¹

1 Introduction↩︎

Exploratory processes are common in biology, such as in the various processes of target search [1]–[5]. One notable example is in the process of mitosis, when chromosomes divide and each copy consequently must be physically separated to different ends of the cell so that each daughter cell will have an exact copy [6]. This process relies on microtubule filaments that search for and bind to chromosomes within the cell [7], [8]. Curiously, microtubules grow and shrink repeatedly during target search, often growing to a very different length each time before shrinking [9]. These dynamics of the microtubules were first discovered by Mitchison and Kirschner in the 1980s, and dubbed the dynamic instability [10].

While the large range of lengths demonstrated during dynamic instability presumably helps microtubules to explore the cellular environment [9], the underlying mechanisms that enable this large variability remain unclear. Notably, the lengths at catastrophe are observed to follow an unusual peaked distribution [11]–[13], which simple single-step models fail to capture [11], [14], [15]. Meanwhile, complicated models [16]–[18] have a large number of free parameters, which are under-constrained by available experimental data. Besides these issues in existing models of microtubules, the complexity of their biochemistry and physical components provides further challenges. Microtubules consist of different tubulin conformations [19], while an additional stabilizing cap at the end is increasingly shown to be important for regulating catastrophes [9], [20], [21]. Moreover, microtubule growth has recently been shown to transiently pause or “stutter” before catastrophes [22], [23], the origin of which has been associated with the cap but remains speculative [24], [25]. Hence, it would be valuable to develop a model for the cap structure that can reproduce key experimental observations, while allowing for experimental comparison without too many free parameters in order to retain transparency.

A novel approach to modeling dynamical biochemical processes that has recently emerged is topology. Topological states produce dimensional reduction such as global cycles [26], [27] or multistable fixed points [28] within large networks of biochemical reactions. Such topological models were first developed in quantum systems [29]–[32] and later extended to classical systems like mechanical lattices [33], [34], photonics [35]–[38], and electrical circuits [39], [40]. In the context of biological systems, topological models have been used to describe the circadian rhythm [27], sensory adaptation during chemotaxis [41], [42], and gene transcription networks [28]. Crucially, the dynamics localize to the edge of a system in a topological phase in a way that is robust to defects and disorder [43]. This framework has also been applied to microtubules, to capture the large range of catastrophe lengths [26] or emergent vibrational or electronic modes that destabilize filament growth [44]–[46]. However, these models only account for very specific features and have not been closely compared to experimental data. For example, microtubules in Ref. [26] shrink from the minus end and mostly at the same catastrophe length, contrary to experimental observations [10], [12], [47].

In this paper, we build upon the simpler model in Ref. [26] with a more biophysically realistic interpretation of the state space based on the cap structure. This cap model simultaneously captures the “stutter” phase [22], [23], [25] and the peaked catastrophe length distribution [11]–[13], which has not been achieved by previous models [16]–[18], [21], [24]. We show that these features arise from unique topological edge currents in our two-dimensional state space, and only requires tuning two free parameters. Further, the topological dynamics are robust to changes in relevant reaction rates, enabling the model to quantitatively reproduce the same features across a wide range of tubulin concentrations [12]. In addition, our model provides analytical conditions for a peaked catastrophe length distribution and several testable experimental predictions.

2 Topological model for microtubule cap↩︎

2.1 Model setup↩︎

We introduce a stochastic model of the microtubule cap, which plays a central role in regulating microtubule dynamics. The cap is located at the microtubule plus end [10], [12]: the top right of Fig. 1 (a) in purple and blue. It enables microtubule growth and stabilizes the microtubule under tubulin addition [48] while its removal immediately triggers catastrophes, i.e., rapid shrinkage from the same end [21], [47] (bottom of Fig. 1 (a)). Based on experimental evidence for a structurally distinct intermediate state in the cap [22], [49], [50], we model the cap with two components: GTP-tubulin dimers (purple) followed by the hydrolysis intermediate GDP-P\(_\text{i}\)-tubulin dimers (blue) [51], [52]. As hydrolysis proceeds, GTP-tubulin is eventually converted into unstable GDP-tubulin (green), which makes up the rest of the filament. We record the number of each of the two dimers in the cap with \((x,y)\), where \(x\) denotes GTP-tubulin and \(y\) denotes GDP-P\(_\text{i}\)-tubulin, while the number of GDP-tubulin dimers is denoted separately with \(z\).

