Multiscale Computational Modeling of Tubulin-Tubulin Lateral Interaction

Abstract

Microtubules are multistranded polymers in eukaryotic cells that support key cellular functions such as chromosome segregation, motor-based cargo transport, and maintenance of cell polarity. Microtubules self-assemble via “dynamic instability,” in which the dynamic plus ends switch stochastically between alternating phases of polymerization and depolymerization. A key question in the field is what are the atomistic origins of this switching, i.e., what is different between the GTP- and GDP-tubulin states that enables microtubule growth and shortening, respectively? More generally, a major challenge in biology is how to connect theoretical frameworks across length- and timescales, from atoms to cellular behavior. In this study, we describe a multiscale model by linking atomistic molecular dynamics (MD), molecular Brownian dynamics (BD), and cellular-level thermokinetic modeling of microtubules. Here, we investigated the underlying interaction energy when tubulin dimers associate laterally by performing all-atom MD simulations. We found that the lateral potential energy is not significantly different among three nucleotide states of tubulin, GTP, GDP, and GMPCPP and is estimated to be $\cong$ −11 k_BT. Furthermore, using MD potential energy in our BD simulations of tubulin dimers confirms that the lateral bond is weak on its own, with a mean lifetime of ∼0.1 μs, implying that the longitudinal bond is required for microtubule assembly. We conclude that nucleotide-dependent lateral-bond strength is not the key mediator microtubule dynamic instability, implying that GTP acts elsewhere to exert its stabilizing influence on microtubule polymer. Furthermore, the estimated lateral-bond strength ($\mathit{\Delta }{G}_{lat}^0\cong$ −5 k_BT) is well-aligned with earlier estimates based on thermokinetic modeling and light microscopy measurements. Thus, we have computationally connected atomistic-level structural information, obtained by cryo-electron microscopy, to cellular-scale microtubule assembly dynamics using a combination of MD, BD, and thermokinetic models to bridge from Ångstroms to micrometers and from femtoseconds to minutes.

Significance

Microtubules are self-assembled polymers in eukaryotic cells that mediate key cellular functions such as chromosome segregation and nerve growth. A key feature of microtubule self-assembly is “dynamic instability,” in which dynamic microtubule ends switch stochastically between alternating phases of polymerization and depolymerization. A major challenge in biology is connecting across length- and timescales, from atoms to cellular behavior and from femtoseconds to minutes or hours. In this study, we developed a multiscale approach in which a small change in tubulin structure, due to the key hydrolysis reaction (GTP hydrolysis) that favors depolymerization, is used to predict microtubule assembly behavior at the cellular level. This multiscale framework provides the ability to potentially link other atom-level perturbations, such as mutations and therapeutic drugs, to cellular-level outcomes.

Introduction

Microtubules are dynamic filaments that facilitate critical cellular functions such as chromosome segregation, intracellular cargo transport, and cell architecture. These filaments are composed of tubulin heterodimers, i.e., tightly associated α- and β-subunits, with a nonexchangeable site for GTP nucleotide binding in the α-subunit and an exchangeable site in the β-subunit where GTP can hydrolyze to GDP, followed by release of inorganic phosphate (Pi). This nucleotide exchange at the heterodimer level confers unique dynamic properties to microtubules, i.e., the stochastic polymerizing and depolymerizing cycles characteristic of dynamic instability (1, 2). The key feature underlying the GTP-tubulin’s greater stability compared to GDP-tubulin, causing alternating growth and shortening phases in microtubule assembly, is yet to be fully understood (3, 4, 5, 6, 7). In addition, microtubule dynamic instability is controlled by several factors such as microtubule-associated proteins (MAPs) (8, 9), microtubule-targeting agents (MTAs) (10, 11, 12), microtubule isotype distribution (13), and tubulin post-translational modifications (14, 15). These interactions enable microtubules to support important cellular functions (16).

Structural studies of microtubules and tubulin in solution have shed light on conformational states of tubulin heterodimer and how that can explain the different behavior of GTP- versus GDP-tubulin (17). Those studies revealed that GDP- and GTP-tubulin structures are curved in solution (18, 19, 20, 21), compared to a straight structure found in microtubule protofilaments (22, 23), and also, GDP- and GTP-microtubule structures differ by lattice compaction and twist (24). This raises the question of whether tubulin nucleotide state dictates the tubulin preference in curvature predominantly and therefore their binding efficiency to the microtubule lattice (25). In addition, with recent advances of cryo-electron microscopy (cryo-EM), high-resolution structures of tubulin bound to various MAPs and MTAs are now available, revealing different drug-binding sites on tubulin (26, 27, 28). Relying only on structural information, the regulation of the assembly dynamics by those agents in the context of physiologically relevant problems cannot be explained. Moreover, acquiring high-resolution structures of dynamic proteins is usually accomplished by using a stabilizing factor (18, 19, 29, 30), which can by itself cause conformational changes to the native structure. By performing molecular dynamics (MD) simulations of tubulin structures, we are able to study the dynamic evolution of this protein in solution and sample the thermal fluctuations through time. For example, MD simulations of curved structures of tubulin have been used to confirm that they preserve their curvature in solution, with no significant difference at intra- versus interdimer bending angles (31, 32). However, using the right timescale and sampling of the ensemble is an important factor in drawing conclusions from the simulation results. In one MD study, using a limited sampling of less than 1 ns, the free energy of tubulin for intradimer bending angle was reported, concluding that αβ-tubulin dimers exist in an intermediate bent conformation (33). However, this study did not demonstrate that such a short sampling would represent converged data of the whole ensemble in the equilibrium state. In another study, pushing the limits of MD simulation timescale by using a time step of 4 fs at the expense of constraining bonds’ fluctuations, multiple 3-μs simulations of both straight and bent tubulin dimers showed the tubulin dimers moved toward a more bent configuration within a dimer, with GTP-tubulin showing a wider range of bending flexibility compared to GDP-tubulin (34). Despite these advances, the question of how two tubulin dimers interact with each other has remained unaddressed by MD simulations.

Considering the limits of current computational resources, capturing kinetic information from MD simulations at a timescale of approximately microseconds remains prohibitive, especially considering that multiple replicates are needed to sample the distribution of initial conditions. By contrast, coarse-grained Brownian dynamics (BD) with approximately millisecond timescales at the cost of atomistic detail and thermokinetic simulations with less detail allowing access to ∼100 seconds timescales together enable recapitulation of the kinetic rates and microtubule tip structures consistent with those found in vitro and in vivo (35, 36, 37, 38). The interactions of particles in the BD and thermokinetic simulations, here being the tubulin dimers, are modeled using an input potential, which can be dissimilar for tubulin as a function of its nucleotide state in different studies (7, 37, 38, 39, 40, 41). However, the interaction energy profiles have been adjusted in these models to match experimental observations of MT tip structure, dynamic assembly rates, and microtubule stiffness. Therefore, this discrepancy in potential of interactions in different BD and thermokinetic models has resulted in incompatible conclusions about various aspects of microtubules’ behavior. For instance, catastrophe—microtubules switching from growth to shortening—has historically been described as stemming from loss of the stabilizing GTP cap due to hydrolysis (37, 42, 43, 44), although recent BD studies (39, 45) suggested that stochastic variations in the number of curled protofilaments are responsible for causing catastrophe, using a different shape and strength for potential of interactions of tubulins from previous models (37, 38). In another study, lateral bond formation has been identified as the limiting step in forming the microtubule lattice with a large entropic component (40), but no direct calculations have been reported for the lateral bond strength or its nucleotide-state dependency even though it is critical for assembly and potentially for dynamic instability, as well.

