Coiled-Coil Response to Mechanical Force: Global Stability and Local Cracking

Abstract

Coiled coils are important structural motifs formed by two or more amphipathic α-helices that twist into a supercoil. These motifs are found in a wide range of proteins, including motor proteins and structural proteins, that are known to transmit mechanical loads. We analyze atomically detailed simulations of coiled-coil cracking under load with Milestoning. Milestoning is an approach that captures the main features of the process in a network, quantifying kinetics and thermodynamics. A 112-residue segment of the β-myosin S2 domain was subjected to constant-magnitude (0–200 pN) and constant-direction tensile forces in molecular dynamics simulations. Twenty 20 ns straightforward simulations at several load levels revealed that initial single-residue cracking events (Ψ > 90°) at loads <100 pN were accompanied by rapid refolding without either intra- or interhelix unfolding propagation. Only initial unfolding events at the highest load (200 pN) regularly propagated along and between helices. Analysis of hydrophobic interactions and of interhelix hydrogen bonds did not show significant variation as a function of load. Unfolding events were overwhelmingly located in the vicinity of E929, a charged residue in a hydrophobic position of the heptad repeat. Milestoning network analysis of E929 cracking determined that the mean first-passage time ranges from 20 ns (200 pN) to 80 ns (50 pN), which is ∼20 times the mean first-passage time of an isolated helix with the same sequence.

Introduction

Coiled coils are important structural motifs formed by two or more amphipathic α-helices that twist into a supercoil. These motifs are found in a diverse array of proteins, including motor proteins, such as myosin (1–3) and kinesin (4,5), and structural proteins, such as intermediate filaments (including keratins, vimentinlike proteins, and lamins) (6,7), tropomyosin (8), and myosin thick filaments (9). In addition, mutations in the coiled-coil β-myosin S2 domain are known to cause familial hypertrophic cardiomyopathy, potentially by blocking the interaction of myosin-binding protein-C with the S2 domain (10,11). Finally, interactions between polar residues on the myosin II head domain and the S2 domain are implicated in the regulation of myosin II motor activity (12).

Coiled coils are stabilized by both interhelix hydrophobic and pairwise interactions (13,14). Both sets of interactions arise from interhelix complementarity due to the characteristic heptad-repeat structure, in which residues are typically labeled a–g. Using the designations of this repeat structure, the classic coiled coil has hydrophobic residues in locations a and d (15,16). These hydrophobic residues arrange themselves in a so-called knobs-in-the-hole packing such that their hydrophobic side chains are buried in close proximity to the corresponding heptad position from the other chain. In addition to the hydrophobic sources of stability, polar residues e and g are solvent-exposed and thought to contribute to interhelical recognition through the formation of pairwise (salt bridges and/or hydrogen bonds) interactions (17,18). The remaining three residues of the repeat (b, c, and f) must also be polar, as they are exposed to the solvent (19,20). These three residues may also contribute to interactions between coiled coils in the formation of multicoil structures such as the myosin thick filament.

Both experimental and computational approaches have been used to test the stability and mechanical strength of coiled-coil domains. Atomic force microscopy (AFM) experiments on the unfolding of various myosin coiled-coil domains reveal a range of unfolding forces, including 20–25 pN for full-length rabbit skeletal myosin dimers (21), 30–50 pN for various myosin II fragments, including the S2 and light chain (22), and 40–80 pN for the tail domains of processive and nonprocessive myosin V molecules (23). Critically, these experiments reveal that unfolding occurs as a molecular extension at a constant plateau force indicative of a structure lacking discrete domains. The lack of discrete domains means that the unfolding process could rapidly reverse such that refolding occurs with the release of the applied load (21).

Computational studies of coiled-coil unfolding have relied on the steered molecular dynamics (SMD) approach to perform an in silico AFM experiment (24). SMD simulation of a 43-amino-acid segment of the myosin S2 domain revealed unfolding forces of ∼500 pN (22). Simulations of vimentin coiled-coil dimer unfolding revealed unfolding forces of ∼350 pN (25) and 200–300 pN (7). In both cases (myosin S2 and vimentin), the lack of a discrete multidomain structure leads to plateau regions of unfolding. Unfolding simulations of an individual fibrinogen domain via SMD revealed transition forces of 100–200 pN (26). In each case, computational limitations required loading rates in excess of those used in the AFM experiments. This dissonance between loading rates has been implicated in the nearly order-of-magnitude difference between experimentally measured and computationally determined unfolding forces (27). It should be noted that these simulations also focus on what will be referred to here as the propagation of unfolding. That is, the spatial scale of unfolding is the whole molecule.

Although previous studies have revealed the mechanics of coiled-coil structures on a molecular level, it is unclear how the application of force alters coiled-coil structure before the unfolding of the whole molecule. This issue is particularly important in biological contexts in which molecular load transfer may not completely unfold the protein but may have implications for intermolecular binding through local disruption to the coiled-coil structure and to mechanical stability of biomolecular carriers under load. In addition, from a mechanical perspective it is unclear what effect load has on the classic coiled-coil stabilization forces (interhelical hydrophobic packing and pairwise interaction). Finally, the coiled coil provides an opportunity to examine the stabilization of local, residue-scale structural disruptions due to the presence of neighboring tertiary structures. It is in this context that previous studies of isolated helix unfolding (28) may provide a useful frame of reference for interpreting the advantages of coiled structures in delaying or preventing the first steps in the helix to extended chain transition, referred to here as the initiation of unfolding.

