such PFs then assembles by lateral interactions to form the MT lattice (Mandelkow et al., 1986; Chrétien and Wade, 1991; Mitchison, 1993; Tilney et al., 1973). MT growth occurs by the addition of tubulin dimers mainly at the plus end, where $\beta $tubulin is exposed. Upon hydrolysis of guanosinetriphosphate (GTP), tubulin subunits undergo a structural conversion that weakens lateral bonds, destabilises the subunit in the MT lattice and converts the relatively straight tubulin state into a state that is bound to guanosinediphosphate (GDP) and is characterised by an increased longitudinal curvature (Wang and Nogales, 2005; Alushin et al., 2014).
MTs are not static assemblies. They can repeatedly and stochastically vary their length by undergoing alternating phases of assembly and disassembly both in vivo and in vitro. This phenomenon is termed ‘dynamic instability’ and it is essential to a number of cellular functions, such as chromosome separation, the remodelling of spatial organisation of the cytoskeleton during mitosis or the exploration of extracellular environment (Mitchison and Kirschner, 1984; Gildersleeve et al., 1992; Cassimeris et al., 1988; Sammak and Borisy, 1988). Understanding the factors that regulate MT dynamic instability is central to cell physiology and disease. Yet a detailed understanding of dynamic instability still remains elusive (Aher and Akhmanova, 2018; Hemmat et al., 2018). This difficulty originates in part from the fact that dynamic instability is the result of several mechanical and kinetic aspects operating at multiple time and length scales (Figure 1).
As such, dynamic instability of MTs has been the focus of extensive experimental and theoretical work (Mandelkow et al., 1986; Chrétien and Wade, 1991; Mitchison, 1993; Tilney et al., 1973; Mitchison and Kirschner, 1984; Fygenson et al., 1994; Gildersleeve et al., 1992; Cassimeris et al., 1988; Sammak and Borisy, 1988; Hemmat et al., 2018; Aher and Akhmanova, 2018; Dogterom and Leibler, 1993; Brun et al., 2009; Antal et al., 2007; Mahadevan and Mitchison, 2005; Nogales et al., 1998; Wang and Nogales, 2005; Alushin et al., 2014; Zovko et al., 2008; Hoog et al., 2011; Mandelkow et al., 1991; Chrétien et al., 1995; Austin et al., 2005; Gardner et al., 2011; Hunyadi and Jánosi, 2007; Rice et al., 2008; Kueh and Mitchison, 2009; Jánosi et al., 1998; Molodtsov et al., 2005; VanBuren et al., 2005; Zapperi and Mahadevan, 2011; Bertalan et al., 2014b; Wu et al., 2009; Duellberg et al., 2016; Seetapun et al., 2012; Cheng et al., 2012; Cheng and Stevens, 2014; Jain et al., 2015; Aparna et al., 2017; Zakharov et al., 2015). Studies initially invoked a kinetic view capturing MT dynamic instability phenomenologically by different rates of polymerisation and depolymerisation depending on the state of the tubulinphosphate complex (Dogterom and Leibler, 1993; Brun et al., 2009; Antal et al., 2007). Evidence of changes in the structural properties of MT subunits during hydrolysis later showed how latticebound tubulin dimers undergo a structural conformational change that increases their curvature (Wang and Nogales, 2005; Alushin et al., 2014; Mahadevan and Mitchison, 2005), suggesting that outward curving tips of hydrolysed MTs can cause such structures to be mechanically unstable (Zovko et al., 2008; Hoog et al., 2011; Austin et al., 2005; Mandelkow et al., 1991; Chrétien et al., 1995; Gardner et al., 2011; Hunyadi and Jánosi, 2007; Rice et al., 2008; Kueh and Mitchison, 2009). Several coarsegrained computer simulations have since then adopted this structuralmechanical view, considering MT elasticity explicitly, to understand different aspects of dynamic instability, including hydrolysisdriven mechanical deformations near the cap (Jánosi et al., 1998), force generation by shrinking microtubules (Molodtsov et al., 2005), or 3D sheetlike/blunt tips (VanBuren et al., 2005), or stochastic microtubule tip configurations and their relation to catastrophe (Zakharov et al., 2015). Few studies, however, have attempted to capture this mechanical view with the aim of emphasizing the qualitative features necessary for dynamic instability and providing phase diagrams that delineate the zones where dynamic instability is seen. Those that exist, for example (Zapperi and Mahadevan, 2011; Bertalan et al., 2014b), focus on the 1D limit, where MTs are modelled as adsorbed chains: the predictions from these models can be qualitatively different from those that correctly account for the 3D geometry of MTs (see ‘Mechanical stability of 3D MTs in the presence of quenched disorder’).
Here, we use a combination of theory and simulations to establish how the kinetics of polymerization and the mechanics associated with the 3D geometry of MTs act together across multiple scales to regulate dynamic instability. We also establish the role of disordered remnants of GDPtubulin in determining the statistics of MT rescue. All together, our study provides a set of qualitative phase diagrams that delineate the regions of parameter space where dynamic instability is seen, consistent with previous observations while providing experimentally testable predictions.
Methods
Computational model
To complement the theory (see ‘A phase diagram for mechanical stability of 3D MTs’), we developed a minimal coarsegrained computational model of MT mechanics and dynamics. To characterize the tubulin heterodimers, we use two patchy spheres linked together by a flexible hinge. We derive the patchy particles from coarsegrained representations of colloidal particles that are decorated by patches on their surface; the patches represent specific anisotropic interactions that promote binding with patches on other particles. In our model, interactions between dimers are described by patches carrying two types of interactions: longitudinal and lateral contacts (Huisman et al., 2008). Longitudinal contacts link dimers headtotail, arranging them into PFs. Lateral contacts connect parallel PFs to form the cylindrical shell of the MT. Each dimer has two longitudinal contacts and four lateral ones, corresponding to three patches per monomer (Figure 2a). Interactions between two patches on monomers $i$ and $j$ are described by the following potential (Feng and Liang, 2012):
We see that there are three distinct contributions associated with the stretching (${V}_{S}$), bending (${V}_{B}$) and twisting (${V}_{T}$) modes, and we choose the following forms for these potentials:
where
Here, $r$ denotes the centertocenter distance between monomers, while the angles ${\theta}_{i}$ and ${\theta}_{j}$ describe the spatial directions of the patches (Figure 2b). The Morse potential term ${V}_{S}$, defined in Equation 2, describes the noncovalent interaction between patches, with $\u03f5$ being the depth of the potential well, ${r}_{0}$ the equilibrium distance between monomers, and $a$ is a parameter that controls the curvature of the potential well and, hence, determines the stretching modulus (see Equation 7). When $r<{r}_{m}$, where ${r}_{m}={r}_{0}\mathrm{log}(2)/a$, the potential ${V}_{S}$ behaves as an isotropic repulsive interaction. In the range ${r}_{m}\le r<{r}_{c}$, where ${r}_{c}=5{r}_{0}$ is the cutoff for ${V}_{S}$ (${V}_{S}$ is set to zero for $r\ge {r}_{c}$), ${V}_{S}$ is modified by multipliers ${D}_{\theta}({\theta}_{i})$ and ${D}_{\theta}({\theta}_{j})$. This yields an anisotropic attractive potential that exists only when the patches are aligned. Indeed, the multipliers ${D}_{\theta}({\theta}_{i})$ and ${D}_{\theta}({\theta}_{j})$, which are defined in Equation 6, weaken the attraction between the patches when these are not aligned (Figure 2c): ${D}_{\theta}$ reaches its maximum value when $\theta =0$. The cutoff of ${D}_{\theta}$ is ${\theta}_{d}=\pi /3$ and limits the influence of the patches within particular range of spatial directions. The potential terms ${V}_{B}$ and ${V}_{T}$, defined in Equations 3 and 4, characterise bending and twisting deformations respectively. They are described as classical harmonic potentials with curvature $b$, respectively, $c$, and are modified by the multipliers ${D}_{r}(r)$, ${D}_{\theta}({\theta}_{i})$ and ${D}_{\theta}({\theta}_{j})$, which are defined in Equations 5 and 6. These multipliers limit the range of ${V}_{B}$ and ${V}_{T}$ to specific spatial locations and directions. The cutoff of ${D}_{r}$ is set as ${r}_{d}=2.7{r}_{0}$, which is smaller than ${r}_{c}$ (the cutoff of ${V}_{S}$). This choice makes ${V}_{B}$ and ${V}_{T}$ shorterrange interactions compared to ${V}_{S}$.