The number of dimers in the cap changes through biochemical reactions illustrated in Fig. 1 (b). Here, we use a simplified description where the microtubule grows helically as a single filament, similar to what has been done in previous models [53]–[55]. GTP-tubulin dimers add to or dissociate from the filament at the plus end ( horizontal arrows that change \(x\)). Once added, GTP-tubulin undergoes hydrolysis in two steps: GTP cleavage converts GTP-tubulin to GDP-P\(_\text{i}\)-tubulin (bottom to top-left arrow that change \(x\) to \(y\)), while phosphate (P\(_\text{i}\)) release converts GDP-P\(_\text{i}\)-tubulin to GDP-tubulin (top-right to bottom arrow that change \(y\) to \(z\)). Such reactions can be primed, e.g., by conformational changes or elastic stress that propagate across the whole cap through allostery [19], [56]. This could lead to P\(_\text{i}\) release after GTP-tubulin addition on the other end of the cap, as shown in the transition \((2,2)_\text{C} \rightarrow (2,1)_\text{A}\) in Fig. 1 (b). We represent these distinct primed states with an internal coordinate s (subscript A, B, or C), so that the full state space of the cap is defined with \((x,y)_\text{s}\).

Besides external transitions that change the number of dimers \(x\) and \(y\), we also consider internal transitions that only change the internal state s while keeping \(x\) and \(y\) fixed (Fig. 1 (c)). For example, state A (primed for GTP cleavage) can convert to state C (primed for P\(_\text{i}\) release) via a conformational expansion at the interface between GDP-P\(_\text{i}\)-tubulin and GDP-tubulin [57]. We denote external and internal transition rates by \(\gamma_\text{ex}^{ij}\) and \(\gamma_\text{in}^{ij}\), respectively, where \(i\) and \(j\) denote the initial and final internal state of the transition with \(i\neq j\).

Microtubules are strikingly out-of-equilibrium, driven by the free energy released from GTP hydrolysis [20], where the energy input biases the system towards biochemical reactions like GTP cleavage over their reverse reactions. Accordingly, we assign larger transition rates to these forward processes, as indicated by larger forward arrows in Fig. 1 (b) and 1 (c). We assume that these reactions happen at specific places in the cap, i.e., GTP cleavage only occurs at the interface between GTP-tubulin and GDP-P\(_\text{i}\)-tubulin, and P\(_\text{i}\) release only occurs at the interface between GDP-P\(_\text{i}\)-tubulin and GDP-tubulin. This is consistent with experimental findings of increased hydrolysis activity at such interfaces [56], and similar to the “vectorial” hydrolysis mechanism in previous models [58]–[60].

As transitions in Fig. 1 (b) and 1 (c) form the basic motifs in our model, we can repeat them over many dimer lengths of \((x,y)_\text{s}\) to obtain a Kagome lattice (Fig. 1 (d)), where the repeated motif is highlighted in blue. The edges of the lattice correspond to cap states with the minimum or maximum number of GTP-tubulin or GDP-P\(_\text{i}\)-tubulin dimers. For example, on the bottom edge the cap contains only GTP-tubulin and no GDP-P\(_\text{i}\)-tubulin, while on the left edge the cap contains only GDP-P\(_\text{i}\)-tubulin where all GTP have been cleaved. Note that this state space only describes the cap dynamics. The total microtubule length, given by \(x+y+z\), is tracked separately by monitoring transitions involving GDP-tubulin \(z\) (green arrows in Fig. 1 (d)). Based on experimental evidence that a single layer of GTP-tubulin is sufficient to stabilize microtubules [61], [62], we assume that catastrophe occurs and filament length shrinks to zero when the cap is entirely lost, which happens when the system returns to the state \((0,0)\) after traversing the state space. Cap loss is assumed to be irreversible, as indicated by a single arrow between \((0,0)\) and \((0,1)_\text{C}\) in Fig. 1 (d).