This work examines a 224-residue, ∼16-nm-long fragment of the β-myosin S2 coiled coil (29) through constant-force molecular dynamics simulations, analysis of coarse-grained variables describing the progression of unfolding, and kinetic postprocessing of both straightforward and enhanced sampling simulations via the Milestoning analysis framework. We reveal the effect of six load levels (0 pN, 25 pN, 50 pN, 75 pN, 100 pN, and 200 pN) on molecular-scale metrics, as well as the kinetics of the initiation of unfolding of a key, relatively unstable residue, E929, known to be involved in intermolecular binding and disease (10). Finally, we compare the initiation of unfolding of a residue within the coiled-coil structure to that of the average residue of an isolated α-helix with the same sequence as the chains of the coiled coil, based on our previous work (28).

Materials and Methods

Simulation conditions

The Protein Data Bank (PDB) structure of the human β-myosin S2 fragment containing a two-helix coiled coil (PDB ID 2fxm) was used as the base structure (29). The length of chain A of the base structure was reduced to provide two chains of equal length, leaving residues A850–L961 for both helices. Forces (0 pN, 25 pN, 50 pN, 75 pN, 100 pN, and 200 pN) were distributed across the 18 residues on either end of both chains (18 × 2 = 36 residues/end). The residues of the loaded groups were A850–E867 and E944–L961. The force magnitudes referenced throughout are the total on both chains. Forces were applied along the nominal longitudinal axis of the coiled coil. See the Supporting Material for a detailed description of the simulation conditions, which were the same as those used in our previous study (28). In this study, we seek cracking events, meaning the changes in a Ψ dihedral of a first amino acid to values >90°.

Milestoning

Basic Milestoning theory and calculation of the mean first-passage time

Milestoning is a tool for calculating the mean first-passage time (MFPT, the mean of the times required for the system to reach a predefined target for the first time). More detailed theories of Milestoning can be found elsewhere (28,30–33), as well as in the Supporting Material. A brief description follows. The atomic simulations are projected into a coarse-grained space that allows the calculation of the two fundamental quantities of Milestoning: the transition kernel, $K_{ij}\left(t\right)$, and the average milestone lifetime, ${\langle \tau \rangle }_i$.

The transition kernel, $K_{ij}\left(t\right)$, is an n × n matrix (where n is the number of milestones) that contains the probability of transitioning from milestone i (given that we start in i) to milestone j. A milestone is an interface between two cells in conformation space where the center of the cell is defined by a single configuration in coarse space called an anchor (see Table 1 and Fig. 1).

Table 1.

Anchors defined by quantity and type of hydrogen bond and Ψ dihedral angle

AA-state designation	Quantity of hydrogen bonds			Ψ angle values (°)
	310	α	π
Ψ > 90°	Any	Any	Any	>90, <−150
90° > Ψ > 0°	Any	Any	Any	0–90
None	0	0	0	<0
310	Any	0	0	<0
310/α2	Any	2	0	<0
310/α1	Any	2	0	<0
α3	0	3	0	<0
α2	0	2	0	<0
α1	0	1	0	<0
π/α3	0	3	Any	<0
π/α2	0	2	Any	<0
π/α1	0	1	Any	<0
π12	0	0	1 or 2	<0
π34	0	0	3 or 4	<0

The keyword Any indicates that any quantity of a given type of hydrogen bond may be present for a given anchor; for example, the 310 state can have any number of 310-helical bonds (for this anchor, that would be either one or two bonds). Note that some anchors allow a variable quantity of hydrogen bonds; for example, the π34 anchor can have either three or four π-helical hydrogen bonds. The full atomically detailed trajectories computed with GROMACS (see Simulation conditions) are mapped to Milestoning trajectories and are used to compute K_ij and 〈τ〉_i.

Figure 1.

Schematic of the three main stages of simulations. The SF simulations are initiated from milestone (MLST) i and allowed to evolve over 20 ns. Although some of the SF simulations may reach the unfolded destination MLST, many of the simulations do not. ES simulations are prepared by identifying a seed for the restrained simulations from an image of the SFs that reached the target, barrier milestone (β). This structure is then restrained to β, and a short simulation with the restraints provides a sampling of β. ES simulations are initiated from structures sampled in the restrained simulations. Each ES seed is first tested to determine whether the structure is a first hitting point by examining whether the random velocities used at the initiation of the ES simulation evolve the structure to another MLST (dashed line) or to a recrossing of β (lower trajectory). Those simulations that recross are discarded. ES structure/velocity combinations that hit another MLST first (dashed line) are used with the negative of the test velocity and allowed to evolve (solid blue), hopefully to MLST f.