In this study, the questions that we address are 1) what role does the lateral bond play in MT assembly and stability and 2) more generally, how do we connect modeling simulations from atoms up to organelles, such as microtubules, at the cellular level? We addressed these questions by developing a multiscale approach to study tubulin-tubulin interaction, building a framework for more complex interactions with MTAs and MAPs. We use high-resolution cryo-EM structures of tubulin for initiating full-atom MD simulations to study the dynamic evolution of different nucleotide states of tubulin in solution. Using the equilibrated structure, we then calculate an energy landscape for tubulin-tubulin lateral interaction in terms of a potential of mean force (PMF), using multiple replicates. To our knowledge, such free-energy calculations on a large globular protein-protein system (MW = 110 kDa per heterodimer), ∼200,000 atoms, using full-atom unconstrained MD simulations have not been reported. This enabled us to probe for possible energetic differences between GTP- and GDP-tubulin in assembly dynamics and whether it is the lateral-bond strength that distinguishes the two states in terms of establishing microtubule stability. Furthermore, we used the PMF obtained as an output from MD to define the input potential energy in BD simulations, as previously developed by Castle et al. (38). Overall, our work provides a multiscale modeling approach to use interaction energy profiles from full-atom MD simulations in BD, which we have previously linked to a thermokinetic model (37), and so we are now able to establish a framework for moving from crystallographic and cryo-EM structures to MT assembly behavior. Assembly dynamics can be used furthermore in cell-level modeling of the microtubules (46) to predict more complex physiologically relevant behavior such as the spatiotemporal distribution of MTs within the cell. Such a multiscale MD-BD-thermokinetic-cell-level framework is able to interconnect protein dynamics at femtosecond timescales and Ångstrom length scales to entire-cell-level behavior with an ensemble of hundreds of microtubules at minutes–hours timescales and micrometer length scales (36, 37, 38).

Methods

Simulation system preparation

Our computational simulations focused on tubulin heterodimers with one lateral neighbor in three different nucleotide states, i.e., GDP, GMPCPP, and GTP. However, tubulin in the GTP-state is unstable because of its tendency to hydrolyze with a rate constant of ∼0.1–1.5 s⁻¹ (36, 41, 47, 48). This makes it difficult to obtain a crystal structure of this state without an additional stabilizing element such as RB3-stathmin-like domain and DARPin protein (18, 19). For our study, the three-dimensional GDP- and GMPCPP-state tubulin structures with one lateral neighbor were extracted from the published cryo-EM dynamic structures of the microtubules by Zhang et al. (29) (Protein Data Bank [PDB]: 3JAS, 3JAT). The structures were obtained in the presence of kinesin head domains decorating the microtubule lattice to distinguish between α- and β-tubulin subunits while presumably having little effect on the microtubule structure. We then built our GTP-state system modifying the initial structure of the GMPCPP-state tubulins and equilibrating the total complex. In each state, we have two tubulin dimers laterally paired, as they would be in a microtubule lattice, with the GTP-associated Mg²⁺ present. The systems were then separately solvated in TIP3P water (49) using an 8 Å clearance from each side, resulting in a periodic cubic box with dimensions of 125 × 90 × 124 Å on average. MgCl₂ ions were added at 2 mM concentration to neutralize the system (31 Mg²⁺ and 2 Cl⁻) based on physiologically relevant salt concentrations. A total number of 128,500–130,000 atoms were used in all the systems.

MD simulations

MD simulations of all three nucleotide systems were run using NAMD 2.10 software (50) with the CHARMM 36 force field (51). The protein complex along with the nucleotides were all parametrized using the CHARMM-GUI interface (52). Each simulation system was initially energy-minimized for 12,000 steps using the conjugate gradient algorithm. The systems were then solvated and neutralized with MgCl₂ at 2 mM. The solvated systems were heated to 310 K for 1 ns using a Langevin thermostat (53) and then run for 7 ns in an NPT ensemble (T = 310 K and P = 1 atm), with the backbone atoms of the proteins being initially constrained with a harmonic potential in all directions, having a spring constant of k = 2 kcal/mol/Ų, and then gradually decreasing the harmonic constraints by dividing the spring constant in half in each 1 ns run to prevent large fluctuations from occurring. The simulations were followed by a total production run of 150 ns for the laterally paired system and 350 ns for the one-dimer system (equilibrating runs). All simulations were run with 2-fs time step and a cutoff radius of 12 Å for van der Waals interactions, using particle mesh Ewald for long-range nonbonded interactions (54). NVIDIA Tesla K40 GPUs (Nvidia, Santa Clara, CA) were used to accelerate the simulations on the Mesabi cluster at the Minnesota Supercomputing Institute, University of Minnesota, and NVIDIA Kepler K80 GPUs were used on Comet, an Extreme Science and Engineering Discovery Environment (55) dedicated cluster at the San Diego Supercomputing Center.

A second set of simulations were run for free-energy calculations. Running our system for ∼200 ns did not yield a full energy landscape of tubulin-tubulin interactions because of insufficient sampling of progressively less-bonded (higher-energy) states and the existence of possible local minima in the energy profile. Because overcoming those energy barriers is beyond that afforded by MD simulations, we employed the umbrella sampling method (56) to sample the ensemble sufficiently and have independent simulations that each can be run for longer sampling times in parallel, considering the large number of atoms. Consequently, this method yields a better convergence compared to the adaptive biasing forces method (57). We obtained a PMF, a free-energy landscape as a function of a specified reaction coordinate. For investigating the lateral potential of interaction, we defined the lateral center of mass to center-of-mass distance of the dimers to be the reaction coordinate, as further verified to be the most probable path of unbinding for two similar tubulin dimers as obtained from BD (see Fig. S1). The bias potential stiffness was tuned to be 10 kcal mol⁻¹ Å⁻² to give sufficient overlap of the histograms of the windows while not being too soft, resulting in a large correlation of dimers’ movements (see Fig. S2). To cover the full range of interdimer interactions, 18 windows were created, each being 1 Å separated from their nearest window. The production run was used to choose 10 equilibrated initial structures for creating replicates of umbrella windows for each system. The structures were selected far enough apart at time points after which the backbone RMSD plateaued in the equilibrium run (∼50 ns) and were separated by the correlation time of the dimers’ movement (10–20 ns for different simulations). Having multiple replicates reassured us that the resultant time-averaged PMFs are converged to the ensemble-averaged interactions. Each window was then equilibrated for 10 ns constrained by the bias potential and followed by a 20-ns sampling run for free-energy calculation. For determining the convergence of the PMF for each replicate, we increased the sampling time incrementally (by 5 ns), calculated the PMF, and compared it to the previously calculated PMF to ensure that the change of energy did not exceed a threshold of 1.5 k_BT, determined by the Monte Carlo bootstrap error analysis of the PMFs (see Fig. S3; Tables S1–S3). We chose well depth, half-force radius, and binding radius as our metrics to compare PMFs’ strength, shape, and range of interaction, respectively. Well depths were defined as the difference between the average of 10 points around the minimum and 10 points around the maximum of the profile because the dimers have thermal fluctuations when bound or unbound. Half-force radius was defined as being where the force (spatial derivative of potential energy) decayed to half of its maximal value. Thus, a softer potential would have a slower decay, and a stiffer potential would have a steeper decay. To evaluate the effect of salt concentration in the solution on the lateral interaction of tubulin, a third set of simulations were run with GDP-tubulin in a neutralized system with 2 mM MgCl₂ and an additional 100 nM of KCl. Three replicates of PMFs were calculated, and the average was compared to the PMF obtained from the initial neutral system obtained from 10 replicates (see Fig. S4).