2.2 Model dynamics↩︎

Our model supports two distinct dynamical regimes, determined by the relative rates of external and internal transitions from the same state \((x,y)_\text{s}\). As shown in Ref. [26], when \(\gamma_\text{ex}^{ij}>\gamma_\text{in}^{ik}\) the state space supports persistent edge currents: once the system encounters the edge it will likely continue along the edge, as can be seen by inspection. The emergence of such edge currents depends on the global pattern of transition rates in state space, which can be captured by a nontrivial topological invariant [26], [43]. We therefore refer to this regime as the topological phase. Conversely, when \(\gamma_\text{ex}^{ij}\lesssim\gamma_\text{in}^{ik}\), the system is in a trivial phase and undergoes random growth and hydrolysis throughout state space via diffusive motion, rather than directed motion along the edges.

Since microtubule growth and hydrolysis (\(\gamma_\text{ex}^{ij}\)) are highly non-equilibrium processes [19], [20] compared to internal conformational changes (\(\gamma_\text{in}^{ik}\)), we expect \(\gamma_\text{ex}^{ij}>\gamma_\text{in}^{ik}\) and focus on the topological phase. We further allow the GTP-tubulin dissociation rate \(\gamma_\text{ex}^{\text{CB}}\) to increase as the cap length \(l=x+y\) grows, e.g., \(\gamma_\text{ex}^{\text{CB}}\propto l^n\) for \(n>0\). This could arise from accumulation of mechanical strain (e.g., from frayed filaments or conformational mismatch between different tubulin states) [63], [64] or structural defects (e.g., missing dimers or changing protofilament number) [21], [44], [65], which weaken tubulin interactions at the tip of a longer cap. The gradually decaying rate introduces a “soft” right boundary in state space (as oppose to a “hard” boundary when transitions are strictly disallowed [26]) and allows for a peaked catastrophe length distribution consistent with [12] as we will show in Sec. 3. This soft boundary separates the topological and trivial phases. To the right of the soft boundary at large cap lengths, the system is in a trivial phase without edge states. To the left, the system is in a topological phase, where edge currents describe cap growth along the bottom edge and cap shrinkage along the left edge (Fig. 1 (e)). Moreover, microtubule growth is paused along the left edge, as no GTP-tubulin addition occurs there. These pauses match experimental observations of brief stutters before catastrophes [22], [25], which are captured naturally by left edge currents in our model.

In between the trivial and the topological phase, the soft boundary permits a mixture of edge and diffusive dynamics, where stochastic trajectories go through directed diffusion from the bottom to the left edge. This corresponds to cap hydrolysis from GTP-tubulin to GDP-P\(_\text{i}\)-tubulin and growth of the microtubule length \(x+y+z\). We illustrate three representative trajectories in Fig. 1 (e) with different colors, where darker circles in the background represent states that are visited more frequently. These dynamics qualitatively capture the different phases of growth, stutter, and shrinkage in dynamic instability. In particular, catastrophes involve a sequence of edge and bulk dynamics in the 2D state space: the cap grows along the bottom edge, followed by a two-step hydrolysis process in the bulk and on the left edge. To further reproduce experimental results quantitatively, we match the transition rates more closely to experimental data below.

3 Validation with experimental data↩︎

To compare our model behavior with experimental data, we perform stochastic simulations using the Gillespie algorithm [66] (details in Appendix A). Our simulations focus on the regime \(\gamma_\text{ex}^{ij}>\gamma_\text{in}^{ik}\), which allows for the edge dynamics shown in Fig. 1 (e). A representative time trace of microtubule length generated by our simulations is shown in Fig. 2 (a), where microtubules grow at a nearly constant speed consistent with measurements [12], [47]. The growth pauses briefly before each catastrophe, as highlighted on the right of Fig. 2 (a), capturing the experimentally observed stutters [22], [25]. In addition, the catastrophe lengths span over one order of magnitude, reproducing the broad range of length scales observed [15], [20], [47].

