Free Energy Landscapes of Biomolecules
Hierarchical Dynamics of Aib9
Biomolecules exhibit structural dynamics on a number of timescales, including picosecond (ps) motions of a few atoms, nanosecond (ns) local conformational transitions and microsecond (μs) global conformational rearrangements. Despite this substantial separation of timescales, fast and slow degrees of freedom appear to be coupled in a nonlinear manner, e.g, there is theoretical and experimental evidence that fast structural fluctuations are required for slow functional motion to happen. To elucidate a microscopic mechanism of this multiscale behavior, Aib peptide is adopted as a simple model system. Combining extensive molecular dynamics simulations with principal component analysis techniques, a hierarchy of (at least) three tiers of the molecule's free energy landscape is discovered. They correspond to chiral left- to right-handed transitions of the entire peptide that happen on a μs timescale, conformational transitions of individual residues which take about 1 ns, and the opening and closing of structure-stabilizing hydrogen bonds which occur within tens of ps and are triggered by sub-ps structural fluctuations. Providing a simple mechanism of hierarchical dynamics, fast hydrogen bond dynamics is found to be a prerequisite for the ns local conformational transitions, which in turn are a prerequisite for the slow global conformational rearrangement of the peptide. As a consequence of the hierarchical coupling, the various processes exhibit a similar temperature behavior which may be interpreted as a dynamic transition.
Hierarchical biomolecular dynamics: Picosecond hydrogen bonding regulates micro- second conformational transitions, J. Chem. Theory Comput. 11, 1330 (2015)
Principal component analysis: Cartesian vs. internal coordinates
Principal component analysis of molecular dynamics simulations is a popular method to account for the essential dynamics of the system on a low-dimensional free energy landscape. Using Cartesian coordinates, first the translation and overall rotation need to be removed from the trajectory. Since the rotation depends via the moment of inertia on the molecule’s structure, this separation is only straightforward for relatively rigid systems. Adopting millisecond molecular dynamics sim- ulations of the folding of villin headpiece and the functional dynamics of BPTI provided by D. E. Shaw Research, it is demonstrated via a comparison of local and global rotational fitting that the structural dynamics of flexible molecules necessarily results in a mixing of overall and internal motion. Even for the small-amplitude functional motion of BPTI, the conformational distribu- tion obtained from a Cartesian principal component analysis therefore reflects to some extend the dominant overall motion rather than the much smaller internal motion of the protein. Internal coordinates such as distances or backbone dihedral angles, on the other hand, are found to yield correct and well-resolved energy landscapes for both examples. The virtues and shortcomings of the choice of various fitting schemes and coordinate sets as well as the generality of these results are discussed in some detail.
Principal component analysis of molecular dynamics: On the use of Cartesian vs. internal coordinates J. Chem. Phys. 141, 014111 (2014)
Hierarchical folding free energy landscape of HP35 revealed by most probable path clustering
Adopting extensive molecular dynamics simulations of villin headpiece protein (HP35) by Shaw and coworkers, a detailed theoretical analysis of the folding of HP35 is presented. The approach is based on the recently proposed most probable path algorithm which identifies the metastable states of the system, combined with dynamical coring of these states in order to obtain a consistent Markov state model. The method facilitates the construction of a dendrogram associated with the folding free energy landscape of HP35, which reveals a hierarchical funnel structure and shows that the native state is rather a kinetic trap than a network hub. The energy landscape of HP35 consists of the entropic unfolded basin U where the prestructuring of the protein takes place, the intermediate basin I which is connected to U via the rate-limiting U → I transition state reflecting the formation of helix-1, and the native basin N containing a state close to the NMR structure and a native-like state that exhibits enhanced fluctuations of helix-3. The model is in line with recent experimental observations that the intermediate and native states differ mostly in their dynamics (locked vs. unlocked states). Employing dihedral angle principal component analysis, subdiffusive motion on a multidimensional free energy surface is found.
Hierarchical folding free energy landscape of HP35 revealed by most probable path clustering, J. Phys. Chem. B 118, 7750 (2014)
Construction of the free energy landscape of peptide aggregation from molecular dynamics simulations
To describe the structure and dynamics of oligomers during peptide aggregation, a method is proposed that considers the intramolecular as well as the intermolecular structure of the multi-molecule system and correctly accounts for its degeneracy. The approach is based on the "by parts" strategy, which partitions a complex molecular system into parts, determines the metastable conformational states of each part, and describes the overall conformational state of the system in terms of a product basis of the states of the parts. Starting from a molecular dynamics simulation of n molecules, the method consists of three steps: (i) Characterization of the intramolecular structure, i.e., of the conformational states of a single molecule in the presence of the other molecules (e.g., β-strand or random coil). (ii) Characterization of the intermolecular structure via the identification of all occurring aggregate states of the peptides (dimers, trimers etc.). (iii) Construction of the overall conformational states of the system in terms of a product basis of the n "single-molecule" states and the aggregate states. Considering as a first application the Alzheimer β-amyloid peptide fragment Aβ16-22, about 700 overall conformational states of the trimer (Aβ16-22)3 are constructed from all-atom molecular dynamics simulation in explicit water. Based on these states, a transition network reflecting the free energy landscape of the aggregation process can be constructed, which facilitates the identification of the aggregation pathways.
Construction of the free energy landscape of peptide aggregation from molecular dynamics simulations, J. Comp. Theo. Chem. 8, 1471 (2012)
Hidden complexity of protein free-energy landscapes
We have proposed a method to analyze molecular dynamics (MD) simulations of protein folding, which is based on a principal component analysis (PCA) of the protein's backbone dihedral angles. The protein is first partitioned into parts (e.g., its secondary-structure elements) for each of which a separate PCA is performed. Based on these PCAs, the free energy landscapes of each part are constructed and used to determine their metastable conformational states. In a second step, the various states of each protein part are employed to construct a product basis, which now represent the metastable conformational states of the full protein. Adopting extensive MD simulations of the villin headpiece by Pande and coworkers, we have shown that this "PCA by parts'' allows us to characterizes the free energy landscape of the protein with unprecedented detail.
Hidden complexity of protein free-energy landscapes revealed by principal component analysis by parts, J. Phys. Chem. Lett. 1, 18, 2769 (2010)