The average lifetime of a milestone, ${\langle \tau \rangle }_i$, which is the time that the trajectory spends after hitting milestone i before crossing (any) other milestone, is defined by ${\langle \tau \rangle }_i={\sum }_j\langle t_{ij}\rangle ={\sum }_j\left[{\int }_0^{\mathrm{\infty }}t\times K_{ij}\left(t\right)dt\right]$. These two fundamental quantities are then used to calculate the MFPT (or ${\langle \tau \rangle }_{if}$) from milestone i to the final (assumed absorbing) milestone, f, by

$${\langle \tau \rangle }_{if}=p_i\sum _j{\left[\mathbf{I}-\mathbf{K}\right]}_{ij}^{-1}{\langle \tau \rangle }_j,$$ (1)

where the initial milestone distribution, $p_i$, is the initial (t = 0) probability density, I is the identity matrix, and $K_{ij}$ is an element of the transition matrix $({\left(\mathbf{K}\right)}_{ij}={\int }_0^{\mathrm{\infty }}{K}_{ij}\left(t\right)dt)$. To model an absorbing boundary condition at f, we set $K_{fi}=0\forall i$.

Stationary flux and net flux calculation

The transition matrix and average milestone lifetimes, modified to express a recirculation condition between the final and initial states, can also be used to calculate the stationary probability distribution, $p_{i,stat}$, of being in a state i (i.e., the last milestone that was passed is i), as well as the stationary reactive flux vector (${\mathbf{q}}_{stat}$). The stationary reactive flux vector is calculated directly from the linear equation below or alternatively as the eigenvector with zero eigenvalue of the matrix I − K:

$${\mathbf{q}}_{stat}\left(\mathbf{I}-\mathbf{K}\right)=0$$ (2)

The stationary probability distribution is a function of both the transition matrix—via the stationary reactive flux vector—and the average milestone lifetime (32):

$$p_{i,stat}={\left({\mathbf{q}}_{stat}\right)}_i\cdot {\langle \tau \rangle }_i,$$ (3)

where the multiplication is of the ith element and is not a vector dot product.

The net flux between any two anchors (β and γ) is defined as the difference in the stationary flux value for the two directional milestones (i and j) linking these two anchors (where i is the interface representing a transition from β to γ and j is the interface representing the γ → β transition). Thus, $\text{Δ}{q}_{\beta \gamma ,stat}=q_i-q_j$.

Incorporation of enhanced sampling transition data

The transition matrix was created from both a set of original straightforward (SF) simulations and a series of enhanced sampling (ES) simulations (see Simulation flow below). The ES simulations were targeted to specific, high-energy-activated milestones that are inadequately sampled in the SF simulations (see below for a description of the ES procedure). The ES simulations make it possible to have a fully connected kinetic matrix. This approach is used extensively in typical Milestoning applications (30,31,34). Each type of simulation produces a quantity of milestone transitions that must be combined to form the overall transition matrix used in the MFPT calculation and the network analysis. We define $n_{ij}^{SF}$ as the quantity of i→j transitions from the SF simulations and $n_{ij}^{ES,\text{Ω}}$ as the quantity of i→j transitions from ES simulation Ω. In a similar way, $n_i^{SF}$ and $n_i^{ES,\text{Ω}}$ are the total number of initial conditions to milestone i in the SF and ES simulations, respectively. With these definitions in hand, the transition matrix is defined as

$$K_{ij}=\sum _{\text{Ω}}\frac{n_{ij}^{SF}+n_{ij}^{ES,\text{Ω}}}{n_i^{SF}+n_i^{ES,\text{Ω}}},$$ (7)

where the summation is over all ES simulations, with the additional criterion that the only milestone transitions included from an ES simulation are the milestone transitions from the milestone targeted by the ES simulation. As described in the Simulation flow section, the first step of a set of ES simulations is a simulation in which the system is restrained to the milestone of interest. This milestone to which the system is restrained is the target milestone, i.

Anchor definitions

The atomically detailed trajectories produced by GROMACS were mapped to a previously defined coarse-grained space (28) in which each state is referred to as an anchor in the language of Milestoning. Note that milestones are defined as the interfaces between these anchors (i.e., a transition domain from one anchor to another). Location within the implemented anchor space is defined based on 1), the quantity and type of hydrogen bonds (where the type of hydrogen bond is a function of the spacing between the donor and acceptor atoms), and 2), the value of the Ψ dihedral angle. Based on previous results, other metrics (such as the Φ dihedral angle) are ignored in the current analysis. Three types of hydrogen bonds that span the residue of interest are considered based on the specific donor and acceptor atoms: 3₁₀, α-, and π-helical. There is a none state, in which no hydrogen bonds are formed but the Ψ angle is <0°. Further, a Ψ angle >90° is used to define when a residue has initiated unfolding—and is therefore at the final state for the present analysis; an intermediate unfolded state is defined by 0° < Ψ < 90°.

Simulation flow

The lack of unfolding across a range of test load levels in a series of SF molecular dynamics simulations revealed the necessity for ES simulations to fully populate the transition matrix and milestone lifetime vectors (Fig. 1).

The simulation steps are:

• 1.A set of 20-ns SF simulations was used to populate states accessible due to thermal fluctuation and load-induced effects and to provide seeds at specific, activated milestones for enhanced sampling;
• 2.A set of structures was extracted from these SF simulations as the ES seeds.
• 3.A series of short (1-ns) simulations were performed for each ES seed with the critical degrees of freedom defining the desired milestones restrained through the application of harmonic restraints.
• 4.Fifty structures were sampled from each restrained simulation and used for even shorter (50-ps) restraint-free ES simulations to populate the transition matrix and milestone lifetime vector entries of the higher-energy milestones selected for ES.