The parameters in our coarsegrained computational model are linked to the mesoscopic mechanical properties of MTs (see section ’Mechanics’) including the interfilament spring stiffness $S$, the filament bending stiffness $B$ and the filament torsional rigidity $K$ as
where $l$ is the length scale of tubulin dimers. $l$ takes different values depending on whether we are calculating longitudinal or lateral properties. In particular, we set $l=2{r}_{0}=8$ nm when calculating longitudinal properties and set $l={r}_{0}=4$ nm for lateral ones (see Appendix 1 and Table 1 for further details on the computer simulation model and a summary of parameter choices). We choose the potentialwell parameters in our simulations such that the resulting mesoscopic mechanical parameters Equation 7 are consistent with the experimentally measured values of the mechanical properties of typical MTs (Gittes et al., 1993; Mickey and Howard, 1995; Felgner et al., 1996; Tolomeo and Holley, 1997; de Pablo et al., 2003; Sept and MacKintosh (2010); Deriu et al., 2010). Simulations of this coarsegrained model of MTs were performed using Molecular Dynamics (MD), as described in Appendix 1.
Results
A phase diagram for mechanical stability of 3D MTs
Our analytical model to study the mechanical stability of 3D MTs is a function of the underlying parameters describing MT mechanics, MT growth kinetics and subunit hydrolysis, which we first consider in the deterministic limit (see ’A phase diagram for mechanical stability of 3D MTs’). This forms the basis for studying the mechanical stability of MTs when there is heterogeneity in the state of tubulin, that is it could be either GTP or GDP bound (see ‘Mechanical stability of 3D MTs in the presence of quenched disorder’), and allows us to investigate the role of GTPremnants (containing random fractions of nonhydrolysed subunits) on rescue (see ‘Role of GTPremnants in rescue’).
The starting point of our mechanical model is that MTs exist as individual polymers with persistence lengths in the O(mm) range, which is much larger than the typical length of MTs (µm range) (Fletcher and Mullins, 2010; Huber et al., 2013). This observation suggests that MTs can be considered to have a welldefined shape that is not affected significantly by thermal fluctuations. Moreover, previous studies indicate that lateral bonds between tubulin dimers are considerably weaker than longitudinal ones (VanBuren et al., 2002; VanBuren et al., 2005; Molodtsov et al., 2005). We model MTs as a set of adherent PFs that have bending stiffness $B$; we approximate lateral interactions between PFs by a series of spring potentials (springs of stiffness $S$) and assume that extending these springs beyond a critical displacement causes MTs to become mechanically unstable and break, potentially leading to dynamic instability.
Kinetics
Previous studies (Wang and Nogales, 2005; Alushin et al., 2014) suggest that tubulin dimers that are part of the MT lattice have different mechanical properties depending on their hydrolysis state. In particular, upon hydrolysis the tubulin dimer undergoes a structural transformation from a relatively straight state to a state with finite curvature. In a 3D setting, the tubulin dimer can be curved both in the longitudinal direction and in the lateral direction. Let $\kappa (0)$ denote the longitudinal curvature of tubulin dimers in their GTPstate and let $\kappa (\mathrm{\infty})$ be the longitudinal curvature in the hydrolysed state. For simplicity, we assume that the hydrolysis reaction affects primarily the longitudinal curvature of the tubulin dimers, such that their curvature $\omega $ in the azimuthal direction can be considered to be constant. This assumption can be relaxed, see Appendix 2. Thus, as a result of GTPhydrolysis, the longitudinal curvature of tubulin dimers, $\kappa $, changes with time, which we assume follows first order kinetics so that
where ${k}_{H}$ is the rate of hydrolysis. While bound tubulin changes its structure, unbound (bound) tubulin can attach (detach) to (from) the free end of the MTs, which we also describe using a minimal first order kinetic law for the evolution of the length of the MT $n(t)$ (expressed in number of subunits) so that
Here ${k}_{G}$ is the net growth rate, ${k}_{+}$ is the elongation rate constant, ${k}_{}$ is the dissociation rate constant and, for simplicity, we have assumed a constant subunit concentration $[m]$ in solution.
Mechanics
In addition to MT growth and subunit hydrolysis, we also need to account for the elastic deformation of the MTs since the geometric state of the assembly is linked to its mechanical state. Individual PFs can bend, but are also constrained by interfilament interactions, so that there are two contributions to elastic energy: (i) curvature energy associated with the bending of PFs, (ii) stretching energy of the springs connecting neighbouring PFs. We capture these energy contributions in a continuum picture that describes a MT as a thin elastic surface of revolution obtained by rotating the function $R(x)={R}_{0}+u(x)$ along the long MT axis ($x$axis), where $R(x)$ is the local radius of MT and ${R}_{0}$ is the natural radius (Figure 2c). In the small gradient approximation, corresponding to ${u}^{\prime}\ll 1$, the total elastic energy can be written as (see Appendix 2 for details):
where $\mathrm{\Sigma}=B{(1+\omega {R}_{0})}^{2}/{R}_{0}^{2}$ and ${}^{\prime}=\partial /\partial x$ denotes derivative with respect to $x$. The first term in Equation 10 is the energy of MT that penalises deviations from its natural curvature $\kappa (t)$ which itself reflects its state of hydrolysis. The second term is a surface energy term that penalises area increase due to the outward curving of the surface. The third term is the stretching energy of the springs. The minimum energy configuration results from a competition between bending energy, which favours a natural curved MT state, and elastic spring energy, which favours a straight cylindrical MT configuration. The overall shape of the axisymmetric tubule is then obtained by solving the EulerLagrange equation associated with Equation 10 (see Appendix 2 for details):
subject to the boundary conditions $u(\mathrm{\infty},t)={u}^{\prime}(\mathrm{\infty},t)=0$ (fixed minus end), ${u}^{\prime \prime}(0,t)=\kappa $ and ${u}^{\prime \prime \prime}(0,t)=0$ (free plus end), and is coupled to the kinetic Equations 8 and 9.
Condition for mechanical stability
There are three natural dimensionless parameters (two mechanical parameters and one kinetic parameter) in our model that read:
The first parameter α describes the effect of longitudinal curvature. The second parameter β pertains to lateral curvature. Coupling these mechanical parameters to the kinetics of subunit hydrolysis and MT growth introduces an additional relevant dimensionless parameter $\gamma $, which is the ratio of the rate of hydrolysis of GTPtubulin dimers to the net rate of addition of GTPsubunits to the MT plus end (see Appendix 2 for details).
These parameters serve as the basis for a phase diagram for the mechanical stability of a 3D MT (Figure 3a). Assuming that a mechanical instability arises when the elastic MT is deformed so that the radial displacement crosses a critical value ($u(0)>{u}_{c}$), we can solve Equation 11 in terms of the maximal deformation $u(0)$ to yield a condition for when the MT is mechanically unstable. Rewriting this in terms of the scaled longitudinal curvature yields a critical value (see the Appendix 2 for details):
above which MTs are mechanically unstable (the transition curve in the αβplane in Figure 3a). This critical value depends on the lateral curvature parameter β, which is related to MT radius through $\beta \simeq 1/{R}_{0}$. In particular, the critical value for $\alpha $ is maximal ($\alpha =1$) when $\beta =0$, that is ${R}_{0}\to \mathrm{\infty}$. This situation corresponds to the limit of a onedimensional MT (Zapperi and Mahadevan, 2011). The critical value for $\alpha $ then decreases with increasing lateral curvature β, that is decreasing MT radius. Overall, these results suggest that MTs with smaller radius are mechanically less stable than MTs with larger radius, consistent with the intuition that increasing azimuthal curvature increases the mechanical strain on the MT and thus makes it more likely to fracture. Furthermore, increasing the rate of MT growth over hydrolysis acts to stabilise MTs mechanically. In the mechanically stable phase, hydrolysis is slower than the addition of GTPtubulin at the plus end, leaving a stabilising GTPcap of size $n$ (see Appendix 2, Video 1). In the mechanically unstable phase, hydrolysis is faster than subunit addition at the plus end. Consequently, the ‘hydrolysis front’ takes over the ‘growth front’, which destabilises MTs. In this case, the PFs curve outward near the plus end, leading to a characteristic morphology of depolymerising MTs that resembles rams’ horns (VanBuren et al., 2005) (see Appendix 2, Video 2). The transition curve separating these fast and slow hydrolysis regimes depends on both the longitudinal and lateral curvatures of the MT.
