Fig 1.
Schematic representation of the SCPS algorithm.
XI represents the initial protein conformation, XF the final target conformation and T1 …N the trajectories connecting XI to XF. Step 1: an ensemble of trajectories starting from XI and successfully reaching XF is generated by employing the rMD, biasing along the pre-defined CV z(X). Since z(X) decreases when the proximity of X to XF increases, the progression along the path is here defined as −z(X). Step 2: the trajectories successfully reaching the target state are then used to compute the mean path ⟨Cij(t)⟩, depicted in purple (see Methods). The mean path is then used to define two new coordinates: sλ, depicted in blue, which value is 1 in the unstructured state and 0 in the target state, therefore (1 − sλ) is used here to define the progress along the mean path; and wλ, depicted in red, that represents the distance to the mean path. Step 3: a modified version of the rMD is employed to generate a new set of trajectories by introducing two biasing forces, acting along sλ(t), and wλ(t) instead of z(X). The trajectories successfully reaching the target state are then used to compute a new mean path (step 2) to perform a new iteration.
Fig 2.
Comparison between plain MD and biased (rMD and SCPS) trajectories.
In each plot, the Kullback-Leibler divergence between the cross-similarity distribution (Ri) of each iteration i and the self-similarity distribution (A) is shown with blue dots. Each cross-similarity distribution Ri represents the path similarity distribution between the biased and the plain MD reactive trajectories. The self-similarity distribution is the path similarity distribution within the folding events sampled with plain MD. The average Kullback-Leibler divergence between the random cross-similarity distributions (Rr) and the self-similarity distribution is depicted as a red line and it is used as a reference. The random cross-similarity distribution was generated by comparing plain MD simulations with randomly sampled sequences of native contact formation. These results show that both rMD and SCPS produce results that are identical to the one generated with plain MD and are different (with the exception of Trp-cage) from the randomly generated events.
Fig 3.
Structure of the HET-s Prion Forming Domain in the amyloid form.
Lateral (A) and top (B) view of a HET-s amyloid trimer retrieved from PDB 2KJ3. Each monomer of the fibril displays a 2-Rung-β-Solenoid conformation. The N-terminal rung (residues 225–245) is depicted in blue, while the C-terminal rung (residues 261–281) is depicted in red. The colored bar at the left represents the polarity (N to C, blue to red) of the fibril.
Fig 4.
Reaction pathways in HET-s rungs formation.
The heat-maps represent the negative logarithm of the probability density calculated from the SCPS reactive paths at different iterations, as a function of the RMSD from the target structure of the N- and T-terminal rungs, i.e. -ln[P(RMSDCT-Rung, RMSDNT-Rung)]. A. Graphs relative to trajectories propagating from the fibril N-terminus. B. Graphs relative to the trajectories propagating from the fibril C-terminus. In both cases, a prominent pathway, consisting of the consecutive formation of rungs starting at the fibril end, begins to appear in the rMD generated trajectories. However, the rMD algorithm also yields pathways with cooperative or inverted rung formation. These alternative events are disappeared after a single SCPS iteration. Of note, a second SCPS iteration does not produce a consistent change in the free energy landscape in both A and B, indicating the convergence of the algorithm.
Fig 5.
Order of β-strand formation along the HET-s propagation pathway.
The median value of the reaction progress variable Q at which each residue of the rungs assumes the β-strand conformation is reported. Blue dots correspond to residues in the N-terminal rung while red dots correspond to residues in the C-terminal rung. Dots shown in transparency indicate residues not achieving a stable β-strand conformation. Horizontal bars span between the first and the third quartile of the distribution. The vertical dashed line delineates the average reaction progress at which half of the rungs-residues are incorporated into β-strand conformation. In the propagation starting from the fibril N-terminus (A) the residues of the C-terminal rung of the converting monomer are incorporated first (mean Q = 0.44), followed by the residues in the N-terminal rung (mean Q = 0.69). The opposite sequence of events is observed when propagation starts from the fibril C-terminus (B): the residues of the N-terminal rung of the converting monomer are incorporated first (mean Q = 0.42), followed by the residues in the C-terminal rung (mean Q = 0.81).
Fig 6.
Principal component analysis of the HET-s propagation trajectories.
Graphs (A) and (D) represent the free energy landscape in the principal component plane of the trajectories propagating from the fibril N- and C-terminus respectively. The letters “U”, “I” and “T” indicate the unstructured, intermediate the final (target) state, respectively. The residue contacts were classified in three categories: contacts between Cα in the same rung (intra-rung), contacts between Cα belonging to different rungs (inter-rung) and contacts between Cα of the converting monomer and Cα of the structured fibril (monomer-fibril). The contribution of these sets of the two principal components is shown in bar plots (B) for the N-terminal propagation and (E) for the C-terminal propagation. Images (C) and (F) show representative protein conformations sampled from the intermediate state, for the N-terminal and C-terminal propagations, respectively.
Fig 7.
Representative scheme for HET-s prion propagation.
A. Propagation scheme starting from the N-terminal end of the fibril. In this process, the C-terminal rung of the converting monomer (depicted in red) is formed by templating onto the structured N-terminal rung of the fibril (shown in transparent grey). Subsequently, the N-terminal part of the converting monomer forms the second rung (depicted in blue); B. Propagation scheme starting from the C-terminal end of the fibril. In this process, the N-terminal rung of the converting monomer (depicted in blue) is formed by templating onto the structured C-terminal rung of the fibril (shown in transparent grey). Subsequently, the C-terminal part of the converting monomer forms the second rung (depicted in red).