Analysis of simulation trajectories

The equilibrium run trajectories were stored every 3000 time steps (6 ps), and the reaction coordinate in free-energy simulations was recorded every 200 time steps (0.4 ps). The stored trajectory files were analyzed for several conformational changes such as RMSD and a detailed interaction energy decomposition. The software VMD 1.9 was used for visualization of the trajectories (58). Weighted histogram analysis method (59, 60) was used to combine the histograms and build the unbiased PMF in a memory-efficient way.

For analyzing the equilibrium trajectories, the buried SASA can be calculated as summing over the SASA of the two dimers separately minus the SASA of the laterally paired tubulins, divided by two, given as

$$\mathrm{Buried}\mathrm{SASA=}\frac{\mathrm{1}}{\mathrm{2}}\left\{\left[\mathrm{SASA}\left(\mathrm{dime}{\mathrm{r}}_{\mathrm{1}}\right)+\mathrm{SASA}\left(\mathrm{dime}{\mathrm{r}}_{\mathrm{2}}\right)\right]\mathrm{-SASA}\left(\mathrm{laterally-paired}\mathrm{dimers}\right)\right\}.$$ (1)

The hydrogen bonds and salt bridges were calculated with the plugins available in VMD.

BD simulations

We examined the kinetics of tubulin dimers’ lateral interaction according to the BD model of Castle et al. (38) with the modification of simulating two dimers in solution as opposed to a single dimer binding to the microtubule lattice. All simulations were run using MATLAB R2018b (The MathWorks, Natick, MA). In the BD simulations, only lateral association/dissociation is possible, and the potential energies are the entropy-corrected PMFs from MD simulations. Simulations in solution were run for three nucleotide cases, and each was run for 500,000 iterations of binding simulations (total time varying from 0.1 μs to 1 ms) and 50,000 iterations of unbinding simulations (total time 0.5 ns–5 μs). For binding runs, a distance criterion was defined based on the input potential of interaction, in which two subunits are considered “bound” laterally if all of the lateral zones’ distances are within the binding radius (r_i ≤ r_b). The binding radius for PMF energy profiles was defined where the slope (interparticle force) reached 50% of its maximal value. Considering the similarity of the PMFs to a Lennard-Jones potential, the force was close to maximal near the minimum of the potential and was very low where the potential plateaus to zero. Therefore, this definition of r_b ensured us that the bound particles felt a force significant enough to hold them together with a high probability (90%). For unbinding simulations, we used a separation distance criterion of R_U = 11 nm, according to (38), where the probability of rebinding is very low (p < 0.01). For BD simulations of tubulin assembly into microtubule protofilaments with one longitudinal and one lateral neighbor, we did not simulate full unbinding events because the dimer was very stable. Instead, we used the method previously described in (38) to estimate the off rate by simulating a series of unbinding events with a cutoff time of 1 ms for each nucleotide case and counting the number of successful unbinding events. Additionally, we ran BD simulations of perfectly aligned tubulin pairs in solution starting from the minimum of the potential interaction energy and investigated the unbinding path, extent of tubulin rotation upon unbinding, and dissociation time to compare to MD simulations’ reaction coordinates and timescale.

Thermokinetic modeling

Dynamic behavior of microtubules was investigated using our previously described thermokinetic model (47) with the parameter set obtained using our MD and BD simulations. We ran the pseudomechanical model, modified by adding on-rate penalties of 2 or 10 for protofilaments having one or two lateral neighbors for the incoming tubulin dimer, respectively, described in (35). All simulations were run using MATLAB R2018b (The MathWorks). The base in vitro parameter set, described in Fig. 5, is used for all simulations. Microtubules are initialized with a seed length of 2 μm, a starting length of 250 tubulin dimers, and a GTP cap consisting of four dimer layers. The energy difference between GDP- and GTP-tubulin (ΔΔG⁰) is imposed either on the lateral bond or the longitudinal bond. Computational code is available upon request.

Figure 5.

Thermokinetic model of microtubule self-assembly as previously described (35, 67) (A) and microtubule net-assembly rate for varying lateral $\left(\mathit{\Delta }{\mathrm{G}}_{\mathrm{lat}}^{\mathrm{0}}\right)$ and longitudinal $\left(\mathit{\Delta }{\mathrm{G}}_{\mathrm{long}}^{\mathrm{0}}\right)$ bond energies (B). The color bar indicates the net-rate values in nm/s. The net rates are obtained in single-state simulations (k_hyd = 0 s⁻¹). Black circle shows our estimate of $\mathit{\Delta }{\mathrm{G}}_{\mathrm{total}}^{\mathrm{0}}$ when v = 0. Orange and cyan circles denote the reference points for GTP- and GDP-tubulin, respectively. The growth (orange) and shortening (cyan) rate contours are consistent with in vitro experiment values. Example microtubule length versus time histories are simulated using the pseudomechanical thermokinetic model in which ΔΔG⁰ is incorporated either on the lateral bond (C) or longitudinal bond (D). In vitro base parameter set (35) is used with the modification of on rate to match BD simulations in water, k_on = 13.2 μM⁻¹ s⁻¹ PF⁻¹, [Tub] = 5.6 μM, k_hyd = 0.2 s⁻¹, ΔΔG⁰ = 2.5 k_BT (on lateral) and 2.8 k_BT (on longitudinal), seed length = 2μm, and on-rate penalties of 2 and 10 for one neighbor and two neighbors. To see this figure in color, go online.

Results and Discussion

Equilibration of tubulin dimers in varying nucleotide states predicts a lack of nucleotide sensitivity of dimer structure in solution

To investigate the nature and strength of the lateral interaction between two tubulin dimers using our multiscale approach, we first built an MD model of two tubulin dimers laterally paired with one of three nucleotide states: GDP-, GMPCPP-, or GTP-tubulin. Similar to earlier MD studies of tubulin heterodimers (32, 33, 61), our simulations (∼200 ns vs. 10–100 ns equilibration in previous studies) indicated that the αβ-tubulin backbone structure is stable and consistent in an aqueous environment regardless of the nucleotide state or lateral neighbor (see Videos S1, S2, and S3). Mean root mean square deviation (RMSD) of backbone atoms was measured from the starting coordinates obtained from the cryo-EM structure as the protein structure stability for three nucleotide states of the dimer. We equilibrated both a single-tubulin dimer in solution for 400 ns and laterally paired tubulins for 200 ns (Fig. 1, A and B). The asymptotic (plateau) behavior of the backbone atoms’ RMSD after ∼50 ns for all conditions without any significant fluctuations reveals that the initial cryo-EM structures are not far from the equilibrium states for single dimers or pairs of laterally associated dimers in solution. In addition, having created the GTP-tubulin structure from GMPCPP-tubulin structure, the backbone RMSD trend confirmed that the estimated GTP structure was stable. To identify the conformational differences between the backbone atoms of different nucleotides, we performed a residue-to-residue RMSD comparison for the average structure using the 150 ns of the production run of different nucleotide states of tubulin (Fig. 1 C). It was observed that the residues in flexible loops have the highest RMSD in each case and the average positions of main helices and β sheets of the backbone atoms remain relatively unchanged among different nucleotide states. Additionally, we examined the flexibility of the residues for the laterally paired tubulins and observed that the M-loop involved in the lateral bond has significantly lower root mean square fluctuations (RMSF) compared to the free M-loop on the neighbor dimer and that the M-loop in α-tubulin is more ordered compared to β-tubulin (see Fig. S6). Hence, the major conformational difference between the laterally paired and a single free dimer was demonstrated to be the ordering of the flexible loops at the lateral interface rather than the backbone structure. Thus, we conclude that the backbone structure of the tubulin heterodimer as measured from cryo-EM measurements is very similar to that predicted from MD at equilibrium for a single dimer or pairs of laterally associated dimers in solution, independent of nucleotide state. This further provides justification for investigating lateral interactions of tubulin by simulating only two tubulin dimers in solution without having to simulate the whole MT lattice.