The lengths of the ES trajectories were sufficient to observe transitions to milestones different from the starting interface, and therefore, we were able to map these trajectory fragments into Milestoning space. The transition data from the ES simulations were then combined with the transition data from the SF simulations to create the transition matrix and milestone lifetime vector data (both means and variances) used for the MFPT-calculation sampling procedure (see Supporting Material for details of the sampling procedure).

SF Simulations

A set of 20 SF simulations was run at each load level (except at 200 pN, for which 10 SF simulations were run) to provide both the seeds for ES simulations and the Milestoning data for the low-energy, easily accessible states. A smaller set of SF simulations (10 simulations) was run for 200 pN based on the greater confidence in the calculated MFPT as seen by the smaller standard deviations at this load level (see Fig. 5). A 2-ns equilibrium (i.e., no-load) simulation was run after equilibration to provide the 10 SF simulation seeds extracted every 200 ps. The same set of SF simulation seeds was used for each load level, with each of the 10 seeds repeated twice and random initial velocities used for each distinct simulation. The SF simulations were run for 20 ns each, providing a total of 400 ns of SF simulation data for each load level and a total of 2.4 μs of SF data.

Figure 5.

MFPT of the initiation of residue 929 unfolding as a function of load. MFPT was calculated with Eq. 1 using the transition matrix and milestone lifetime vector built from both the SF and the ES simulations. Data points are the average of the MFPT calculation sampling procedure, and error bars indicate the mean ± SD of the calculated MFPT distributions.

Identification of ES seeds

To better sample the relatively rare transitions associated with unfolding, a set of structures was simulated with distance and dihedral angle restraints to provide ES. Table 2 lists the target milestones, the number of simulations initiated per load at each target milestone, and statistics for the sampling enhancement of the 0 pN ES simulations. As the 20-ns SF simulations provided good sampling of the milestones in the neighborhood of the folded conformations, all five ES milestones were selected because they were likely to sample the unfolded states based on both their explicit proximity to unfolding and knowledge of the unfolding pathways for individual α-helices.

Table 2.

Milestones used for enhanced sampling

Target milestone	Simulations/load	Recrossing events (0 pN)	Visits to Ψ > 90° (0 pN)
310: α1	250	13	6
α1: none	500	33	2
None: 90° > Ψ > 0°	500	53	113
310: 90° > Ψ > 0°	500	62	109
α1: 90° > Ψ > 0°	500	42	90

Target milestones of the ES simulations, number of ES simulations initiated per load, data for the number of recrossing events that were discarded for the 0-pN ES simulations (column 3), and number of unfolding events from the 0-pN ES simulations (column 4). As noted in the text, structure/velocity combinations leading to recrossing events are assumed not to be part of the first-hitting distribution of the target milestone and are therefore discarded (30,31).

All ES seeds must first satisfy the desired milestone definition (i.e., the last milestone passed corresponded to the target milestone). In addition, the seeds selected had two of three α-helical hydrogen-bond distances that were within one standard deviation of the average hydrogen-bond distances for the desired milestone. An analogous procedure was implemented for the Ψ dihedral angle. Candidate ES seeds had to meet both the hydrogen bond and Ψ dihedral angle criteria to be considered. This pool of candidate restrained seeds was further filtered by eliminating any candidates whose neighboring two residues in the C- and N-terminal directions had positive Ψ angles. This filter on neighboring residues ensured that the seeds selected for restrained simulations were unfolding initiation events rather than propagation events.

Restrained simulations

Restraint to the target milestone was enforced through the application of three sets of restraints: 1), distance restraints for the three α-helical hydrogen bonds; 2), distance restraints on the two 3₁₀-helical hydrogen bonds; and 3), a Ψ dihedral angle restraint (see Supporting Material for more details of the restraint selection and definition). The set of selected ES seeds was used for all load levels. The 1-ns restraint simulations were run in the presence of the applied loads to ensure that any effects of the loads on the structure were present when the ES simulations were initiated. A total of 50 postrestraint ES seeds were selected from each restrained simulation at 20-ps increments.

ES simulations

The atoms of each ES seed selected from the restraint simulations were given a random test initial velocity before the ES simulation. To determine whether the current phase space point is sampled from a first-hitting-point distribution, each ES seed was run for 20 ps to determine whether a milestone other than the target milestone was hit before any recrossing of the target milestone. The time of 20 ps could have been made shorter (in case a crossing event was detected earlier), but for convenience we left the length fixed. If recrossing occurred before another milestone was reached, then the combination of the ES structure and velocities was not considered part of the first hitting distribution (FHD) and this structure/velocity pair was discarded without replacement. If another milestone was reached before recrossing occurred (or if recrossing did not occur), then the structure/velocity combination is assumed to be part of the FHD (30,31). The full ES simulation is then run with the negative of the test velocities, with the interpretation that the test velocities represent a backward integration to determine which milestone the ES structure/velocity combination last passed. ES structure/velocity combinations that were determined to be part of the FHD were run for 50 ps, which was sufficient to ensure crossing of at least one other milestone.