In Figure 3b, we show that crossing the transition curve given by Equation 13 causes a switch from the mechanically stable phase into the mechanically unstable phase. This could result from variations in either $\alpha $, $\beta $ or $\gamma $. For MTs in the mechanically stable phase, catastrophic failure can still occur via thermal activation. In this regime, the rate of catastrophe follows Arrhenius’ law ${r}_{c}\simeq \mathrm{exp}(\mathrm{\Delta}E/{k}_{B}T)$, where ${k}_{B}T$ is the thermal energy and $\mathrm{\Delta}E$ is the energy barrier given by $\mathrm{\Delta}E\propto \alpha 1+{\beta}^{2}$ is a measure of the ‘distance’ from the transition curve Equation 13 in the phase diagram of Figure 3b; the further away a MT is from this transition curve, the less likely it is to undergo catastrophe. The dependence of the rate of catastrophe on MT radius is through the parameter β, such that $\mathrm{ln}{r}_{c}\propto 1/{R}_{0}^{2}$. Thus, at constant temperature and at fixed values of the mechanical parameters, the rate of catastrophe increases with decreasing MT radius or, equivalently, decreasing PF number.
Comparison with computer simulations
We used our computer simulations to test the prediction from Equation 13 for how the critical value of $\alpha $ varies with β, which is a function of MT radius ${R}_{0}$ that is controlled by changing the number ${N}_{\mathrm{f}}$ of PFs in the MT. The results (Figure 3b) show that the critical value for $\alpha $ is maximal when $\beta =0$, and decreases with increasing β, in agreement with the theoretical prediction of Equation 13 (solid line).
Mechanical stability of 3D MTs in the presence of quenched disorder
Having considered the deterministic limit where growth kinetics, hydrolysis and longitudinal/lateral curvatures characterise the mechanical stability of a 3D MT, we now consider the role of randomness by including a random fraction of GTPtubulin dimers in their lattice. We can model this situation by introducing quenched disorder in the state of the tubulin subunit. Quenched disorder describes the general situation when certain parameters in the system become random variables; disorder can be considered to be ‘quenched’ when the probability distribution of parameter values either does not vary with time or it varies with time slowly compared to some underlying fast dynamics, and thus cannot be described solely using equilibrium statistical mechanics. In the context of MTs, such a separation of timescales emerges very naturally when comparing fast polymerization/depolymerization kinetics and the comparatively slower GTP turnover.
Since the primary mode of MT instability is due to the breaking of lateral bonds, a natural parameter for discussing the role of disorder is the spring stiffness $S$. The underlying motivation for this choice is that mechanical forces can influence the rates of chemical reactions. Indeed, mechanical work contributes to the free energy, which in turn determines the rates of a chemical reaction (Howard, 2001). In our context, GDPtubulins within the MT lattice experience mechanical stresses due to the strong curvature. These mechanical stresses can shift the polymerisationdepolymerisation equilibrium and favour free monomers. This is consistent with the idea that GDPtubulins are less tightly bound to MTs than GTPtubulins (Wang and Nogales, 2005; Alushin et al., 2014), even if the chemical bonds are identical.
Thus, instead of having a welldefined spring constant $S$ throughout the MT, we consider a MT with varying $S$. Each lateral interaction is characterised in principle by a different spring constant $S$, which is drawn from a time–independent probability distribution $p(S)$ of spring constants. For convenience, we choose the Gammadistribution
where $\mathrm{\Gamma}(x)$ is the Gamma function, $\u27e8S\u27e9$ is the average spring stiffness, and the parameter $1/k=\sigma /\u27e8S\u27e9$, with $\sigma $ being the standard deviation of the distribution, is the coefficient of variation that describes the degree of disorder in the system. The choice of the Gammadistribution admits a simple parameterisation in terms of the coefficient of variation that allows us to explore a range of different extreme value statistics. For instance, for $k=1$ the Gamma distribution $p(S;1,\u27e8S\u27e9)$ yields the exponential distribution with intensity $\lambda =1/\u27e8S\u27e9$, while for $k\gg 1$ it yields a normal distribution with mean $\mu =k\u27e8S\u27e9$ and variance ${\sigma}^{2}=k{\u27e8S\u27e9}^{2}$. We distinguish two limiting modes for disorder: lateral (Figure 3c) and longitudinal (Figure 3d). Any realisation of disorder can then be decomposed into a combination of these two limiting modes.
Lateral disorder
In the presence of disorder in $S$, lateral interactions are characterised by variations in $S$ azimuthally but not longitudinally. In this case, whether the MT undergoes catastrophe depends on the breaking of the weakest lateral bond along the circumference of the MT, resulting in a MT that is ‘cutopen’ along the longitudinal direction (Figure 3c and Appendix 2, Video 3). This situation is fully analogous to what happens when pulling a onedimensional chain by its ends: the chain will break as soon as its weakest link breaks. The mechanical stability of a MT with lateral disorder is thus equivalent to the mechanical stability of a MT with uniform spring stiffness $\u27e8{S}_{\mathrm{min}}\u27e9$, where $\u27e8{S}_{\mathrm{min}}\u27e9$ denotes the average value of the weakest spring stiffness along the MT circumference. This replacement maps the study of the mechanical stability of a MT with lateral disorder onto a problem of extreme value statistics (Zapperi and Mahadevan, 2011; Bertalan et al., 2014b): the determination of $\u27e8{S}_{\mathrm{min}}\u27e9$ for a system of $N$ independent and identically distributed links with spring constants ${S}_{1},\mathrm{\cdots},{S}_{N}$. In Appendix 2, we show that $\u27e8{S}_{\mathrm{min}}\u27e9$ can be calculated from Equation 14 using extremevalue statistics, yielding:
The condition for mechanical instability of a MT with lateral disorder in $S$ is thus obtained by replacing $\u27e8S\u27e9$ by $\u27e8{S}_{\mathrm{min}}\u27e9$ in Equation 13, yielding:
Since $\u27e8{S}_{\mathrm{min}}\u27e9<\u27e8S\u27e9$, Equation 16 predicts that the transition curve between mechanically stable and unstable MT regions shifts towards the instability region. Thus, lateral disorder weakens MTs.
Longitudinal disorder
A different situation arises when quenched disorder is distributed longitudinally. Here, the strongest lateral bond determines MT stability. In fact, longitudinal disorder leads to the presence of ‘rings’ of particularly strong bonds that prevent the MT from depolymerising completely (Figure 3d and Appendix 2, Video 4). The mechanical stability of a MT with longitudinal disorder is thus equivalent to that of a MT with uniform spring stiffness $\u27e8{S}_{\mathrm{max}}\u27e9$, where $\u27e8{S}_{\mathrm{max}}\u27e9$ is the expected value of $S$ associated with the strongest lateral bond. Using extremevalue statistics, one finds (see Appendix 2) $\u27e8{S}_{\mathrm{max}}\u27e9/\u27e8S\u27e9=({\gamma}_{e}+\mathrm{log}N)/k$, where ${\gamma}_{e}\approx 0.5772$ is the EulerMascheroni constant (Taloni et al., 2018; Bertalan et al., 2014b; Zapperi and Mahadevan, 2011). Hence, the curve separating mechanically stable and unstable regions in the presence of longitudinal disorder is Zapperi and Mahadevan (2011):
Since $\u27e8{S}_{\mathrm{max}}\u27e9>\u27e8S\u27e9$, longitudinal disorder in $S$ reinforces MTs.
It is important to note that longitudinal disorder is the only mode of disorder present in a 1D MT (Zapperi and Mahadevan, 2011). Lateral disorder is thus a defining feature of the 3D geometry of MTs. Our results thus reveal a fundamental role of MT dimensionality: while in a 1D setting quenched disorder stabilises MTs mechanically, in a 3D setting it can destabilise MTs.
Comparison with computer simulations
We have tested the theoretical predictions of Equation 16 and Equation 17 using our coarsegrained simulations (Figure 3c,d). Quenched disorder was realised using Equation 14 with disorder parameter $k=50$. These simulations confirm that longitudinal quenched disorder increases the mechanical stability of MTs (Figure 3c), whereas the effect of quenched disorder in the lateral direction is to destabilise MTs mechanically (Figure 3d).