Figure 1.

Stability of the backbone atom structure when equilibrated with water and ions is shown as the average root-mean-square deviation (RMSD) of the protein system, for a tubulin dimer (A) and tubulin-tubulin complex (B). RMSD per residue for the comparison of different nucleotide’s average structure is shown in (C) for GMPCPP compared to GDP, GTP to GDP, and GTP to GMPCPP respectively. The color bar indicates the value of RMSD in Angstrom. To see this figure in color, go online.

Video S1. MD Simulation of Equilibration of Laterally Paired GDP-Tubulins with Water Molecules and Ions and Key Interacting Lateral Residues

Simulation length, 200 ns. Protein structure is aligned with respect to the initial crystal structure to eliminate rigid body displacements and rotations. Water, nucleotides and ions are not shown for better visualization. Outside view of the microtubule is shown. The two key lateral interactions are highlighted (right), 1) H3 helix: H9-S10 loop (light purple), and 2) M-loop: H1’-S2 loop and H2-S3 loop (dark purple).

Video S2. MD Simulation of Equilibration of Laterally Paired GMPCPP-Tubulins with Water Molecules and Ions and Key Interacting Lateral Residues

Simulation length, 200 ns. Protein structure is aligned with respect to the crystal structure to eliminate rigid body displacements and rotations. Water, nucleotides and ions are not shown for better visualization. Outside view of the microtubule is shown. The two key lateral interactions are highlighted (right), 1) H3 helix: H9-S10 loop (light purple), and 2) M-loop: H1’-S2 loop and H2-S3 loop (dark purple).

Video S3. MD Simulation of Equilibration of Laterally Paired GTP-Tubulins with Water Molecules and Ions and Key Interacting Lateral Residues

Simulation length, 200 ns. Protein structure is aligned with respect to the crystal structure to eliminate rigid body displacements and rotations. Water, nucleotides and ions are not shown for better visualization. Outside view of the microtubule is shown. The two key lateral interactions are highlighted (right), 1) H3 helix: H9-S10 loop (light purple), and 2) M-loop: H1’-S2 loop and H2-S3 loop (dark purple).

Interaction energy decomposition at the tubulin-tubulin lateral interface indicates that lateral interactions are largely independent of dimer nucleotide state

To analyze the contribution of different types of noncovalent interactions predicted to mediate the lateral bond between adjacent tubulins, we decomposed the lateral interdimer interaction energy into H-bond, ionic, and hydrophobic interactions. The equilibrium simulations of each nucleotide state were analyzed for each of these three types of interaction, accounting for the autocorrelation time between each data point of the trajectories to ensure that we obtained temporally independent samples (Fig. 2). On average, ∼6 interdimer H-bonds were found among all three cases (Fig. 2 A). The number of interdimer salt bridges that were present for more than half of the time trajectory was calculated to be ∼2 on average (Fig. 2 B). GMPCPP- and GTP-tubulin showed slightly stronger ionic interaction at the dimer interface compared to GDP-tubulin. For the last component, hydrophobic interactions, we calculated the interdimer buried solvent-accessible surface area (SASA) as the hydrophobic pocket that would have weak interactions with water. Fig. 2 C shows that unlike the ionic interactions, GDP-tubulin has a slightly higher interdimer hydrophobic interaction compared to the other two states. Important residues involved more than 25% in lateral interaction are shown in Table 1. There are two main favorable interactions found within the lateral interface indicated in Fig. 2 D (see Videos S1, S2, and S3). A lock-and-key interaction between M-loop (the key) on one side and H2-S3 and H1′-S2 loops (the lock) on the other side in both subunits, which was also identified as the main lateral contact in the cryo-EM study (29), and an interaction between H3 on one side and H9 and H9-S8 loop on the other side, which was not found within the cryo-EM structure, are identified from our results. Our analysis indicates that the interaction between H3 and the H9-S8 loop is the strongest in all nucleotide cases and the interaction between the M-loop and H1′-S2 is the weakest (<25% in all states). This finding highlights the importance of studying the interactions over the course of a dynamic simulation in which side chains can thermally fluctuate and interact with the neighbors, in contrast to a vitrified protein structure used to obtain cryo-EM structures. Because the M-loop is relatively ordered in the laterally bonded state, compared to a single dimer in solution, we conclude that the decreased entropy upon binding acts to destabilize the lateral bond (40). Besides, the decreased fluctuations due to the lateral bond are more pronounced in the M-loop in β-tubulin, highlighting a key lateral role for the M-loop in β-tubulin compared to α-tubulin (see Fig. S6) (62). In addition, through the interdimer interaction decomposition analysis we found that different nucleotide states have only modestly dissimilar contributing energy components in stabilizing their lateral bond. Even so, the total strength of the interaction components taken together as a lateral bond remains to be calculated and cannot be determined with confidence through equilibrium simulations only.

Figure 2.

Interdimer lateral energy decomposition does not depend on nucleotide state over the production run (150 ns), including mean number of interdimer hydrogen bonds (A), mean number of interdimer salt bridges (B), and mean buried solvent-accessible surface area (SASA) (C). The main interface residues involved in the lateral interaction are shown in (D). Error bars indicate standard error of the mean (SEM) of 100 bootstrap sets of data considering the correlation time of the time series. P-values smaller than 0.05 are marked with a ^∗.To see this figure in color, go online.

Table 1.

Important Interface Residues Identified in Lateral Interaction of the Tubulin Dimers

Interdimer Interaction	Important Involved Residues (>25% Occupancy)	Secondary Structure	Tubulin State (>25% Occupancy)
H-bonds	LYS124-ASP297 (β)	H3–H9	all
	LYS299-ASP120 (β)	H3–H9-S8	all
	SER128-GLU290 (β)	H3–H9-S8	GDP (40%)
	ARG308-ASP116 (β)	H3–H9-S8	GMPCPP (26%)
	ARG308-ASP127 (α)	H3–H9-S8	GMPCPP (30%)
	TYR283-GLN85 (β)	M-loop–H2-S3	GDP (27%), GMPCPP (26%)
	ARG123-GLU297 (α)	H3–H9-S8	GMPCPP (26%), GTP (25%)
Salt bridges	ASP120-LYS299 (β)	H3–H9-S8	all
	GLU127-LYS338 (β)	H3–H10-S9	all
	ASP127-LYS338 (β)	H3–H10-S9	GMPCPP (93%), GTP (31%)
	GLU90-LYS280 (α)	M-loop–H2-S3	GDP (31%), GTP (28%)
	GLU290-LYS124 (β)	H3–H9	GMPCPP (40%)
	ASP116-LYS299 (α)	H3–H9-S8	GMPCPP (30%)
	GLU284-LYS124 (α)	M-loop–H3	GMPCPP (26%)
	GLU297-LYS124 (α)	H3–H10-S9	GMPCPP (35%)

Tubulin dimers exhibit only modest intradimer bending as a function of nucleotide state

Because our starting structure for our simulations was obtained from straight protofilaments through cryo-EM (29), we were interested to see whether the structure of the tubulin dimers deviated from the lattice structure once they were equilibrated in solution without any constraints. In particular, microtubule stability has been proposed to depend on the straight conformation of tubulin dimers, whereas instability has been proposed to result from bent tubulin. This led us to calculate the conformational difference of the three nucleotide states in terms of the intradimer angle between the individual α- and β- subunits (18, 19, 33).