To quantitatively connect our model to experiments, we define the transition rates using biophysical parameters from the literature. In principle, each transition rate in Fig. 1 (d) can be different, yielding as many as 12 parameters: a global scaling factor that sets the overall timescale, and 11 ratios between transition rates. We determine the global scaling by tuning a single base rate \(\gamma_\text{ex}^{\text{BC}}\), while other rates scale proportionally based on the ratios. Three out of the 11 ratios can be constrained by previous experimental measurements. Where no experimental data are available, we simplify the model by taking certain ratios equal to the constrained ratios, as explained below. Such simplifications reduce the number of free parameters to two, which we fit to observed catastrophe length distributions [12].

We begin by defining the GTP-tubulin dissociation constant \(k_\text{d}=\gamma_\text{ex}^{\text{CB}}(l=1)/\gamma_\text{ex}^{\text{BC}}\), which characterizes GTP-tubulin binding affinity. Experimental measurements of microtubule growth report a dissociation rate of 44 s\(^{-1}\) and an association rate of 8.9 \(\mu\)M\(^{-1}\) s\(^{-1}\) [47]. To define \(k_\text{d}\), we assume a constant tubulin concentration of 12 \(\mu\)M, consistent with the catastrophe length data considered later [12]. This yields \(k_\text{d} = \frac{44}{8.9\times12}\approx0.41\). Further, we need to specify how the dissociation rate depends on cap length. For now, we assume a simple linear relationship \(\gamma_\text{ex}^{\text{CB}}\propto l\), and discuss more general length dependencies in Sec. 4.

Next, we define the GTP cleavage and P\(_\text{i}\) release rates by \(s_\text{st} = \gamma_\text{ex}^{\text{CA}}/\gamma_\text{ex}^{\text{BC}} = \gamma_\text{ex}^{\text{AB}}/\gamma_\text{ex}^{\text{BC}}\). In the absence of direct experimental measurements, we assume that the two rates are equal. The ratio \(s_\text{st}\) scales the rate \(\gamma_\text{ex}^{\text{CA}}\) and also \(\gamma_\text{in}^{\text{AC}}\) defined below, which together control the duration of stutters. We tune \(s_\text{st}\) to match experimental measurements of average stutter times at \(\sim 7\) seconds [22], [23], [67]. Since \(s_\text{st}\) only defines relative rates and not absolute stutter times, we tune it together with the global scaling factor (discussed later) that sets the absolute timescale, yielding \(s_\text{st}=16.5\).

To capture the out-of-equilibrium nature of the transitions, we define the slower reverse rates \(\gamma_\text{ex}^{ji}\) and \(\gamma_\text{in}^{ji}\) by \(e^\mu=\gamma_\text{ex}^{ij}/\gamma_\text{ex}^{ji}=\gamma_\text{in}^{ij}/\gamma_\text{in}^{ji}\), where \(\mu\) (in units of \(k_\text{B} T\)) measures the energy input that drives the system out of equilibrium [68]. In our model, GTP cleavage (\(\gamma_\text{in}^\text{AB}\)) and P\(_\text{i}\) release (\(\gamma_\text{in}^\text{CA}\)) are powered by GTP hydrolysis, which releases free energy on the order of \(20\,k_\text{B} T\) [19], [69]. We thus set \(\mu=20/2=10\), assuming that the driving is equally distributed between the two reactions. We further assume that internal transitions are also driven and set a similar value of \(\mu=10\) here.

To finally obtain the length distributions, we specify the topology of the model by two ratios: \(r=\gamma_\text{ex}^{\text{AB}}/\gamma_\text{in}^{\text{AC}}=\gamma_\text{ex}^{\text{BC}}/\gamma_\text{in}^{\text{BA}}\) (set equal for simplicity) and \(r_\text{P}=\gamma_\text{ex}^{\text{CA}}/\gamma_\text{in}^{\text{CB}}\). These ratios measure the competition between forward external and internal transitions. As we have not found experimental data that constrains these ratios, we take them as free parameters to fit the experimentally observed catastrophe length distribution [12].