Role of GTPremnants in rescue
Using our theoretical model of mechanical stability of 3D MTs in the presence of quenched disorder, we are now in the position to investigate the role of remnants of GTPtubulin in rescue. Rescue refers to the transition from depolymerisation to polymerisation during MT dynamic instability but is still poorly understood (Brouhard, 2015). Experiments indicate that GTPtubulin addition at the plus end is not critical for rescue (Gardner et al., 2013; Walker et al., 1988), but that the presence of remnants of GTPtubulin along the MT lattice in socalled ‘GTPislands’ can lead to MT rescue (Tropini et al., 2012; Dimitrov et al., 2008; Aumeier et al., 2016; Gardner et al., 2013; Vemu et al., 2018). In particular, experiments in vivo have revealed a strong correlation between rescue probability and the presence of remnants of GTPtubulin in older parts of the MTs, suggesting that these ’GTPremnants’ can function as rescue sites (Dimitrov et al., 2008). This view was further supported by the observation that the presence of a slowly hydrolyzable analogue of GTP bound to tubulin subunits contributes to MT rescue (Tropini et al., 2012. Aumeier et al., 2016) also demonstrated the possibility to generate GTPislands along the MT lattice in a controlled manner by means of laser damaging and subsequent repair of the damaged site by incorporation of GTPtubulin from solution (Schaedel et al., 2015): rescue occurred at laserdamaged sites in the presence of free GTPtubulin (Aumeier et al., 2016). Separately, recent studies (Vemu et al., 2018; Vemu et al., 2019) reported of a damagerepair mechanism that stabilises MTs mediated by the enzymes spastin and katanin. Overall, these studies suggest that disordered GTPislands in an otherwise structurally periodic lattice are involved in rescue regulation. Since these GTPremnants are characterised by a random mixture of different states of tubulin, we ask if our framework might help to quantify these observations.
Computer simulations
We first used our coarsegrained simulations to study the role of disordered GTPremnants in MT rescue. We generated reinforcing islands by inserting, in the middle of a fully hydrolysed, depolymerising MT, a ring consisting of several layers of GTPtubulin dimers (Figure 4a). We then observed whether the reinforcing GTPislands were able rescue the depolymerising MTs as a function of two parameters: 1) the length of the GTPisland ${N}_{\mathrm{rf}}$ (defined here as the number of layers in the island) and 2) the fraction $\varphi $ of GTPtubulin in the island. The results of these simulations (Video 5) are shown in Figure 4b. Note that the parameter $\varphi $ controls the amount of disorder present in the island at the level of GTPhydrolysis. This mimics both the scenario when rescue islands are formed because not all GTPtubulin is able to hydrolyse, as suggested in Dimitrov et al. (2008), or when rescue islands result from the incorporation of GTPtubulin during the repair process of a damaged site, as suggested in Aumeier et al. (2016). If disorder varies slowly over time compared to the characteristic timescale of polymerisation/depolymerisation, we can model slow changes of MT mechanical properties by making the relevant mechanical parameters explicit functions of time. For the parameters in our simulation, we find that when the reinforcing island is one layer long (${N}_{\mathrm{rf}}=1$), the probability of rescue is close to zero, irrespective of the GTPfraction in the reinforcing island. Interestingly, when ${N}_{\mathrm{rf}}>1$, we observe that rescue probability ${p}_{\text{rescue}}$ increases with $\varphi $ in a highly nonlinear manner. Specifically, ${p}_{\text{rescue}}$ is either close to zero or close to one for most values of $\varphi $, with a sharp increase in the transition region.
Percolation model of rescue
To qualitatively understand the observed nonlinear behaviour of rescue probability with GTPfraction in the reinforcing island $\varphi $, we propose a site percolation model of rescue (Figure 4c). Site percolation is concerned with the following question: given a random graph, in which each site is (independently) occupied with probability $q$ or empty with probability $1q$, what is the probability that a connected path of occupied sites exists between the boundaries of the graph? In our percolation model of rescue, each site of the reinforcing island is occupied by a GTPtubulin dimer with probability $\varphi $, while it is occupied by a GDPtubulin dimer with probability $1\varphi $. In Sec. ’Role of quenched disorder in mechanical stability of MTs’, we have shown that the presence of randomly distributed weak lateral bonds (mediated by GDPtubulin) can destabilise MTs mechanically when disorder is longitudinal. As such, a MT will be mechanically unstable when a connecting path of GDPtubulins runs longitudinally through the reinforcing island (Figure 4c). The question of whether a reinforcing island with GTPfraction $\varphi $ is able to rescue a depolymerising MT is thus analogous to site percolation with $q=\varphi $. The rescue probability ${p}_{\text{rescue}}$ thus relates to $1{p}_{\mathrm{perc}}$, where ${p}_{\mathrm{perc}}$ is the probability of percolation of a longitudinal path of GDPtubulin subunits through the length of the reinforcing island. Figure 4d shows that the results of site percolation on a square lattice of dimensions $13\times {N}_{\mathrm{rf}}$ with varying $\varphi $ are in qualitative agreement with simulated rescue probabilities (Figure 4b).
Discussion
Our multiscale approach to dynamic instability incorporates the mechanics and 3D geometry of MTs, the kinetics of tubulin addition and GTPhydrolysis, and quenched disorder in the state of the tubulin subunit. Our results provide a series of phase diagrams for the presence of dynamic instability, revealing the dimensionless mechanical and kinetic parameters controlling the problem. Compared to previous analytic studies of dynamic instability, our results reveal the key role of the 3D geometry of MTs. In particular, we find that the mechanical stability of a MT is strongly affected by its radius (Mahadevan and Mitchison, 2005); a MT with a smaller radius is mechanically less stable than an identical MT with a larger radius.
Since MT radius has been shown to vary because the number ${N}_{\mathrm{f}}$ of PFs composing MTs typically ranges between 9 to 16 (Chrétien et al., 1992; Chalfie and Thomson, 1982; Chaaban and Brouhard, 2017; Cueva et al., 2012; Díaz et al., 1998), our study suggests quantitative experimental tests via measurements of catastrophe rates ${r}_{c}$ as a function of PF number ${N}_{\mathrm{f}}$ to verify the theoretical prediction for the rate of catastrophe $\mathrm{log}({r}_{c})\propto 1/{N}_{\mathrm{f}}^{2}$. Another key prediction from our study is that the rescuing power of GTPislands displays a sharp drop at intermediate values of GTPfraction. In particular, the percolation model predicts that there is a critical point for the GTPfraction, $\varphi ={\varphi}_{c}$, below which reinforcing islands lose their ability to rescue MT disassembly. The numerical value of the threshold ${\varphi}_{c}$ depends on the thickness of the reinforcing island as well as on the MT lattice structure. This critical GTPfraction could be determined experimentally and compared to theory by using nonhydrolyzable analogs of tubulin (Tropini et al., 2012), to control the amount of disorder at the level of the state of tubulin subunit in the island, in combination with superresolution microscopy (Huang et al., 2009) to establish island length. Finally, we note that our assumption of the form of the quenched disorder in terms of the Gamma distribution is just that  an assumption. Further experimental work will be required to solve the inverse problem of estimating the average GTPfraction in GTP remnants from rescue probabilities determined experimentally (Dimitrov et al., 2008; Aumeier et al., 2016; Vemu et al., 2018) to see if we might determine both the form of the disorder and the intrinsic parameters characterising it, thus allowing future research to address the question of how to control dynamic instability.
Appendix 1
Computer model
We assign a local coordinate system to each dimer to describe its overall rotation, as well as the relative position of the patches within the dimer. To describe the whole dimer, we need three basis vectors $t,s{\textstyle \text{and}}n$ is directed along the long dimer axis and points from the $\mathit{\bm{\alpha}}$tubulin to the $\mathit{\bm{\beta}}$tubulin. $\mathit{\bm{s}}$ and $\mathit{\bm{n}}$ lie in the plane perpendicular to $\mathit{\bm{t}}$. This coordinate system is particularly convenient for describing the rotation of a dimer as a whole. However, it cannot capture the relative rotation of the $\mathit{\bm{\alpha}}$ and $\mathit{\bm{\beta}}$tubulin monomers in the dimer. This inner rotation can only be along the axis $\mathit{\bm{n}}$, and thus two patches coordinate systems (one for each tubulin in the dimer) can be obtained by rotating the local coordinate system around $\mathit{\bm{n}}$ as:
where $\xi $ is the angle between the vectors ${\mathit{\bm{t}}}_{\mathit{\bm{\alpha}}}$ and ${\mathit{\bm{t}}}_{\mathit{\bm{\beta}}}$ (the axes of the two monomers), as shown in Appendix 1—figure 1a.
There are two types of connections between dimers: longitudinal and lateral contacts. Longitudinal contacts link dimers head to tail and arrange them into PFs. Lateral contacts connect parallel PFs arranging them into a cylindrical MT shell. Each dimer has two longitudinal contacts and four lateral ones, that is there are six patchy points on each dimer. These are numbered as in Appendix 1—figure 1b or Figure 2a of the main text. The coordinates of the six patchy points are described by the following equations:
The values of the angles ${\chi}_{chain},{\chi}_{side1},{\chi}_{side2}$ are given in Table 1.