To assess whether nucleotide state and the absence of the microtubule lattice influences intradimer bending angle, we calculated intradimer bending angle for equilibrated dimers as the relative angle of the β-subunit to the α-subunit from the last 350 ns of the MD production run after equilibrium. The rotation angle was calculated as the angle between the vector connecting the center of mass (COM) of α- and β-subunits at time zero (crystal structure) and time t, after superimposing α-subunit backbone atoms (Fig. 3 A), similar to the methodology in previous studies (31, 33, 63). We decomposed the angles of rotation into three perpendicular bending modes (Euler angles) rather than calculating the dot product of the two vectors because we can obtain more information about the directions of bending using this approach. The angle decomposition into x-, y-, and z-directions (Fig. 3, B–E) indicates that there is almost no intradimer twist (θ_z), only a small tangential bending, and the majority of rotation happens around bending radially outward (θ_x). This result is consistent with previous MD simulation of straight tubulin dimer bending both tangentially and radially outward in solution, nucleotide-independent (31, 34, 63) (referred to as “twist-bend” mode in (34)). In addition, bending angles calculated for the laterally paired heterodimers in solution revealed that the presence of a lateral bond diminished the intradimer bending of tubulins independent of nucleotide (see Fig. S7), in agreement with the results of Peng et al. (33). This result confirms that our starting structures for free-energy calculations are not far from equilibrium.

Figure 3.

Intradimer rotation angles of tubulin dimer in solution show both GDP- and GTP-tubulin move toward a slightly bent configuration. Bending angles are defined as the Euler angles required to superimpose vectors connecting the center of mass of monomers (β-tubulin to α-tubulin) to its reference crystal structure coordinates after aligning the whole dimer based on the α-subunit (A). Top view of tubulin (B), side view of tubulin (C), bottom view of tubulin (D), and quantitative values of the angles decomposed in the x, y, and z axes (E) are shown. Cyan, purple, and orange indicate GDP-, GMPCPP-, and GTP-tubulin average structure, respectively, from MD simulation, and gray is the GDP-tubulin crystal structure. Error bars indicate SEM of 100 bootstrap sets of data considering the correlation time of the time series. To see this figure in color, go online.

Although the structures of both unpolymerized GDP- and GTP-tubulin move slightly toward a bent conformation and are significantly different from the crystal structure, the extent of the outward bending (∼3°) does not match the 9–12° intradimer rotation observed when tubulin is bound to stathmin domain and DARPin protein or in γ-tubulin structure obtained from small-angle X-ray scattering (18, 19, 64). We conclude that the discrepancy of bending angle between our simulations and stathmin- and DARPin-bound tubulin may partially stem from the fact that this degree of freedom is slow to relax (high correlation time of ∼10–30 ns during 350 ns of the production run), and further sampling time is required to reach a conclusion about the final converged mode of bending. To further compare our data with the previously published MD study of the straight tubulin dimer in solution (PDB: 3JAS, 3JAT) (34), we analyzed their trajectory, kindly provided by Igaev et al., for the Euler angles of the straight conformation of GDP- and GTP-tubulin (see Fig. S8, A–C). We observe high correlation of movement and fluctuation in the angles even after 3 μs, indicating that bending angles’ convergence require rigorously long sampling. This is further supported by the data in the coarse-grained simulations of protofilaments in Grafmüller et al. (31), suggesting that both the angles’ direction and values fluctuate significantly, and further simulation time is required for convergence of the angle distribution. Even so, having simulated tubulins for several microseconds and using a Cartesian metric for the bending reaction coordinate, Igaev et al. (34) did not see a bending convergence between the curved and straight conformations of unpolymerized tubulin, indicating that experimentally obtained curved conformations do not fully reveal the bending mode of pure unpolymerized αβ-tubulin.

We were also curious to see where our equilibrated structures of single-tubulin dimers are located in Igaev et al.’s PMF of the intradimer bending reaction coordinate, defined by a collective motion of atoms in Cartesian coordinates instead of bending angles (Fig. 6 in (34)). This would potentially show us how far our equilibrated structures are from the ultimate “bent” conformation, starting from a straight tubulin structure. Thus, providing two equilibrated structures from our trajectories (one at the end of 400 ns and the other averaged through the last 100 ns) for both GDP- (3JAS) and GTP-tubulin (3JAT), Igaev et al. calculated the location of our structures along their PMF of bending RC (see Fig. S8 D). The resulting locations indicate that 1) GTP-tubulin does not deviate significantly from the straight MT-like conformation, as expected from the shallow PMF, and 2) as we progress through our 400 ns trajectory for GDP-tubulin, i.e., from the average structure to the 400 ns structure, we move toward the potential minimum, i.e., the bent conformation defined by the bending RC.

Different nucleotide-state tubulins do not have significantly different lateral-bond free-energy landscapes

We expect that lateral free-energy calculations of tubulin could reveal information on how strong the lateral potential energy of tubulin in solution is, what profile the lateral free energy has, and whether the lateral bond is nucleotide-dependent. Equilibrium MD simulations and simple Boltzmann sampling are not efficient approaches to estimate the total energy landscape as two dimers separate laterally. Therefore, we sought to calculate the total lateral interaction energy by performing free-energy calculations to obtain a PMF as a function of nucleotide state (Fig. 4 A). We performed the umbrella sampling method (56) for a center of mass to center of mass distance between dimers as reaction coordinate, which we determined is the most efficient binding path according to BD simulations (Fig. S1). As shown in Fig. 4 A, we observe that there is no energy barrier found between the bound and unbound states in any of our PMF replicates, and it is unlikely to exist because there is no significant desolvation of the binding interface, in contrast to the assumption made in some previous BD studies (39, 40). For comparison, the PMF obtained by MD is plotted along with previously published BD estimates of the lateral-bond strength (38, 39) (see Fig. S9). A statistically significant (p < 0.005) minimum shift is observed between the GDP-state being at 53.7 Å and the average minimum of GMPCPP- and GTP-states being at 52.4 Å. This modest shift is only important when there is incongruity in the lattice, i.e., an ensemble of GDP- and GTP-tubulins with different preferred lateral distance, which can appear as an existing mechanical strain stored in the lattice (65). In a heterogeneous lateral interface, both GDP- and GTP-tubulin will settle down on an intermediate minimum distance, which will be equal to ∼1 k_BT energy change in the lateral-bond strength according to both PMFs. Moreover, we incorporated this minimum shift into our BD simulations of tubulin dimers both in the solution and in the lattice (using the lateral potential from our MD simulations and our previously estimated longitudinal energy profile for the lattice assembly (38)) to investigate whether this shift affects the kinetics of the dimers (see Fig. S10; Table S4). No significant change was detected between the associations of GDP- and GTP-tubulin in solution, and the dissociation rate was found to be high (∼10⁷ s⁻¹) in both cases, with GDP-tubulin having slightly weaker lateral-bond energy $\left(\mathit{\Delta }{\mathrm{G}}_{\mathrm{lat}}^{\mathrm{0}}\right)$. We conclude from the BD results that this minimum shift (∼1 Å) appears only in the ensemble of GDP- and GTP-tubulin lateral interface and is tolerated within the MT lattice.

