Atomistic Basis of Microtubule Dynamic Instability Assessed Via Multiscale Modeling

Microtubule “dynamic instability,” the abrupt switching from assembly to disassembly caused by the hydrolysis of GTP to GDP within the β subunit of the αβ-tubulin heterodimer, is necessary for vital cellular processes such as mitosis and migration. Despite existing high-resolution structural data, the key mechanochemical differences between the GTP and GDP states that mediate dynamic instability behavior remain unclear. Starting with a published atomic-level structure as an input, we used multiscale modeling to find that GTP hydrolysis results in both longitudinal bond weakening (~ 4 kBT) and an outward bending preference (~ 1.5 kBT) to both drive dynamic instability and give rise to the microtubule tip structures previously observed by light and electron microscopy. More generally, our study provides an example where atomic level structural information is used as the sole input to predict cellular level dynamics without parameter adjustment. Supplementary Information The online version of this article (10.1007/s10439-020-02715-6) contains supplementary material, which is available to authorized users.


Tubulin bending angle and flexibility analysis
Tubulin dimer bending angles were calculated based on the methodology described in 3 , similar to 2,5 . Two bending motions (radial and tangential to microtubule wall) were identified throughout constrained and unconstrained simulations for both nucleotide states (Fig S1A, S1B). To further investigate the effect of the lattice on the flexibility of the dimers, we calculated the root-mean-square fluctuation (RMSF) of the backbone residues averaged over the last 300ns of the trajectories, for GDP-and GTP-tubulin (Fig S1C,  S1D). As indicated in the bending motion, when we get farther from the lattice constraints, motion fluctuations of the residues are higher on average (red color in the RMSF map). When the RMSFs of the two nucleotide states are compared for each residue (Fig S1E,  S1F), the lattice constraints confine GTP-tubulin dimer motions to a higher degree as compared to GDP-tubulin, while the free GTP-dimer fluctuates more than GDP-tubulin.
For intradimer bending angle calculation -tubulin subunit of the dimer was first aligned and then the angle between the vector connecting the COMs --subunits and dot). For interdimer bending angle, we employed a similar method to the one applied to the intradimer interface, with the vectors connecting COM of the first dimer to the second dimer. We further decomposed the angles to 3 perpendicular components to identify the dominant direction of bending and their autocorrelation times, as seen in Fig S2 and S3 for the unconstrained and constrained dimers.
To compare the average values and distribution of the angles for the two nucleotide states, we calculated the mean of 100 bootstrapped data points, selected far apart (twice the autocorrelation characteristic time) to avoid correlation, from the last 300ns of the trajectories, summarized in Table S1A, S1B, S2A and S2B. To characterize the angle fluctuations magnitude, flexural stiffness was calculated based on the equipartition theorem 4 , where thermal motion variance is inversely related the flexural stiffness: Where kB is Boltzmann constant, T is absolute 2 is the variance. The autocorrelation function is calculated by autocorr function in MATLAB, and the characteristic time ACF) is obtained via fitting an exponential decay function.
We believe that these bending motions are strongly auto-correlated (Table S1B, Table  S2B) and highly variable among different simulation replicates (as observed in Fedorov et al. 1 ). The bending angles for both GDP-and GTP-tubulins showed that, compared to the unconstrained simulations, the tangential and radial bending motions of the constrained dimers are limited, and they maintain a straighter conformation, as expected ( Fig S1B, S3A, S3B). We found that t fluctuations for the bottom dimer are significantly decreased due the lattice constraints, while for the top dimer the lattice constraint effects are less significant ( Fig S3B).
Interestingly, we found that GTP-dimer bending motions are more confined in the lattice than GDP-dimer, whereas at the protofilament tip where there are no neighbor constraints, GTP-tubulin exhibits a higher range of motion (Table S2A). This result suggests that the presence of the microtubule lattice modulates bending stiffnesses.

Optimization of umbrella sampling method
The collective variable sampled in our umbrella sampling method was chosen as the COM-to-COM distance of the two dimers. To ensure that our reaction path is the most probable path taken by the dimers to bind to each other, we ran BD simulations of separate dimers, initially located randomly in a sphere of 1.5 nm radius, binding to each other and inspected the binding efficiency as a function of the initial relative rotation angles ( Fig S5). The results show that only a limited range of angles (<4 degrees) has a high probability of binding, meaning that separating the dimers longitudinally without a major rotation is not an unreasonable choice of the reaction coordinate.
Supporting Tables   Table S1A. Analysis of interdimer and intradimer bending angles of unconstrained tubulin dimers in GDP-and GTP-state   Lateral bond free energy -5 kBT       shown when GDP-tubulin is stiffer than GTP-tubulin with EI GDP =4.7x10 -24 Nm 2 , EI GTP =2.3 x10 -24 Nm 2 , and both dimers prefer a radially outward bending of 22°. Abnormal dynamics are observed. (E) Microtubule length-time is shown when GDPtubulin is stiffer than GTP-tubulin with EI GDP =4.7 x10 -24 Nm 2 , EI GTP =2.3 x10 -24 Nm 2 , and both dimers prefer to be straight. Microtubule growth is observed. (F) Microtubule length-time is shown when GDP-tubulin is stiffer than GTP-tubulin with EI GDP =4.7 x10 -24 Nm 2 , EI GTP =2.3 x10 -24 Nm 2 , and GDP dimer has a radial bending preference of 22°. Growing microtubule is shown. Bold borders show mechanisms consistent with experimental results.

Supporting Movies
Movie S1. Radial bending motion of unconstrained tubulin oligomers in GDP-(left) and GTP--tubulin is depicted in silver a -tubulin is shown in cyan and orange for GDP-and GTP-states respectively. GTP-and GDP-nucleotide are represented in yellow and purple, respectively. Side view of a microtubule is shown.
Movie S2. Tangential bending motion of unconstrained tubulin oligomers in GDP-(left) and GTP---tubulin is shown in cyan and orange for GDP-and GTP-states respectively. GTP-and GDP-nucleotide are represented in yellow and purple, respectively. Outside view of a microtubule is shown.