When fitting these parameters, we focus on the regime where external transitions are faster than internal transitions (\(r>1\) and \(r_\text{P}> 1\)), which produces the edge dynamics in Fig. 1 (e). In this regime, \(r\) and \(r_\text{P}\) control the length of edge currents and the tendency to move towards the left edge during directed diffusion. To obtain the best parameters, we record the microtubule lengths at catastrophe and fit the resulting distribution to experimental data at 12 \(\mu\)M tubulin [12], which yields \(r=220\) and \(r_\text{P}=200\). The simulation results, shown on the upper panel of Fig. 2 (b), closely match the peaked distribution observed [12]. This peaked distribution results from edge dynamics at the bottom of the state space, which governs the catastrophe lengths (more in Sec. 4).

Furthermore, we set the overall timescale of the model by scaling the base rate \(\gamma_\text{ex}^{\text{BC}}\) to match the average microtubule lifetime from experimental data [12]. This yields \(\gamma_\text{ex}^{\text{BC}}=34.7\) s\(^{-1}\), consistent with single-molecule measurements of the GTP-tubulin association rate at \(40.8\pm19.2\) s\(^{-1}\) [70]. The resulting lifetime distribution is shown in the bottom panel of Fig. 2 (b), which is also peaked and matches experimental results [12].

More generally, our model captures dynamic instability across a wide range of tubulin concentrations \([c]\). To model changing concentrations, we set both GTP-tubulin association and dissociation rates (\(\gamma_\text{ex}^\text{BC}\) and \(\gamma_\text{ex}^\text{CB}\)) proportional to \([c]\) based on experimental measurements [47], [71]. Past experiments reported dynamic instability for concentrations from 5 to 30 \(\mu\)M [11], [12], [15], [47]. Across this broad range of concentrations, our model consistently reproduces dynamic instability in contrast to previous single-filament models [21], [54], [55], [60], as the topological edge currents remain robust to changing reaction rates from different concentrations.

We further examine how the model responds to changing tubulin concentration. As \([c]\) increases, both the average catastrophe length and microtubule lifetime increase, consistent with experimental results [12] as shown in Fig. 2 (c). Across the full range of concentrations considered, the model consistently generates peaked catastrophe length distributions, in agreement with experimental data [12]. These results highlight the generality of our model, which reproduces key experimental features across different tubulin concentrations.

4 Analytical condition for peaked distribution↩︎

To understand when we expect a peaked catastrophe length distribution, we derive an analytical condition for when a peak emerges. Our analysis focuses on the early stage of stochastic trajectories along the bottom edge of the state space. As shown in Fig. 1 (e), we define the coordinates at which the system first leaves the bottom edge to be \((x_\text{hydr},0)\), where \(x_\text{hydr}\) corresponds to the cap length at the onset of cap hydrolysis. In our model, catastrophe length increases monotonically with \(x_\text{hydr}\) (Fig. 3 (a)), which is intuitive because a longer cap leads to a more stable microtubule that grows for longer before catastrophe. As a result, the catastrophe length distribution is peaked whenever the \(x_\text{hydr}\) distribution, denoted by \(P(x)\), is peaked (e.g., see Fig. 3 (b)). This observation allows us to focus on the simpler distribution \(P(x)\), where a peak emerges when the derivative \(P'(x)=0\) at some positive \(x=x^*\) that satisfies this condition.

We calculate \(P(x)\) by analyzing stochastic trajectories along the bottom edge (Fig. 3 (c)). At each \(x\) coordinate, a trajectory either enters the bulk of the state space (blue) with probability \(p(x)\), or continues along the edge (orange) with probability \(1-p(x)\), as we neglect higher-order effects such as less likely trajectories. \(P(x)\) is then the probability that a trajectory continues along the edge at all previous \(x-1\) steps, and enters the bulk at \(x\), i.e., \[\label{eqn95Px} P(x) = \left\{\prod\limits_{x'=1}^{x-1}\left[1-p(x')\right]\right\}p(x).\tag{1}\] This expression shows that length dependence is required for a peaked \(P(x)\) distribution. If \(p(x)\) is constant, \(P(x)\) follows a geometric distribution and decays exponentially.