Figure 4.

Mean potential of mean force (PMF) of tubulin-tubulin lateral interaction is found to be nucleotide-independent for three nucleotide states, GDP- (blue), GMPCPP- (purple), and GTP-tubulin (orange) (A). Dashed lines are showing the lower and upper estimates of lateral interaction well depth (U_lat) from previous publications (39, 45). The error bars are the SEM of 10 replicates for each nucleotide. Average lateral PMF of the three nucleotide states is shown in (B). The error bars are SEM of the three states. Interaction potential (U_BD) input for GDP-tubulin from MD simulations used as lateral zone interaction energies in BD simulations of dimers in solution (C) is shown. Dashed line shows the Shannon-entropy-corrected energy profile, and solid line shows the initial PMF resulted from MD simulation. Lateral input potential from MD simulations and harmonic longitudinal input potential from previous study (38) used as lateral and longitudinal zones’ interaction energies for BD simulation of tubulin assembly into the MT lattice (D) are shown. Distance is measured from surface to surface of the rigid-body dimers in BD simulations. To see this figure in color, go online.

To more specifically examine the difference between the PMFs for GDP-, GMPCPP-, and GTP-tubulins, we analyzed PMFs for their well depth, shape (energy’s first spatial derivative during unbinding), and binding radius (Table 2). We defined a half-force radius as an indicator of where the derivative of the energy profile (force) decays to 50% of its maximal value when unbinding occurs (Methods). The binding radius at which we see significant interaction between the dimers is around 0.5 nm (half-force radius), which is close to the value of Castle et al. (38). The well-depth values are found to be higher than the harmonic potential strengths previously used in Castle et al.; however, considering the shape of our PMFs being closer to a Lennard-Jones potential, with a slow decay rate to zero compared to a steep decay of a harmonic potential, the ultimate kinetic rates are found to be well-aligned. Interestingly, by this metric, the well depth, binding, and half-force radius of the PMFs were not statistically different from each other as a function of the nucleotide state, and therefore, we are able to calculate one final average PMF for the lateral interaction of tubulin regardless of the nucleotide state using all 30 replicate simulations (Fig. 4 B). To check whether the strength of our PMFs is affected by the ionic strength in our simulations, we first neutralized the system, then increased the salt concentration in the solution and ran three replicates of GDP-tubulin solvated with an additional 100 nM KCl (Fig. S4). The analysis of the PMF profile indicates a slight but statistically significant (p < 0.02) minimum shift from 5.39 to 5.19 nm in the added KCl solution. All other properties of the profiles, including the well depth, binding, and half-force radius were not statistically significant (Table S5). We conclude that adding more salt in the solution mediates more interactions at the interface, bringing the subunits closer together at the cost of disrupting previous interactions such that the total lateral-bond strength remains the same. Thus, our previously calculated PMF values in a neutralized simulation system remain valid. Our analysis of the PMFs together shows that the thermodynamics of the lateral association of tubulin dimers is nucleotide-independent.

Table 2.

Potential Well-Depth Mean Values ± SE of 10 Replicates for each Nucleotide State, Potential Minimum Location, and Calculated Binding and Half-Force Radii for Three Simulation Cases

Nucleotide State	Well Depth (kBT) ± SE	Binding Radius (nm) ± SE	Half-Force Radius (nm) ± SE	Potential Minimum (nm) ± SE
GDP	10.95 ± 1.33	0.90 ± 0.06	0.42 ± 0.03	5.39 ± 0.02
GMPCPP	10.7 ± 1.46	0.80 ± 0.22	0.37 ± 0.02	5.27 ± 0.01a
GTP	11.5 ± 1.60	0.71 ± 0.05	0.37 ± 0.02	5.28 ± 0.02a

Values are calculated based on the average of 10 data points around the absolute maximum or minimum due to thermal fluctuations. Kruskal-Wallis combined with multiple comparison in MATLAB have been used as statistical tests.

^ap-values < 0.02 compared to GDP-state.

BD simulations of the lateral-bond potential obtained via MD imply a weak lateral bond

The BD simulations used here were modified from a previously developed model (38) to simulate free tubulins instead of a microtubule lattice. Lateral association was only allowed for the dimers through two lateral zones, similar to the previous model. Instead of using a simple harmonic potential for the lateral interaction, the estimated lateral energy landscape from MD simulations was used.

Upon binding of a tubulin subunit, the dimer loses rotational and translational entropy $\left(\mathit{\Delta }{\mathrm{G}}_{\mathrm{S}\mathrm{rigid-body}}^{\mathrm{0}}\right)$ as well as atomic-level entropy of the flexible side chains and loops in the protein $\left(\mathit{\Delta }{\mathrm{G}}_{\mathrm{S}\mathrm{atomistic}}^{\mathrm{0}}\right)$. MD simulations can mainly sample the atomic-level entropy reduction because of time limitation in each umbrella window sampling (∼30 ns). By contrast, in BD simulations, the atomistic entropy is incorporated in the input potential, and the rigid-body-entropy cost of the coarse-grained subunits is calculated as an output because of longer simulation times (∼1 ms) (Eqs. 2 and 3). To calculate $\mathit{\Delta }{\mathrm{G}}_{\mathrm{lat}}^{\mathrm{0}}$ from MD simulations, multiple PMFs restraining different degrees of freedom (rotational and translational) must be run for both bound and unbound proteins (66), which is computationally expensive. Alternatively, using BD simulations, we can estimate the full entropy cost of binding and therefore obtain $\mathit{\Delta }{\mathrm{G}}_{\mathrm{lat}}^{\mathrm{0}}$. We chose the latter and calculated a set of $\mathit{\Delta }{\mathrm{G}}_{\mathrm{lat}}^{\mathrm{0}}$ for different nucleotide states. To fully relate the interaction energy outputs (PMFs) of MD simulations to BD simulations, we calculated and subtracted rigid-body entropy as a function of the reaction coordinate in MD simulations so that the PMF would be closer to the input potential of interaction of the BD simulations (U_BD-input), containing zero rigid-body entropy. For rigid-body-entropy correction of the PMFs, Shannon entropy was calculated for each MD simulation as a function of rotational and translational degrees of freedom (Figs. 4 C and S11; Methods).

$$\mathit{\Delta }{\mathrm{G}}_{\mathrm{MD}}^{\mathrm{0}}=\mathrm{U-T}\left(\mathit{\Delta }{\mathrm{S}}_{\mathrm{rigid-body}}+\mathit{\Delta }{\mathrm{S}}_{\mathrm{atomistic}}\right)$$ (2)

$${\mathrm{U}}_{\mathrm{BD-input}}\mathrm{=U-T}\left(\mathit{\Delta }{\mathrm{S}}_{\mathrm{atomistic}}\right)$$ (3)

Previous studies modeling MT dynamics have used different potentials of interaction for tubulin (7, 37, 38, 39, 41, 47, 67, 68). We reviewed the values found for lateral and longitudinal bond strengths and the energy difference distinguishing the GDP- and GTP-states (ΔΔG⁰) in Table 3. Note that in BD simulations, there is a difference between the intrinsic bond energies (U_input-BD) and the total bond energy $\left(\mathit{\Delta }{\mathrm{G}}_{\mathrm{B}}^{\mathrm{0}}\right),$ mainly because the interaction zones are not all perfectly aligned when two subunits approach and bind to each other. Therefore, $\mathit{\Delta }{\mathrm{G}}_{\mathrm{B}}^{\mathrm{0}}$ would be the time average of lateral interaction energy and not the maximal intrinsic energy (well depth) $\left(\mathit{\Delta }{\mathrm{G}}_{\mathrm{B}}^{\mathrm{0}}<\mathrm{U}\right),$ a distinction described previously by Castle et al. (38).

