(Top) Helical structure (2fxm.pdb chain A) within simulation box filled with explicit water. (Bottom) Helical structure illustrating two groups (red and blue) comprised of the 18 residues on either end. Constant force loads are applied to these two groups in a constant direction (dashed black line) aligned with the long helical axis.
Schematic illustration of a set of anchors that are coarse grained descriptions of atomic space (represented here by the α3, α1, [none], Ψ > 90, and π34 anchors) and a set of milestones that are the interfaces between these anchors (represented here by the α3:α1, α1:α3, α1:[none], and [none]:Ψ > 90 designations). For more details, see Sec. II D , “Coarse graining approach,” below. Briefly, the “α” designation in the anchor names indicates the existence of α-helical hydrogen bonds while the numeral indicates the number of α-helical bonds currently formed (similar for the π34 designation in which 4 π-helical bonds are shown). The [none] and Ψ > 90 anchors lack intact hydrogen bonds and have Ψ dihedral angles <0 and >90, respectively. Note the use of directional milestones as illustrated by the two distinct milestones separating the α3 and α1 anchors, each with a unique directionality relative to the two anchors.
Simulation time of first observed initiation of helical unfolding (“MD MFPT”) in straightforward MD simulations and MFPT of helical unfolding initiation (“Milestoning MFPT”) calculated via Milestoning analysis of the simulation trajectories. Observed simulation times are computed directly from multiple molecular dynamic trajectories at a given load by recording the first observation of Ψ > 90°; data points are the averages and error bars are the standard deviations of these observations. Data points and error bars for the Milestoning MFPT are the averages and standard deviations from the Milestoning sampling protocol.
Milestoning probabilities of networks of different loads are provided. The plot (a) includes the network of probabilities of different edges for zero load, which is also used as the reference distribution. The rest of the networks are for differences in probabilities of edges of networks of a high and low (reference) load network. Positive values (red) mean that the probability is increased for the higher load level. Only probabilities above 0.001 are presented; others are omitted for clarity. Line thickness for (a) is determined by the log10 of probabilities; line thickness for (b)–(d) is determined by the absolute difference in probability without any logarithmic adjustment. Each edge is the sum of the probabilities of the two directional milestones connecting the two relevant anchors. Note that as the load increase from zero to 25 pN the contribution of the π helix to the network is increasing, while it is reduced when the load grows to 70 pN. Turning on and off the contribution of the off the pathway π helix explains the non-monotonic behavior of the MFPT. The absence of connections to the Ψ > 90° anchor reflects the fact that the analysis of the simulations only considered the trajectory up to the first transition to this anchor. Therefore, the milestones connecting to this final anchor have an undefined lifetime, represented here as a lifetime of zero, (see Fig. 6 ). The milestone probability is also zero despite the presence of significant fluxes through these milestones (see Fig. 5 ).
Net flux networks as a function of applied load magnitude. The net fluxes are calculated as the difference between the directional milestone fluxes for a given pair of anchors. The directional milestone fluxes are calculated via solution of q stat (I − K) = 0. The edge thickness corresponds to the magnitude of the net flux, with the edges of each network normalized by the highest net flux for a given load level. The direction of the edges corresponds to the directionality of the net flux. Edges with normalized values less than 10% of the maximum net flux are omitted for clarity.
The average milestone lifetimes for the 0 pN (a) network as well as comparisons between to 0 pN–25 pN (b), 25 pN–70 pN (c), and 70 pN–100 pN (d). The comparisons made in (b)–(d) are the arithmetic differences between the milestone lifetimes of the two load levels. For example, in (b) the edge values plotted are calculated as . The average milestone lifetime values graphed here are the vector elements ⟨τ⟩ i . For clarity, only milestone lifetimes above 0.5 ps are presented. Note that the lifetimes of the milestones are presented as the edges between anchors, which serve as the nodes of the network. Line thickness corresponds to the magnitude of the milestone lifetime; where relevant, the color corresponds to increasing or decreasing lifetime with respect to load. Dashed lines are renderings of relatively thin, and therefore short-lived, edges. The absence of connections to the Ψ > 90° anchor reflects the fact that the analysis of the simulations only considered the trajectory up to the first transition to this anchor. Therefore, any milestones representing connections to this final milestone have an undefined lifetime, represented here as a zero lifetime.