Appendix 2
Theoretical model
Here we provide additional mathematical details pertaining to the theoretical model discussed in the main text, in terms of the elastostatics of the structure and kinetics of polymerisation/depolymerisation and hydrolysis. Finally, we study this model in the presence of quenched disorder.
Mechanical stability of 3D MTs (statics)
Our model considers two contributions to the total elastic energy of MTs: 1) a curvature energy term describing the bending of PFs in the MT and 2) the stretching energy of the springs that connect neighboring PFs. We now consider these two energy contributions in detail.
Curvature energy
We describe the curvature energy term by means of the Helfrich (1973):
where ${k}_{1}$ and ${k}_{2}$ are bending rigidities, $H=({\kappa}_{1}+{\kappa}_{2})/2$ is the mean curvature, $K={\kappa}_{1}{\kappa}_{2}$ is the Gaussian curvature, ${\kappa}_{1}$ and ${\kappa}_{2}$ are the principal curvatures, and $dA$ is the surface area element. We describe the MT as a continuous elastic sheet modelled as a solid of rotation obtained by rotating the function $R(x)$ along the $x$axis, which describes the microtubule growth axis (Appendix 2—figure 1a). We assume that the MT possesses a natural radius ${R}_{0}$ (when flat) and, for convenience, we write $R(x)={R}_{0}+u(x)$, where $u(x)$ describes the deviation of local radius $R(x)$ from ${R}_{0}$. The MT surface is thus parameterised as $(x,R(x)\mathrm{cos}\phi ,R(x)\mathrm{sin}\phi )$ where $x\in [0,\mathrm{\infty})$ and $\phi \in [0,2\pi )$. In this parametrisation, $x=0$ corresponds to the MT plus end, while $x=\mathrm{\infty}$ corresponds to the minus end. We then use the following expressions for the mean and Gaussian curvatures of a surface of revolution Gray, 1997:
where ${}^{\prime}=d/dx$ denotes derivative with respect to $x$, and the surface area element is $dA=2\pi R(x)\sqrt{1+{R}^{\prime}{(x)}^{2}}dx$. From these expressions, we note that the Gaussian curvature term, $K(x)dA={R}^{\prime \prime}(x)/{[1+{R}^{\prime}{(x)}^{2}]}^{3/2}dx$, is fully integrable, that is it gives rise to boundary terms only. We thus focus on the mean curvature term only. To this end, we consider a small gradient approximation, which corresponds to ${u}^{\prime}(x)\ll 1$. We then write $R(x)={R}_{0}+u(x)$ in Equation 31 and after keeping only the leading order terms, we find
where $\mathcal{R}$ stands for higher order terms in ${u}^{\prime}(x)$ or for boundary terms (i.e. terms that are fully integrable and thus, after integration, simply shift the curvature energy by a constant value).
Our calculation so far has accounted for bending energy relative to a flat cylindrical MT. To account for the natural longitudinal and lateral curvatures of subunits ($\kappa $ and $\omega $, respectively), we extend the Helfrich Hamiltonian as follows:
where ${H}_{0}=(\kappa +\omega )/2$ is the natural (or spontaneous) mean curvature. In the small gradient approximation (${u}^{\prime}\ll 1$), the leading contribution to the curvature energy in the presence of natural curvatures $\kappa $ and $\omega $ is found to be
Thus, the curvature energy is
where $B=2\pi {R}_{0}{k}_{1}$ and $\mathrm{\Sigma}=B{(1+\omega {R}_{0})}^{2}/{R}_{0}^{2}$.
Elastic spring energy
The second contribution to the total elastic energy of MTs is due to the stretching energy of the springs connecting neighbouring PFs. To construct this energy contribution, we model the cross section of the MT as a set of PFs connected by ${N}_{\mathrm{f}}$ harmonic springs with stiffness ${k}_{s}$ (Appendix 2—figure 1b). The stretching energy is proportional to the square of the deviations of the distance $d$ between PFs from their equilibrium position ${d}_{0}$. By symmetry in the $\phi $ direction, springs are stretched by $d(x)\simeq {\theta}_{N}R(x)=2\pi R(x)/{N}_{\mathrm{f}}$ (${\theta}_{N}$ is defined in Appendix 2—figure 1b). The rest length of springs is ${d}_{0}={\theta}_{N}{R}_{0}=2\pi {R}_{0}/{N}_{\mathrm{f}}$. Thus, the stretching energy per unit length of the system of ${N}_{\mathrm{f}}$ springs can be estimated as:
Thus, writing $R(x)={R}_{0}+u(x)$, the stretching energy is found to be:
where $S=4{\pi}^{2}{k}_{s}/{N}_{\mathrm{f}}$.
Total elastic energy
In summary, the total elastic energy of a 3D MT can be written in terms of $u(x)$ by combining the contributions from curvature energy and spring energy:
The first term describes the longitudinal bending energy of the MT away from the natural curvature $\kappa $. The second term comes from the lateral curvature. At leading order, this term is a surface energy term that penalises the increase in surface area when the MT curves out; the parameter $\mathrm{\Sigma}$ plays the role of a surface tension. Finally, the third term is the stretching energy of the springs, which gives rise to an energy density contribution proportional to $u{(x)}^{2}$.
EulerLagrange equation and minimum energy configuration
Having defined the elastic energy functional for the problem, ${\mathcal{E}}_{tot}[u]={\int}_{0}^{\mathrm{\infty}}\mathscr{H}(u,{u}^{\prime},{u}^{\prime \prime})\mathit{d}x$ (Equation 38), we now consider the associated EulerLagrange equation that describes the minimal energy configuration of MTs:
Using Equation 38, Equation 39 is found to be:
Subject to the following boundary conditions
the analytical solution to Equation 40 reads:
where
and the two relevant length scales in the problem are
The first length scale is associated with the lateral curvature of the MT. In fact, from $\mathrm{\Sigma}=B{(1+\omega {R}_{0})}^{2}/{R}_{0}^{2}$ it follows ${\mathrm{\ell}}_{\alpha}={\left(B/\mathrm{\Sigma}\right)}^{1/2}\simeq {R}_{0}$.
Condition for mechanical stability/instability of 3D MTs
To study the mechanical stability/instability of MTs as a function of its mechanical parameters, we assume that the springs connecting PFs break if their extension exceeds a critical value ${u}_{c}$. From Equation 44 we thus obtain the following condition for mechanical instability:
To rewrite this in a more transparent way in terms of the dimensionless parameters, we first define the scaled longitudinal and lateral curvatures:
so that the parameters $\alpha $ and $\beta $ are related to the characteristic length scales defined in Equation 46 through
where ${\mathrm{\ell}}_{c}=\sqrt{{u}_{c}/\kappa}$ is a length scale corresponding to the geometric average of the longitudinal radius of curvature and the critical extension ${u}_{c}$.
The condition Equation (47) can be reformulated most conveniently as
The resulting curve is shown in a phase diagram in Appendix 2—figure 4a.
Scaling behaviour of critical curvature with mechanical parameters

Appendix 2—figure 3—source data 1
 https://cdn.elifesciences.org/articles/54077/elife54077app2fig3data1v1.xls
Equation (50) predicts that there is a critical value for the longitudinal curvature, ${\kappa}_{c}$, above which MTs are mechanically unstable. We now study the scaling behaviour of ${\kappa}_{c}$ with system parameters such as the bending stiffness $B$ and spring stiffness $S$. To this end, we performed a series of computer simulations by varying the longitudinal bending stiffness $B$ of PFs and the lateral connecting strength $\u03f5$, which in turn determines the spring stiffness $S$. The results are shown in Appendix 2—figure 3a. We find that ${\kappa}_{c}$ follows a scaling law with the longitudinal bending stiffness $B$ and $S$, ${\kappa}_{c}\sim {(B/S)}^{\sigma}$. This scaling behaviour is shown in the doublelogarithmic plots in Appendix 2—figure 3a, where the slope corresponds to the scaling exponent $\sigma \sim 0.53$. The observed scaling behaviour can be rationalised using (50), which can be rewritten as
where $\sigma $ is the scaling exponent, given by
After rewriting Equation 47 as
we find
where in the second step we have used ${\mathrm{\ell}}_{\beta}={(B/S)}^{1/4}$ (Equation 46). Finally, using $\partial {x}^{a}/\partial \mathrm{log}x=a{x}^{a}$, we arrive at:
The scaling exponent $\sigma $ interpolates between −0.5 for ${\mathrm{\ell}}_{\beta}\ll {\mathrm{\ell}}_{\alpha}$ (limit of a singlefilament model of MT Zapperi and Mahadevan, 2011) and −1 for ${\mathrm{\ell}}_{\beta}\gg {\mathrm{\ell}}_{\alpha}$ (limit of strong lateral curvature), see Appendix 2—figure 4b. This scaling behaviour for ${\kappa}_{c}$ has been verified using our coarse grained simulations of depolymerising MTs in Appendix 2—figure 3a of the main text. Representative values for the mechanical parameters in the simulations are $B=1.23\times {10}^{26}$ Nm^{2}, $S=91$ MPa and $R=10$ nm. These values give ${\mathrm{\ell}}_{\beta}={(B/S)}^{1/4}\simeq 3.4$ nm, ${\mathrm{\ell}}_{\alpha}=10$ nm and, therefore, $\sigma \simeq 0.55$, in close agreement with the simulations in Appendix 2—figure 4a.