Table 3.

Estimates of Tubulin Lateral and Longitudinal Bonds and the GDP-GTP-Associated Energy Difference of Previous Published BD and Thermokinetic Models

Study	Ulat	Ulong	$\mathit{\Delta }{\mathit{G}}_{\mathrm{lat}}^{\mathrm{0}}$	$\mathit{\Delta }{\mathit{G}}_{\mathrm{long}}^{\mathrm{0}}$	ΔΔG0	Probability
VanBuren et al. (47)			−3.2 to −5.7	−6.8 to −9.4	2.1–2.5	<0.02
VanBuren et al. (37)			−3.2 to −5.7	−9.4 to −6.8	3.7–4.2	<0.02
Gardner et al. (67)			−4.5 to −5	−9.5	N/A	N/A
Margolin et al. (7)			−0.4 to 1.4	−11 to −18	1.75a	0.18
Coombes et al. (68)			−5.7	−7.2	3.3	<0.02
Castle et al. (38)	−6.8	−20.4	N/A	N/A	N/A	N/A
Zakharov et al. (39)	−9.1	−15.5	N/A	N/A	5.5	<0.02
Piedra et al. (41)	N/A	N/A	N/A	−5.8	3.9b	<0.02
Castle et al. (35)	N/A	N/A			3.6	<0.02
McIntosh et al. (45)	−5.3	−16.6	N/A	N/A	1	0.31

All energy values are in k_BT. N/A, not applicable

^aThe maximal value of $\mathit{\Delta \Delta }{\mathrm{G}}_{\mathrm{lat}}^{\mathrm{0}}$ found within different cases of lateral neighbor in this study.

^bThis value is obtained by converting a 50-fold weakening effect in the kinetic rates to energy.

Because MD energy values have an SD around the mean because of variability in the initial conditions sampled from equilibrated structures, random sampling within the standard error of the mean is required. We sampled through entropy-corrected PMFs within their standard error of the mean as the input for our BD simulations and calculated $\mathit{\Delta }{\mathit{G}}_{\mathrm{lat}}^{\mathrm{0}}$ for GDP- and GTP-state and a series of ΔΔG⁰ values. Because we have a distribution of ΔΔG⁰ values, we calculated the probability at which a specific ΔΔG⁰ used in previous models can be found within the distribution. The analysis indicates that a ΔΔG⁰ larger than 2 k_BT is not consistent (>95% confidence) with the $\mathit{\Delta }{\mathit{G}}_{\mathrm{lat}}^{\mathrm{0}}$ values from the two energy landscapes obtained in MD simulations. Based on the calculations of VanBuren et al. (47), a ΔΔG⁰ value of larger than 2.1 k_BT is required to maintain the MT dynamic instability observed in vitro and in vivo. Thus, our results imply that the GTP- and GDP-state-dependent energetics driving the dynamic instability behavior of microtubules are not solely related to nucleotide-dependence of the lateral interactions. This is consistent with the observation from the cryo-EM structure comparison of GDP and GMPCPP protofilaments (29) that the main structural difference between the two nucleotide states resides in the longitudinal compaction of the lattice rather than at the dimers’ lateral interface.

Using BD simulations of tubulin dimers in solution, kinetic rates and full entropy costs were calculated for different nucleotide states (Tables 4 and 5), indicating that the lateral bond, by itself, is weak, unfavorable, and shortly broken after ∼100 ns. This is in line with the results previously obtained (38) that longitudinal association is necessary to make the lateral bond stable. To demonstrate the connection between MD simulations, BD simulations, and thermokinetic modeling, we used our previously estimated harmonic longitudinal bond (38) along with our MD estimated lateral-bond potential as inputs for BD simulations of microtubule lattice assembly (Fig. 4 D). The kinetic outputs are indicated in Table 5, for the association to the tip of a protofilament without any lateral neighbor and with one lateral neighbor (a penalty of ∼2 is observed in the on rate). The results for the assembly to a protofilament with two lateral neighbors is not calculated because of the stability of the subunits with long unbinding times (>50 s). We found a $\mathit{\Delta }{\mathrm{G}}_{\mathrm{long}}^{\mathrm{0}}$ value of ∼−6.4 k_BT for the longitudinal bond and a $\mathit{\Delta }{\mathrm{G}}_{\mathrm{lat}}^{\mathrm{0}}$ value of ∼−5.3 k_BT for GDP-tubulin and ∼−5.2 for GTP-tubulin. These estimates suggest that our PMFs are well-aligned with the previous estimates of $\mathit{\Delta }{\mathrm{G}}_{\mathrm{lat}}^{\mathrm{0}}$ (35, 67), and the kinetic rates of assembly are very fast when no lateral neighbor is present.

Table 4.

BD Model Outputs for a Dimer in Solution with One Lateral Neighbor for Three Different Potential Inputs

Lateral Bond Only in Solution (One Lateral Neighbor)
Model Estimated Parameters	GDP	GMPCPP	GTP
kon (μM−1 s−1)	4.54 ± 0.40	2.48 ± 0.39	2.53 ± 0.51
koff (s−1)	1.64 × 107	1.63 × 107	1.62 × 107
$\mathit{\Delta }{\mathit{G}}_{\mathit{B}}^{\mathrm{0}}$( kBT)	−0.45	−0.46	−0.49
ΔG0(kBT)	+1.28	+1.89	+2.57
$\mathit{\Delta }{\mathit{G}}_{\mathit{S}}^{\mathrm{0}}$( kBT)	+1.73	+2.34	+3.06

Table 5.

BD Model Outputs for Dimer Assembly into Microtubule Lattice Protofilaments with Zero and One Lateral Neighbor for Three Different Potential Inputs

Lateral Bond + Longitudinal Bond in MT Lattice
Model Estimated Parameters	Longitudinal Bond Only	One Lateral Neighbor
		GDP	GTP
kon (μM−1 s−1 PF−1)	13.20 ± 0.39	8.63 ± 0.41	7.65 ± 0.31
koff (s−1)	23 × 103	75	72
$\mathit{\Delta }{\mathit{G}}_{\mathit{B}}^{\mathrm{0}}$( kBT)	−17.46	−29.72	−28.86
ΔG0(kBT)	−6.36	−11.66	−11.58
$\mathit{\Delta }{\mathit{G}}_{\mathit{S}}^{\mathrm{0}}$( kBT)	+11.10	+18.06	+17.28

We note that the per protofilament on-rate constant of ∼13 μM⁻¹ s⁻¹ PF⁻¹ for the zero lateral neighbor case is much higher than what was measured for the tubulin-bound gold nanoparticles association (69). We believe that because of a temporal resolution limitation of 10 ms in the nanoparticle experiments, this previous study was not able to observe fast binding and/or unbinding events that we predict here to be around 0.04–0.05 ms and distinguish them from the slow diffusion of gold particles. Thus, the nanoparticle-based estimate is best regarded as a lower bound, and our BD estimates are above this estimate. We note that our BD calculated on rates here are slightly higher (∼2-fold) than the rates reported for in vitro experiments (67) because of simulating the dimers in water, which has a lower viscosity than cytoplasm.