Results

SF simulations

Analysis of the SF simulations (Table 3) reveals the stability of the coiled coil, even in the presence of loads many times larger than the force-generating capability of individual myosin heads (∼5 pN). The stability is marked both by the relatively long simulation times required for unfolding (>10 ns) as well as a strong propensity of unfolding events to refold at low (<100 pN) load levels. The stability demonstrated over 20-ns simulation times provides motivation for the use of enhanced sampling procedures to better probe the structural transitions occurring between high-energy, activated states of the system. ES results are presented below. Note that we measure the first cracking event, which does not imply complete unfolding. Unfolding of the full coiled coil may require significantly longer times.

Table 3.

SF simulation event summary

Load (pN)	No. SF sims	No. of unfolds	No. of refolds	No. of hops	t Unfold (ns)	Residue no.
0	20	7	3	0	6.1, 18.3, (14.1), 2.8, (1.8), 17.8, (2.7)	929, 929, (829), 933, (926), 929, (929)
25	20	4	3	0	(13.0), (1.2), (14.9), 7.3	(924), (931), (892), 929
50	20	7	5	0	(13.1), (12.8), (9.6), (5.8), (7.6), 1.8, 10.0	(925), (926), (930), (929), (929), 929, 929
75	20	4	4	0	(17.3), (1.6), (8.9), (11.4)	(928), (929), (892), (897)
100	20	11	3	1	16.1, 19.0, 6.8, 2.6, (3.5), 4.7, 6.1, 14.6, (0.9), (17.0), 3.0	929, 929, 891, 929, (931), 929, 929, 894, 924, 924, 929
200	10	9	3	6	15.6, 13.3, 4.2, 9.4, (12.4), 12.0, 8.9, (4.4), (19.1)	929, 929, 930, 929, 924, 929, 929, 893, 931

Unfolding events (unfolds) are defined by the first instance within a simulation of a Ψ dihedral angle >90°; the last column lists the residue of this first instance. Refolding events (refolds) are defined by the residue of the initial unfolding event having an AA state with an intact hydrogen bond at the end of the 20-ns simulation. Hopping events are determined by the appearance of unfolding events on both chains at locations within five residues of each other. The residue of unfolding is from either chain A or chain B. Data relating to refolding events (t unfold and residue number) are shown in parentheses. Sims, simulations.

In addition, unfolding events at these lower load levels do not show a strong proclivity to hop between helices of the coiled coil, indicating that any lasting instability that occurs remains localized. Only under the highest load level tested (200 pN) does the interstrand propagation of unfolding become apparent, and even at this highest load, hopping is not guaranteed (4 of 10 simulations either did not crack or did not propagate within the 20-ns simulation time).

The cracking events that did occur were noticeably centered on residues 928–931 on either chain A or chain B (note that each chain has the same sequence). Of 42 unfolding events registered during the initial simulations, 27 occurred in this residue range. In addition, 21 of these 27 occurred on residue 929. For this reason, residue 929 is used as the focus of the ES simulations described below. These residues are found within a larger, negatively charged block of residues from 921–935 that has been implicated in binding to other proteins (10,11,29). The propensity of unfolding events to occur in residues 928–931 is noteworthy, as this block of residues has a highly charged sequence of DEEE. Notably, residue 929 is in the normally hydrophobic d position of the heptad repeat structure. Although there are four other instances of a charged residue occupying either the a or d hydrophobic residue of the heptad repeat, residue 929 is the only instance in the studied sequence in which there is not a hydrophobic residue either immediately before or after the a or d position. Thus, the 928–931 residue range represents a clear disruption to the hydrophobic complementarity that is a hallmark of coiled coils, and this disruption is highly correlated with the initiation of unfolding events seen here.

Heptad repeat hydrophobic residue exposure

The solvent-accessible hydrophobic surface area (SASA) of the uncharged a and d heptad repeat residues was determined throughout the SF simulations to test the role of the burial of these hydrophobic residues in stabilizing the coiled-coil structure. All uncharged residues (both hydrophobic and polar/uncharged in nature) occupying the a and d positions were included in the calculation, which was performed using the GROMACS g_sas utility (35). The resulting hydrophobic surface area trajectories were used to generate probability distributions for the SASA, p(SASA), for all simulation time before the initiation of unfolding. These probability distributions were converted to an energy landscape via Boltzmann inversion. Data from the 200-pN load level are omitted, as the rapid cracking events that appeared in the SF simulations at this load led to poor SASA statistics relative to the other loads (Fig. 2).

Figure 2.

Energy landscape of the total SASA of uncharged heptad repeat a and d residues as a function of load. The total SASA of the uncharged a and d residues of the analysis section was calculated at each image of the SF simulations. Probability distributions of the SASA trajectories were calculated over all 20-ns SF simulations at each load level. Data after the first unfolding event for each simulation were discarded. The energy landscape was calculated through a Boltzmann inversion of the SASA probability distribution: U(SASA) = −ln[p(SASA)].

It is clear from the energy landscapes that the energy minima of the SASA are independent of load magnitude. Thus, applied loads do not significantly destabilize the interhelix interactions that result from hydrophobic surface complementarity. This indicates that the unfolding mechanism is not monotonic with exposed surface area, or that exposed hydrophobic surface area is not necessarily a good reaction coordinate, at least for coiled-coil cracking. In addition, interhelix distances between the α carbons of the residues used in the SASA calculation do not reveal any load dependence (see Fig. S1 in the Supporting Material).