To obtain an analytical expression for \(P(x)\), we express \(p(x)\) in terms of transition rates in our model. \(p(x)\) is given by the probability of a family of trajectories that enters the bulk at \(x\), which involve different rounds of association and dissociation reactions between \(x\) and \(x+1\), as illustrated in Fig. 3 (d). To obtain the probability of such trajectories, we first define \(p_\text{BA}\), the probability for the \((x,0)_\text{B}\rightarrow(x,0)_\text{A}\) internal transition: \[\label{eqn95pBA} p_\text{BA} = \frac{\gamma_{\text{in}}^{\text{BA}}}{\gamma_{\text{in}}^{\text{BA}} + \gamma^{\text{BC}}_{\text{in}} + \gamma_{\text{ex}}^{\text{BC}}}.\tag{2}\] Next, we define \(p_\text{a}\) and \(p_\text{d}(x)\), the probabilities for GTP-tubulin association or dissociation respectively, at cap length \(l=x\): \[\begin{align} p_\text{a} &= \frac{\gamma_{\text{ex}}^{\text{BC}}}{\gamma_{\text{in}}^{\text{BA}} + \gamma^{\text{BC}}_{\text{in}} + \gamma_{\text{ex}}^{\text{BC}}}, \\ p_\text{d}(x) &= \frac{\gamma_{\text{ex}}^{\text{CB}}(x)}{\gamma^{\text{CA}}_{\text{in}} + \gamma_{\text{in}}^{\text{CB}} + \gamma_{\text{ex}}^{\text{CB}}(x)}. \label{eqn95pa} \end{align}\tag{3}\] Each round of association and dissociation contributes a factor of \(p_\text{a}p_\text{d}(x+1)\) to the trajectory probability, as shown in Fig. 3 (d). We neglect higher-order trajectories with two or more steps forward in \(x\) before returning, as they go through backward internal transitions \(\gamma_\text{in}^\text{BC}\) and have negligible probabilities.

Then \(p(x)\) is given by the sum \[\begin{align} \label{eqn95px} \begin{aligned} p(x) =& p_\text{BA} \sum\limits_{m=0}^\infty \left[p_\text{a} p_\text{d}(x+1)\right]^m \\ =& \frac{p_\text{BA}}{1-p_\text{a} p_\text{d}(x+1)}. \end{aligned} \end{align}\tag{4}\] Substituting Equations (2 4 ) into Equation (1 ), we get the analytical expression for \(P(x)\), which agrees with simulation results as shown by the blue curve in Fig. 3 (b).

From the expression of \(P(x)\), we obtain the analytical condition for a peak in both \(P(x)\) and the catastrophe length distribution. Note that \(P(x)\) is only defined at discrete values \(x_\text{hydr}=1,2,...\), which represents the cap length at the onset of hydrolysis (\(x=0\) is excluded, consistent with typical length measurements from experiments [11], [12]). Therefore, \(P(x)\) is peaked only when its maximum \(P(x^*)\) occurs at \(x^*>1\). This condition imposes a minimal rate at which the dissociation rate has to increase with \(l\), i.e. \(\gamma_{\text{ex}}^{\text{CB}}\) has to increase as \(l^{0.1}\) or more quickly in order to have a peak, as shown in Fig. 3 (e). This threshold can be calculated analytically by noting where \(P'(x^*)=0\) and \(x^*>1\) (see Appendix 8). To conclude, we have shown that a cap-length-dependent dissociation rate \(\gamma_{\text{ex}}^{\text{CB}}\) is essential for generating a peaked distribution in our model, but the required dependence can be more general and weaker than the linear form we currently assume.

5 Single-component cap model fails to capture catastrophe behavior↩︎

Thus far, we have modeled the microtubule using a two-component cap. To investigate whether both components are necessary, we compare our model with a reduced single-component cap model. Following earlier work [55], [72], [73], we assume that GTP cleavage is fast relative to other timescales and coarse-grain out the reaction, treating GTP-tubulin and GDP-P\(_\text{i}\)-tubulin as the same species. This merges internal states A and B, reducing our state space from a 2D Kagome lattice to a 1D chain with two internal states (Fig. 4 (a)), where \(x\) now denotes the cap length.