Multiscale modeling using MD-BD outputs as inputs to thermokinetic modeling

We ran our previously described thermokinetic model (35, 47) (Fig. 5 A) with the output rates and energies resulted from BD simulations (Fig. 4 D). ΔΔG⁰ can be incorporated on the longitudinal bond or the interdimer bending preference of the dimers or both, as further MD simulations will clarify this issue in the future. Because in the thermokinetic model, we do not have an explicit term for the bending mechanical energy in the dimers, we can impose the energy difference on the lateral bond, as shown in Fig. 5 B, as the presence of a lateral bond imposes a strain on a dimer with a preferential curled conformation. Thus, a dimer with a preferred curled conformation would break the lateral bond first, compared to a dimer that favors a straight conformation, although the lateral-bond strength itself is the same. In here, we implemented ΔΔG⁰ on the lateral or longitudinal energy (Fig. 5, C and D), as our demonstration of how to connect our BD results to our thermokinetic model, and found that a $\mathit{\Delta }{\mathrm{G}}_{\mathrm{lat}}^{\mathrm{0}}$ of ∼−5 k_BT requires a $\mathit{\Delta }{\mathrm{G}}_{\mathrm{long}}^{\mathrm{0}}$ of ∼−7 k_BT, which are pretty consistent with our estimates. We note that our estimate of longitudinal bond energy ($\mathit{\Delta }{\mathrm{G}}_{\mathrm{long}}^{\mathrm{0}}$ = −6.4 k_BT) is lower than what is obtained from thermokinetic model. This is due to our input longitudinal potential being an estimated harmonic potential and can be adjusted by performing further MD simulations to identify a more accurate longitudinal potential as a function of nucleotide. The exact origin and value of the inherent energy difference at hydrolysis (ΔΔG⁰) can also be calculated via future MD simulations.

Hence, the combination of the lateral bond along with the longitudinal bond makes assembly dynamics feasible in a manner previously described in thermokinetic terms that predict cellular-level (micrometer-scale) dynamics at timescales of minutes (35, 47). The thermokinetic modeling outputs such as MT growth rate, shortening rate, catastrophe frequency, and rescue frequency can be then used as inputs to cell-level models to predict dynamics and microtubule distributions associated with cell functions (46, 70, 71, 72, 73, 74, 75), which allows us to now connect MD to BD to thermokinetic to cell-level models.

Conclusions

A key challenge in biology is linking atomistic-level information obtained by structural analysis to cellular-level dynamics and function. In this study, we established a final link in an integrated multiscale modeling chain that intertwines MD-BD-thermokinetic-cell-level modeling approaches for studying microtubules that connects information from atoms in crystal structures to microtubule dynamics in cells. Thus, we can interconnect from Ångstrom length and femtosecond timescales to micrometer length and minute timescales, a span of five orders of magnitude in the spatial domain and 17 orders of magnitude in the temporal domain. We used this approach to address a key question about the role of the lateral bond in MT stability. We found that tubulin dimers interact with each other laterally with a non-negligible force (∼30–100 pN) at an edge-to-edge distance of 0.5 nm with an average potential strength of ∼11 k_BT, with previously estimated stiffness of ∼0.2–0.5 pN/nm, regardless of nucleotide state. BD simulations revealed that the lateral bond is too weak, with a high rate of dissociation, ∼10⁷ s⁻¹, to be a significant pathway to MT assembly by itself. In fact, the BD simulations suggest that, given the full entropy cost of lateral binding, the total standard Gibbs free-energy change of binding is unfavorable for the lateral bond by itself. This is consistent with the hypothesis that lateral-bond formation is accompanied by ordering of the flexible lateral loops such as M-loop and is entropically unfavorable despite a number of favorable noncovalent interactions (40). Because dynamic instability is a key feature of microtubule assembly and vital to microtubule function, we investigated whether the lateral-bond strength was nucleotide-dependent and found that it was not (<2 k_BT). We propose that there are two possible scenarios. Either the energetic difference driving the dynamic behavior of microtubules can be incorporated into the longitudinal bond or else into the preferred bending angle in the lattice. Further MD simulations along those reaction coordinates can help us discriminate between these two possibilities.

The MD simulation findings in this study are supported by previous MD simulations of tubulin showing that the backbone structure of tubulin is significantly stable (simulated up to ∼5 μs) and the straight tubulin structure with zero lateral neighbors has a tendency for intradimer bending, independent of its nucleotide state (31, 33, 34, 63). The energy profiles from MD simulations in this study are remarkably well-aligned with our previous estimates based on integrated BD and thermokinetic modeling constrained by experimental kinetic rates and MT tip structure observed from assembly dynamics in vitro and in vivo (35, 37, 67). Given that the previous work started with thermokinetic modeling at the molecular-organelle level and from there led to BD modeling, the consistency across studies increases confidence in these estimates and further supports our conclusion that there is sufficient sampling in our MD simulations. The consistency between studies with different scales allays two concerns about the MD calculations. First, our simulation results used as a starting point a tubulin structure decorated with monomeric construct of kinesin-1, which later was found to have some minor effects on compacting the longitudinal spacing of the GMPCPP-lattice while not affecting the already-compacted GDP-lattice, using the same cryo-EM buffer and conditions (76). This potentially conflicts with the in vitro results of Peet et al., who found that the strong binding state of kinesin-1 (ATP) extends GDP-tubulins by 1.6% (77). We note, however, that the modest effects of kinesin are all described at the longitudinal spacing rather than the lateral interaction, and the equilibration MD runs reached steady state in ∼50 ns. Second, the umbrella sampling is performed on a specific chosen reaction coordinate, for which there is some uncertainty. We note, however, that the BD sims of tubulins unbinding largely follow the reaction coordinate used in the PMF calculation.

Additionally, with further MD calculations to incorporate the longitudinal bond with the lateral bond, the resulting PMFs can be combined to perform BD simulations whose outputs are k_on and k_off for single-tubulin addition and loss in a thermokinetic model of microtubule assembly. The thermokinetic model can then be used to investigate microtubule tip structure and assembly rates in growth and shortening phases. For example, assembly rates can be used to investigate the cell-level distribution of MTs in cells such as neurons (46), consisting of numerous MTs stochastically undergoing dynamic instability. Hence, thermokinetic modeling combined with cell-level modeling can establish a direct connection with in vivo experimental results and large-scale cellular processes. In conclusion, our multiscale approach provides a systematic model in which a small change in tubulin structure, due to mutation, post-translational modification, binding of a microtubule-targeting drug, or MAPs, can be connected to microtubule dynamics. Thus, the approach outlined here creates a spatiotemporal connection between MD, BD, thermokinetic, and cell-level modeling to scale from Ångstroms to micrometers and femtoseconds to minutes to address longstanding questions regarding the origin of MT dynamic instability.

Author Contributions

Conceptualization, M.H. and D.J.O.; methodology, M.H., J.N.S., and D.J.O.; software, M.H. and B.T.C; analysis, M.H. and D.J.O.; writing: original draft, M.H. and D.J.O.; writing: review and editing, all authors; visualization, M.H.; supervision, J.N.S. and D.J.O.; and funding acquisition, J.N.S. and D.J.O.

Editor: David Sept.

Supporting Citations

References (78, 79, 80, 81, 82, 83, 84, 85, 86) appear in the Supporting Material.