Coil stabilization via interhelix hydrogen bonds

In addition to stabilization via the burial of the a and d hydrophobic residues, pairwise interactions (hydrogen bonds) between the e and g heptad locations between helices are thought to contribute to stability. To assess the stabilization provided by such interactions, as well as the role of applied load in governing these interactions, the quantity of interhelix hydrogen bonds was determined using the GROMACS utility g_hbond. All possible interhelix hydrogen bonds were evaluated at each saved image of the simulations before the first unfolding event. The resulting trajectories of hydrogen-bond quantity were converted to a probability distribution for each load level (see Fig. S2).

The average number of intact hydrogen bonds (∼5) represents only a fraction of the possible hydrogen bonds that could form between the e and g positions of the coiled-coil section analyzed here. The analysis section examined includes 10 heptad repeats, and each heptad repeat classically supports two interstrand hydrogen bonds. In addition, note that the quantification of the hydrogen bonds was between any two interhelix residues of the coil and therefore not limited to the e and g positions. Furthermore, note that the hydrogen-bond quantity distributions are independent of load magnitude. The load independence suggests that these interhelix bonds are not responsible for load sharing between the helices, as such load sharing would be expected to result in fewer intact bonds at greater load magnitudes.

Residue stability

The stability of specific residue locations along the length of the coil was assessed by direct computation (i.e., without using the Milestoning postprocessing analysis) of the anchor probability of each residue. Anchor probabilities were determined by first converting the atomic trajectory to the coarse-grained AA-state space of the anchor definitions (see Table 1), assigning each residue to a specific anchor (AA state) at each simulation image. These coarse-grained trajectories were then used to determine the probability distribution of each residue before the first cracking event (Fig. 3).

Figure 3.

Probability of chain A residues in the α3 (upper) and 3₁₀ (lower) AA states as a function of position and load magnitude. AA-state probabilities were calculated over all images of the ten 20-ns simulations for each load. Note that the scale of the p(3₁₀) distribution (lower) is one-tenth that of the p(α3) distribution (upper). Chain B probability distributions are indistinguishable from chain A distributions and are therefore omitted. The chain A sequence is reprinted below the probability distributions, with positive amino acids in bold, negative amino acids in italic, and noncharged amino acids in gray. The boxed regions refer to heavily charged domains thought to be involved in protein binding and disease (10).

The clear preference (p > 70%) of all positions along the coil was to remain in the native α3 AA state in which all three α-helical hydrogen bonds spanning the residue are simultaneously intact. The α2 AA state in which only one native α-helical hydrogen bond is broken was the second most populated AA state, accounting for an additional 10–20% of the probability. This overwhelming preference for the native state is in agreement with the unfolding results from the initial simulations in demonstrating the clear stability of the coiled coil over the tested simulation duration (20 ns), even in the presence of relatively large applied forces. In addition, the π-helical states are not significantly populated (p < 1%) for all residues and at all load levels, in noted contrast to the results from isolated α-helices.

The probability distributions shown in Fig. 3 illustrate the independence of the overall coil stability to the applied load magnitude before the initiation of localized unfolding events. Although there is an extremely slight increase in the probability of occupying an AA state in which a 3₁₀-helical hydrogen bond is formed at the highest load level (200 pN), the α3 distribution is independent of applied load. In addition to the independence of AA state with respect to load, the probability distributions were independent of the specific helix examined (not shown), as expected for the coil examined in which the two helices have the same sequence.

Comparison of the AA-state probabilities along the length of the coil indicates a clear dependence on residue location. Of particular interest are the two negatively charged regions, residues 894–906 and 921–935. The first region shows stability as demonstrated by the relatively low p(3₁₀) values for the majority of the region, with the noticeable exception of residue 898. In contrast, the second region shows both a diminished p(α3) as well as an increased p(3₁₀). Note that this second region is the location of the majority of unfolding events directly observed in the SF simulations.

In addition, the periodic spiked appearance of the p(3₁₀) distribution aligns with the heptad repeat structure. The spikes indicate that the residues in the a and d positions, whose hydrophobic side chains are prototypically buried, are more likely to form 3₁₀-helical hydrogen bonds. Thus, the formation of 3₁₀-helical bonds appears to be dependent on the position of a given residue relative to the larger coiled-coil structure.

Interhelix unfolding propagation

The ability of localized unfolding events to propagate between helices of the coil was assessed by examining the AA-state probabilities of residues 927–931 on chain B at three stages of unfolding of residue 929 on chain A (r929-A): 1), before unfolding of r929-A, 2), after only r929-A has unfolded, and 3), after r929-A and at least one neighbor (±2 residues) of r929-A has unfolded (Fig. 4). Due to the lack of interhelix unfolding propagation at low load levels, only data from the 200-pN simulations were used; only the SF simulation data were included. Only chain B data before the initiation of chain B unfolding were included in the probability calculation. Probabilities were directly computed (i.e., without using Milestoning postprocessing analysis) from the AA-state trajectories of residues 927–931 of chain B.

Figure 4.

AA-state probability of residues 927–931 of chain B at various stages of chain A residue 929 unfolding. The three stages of chain A residue unfolding are no unfolding events (upper), only residue 929 of chain A unfolding (middle), and residue 929 and at least one of its neighbors (from residues 927, 928, 930, and 931) unfolding. All data are from the 200-pN simulations. The AA-state probability of each residue was calculated for all images before unfolding of any of the residues 927–931 of chain B (therefore, there is no probability associated with the Ψ > 90° AA state). All AA states containing a π-helical hydrogen bond or 3₁₀-helical bonds, as well as the α2 and α1 AA states, are omitted for clarity.