Coupling mechanical stability with MT kinetics
We now combine our static calculation of mechanical stability of MTs with the dynamics of subunit hydrolysis and MT polymerisation/depolymerisation. This allows us study how kinetics affect the phase diagram of Appendix 2—figure 3a. Let ${k}_{G}$ be the rate of addition of subunits to the MT end and let ${k}_{H}$ be the rate of hydrolysis of GTPtubulin dimers. With these parameters we can construct a dynamic dimensionless parameter as the ratio between the times for hydrolysis and growth
To understand how the dynamic parameter $\gamma $ modifies the static stability of the MTs, our starting point are the kinetic equations in the main text:
Equations 57 and 58 can be combined together by expressing time as $t=n/{k}_{G}$ (follows directly from the solution to Equation 58). The solution to Equation 57 can thus be expressed as
Using this parametrisation, we can express the condition for mechanical instability, Equation 47, as
This condition can be solved with respect to $\alpha $ to yield the following expression for the phase boundary:
The resulting phase diagram as a function of $\gamma $ is shown in Appendix 2—figure 4b. We see that increasing $\gamma $ increases the region of mechanical stability of MTs. The physical interpretation of this result is that MTs remain mechanically stable as long as the transition curve, Equation 47, is reached after a stabilising cap of length $n$ is added onto the MT end. Increasing the rate of MT growth, ${k}_{G}$, over subunit hydrolysis, ${k}_{H}$, favours the formation of the stabilising cap. In the limit when $\omega $ does not vary with time, Equation 61 reduces to
which can be solved to yield Equation 13 of the main text.
Mechanical stability of MTs in the presence of laterallydistributed quenched disorder
As argued in the main text, in the presence of lateral disorder, the mechanical stability of a MT is controlled by its weakest link. To understand this quantitatively, we replace the MT with lateral disorder with an equivalent one with uniform spring stiffness $\u27e8{S}_{\mathrm{min}}\u27e9$, where $\u27e8{S}_{\mathrm{min}}\u27e9$ denotes the average value of the weakest spring stiffness. This replacement maps the study of the mechanical stability of a MT with lateral disorder onto a problem of extreme value statistics (Zapperi and Mahadevan, 2011; Bertalan et al., 2014a): the determination of $\u27e8{S}_{\mathrm{min}}\u27e9$. To this end, consider $N$ independent and identically distributed links with spring constants ${S}_{1},\mathrm{\cdots},{S}_{N}$. We assume that the values of spring constants are random and distributed according to a Gammadistribution
where $\u27e8S\u27e9$ is the average spring stiffness (Appendix 2—figure 4a). The ratio between the standard deviation $\sigma $ and the average $\u27e8S\u27e9$
is the coefficient of variation, a key parameter which we use to describe the degree of disorder in the system; small values of $k$ correspond to nearly ordered system while large values correspond to a strongly disordered system.
With $S}_{\mathrm{m}\mathrm{i}\mathrm{n}}=\underset{i}{min}{S}_{i$ being the smallest value of spring constants and letting
be the cumulative probability distribution for $S}_{\mathrm{m}\mathrm{i}\mathrm{n}$ over the $N$ links, we can calculate ${P}_{N}(S)$ directly from the definition of ${P}_{N}(S)$, yielding
where
is the cumulative probability distribution of $S$. For large $N$, we can approximate the exact expression in Equation 67 as
The interesting behaviour of ${P}_{N}(S)$ happens for small values of $S$,when ${P}_{N}(S)$ is controlled by the lowvalue tail of $P(S)$. The cumulative distribution function $P(S)$ can then be calculated explicitly from Equation 64 as
where $\gamma (k,x)$ is the incomplete gamma function. Since we are interested in the lowvalue tail of $P(S)$, we can use the small $x$ expansion of $\gamma (k,x)$
to arrive at the following expression
which is valid for $S\to 0$ (Appendix 2—figure 4b). The cumulative probability distribution for ${S}_{\mathrm{min}}$ then converges to a Weibull distribution
The average value for the weakest spring constant ${S}_{\mathrm{min}}$ is therefore
Having solved the extremevalue statistics problem of determining $\u27e8{S}_{\mathrm{min}}\u27e9$ allows us to estimate the transition curve separating regions of mechanical stability and instability for a MT exhibiting lateral disorder. The curve separating these zones is obtained by replacing $\u27e8S\u27e9$ by $\u27e8{S}_{\mathrm{min}}\u27e9$ in Equation 50, yielding:
where we note that $\u27e8{S}_{\mathrm{min}}\u27e9<\u27e8S\u27e9$. Thus, by comparing Equation 75 with the deterministic result Equation 50, we see that, in the presence of lateral disorder, the curve separating mechanically stable from mechanically unstable MTs shifts in such a way as to increase the region of mechanical instability. Lateral disorder thus weakens MTs, making them more susceptible to catastrophic failure.
Data availability
All data generated or analysed during this study are included in the manuscript and supporting files.
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Decision letter

Raymond E GoldsteinReviewing Editor; University of Cambridge, United Kingdom

Detlef WeigelSenior Editor; Max Planck Institute for Developmental Biology, Germany

Jennifer RossReviewer; University of Syracuse, United States
In the interests of transparency, eLife publishes the most substantive revision requests and the accompanying author responses.
Acceptance summary:
One of the most characteristic features of microtubules is their stochastic growth and shrinkage. The present article provides a microscopic analysis of this process which incorporates the bending elasticity of free ends of filaments that curve away from the microtubule axis along with the effects of disorder in the distribution of hydrolysed tubulin. In bringing together ideas from mechanics and statistical mechanics this work provides new insights into microtubule dynamics.
Decision letter after peer review:
[Editors' note: the authors submitted for reconsideration following the decision after peer review. What follows is the decision letter after the first round of review.]
Thank you for submitting your work entitled "Mechanics and kinetics of dynamic instability" for consideration by eLife. Your article has been reviewed by two peer reviewers, and the evaluation has been overseen by a Reviewing Editor and a Senior Editor. The following individual involved in review of your submission has agreed to reveal their identity: Jennifer Ross (Reviewer #1).
Both reviewers found the work to be interesting and solid, but they raised
a number of issues regarding the novelty of the research, its connection with
previous work, its predictive nature, and its presentation. Based on consultation between the reviewers and editors, we regret to inform you that your work in its present form will not be considered further for publication in eLife. Please see below for the individual reviews. We would in principle be prepared to consider a new submission if the concerns of the reviewers are fully addressed. The paper and the pointbypoint rebuttal would then most likely be sent to the same reviewers.
Reviewer #1:
The manuscript "Mechanics and kinetics of dynamic instability" from Michaels et al., is a theoretical study of the fundamental causes behind microtubule dynamic instability. The manuscript specifically focuses on the effect of GTPislands, locations within the lattice where the tubulin has not fully hydrolyzed yet. They use the assumption that the lateral and longitudinal intertubulin binding interactions and the conformation of the tubulin dimers both depend on the GTPhydrolysis state of the tubulin dimer to control the stability of the microtubule structure. Overall, I found the model to be clear (although laced with too much undefined physics jargon) and interesting.
The positive aspects of the manuscript are:
1) The model is based on prior experimental and theoretical results that all seem reasonable.
2) They bring in the concept of frozenin (quenched) states where the GTPhydrolysis is stuck in some state to allow the presence of GTPislands that have been experimentally observed.
3) They also use the concept of percolation (connectedness) of these stable sites to understand why the phase change from growth to shrinkage would be so dramatically sensitive to the size and shape of the GTPisland.
Some problematic aspects of the manuscript:
1) The model put forward is fairly derivative as it is based on prior theory and models. It is entirely possible that I am mistaken, but if so, the authors should explain the novelty of their model, which is not wellexplained. What does this model bring to the table that prior models have not? This will help readers and people who might want to use it know when it would be appropriate to use or compare to their experiments.