The single-component cap model fails to reproduce the peaked length distribution and other key features of catastrophe. To compare with data, we refit the free parameters \(r\), \(r_\text{P}\) and the base rate \(\gamma_\text{ex}^{\text{BC}}\) to the same catastrophe length and lifetime data from [12], keeping all other ratios fixed. The best fit yields \(r=1.05\times10^5,r_\text{P}=9\times 10^4,\gamma_\text{ex}^{\text{BC}}=34.8\) s\(^{-1}\). In this 1D model, microtubules remain short for extended periods after catastrophe (Fig. 4 (b)), inconsistent with experimental observations [12], [15], [47]. Moreover, growth transitions directly to shrinkage without a stutter phase. The resulting catastrophe length and lifetime distributions also show poor agreement with experimental data [12], as shown in Fig. 4 (c). Such discrepancy with experiments shows that dynamics along the 1D edge is not sufficient to generate a peaked distribution, and points to the the necessity of a two-component cap model that describes two-step hydrolysis in a 2D state space.

6 Discussion↩︎

In this study, we develop a topological model of the microtubule cap that captures three key features of dynamic instability: a dynamical cap structure [9], [19], repeated cycles of growth-stutter-shrinkage [25], and a peaked catastrophe length distribution [11]–[13]. In our model, the stutter phase before catastrophe arises from left edge currents, where microtubule growth from GTP-tubulin addition is paused. Meanwhile, the peaked catastrophe length distribution is governed by the dynamics of bottom edge currents in the two-dimensional state space. These topological edge currents are robust to changes in system parameters, allowing dynamic instability and the peaked distribution to persist in a wide range of tubulin concentrations. Our model further identifies the conditions required to produce the peaked catastrophe length distribution. Analytical results show that a cap-length-dependent GTP-tubulin dissociation rate is needed, suggesting the role of increasing cap instability in regulating catastrophe dynamics. Moreover, we find that a two-component cap model, which describes a two-step hydrolysis process in a 2D state space, is necessary to capture the experimental features, while a reduced single-component cap model with a 1D state space is insufficient.

The topological dynamics in our model yields several testable experimental predictions. First, we expect the catastrophe length distribution to show a peak only above some minimal rate at which dissociation becomes faster with cap length. Interestingly, as the rate increases (e.g., by increasing strain or using mutants), the cap length at which the peak occurs increases at first but decreases past a certain point (see Fig. 3 (e)). This is due to competing effects whereby a minimal dissociation rate increase is needed for a peak, but when this dissociation rate grows too fast, it makes hydrolysis more likely even at small cap lengths. Second, we expect cap length and catastrophe length to be positively correlated across individual catastrophe events, with both following peaked distributions. The correlation arises because the catastrophe length is governed by the cap length \(x_\text{hydr}\) at the onset of hydrolysis – larger caps allow microtubules to grow longer. Experimentally, the cap length can be estimated by tracking EB1, a protein thought to bind to the cap [22], [74]. Third, mixing wild-type tubulin with increasing amounts of polymerization-blocking mutants [70], [75] should shift the catastrophe length distribution from peaked to exponential. This is because the mutants set the GTP-tubulin association rate \(\gamma_\text{ex}^\text{BC}\) to 0 after binding, imposing a hard boundary in place of the soft boundary in state space. Stochastic trajectories move along this new boundary to reach the left edge, bypassing directed diffusion, which leads to premature catastrophes and an exponential distribution. Lastly, we predict that the stutter time distribution is peaked under constant tubulin concentration \([c]\), while its average increases with \([c]\). This is because the stutter time in our model is determined by the cap length when trajectories reach the last edge, which follows a peaked distribution similar to \(x_\text{hydr}\) and grows with \([c]\).

By connecting to concrete biochemical reactions, we demonstrate how topology promotes microtubule exploration through protected edge currents in state space. More broadly, our work suggests a new mechanism that utilizes topological dynamics for exploratory behavior over a wide range of length scales. This topological mechanism can provide insight on other exploratory structures [76], e.g., filopodia of neuronal growth cones [1], [11], that similarly rely on cycles of growth and shrinkage for search and movement.

Acknowledgments↩︎

We thank Dan Needleman, Jane Kondev and Rob Raphael for helpful comments. We gratefully acknowledge support from the NSF Center for Theoretical Biological Physics (PHY-2019745), the NSF CAREER Award (DMR-2238667), and the Chan-Zuckerburg Foundation.

References↩︎