As illustrated by the agreement between the upper two parts of Fig. 4, interhelix unfolding propagation does not occur when only a single residue (r929-A) is unfolded. However, there is a deleterious effect on the stability of chain B when more than one residue of chain A unfolds, as illustrated in the bottom part of Fig. 4. Whereas the native α3 AA states dominate in stages 1 and 2 of r929-A unfolding (i.e., chain B remains folded in the native conformation), a marked reduction in α3 occupancy is accompanied by an increase in the none and 90° > Ψ > 0° AA states once more than one residue in the vicinity of r929-A unfolds (Fig. 5, lower).

These results suggest that coiled-coil unfolding initiates as a local event in one helix and only hops to the second helix once propagation of the local event to neighboring residues has occurred in the initial helix. Thus, coil instability requires loads capable of propagating a local unfolding event that occurs in a single helix. This observation is supported by the SF simulation results (Table 3) in which hopping rarely occurs at low load levels and refolding of initial cracking events is a dominant feature of unfolding at low load levels. Such refolding events prevent the interhelix propagation of the initial crack and therefore keep the instability localized to one helix.

MFPT of residue 929 unfolding initiation

The MFPT of the initiation of unfolding of residue 929 was calculated with the Milestoning equations using a combination of both the SF simulations and the ES simulations (Fig. 5). An iterative sampling procedure was used to establish the degree of confidence in the calculated MFPT values for each load level based on the quality of sampling of individual milestone transitions. Residue 929 was chosen as the focus based on the observation from the SF simulations that nearly half (11 of 25) of all unfolding events observed occurred at this residue, whereas another six unfolding events occurred on a neighboring (±2) residue.

The MFPT of the initiation of unfolding is at least an order of magnitude longer for an individual residue in a coiled coil than for a residue of an isolated helix with the same sequence (which was previously found to be ∼2–5 ns (28)). This order of magnitude slow-down due to the presence of the second helix underscores the relative stability of the coiled-coil structure. Note that the calculated MFPT values presented here are in qualitative agreement with the results from the SF simulations. At the lower load levels (≤75 pN), less than half of the simulations experienced unfolding events, and those that did typically occurred after 10 ns, in agreement with an MFPT distribution in which only the fastest cracking times are in the vicinity of 20 ns, particularly for the intermediate load levels.

Intriguingly, the MFPT profile shows a noticeable, albeit not statistically significant, slow-down in the initiation of unfolding at intermediate load levels (25 pN and 50 pN) relative to the zero-load condition. This catch-bond-like behavior is in agreement with the profile of the isolated α-helix in which intermediate forces (20–30 pN) showed a similar peak in the MFPT of unfolding initiation. Note that the force levels referenced in this study are the total forces across both helices, and therefore a force here of 50 pN corresponds to a 25-pN load on each helix. Thus, the force magnitude of the peaks is in rough quantitative agreement. However, as discussed above with respect to the AA-state probabilities, and in further detail below, the mechanism of this catch-bond-like behavior in the isolated helix was demonstrated to involve transitions through π-helical states that are not heavily visited for residue 929 of the coiled coil.

MaxFlux pathways of unfolding

A MaxFlux pathway connecting two predetermined states is defined as the sequence of connected edges that carries maximum flux between any two points along the path (28,30,36,37). MaxFlux pathways of the initiation of unfolding were calculated, reflecting the pathway through the net flux networks with the highest transition edge. The transition edge refers to the edge of the pathway with the lowest net flux. Thus, the MaxFlux calculation determines the max/min pathway through the net flux networks calculated from the stationary fluxes ($q_{i,stat}$) via Eq. 2. The top MaxFlux pathways were calculated for each sampled transition matrix, $K_{ij}$, and milestone lifetime vector, ${\langle \tau \rangle }_i$, used in the iterative calculation of the MFPT (see Supporting Material for a discussion of the MFPT sampling procedure).

The MaxFlux pathway distributions illustrate that the intermediate load level (50 pN), which corresponds to the slowest MFPT for the initiation of unfolding, has fewer available unfolding channels. These limited options are seen in the relative sparseness of the MaxFlux network relative to the 0-pN and 200-pN networks, particularly in the neighborhood of the 90° > Ψ > 0° AA state. An interesting feature of this 50-pN network is the fact that by far the most probable connection (and therefore darkest red edge) to an AA state with a positive Ψ dihedral angle occurs through the α1:90° > Ψ > 0° edge. The lack of additional connections to the penultimate 90° > Ψ > 0° AA state is due to a negative net flux from both the α1 and 3₁₀ to the (none) AA state (the negative net flux values are manifested in the lack of connections in the MaxFlux network). This negative net flux implies that the 50-pN system must both break the final hydrogen bond and rotate the Ψ dihedral angle in a single step rather than following the sequential patterns exhibited by the 0-pN and 200-pN networks, in which the (none) AA state is the predominant feeder to the 90° > Ψ > 0° AA state.