The correlations of individual transition element values with the calculated MFPT. Correlations are calculated between the set of values for a given transition element and the set of calculated MFPTs during the sampling procedure used for the MFPT calculation at each load level. The correlations are displayed as the outer anchor connection between two milestones; the transition anchor that links the two milestones together is omitted for clarity. Only correlations that are statistically significant (p < 0.05) are displayed. The thickness of the lines corresponds to the correlation coefficient.
Correlation between the sampled K ij element value and calculated MFPT for the transition from the 310:90° > Ψ > 0° milestone to the 90° > Ψ > 0°:Ψ > 90° milestone at 0 pN. The correlation value for this set of data was −0.41.
Transition matrix elements with strong negative correlations with the calculated MFPT; these elements represent the final transition to the unfolded state. Values are the averages determined from the post-processing analysis of the simulations and represent the mean values used in the MFPT calculation sampling procedure for each load level; error bars represent the standard deviation of these transition values throughout the sampling procedure. Note that this plot does not explain the non-monotonic behavior of the MFPT.
Sensitivity (Ω ij ) of the MFPT calculation to changes in the transition matrix elements with strong negative correlations to the MFPT. The sensitivity is numerically calculated as the slope of the linear regression between MFPT and K ij element values: . Negative sensitivities indicate that the MFPT decreases as the K ij element increases.
Sensitivity (Ω ij ) of the MFPT calculation to changes in transition matrix elements associated with breaking α-helical hydrogen bonds. The sensitivity is numerically calculated as the slope of the linear regression between MFPT and K ij element values: . Negative sensitivities indicate that the MFPT decreases as the K ij element increases. Note that the filled symbols represent successive bond breakage while the open symbols represent hydrogen bond formation.
MaxFlux pathway distributions representing the probability of a given edge being part of one of three MaxFlux pathways calculated for each sampled K ij . Color represents the probability of an edge being part of a MaxFlux pathway; thickness is the average weight of that edge. The results explored 1000 sampled networks at each load. See text on the sensitivity analysis of the networks for more details.
Average weight (net flux) of the transition edge for the three pathways calculated for each iteration of the MFPT calculation at a given load level. The transition edge is defined as the edge along a given pathway with the lowest net flux, and therefore represents the rate-limiting transition for that pathway. Data points are the averages over all 1st, 2nd, or 3rd pathways calculated during each iteration of the MPFT calculation sampling procedure for a given load level; error bars are the standard deviation of the distribution of these weights.
Edge weights (net fluxes) of four key transitions to the 90° > Ψ > 0° state. Data points are the averages of all samples of the K ij matrix for the MFPT calculation; error bars are the standard deviation of the MFPT computed from the samples.
Ψ angle distributions for four critical anchors whose interfaces with the 90° > Ψ > 0° state form the majority of transition edges for the MaxFlux pathways. Energies are calculated from the Boltzmann inversion of the Ψ dihedral angle distribution for each anchor (E anchor (ψ) = −ln [p(ψ)]). Distributions of the Ψ angle were determined in the zero load case; our previous work demonstrated that the Ψ angle distributions for these anchors were not qualitatively different at higher loads. Distributions are extracted from our previous simulations. 15
Number of simulations for each load level.
Anchor definitions as a function of the quantity and type of hydrogen bond spanning the residue of interest and the Ψ dihedral angle. 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 1 or 2 bonds). Note that certain anchors allow a variable quantity of hydrogen bonds; for example, the π34 anchor can have either 3 or 4 π-helical hydrogen bonds. These variable-bond-quantity anchors are not to be confused with milestones, which are defined to be the interface between domains of any two anchors listed here. The full atomically detailed trajectories computed with GROMACS addressed in Sec. II A are mapped to Milestoning trajectories and are used to compute K ij and ⟨τ⟩ i .
Article metrics loading...
Full text loading...