2) The model suggests at the end that it could make predictions, but it actually fails to make those predictions. Actual, specific predictions would make the manuscript far more useful to experimentalists. For instance, what if you have a 9protofilament microtubule? How sensitive is it to protofilament number? Can it tell the difference between 9 and 10 protofilament microtubules? In the same field of view?
3) The model does not explain some important aspects of dynamic instability. Specifically, it does not explain the switch from growing phase to shrinkage phase. Or if it does, that was not articulated well. It does a better job of explaining rescue, but with the limitation that the GTPislands are frozen states. Although it is nice to confirm that this is true with a model, the model is not as realistic as it should be because it is unlikely that the GTPislands are actually frozenin. More likely, there is a rate of GTP turnover that is statistical and probabilistic, so the locations of the islands are dynamic and tend to break the connectedness (percolation) as time proceeds. I understand the importance of initial toy models, but this model, especially given the inarticulation of specific predictions, could do so much more.Reviewer #2:The manuscript by Michaels and coauthors develops theory and simulations of the dynamic instability of microtubules. The authors use two approaches: (1) a coarsegrained simulation model of a tubulin dimer with variable curvature and interacting patches to represent lateral and longitudinal interactions and (2) a continuum mechanical model of a microtubule as an elastic sheet. The authors use these models to show several results. First, they demonstrate that stronger lateral interactions and lower longitudinal interactions stabilize microtubules (favor growing over shrinking). Second, they introduce, without physical motivation, random variation in the spring constant of longitudinal and lateral interactions, and show that random variation of lateral interactions destabilizes microtubules (as expected, because breaking of lateral bonds is required for shrinking), but random weakening of longitudinal interactions strengthens microtubules (again as expected, because a single strong ring can keep the microtubule together). Third, the authors consider how hypothesized GTP islands could allow rescue, by introducing into the model GTP islands with varying properties into their simulations and studying the rescue probability. They find that the GTP island length and the GTP fraction of dimers in the island must be high enough for GTP tubulin to percolate the island to achieve a high probability of rescue.
While this paper is technically sound, in my opinion it does not meet the novelty standard of eLife. The results are exactly what one would intuitively expect, so the paper reads as using mathematics to demonstrate previously expected results on dynamic instability. The nature of the modeling and the results obtained are similar to many previously found. Previous related work is not cited, including Zakharov et al., 2015. In particular, the model in this manuscript is quite similar to that of Zakharov et al., so what new insight is gained from the current work is unclear.
Additionally, this manuscript does not cite recent work suggesting that tubulin dimers and protofilaments have intrinsic curvature in the GTPbound state, which would change the nature of the model and possibly the conclusions.
In addition, the authors have not made sufficient effort to use language consistent with that used in the field, or to explain technical terms from other fields. For examples, the authors refer to the rams' horns of shrinking MTs as a flowerlike shape. Similarly, the authors use the term quenched disorder without explaining clearly that it simply means random variation in a property of the model, which will confuse readers who are not already familiar with the condensedmatter usage of quenched disorder.
Therefore I recommend that the paper be published in a more specialized journal (such as the Biophysical Journal, which would be a good fit for this paper) and significantly rewrite the paper to clarify the presentation and the connection to previous work.
[Editors' note: further revisions were suggested prior to acceptance, as described below.]
Thank you for submitting your article "Mechanics and kinetics of dynamic instability" for consideration by eLife. Your revised article has been reviewed by
one of the original reviewers, and the evaluation has been overseen by a Reviewing Editor and Detlef Weigel as the Senior Editor. The following individual involved in review of your submission has agreed to reveal their identity: Jennifer Ross (Reviewer #1).
As you can see from the review below, there are only minor changes needed to the manuscript in order for it to be acceptable for publication. Please make these at your earliest convenience.
Reviewer #1:
The manuscript "Mechanics and Kinetics of Dynamics Instability" by Michaels, et al., is a revision of a prior submission to eLife. The goal of the manuscript is to understand the mechanisms of dynamic instability of the 3D model. After having read the new version and the response to reviews, I feel the manuscript is a good contribution to the literature. In particular, I would like to commend the authors on their ability to communicate the physical concepts in much clearer terms. It is clear that they spent time and effort to make the manuscript more accessible. In so doing, they made the manuscript more impactful. It was wellworth the effort. Thank you also for clarifying the predictions of this model.
The authors state that, "MT growth… is powered by the hydrolysis of GTP." This is not true. Dynamic instability is powered by GTP hydrolysis, since the hydrolysis is required for depolymerization, but growth simply needs the GTP binding – not hydrolysis. Perhaps this sentence, which is a compound sentence, meant something else, when it said, "'it' is powered by the hydrolysis of GTP." Whatever, the way it reads currently is that the authors are stating that growth is powered by GTP hydrolysis, which is opposite to currently accepted mechanisms.
https://doi.org/10.7554/eLife.54077.sa1Author response
[Editors' note: the authors resubmitted a revised version of the paper for consideration. What follows is the authors' response to the first round of review.]
Reviewer #1:
… Some problematic aspects of the manuscript:
1) The model put forward is fairly derivative as it is based on prior theory and models. It is entirely possible that I am mistaken, but if so, the authors should explain the novelty of their model, which is not wellexplained. What does this model bring to the table that prior models have not? This will help readers and people who might want to use it know when it would be appropriate to use or compare to their experiments.
We would like to respectfully disagree with the statement that our model is fairly derivative. Previous theoretical work on the mechanics of dynamic instability has been almost exclusively computational, with the number of parameters often exceeding the number of predictions. The few studies that have combined theory and computer simulations to model dynamic instability mechanically (e.g. Zapperi, LM 2011) have been limited to considering MTs as onedimensional chains. To the best of our knowledge, our work is the first to bring together mechanics and kinetics of dynamic instability within a 3D setting which is not incremental.
Accounting for the full 3D geometry of MTs fundamentally changes the predictions relative to the 1D models qualitatively. In 1D, the presence of quenched disorder at the level of GTPhydrolysis always stabilises MTs. In a 3D setting the quenched disorder acts to destabilise MTs. This difference has profound consequences on the model predictions regarding the stabilising role of GTP remnants. Within a 1D theory, all remnants would be stabilising, i.e. we would predict that the rescue frequency of such remnants would be equal to 1. This is however not what is observed in experiments. Our theory suggests that this might be related to the 3D nature of the remnants, as captured by our percolation model, which we may be the first to introduce to the study of this problem. Our resulting phase diagram which combines two geometric parameters and one kinetic parameter summarizes the role of the tubular structure, the kinetics of hydrolysis and the role of disorder is new, but perhaps this did not come through in our earlier version. We have taken the reviewer's criticism to heart and have revised the main text to better explain the key contributions of our work (see e.g. revised Abstract and Introduction).
2) The model suggests at the end that it could make predictions, but it actually fails to make those predictions. Actual, specific predictions would make the manuscript far more useful to experimentalists. For instance, what if you have a 9protofilament microtubule? How sensitive is it to protofilament number? Can it tell the difference between 9 and 10 protofilament microtubules? In the same field of view?
These are good suggestions. It may be possible for example to measure unbinding velocities of MT at constant temperature as a function of MT radius or number of PFs. A direct prediction from our theory is that unbinding velocity increases with decreasing number of PFs. The dependence is exponential (through the thermal energy); as such, differences in unbinding velocities might be sufficiently large to be detected experimentally. We have included a paragraph describing our predictions for catastrophe as a function of MT radius in the revised manuscript (see subsection 'Condition for mechanical stability').
3) The model does not explain some important aspects of dynamic instability. Specifically, it does not explain the switch from growing phase to shrinkage phase. Or if it does, that was not articulated well. It does a better job of explaining rescue, but with the limitation that the GTPislands are frozen states. Although it is nice to confirm that this is true with a model, the model is not as realistic as it should be because it is unlikely that the GTPislands are actually frozenin. More likely, there is a rate of GTP turnover that is statistical and probabilistic, so the locations of the islands are dynamic and tend to break the connectedness (percolation) as time proceeds. I understand the importance of initial toy models, but this model, especially given the inarticulation of specific predictions, could do so much more.
