Exploring the free energy landscape: from dynamics to networks and back

Abstract

Knowledge of the Free Energy Landscape topology is the essential key to understanding many biochemical processes. The determination of the conformers of a protein and their basins of attraction takes a central role for studying molecular isomerization reactions. In this work, we present a novel framework to unveil the features of a Free Energy Landscape answering questions such as how many meta-stable conformers there are, what the hierarchical relationship among them is, or what the structure and kinetics of the transition paths are. Exploring the landscape by molecular dynamics simulations, the microscopic data of the trajectory are encoded into a Conformational Markov Network. The structure of this graph reveals the regions of the conformational space corresponding to the basins of attraction. In addition, handling the Conformational Markov Network, relevant kinetic magnitudes as dwell times and rate constants, or hierarchical relationships among basins, completes the global picture of the landscape. We show the power of the analysis studying a toy model of a funnel-like potential and computing efficiently the conformers of a short peptide, dialanine, paving the way to a systematic study of the Free Energy Landscape in large peptides.

Figure 1. SSD algorithm applied to a synthetic funnel-like potential.

(A) 2D funnel-like potential. (B) A stochastic trajectory is translated into a CMN where 6 sets of nodes (corresponding to different color) are the result of the SSD algorithm. (C) Recovering the spatial coordinates, the stationary probabilities of each node are shown in color code. The 6 basins detected are represented as color striped regions. (D) A coarse-grained CMN is built where new nodes take the role of the basins.

Figure 2. Hierarchies of the basins detected for the funnel-like potential.

(A) Free Energy hierarchy: based on the relative free-energy of the nodes. (B) Temporal hierarchy: number of basins defined by SSD for the different networks built by Eq. (7). The original basins merge in function of time. Both hierarchies reveals a coarse-grained behavior of two macro-states: and .

Figure 3. Free energy basins of the Alanine dipeptide.

(A) The dialanine dipeptide with the angles and . (B) Plot of the CMN generated. The 6 sets of nodes (corresponding to different colors) are the result of the SSD algorithm. (C) Left: Ramachandran plot with the probability of occupation of the cells used to build the CMN. The boundaries of the free energy basins are shown in white. Right: the 6 basins represented as regions of different color. (Color code: , , , , and ).

Figure 4. Dendogram based on the relative Free Energy of the CMN nodes.

Two sets of basins are clearly distinguished with a high free energy barrier in between: (, , , ) and (, ). Note that looks like the conformer with the largest dwell time, in agreement with data in Table 1.

Figure 5. Dendogram based on the temporal hierarchy of basins.

In around 100 ps the peptide finds the way to reach the global minimum, conformer , from any basin.