The average flux of transition edges provides further explanation for the catch-bond-like profile of the MFPT. The strong negative correlation, with a Pearson correlation coefficient of −0.85, between the flux of the transition edge and the overall MFPT underscores the importance of the transition edge in throttling the overall process. Critically, by far the most likely transition edge for all three of these MaxFlux distributions is the final 90° > Ψ > 0°:Ψ > 90° transition, appearing as the transition edge in 80% of 0-pN pathways, 99% of 50-pN pathways, and 92% of 100-pN pathways. Therefore, the transition-edge profile with respect to load reflects the effect of load on this final transition. Thus, the MaxFlux distribution networks illustrate that the initiation of unfolding under the 50-pN load level is retarded by a combination of limited connectivity to the 90° > Ψ > 0° AA state and a lower net flux of the connection from this 90° > Ψ > 0° AA state to the final Ψ > 90° state.

Discussion

Perhaps the most experimentally relevant prediction of these simulations is the observation that the MFPT of the initiation of unfolding (Fig. 6) is on the order of 20–100 ns for the coil. The Milestoning analysis approach implemented is in agreement with the direct computational results from the SF simulations (Table 3) in which unfolding was very rare for the lower load levels tested. We note that this unfolding time is roughly a factor of 20 slower than the MFPT of unfolding initiation for an isolated helix with the same sequence, implying that by design the coiled-coil structure is an exceptionally strong wire that fits well the goal of stable transmission of mechanical load.

Figure 6.

MaxFlux pathway distributions for 0 pN (A), 50 pN (B), and 200 pN (C) and the milestone edge weight for the transition edge of the MaxFlux pathways (D). The transition edge of a pathway is defined as the edge with the smallest flux along the path computed from the Milestoning (Eq. 2). The error bars in D reflect the standard deviation of the value of the transition-edge weight (flux). The top MaxFlux pathway is calculated for each transition matrix, $K_{ij}$, and milestone lifetime vector, ${\langle \tau \rangle }_i$, sampled in the statistical analysis of Milestoning network (see Supporting Material). Pathway distributions reflect the probability of an edge being along the MaxFlux pathway for a given load level; the color of the edge reflects this probability. The thickness of an edge reflects the average weight of the edge. Thus, transition edges are the thinnest lines.

The stabilization of the coiled-coil structure evidenced by the decrease in MFPT was associated with limitations in the unfolding mechanisms as characterized by milestone probability networks (Fig. S3), net flux networks between AA-states (Fig. S4), and MaxFlux pathways of unfolding (Fig. 6). As discussed for the intermediate, 50-pN load level (Fig. 6), the progress of cracking was limited by effective hydrogen-bond reformation and the subsequent closure of a set of cracking channels that was more accessible in the 0-pN and 200-pN networks. The importance of hydrogen-bond reformation in delaying unfolding is also seen in the net flux networks (Fig. S2), which show that the edges between the high-energy states (i.e., those closest to unfolding) have a much lower magnitude of net flux, even for the structure in the absence of load. The lower net flux values indicate that rare transitions leading to unfolding are easily reversed in the coil relative to the isolated helix.

In contrast to the one-dimensional (1D) description of loading, the network analysis is consistently richer. In 1D, any significant disruption along a pathway can have a significant impact on the outcome. In contrast, a network may be more stable against perturbation by making it possible to use alternative routes to a prime reaction coordinate that may circumvent disruptions to an individual pathway. Moreover, by opening (as a function of load) off-pathway courses, nonmonotonic kinetic behaviors can be observed. This catch-bond-like behavior cannot be easily captured in a 1D picture. This phenomenon was particularly striking in the cracking of the isolated helix (28) and is reduced in scope for the coiled-coil system.

It is interesting to note that both the network descriptions and the directly calculated AA-state probabilities indicate that the coiled-coil structure does not visit the nonnative π-helical states that are significantly populated in the isolated helix. As the formation of π-helical hydrogen bonds requires a rotation of the Ψ dihedral angle in the negative direction, the lack of occupancy of these AA states and milestones suggests that the presence of the second helix in the coiled-coil structure limits the conformational flexibility of the backbone.

The overall stability of the coiled-coil structure is also supported by the independence of the SASA of buried hydrophobic residues (Fig. 2), the indifference of the quantity of interhelix hydrogen bonds to load magnitude (Fig. 3), and the independence of the AA-state probabilities on load magnitude (Fig. 4). Indeed, the SASA appeared to show a counterintuitive stabilization at the highest load level tested (Fig. 2), suggesting that the overall orientational effects of applied loads assist in maintaining hydrophobic surface complementarity. This stabilization may be the result of the higher load level limiting large-scale fluctuations that provide opportunities for the hydrophobic side chains to escape their local pockets.

Conclusion

Coiled-coil stability under applied mechanical forces is maintained through a combination of previously identified hydrophobic forces and a strong tendency to refold local cracking events. Helical cracking localized to E929 occurs at times (MFPTs) in excess of 50 ns at load levels experimentally shown to unfold coiled-coil structures (≤75 pN). Notably, SF simulations demonstrated that at these lower loads, helical cracking overwhelmingly resulted in healing of the coil and a noted inability to propagate the crack beyond the initial residue. The use of Milestoning networks enabled the quantitative study of coiled-coil cracking. As a result, molecular unfolding at these loads is expected to require timescales significantly longer than the MFPT of an individual residue. The results presented here provide guidelines for interpretation and planning of both experimental and computational unfolding of coiled structures.