As discussed in the response to the previous comment, we can use our model to account for the switching to depolymerisation. MTs in the mechanically unstable region of the phase diagram in Figure 3 undergo catastrophe. MTs below the stability line can undergo instability in the presence of thermal fluctuations. The catastrophe rate depends exponentially the 'distance' from the stability line, which represents an activation energy barrier. Following the reviewer's comment, we have articulated this explicitly in the revised manuscript (subsection 'Condition for mechanical stability'). Regarding the choice to model GTP using quenched disorder, it is important to appreciate that the term 'frozen'does not imply static. Instead, it means 'slowlyvarying' compared to some underlying fast dynamics. In the context of MTs, the use of slowly hydrolyzable analogues of GTPtubulin contributes to rescue (Tropini et al., 2012). This suggests that the fraction of GTPtubulin in remnants is random but its composition varies slowly with time, compared to the fast kinetics of polymerisation and depolymerization. Mathematically, this means that there is a separation of timescales between fast polymerization/depolymerization and the slow variation of island properties. According to the framework of nonlinear dynamics/asymptotic analysis we can consider polymerization/depolymerization kinetics separately from the island kinetics (in the nonlinear dynamics language this is termed a 'slow manifold'). In practice, the parameters describing island composition can be simply replaced by time dependent ones, which makes our theory applicable to the situation highlighted by the reviewer. As long as disorder varies slowly compared to the characteristic timescale of polymerization/depolymerization, one can make the amount of disorder a function of time. We have now made this point explicit in the main text (subsections 'Mechanical stability of 3D MTs in the presence of quenched disorder' and 'Computer simulations').
Reviewer #2:
... While this paper is technically sound, in my opinion it does not meet the novelty standard of eLife. The results are exactly what one would intuitively expect, so the paper reads as using mathematics to demonstrate previously expected results on dynamic instability. The nature of the modeling and the results obtained are similar to many previously found. Previous related work is not cited, including Zakharov et al., 2015. In particular, the model in this manuscript is quite similar to that of Zakharov et al., so what new insight is gained from the current work is unclear.
Additionally, this manuscript does not cite recent work suggesting that tubulin dimers and protofilaments have intrinsic curvature in the GTPbound state, which would change the nature of the model and possibly the conclusions.
We thank the reviewer for giving us the opportunity to better explain the novelty of our results. Previous theoretical work on the mechanics of dynamic instability has been almost exclusively computational. The few studies that have combined theory and computer simulations to model dynamic instability mechanically have been limited to considering MTs as onedimensional chains. To the best of our knowledge, our work is the first to bring together mechanics and kinetics of dynamic instability within a 3D setting which is not incremental.
Accounting for the full 3D geometry of MTs fundamentally changes the predictions relative to the 1D models qualitatively. In 1D, the presence of quenched disorder at the level of GTPhydrolysis always stabilises MTs. In a 3D setting the quenched disorder acts to destabilize MTs. This difference has profound consequences on the model predictions regarding the stabilizing role of GTP remnants. Within a 1D theory, all remnants would be stabilizing, i.e. we would predict that the rescue frequency of such remnants would be equal to 1. This is however not what is observed in experiments. Our theory suggests that this might be related to the 3D nature of the remnants, as captured by our percolation model, which we may be the first to introduce to the study of this problem. Our resulting phase diagram which combines two geometric parameters and one kinetic parameter summarizes the role of the tubular structure, the kinetics of hydrolysis and the role of disorder is new, but perhaps this did not come through in our earlier version. We have taken the reviewer's criticism to heart and have revised the main text to better explain the key contributions of our work. and used mathematics to translate these results into testable predictions.
In this context, we have extended the Discussion at the end, which now includes a discussion of explicit predictions that could be tested experimentally. We also thank the reviewer for pointing out the reference Zakharov et al., 2015, which is indeed relevant to our work and we have now appropriately cited in the revised manuscript. The key difference between our work and Zakharov et al. is that our work provides an analytical theory, while Zakharov et al. is a simulational study. Moreover, Zakharov et al. studies the MT tip, while our work we have focused on reinforcing islands. There are however important similarities. In particular, Zakharov et al. studied the role disorder at the level of the MT tip composition varies over long timescales compared to growth/shrinkage; this situation is entirely analogous to our consideration of quenched disorder at the level of the reinforcing island's composition. We have also followed the reviewer's suggestion to cite further recent literature on the curvature of the tubulin dimer in its GTP and GDPbound forms. In this context, we would like to point out that our model already accounts for a slight intrinsic curvature of the GTPtubulin dimer (see Table 1); we have clarified this point in the revised manuscript.
In addition, the authors have not made sufficient effort to use language consistent with that used in the field, or to explain technical terms from other fields. For examples, the authors refer to the rams' horns of shrinking MTs as a flowerlike shape. Similarly, the authors use the term quenched disorder without explaining clearly that it simply means random variation in a property of the model, which will confuse readers who are not already familiar with the condensedmatter usage of quenched disorder.
Therefore I recommend that the paper be published in a more specialized journal (such as the Biophysical Journal, which would be a good fit for this paper) and significantly rewrite the paper to clarify the presentation and the connection to previous work.
We have taken the reviewer's comment to heart and have put significant effort to revise our manuscript and improve accessibility. In particular, we have made sure to use technical language consistent with the MT field; moreover, we have explained technical terms such as, for example, quenched disorder (see Section 'Mechanical stability of 3D MTs in the presence of quenched disorder') to make the paper more accessible to an interdisciplinary readership that might not be familiar with the physics literature (see also comment of reviewer 1). We hope that the revised manuscript has clarified these points.
[Editors' note: what follows is the authors' response to the second round of review.]
Reviewer #1:
... The authors state that, "MT growth... is powered by the hydrolysis of GTP." This is not true. Dynamic instability is powered by GTP hydrolysis, since the hydrolysis is required for depolymerization, but growth simply needs the GTP binding – not hydrolysis. Perhaps this sentence, which is a compound sentence, meant something else, when it said, "'it' is powered by the hydrolysis of GTP." Whatever, the way it reads currently is that the authors are stating that growth is powered by GTP hydrolysis, which is opposite to currently accepted mechanisms.
We thank the reviewer for raising this comment. We have rephrased the sentence accordingly (see the Introduction).
https://doi.org/10.7554/eLife.54077.sa2Article and author information
Author details
Funding
Swiss National Science Foundation
 Thomas CT Michaels
National Natural Science Foundation of China (11272303)
 Shuo Feng
 Haiyi Liang
National Natural Science Foundation of China (11072230)
 Shuo Feng
 Haiyi Liang
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
Acknowledgements
We acknowledge financial support from the Swiss National Science foundation (TCTM), the National Natural Science Foundation of China (11272303,11072230) (SF, LH).
Senior Editor
 Detlef Weigel, Max Planck Institute for Developmental Biology, Germany
Reviewing Editor
 Raymond E Goldstein, University of Cambridge, United Kingdom
Reviewer
 Jennifer Ross, University of Syracuse, United States
Publication history
 Received: November 30, 2019
 Accepted: April 4, 2020
 Version of Record published: May 11, 2020 (version 1)
Copyright
© 2020, Michaels et al.
This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.
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Background:
Postoperative knee instability is one of the major reasons accounting for unsatisfactory outcomes, as well as a major failure mechanism leading to total knee arthroplasty (TKA) revision. Nevertheless, subjective knee instability is not well defined clinically, plausibly because the relationships between instability and implant kinematics during functional activities of daily living remain unclear. Although muscles play a critical role in supporting the dynamic stability of the knee joint, the influence of joint instability on muscle synergy patterns is poorly understood. Therefore, this study aimed to understand the impact of selfreported joint instability on tibiofemoral kinematics and muscle synergy patterns after TKA during functional gait activities of daily living.
Methods:
Tibiofemoral kinematics and muscle synergy patterns were examined during level walking, downhill walking, and stair descent in eight selfreported unstable knees after TKA (3M:5F, 68.9 ± 8.3 years, body mass index [BMI] 26.1 ± 3.2 kg/m^{2}, 31.9 ± 20.4 months postoperatively), and compared against 10 stable TKA knees (7M:3F, 62.6 ± 6.8 years, 33.9 ± 8.5 months postoperatively, BMI 29.4 ± 4.8 kg/m^{2}). For each knee joint, clinical assessments of postoperative outcome were performed, while joint kinematics were evaluated using moving videofluoroscopy, and muscle synergy patterns were recorded using electromyography.
Results:
Our results reveal that average condylar AP translations, rotations, as well as their ranges of motion were comparable between stable and unstable groups. However, the unstable group exhibited more heterogeneous muscle synergy patterns and prolonged activation of knee flexors compared to the stable group. In addition, subjects who reported instability events during measurement showed distinct, subjectspecific tibiofemoral kinematic patterns in the early/midswing phase of gait.
Conclusions:
Our findings suggest that accurate movement analysis is sensitive for detecting acute instability events, but might be less robust in identifying general joint instability. Conversely, muscle synergy patterns seem to be able to identify muscular adaptation associated with underlying chronic knee instability.
Funding:
This research received no specific grant from any funding agency in the public, commercial, or notforprofit sectors.