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Hierarchical analysis of conformational dynamics in biomolecules: Transition networks of metastable states
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10.1063/1.2714539
/content/aip/journal/jcp/126/15/10.1063/1.2714539
http://aip.metastore.ingenta.com/content/aip/journal/jcp/126/15/10.1063/1.2714539

Figures

Image of FIG. 1.
FIG. 1.

(Color) Illustration of the PCCA clustering procedure on a sample potential. (a) Potential in units of , defined over a discrete coordinate with 100 boxes. (b) Transition matrix, , for a Markov chain sampling the potential, using a lag time of steps. Each matrix entry represents the transition probability from cell to cell within time (blue: , red: ). The Markov chain was generated by using a Metropolis Monte Carlo where in each step only jumps to the current and the adjacent boxes were considered. The nearly block-diagonal structure of is apparent. (c) Left eigenvectors of used to identify metastable states. The first eigenvector gives the stationary distribution. The sign structure of the second eigenvector decomposes the state space into two metstable states (thick magenta line). The sign structure of the third eigenvector further splits the right metastable state (thin magenta line), obtaining three metastable states. (d) The eigenvalue spectrum of , indicating how many states are metastable. There are clear gaps after two and three eigenvalues.

Image of FIG. 2.
FIG. 2.

Mean transition time for state decompositions using the and dynamics.

Image of FIG. 3.
FIG. 3.

Examples of decay curves of metastable state populations. The brown circles and crosses show the fraction of the initial population remaining in a given state for a given time. The selected states are the global energy minima of following metastable state partitions: : ten states for torsion microstates and five states for hydrogen-bond microstates; : eight states for torsion microstates and five states for hydrogen-bond microstates.

Image of FIG. 4.
FIG. 4.

(Color) Hierarchical transition network analysis for the peptide based on backbone torsion rotamer microstates. Each bullet and the structure next to it corresponds to one metastable set of backbone torsion rotamer patterns. The bullets contain the state name (a letter), the free energy in kcal/mol (upper number), and the mean lifetime in picoseconds (lower number). Each structure is shown by a few representative tubes and an overlay of 100 examples randomly drawn from the ensemble of structures of each state, shown as line representations of the backbone. A pair of states is connected if at least one transition between these states was observed in the trajectory. The hierarchical relationship between the three networks for , 6, and 10 metastable states is indicated by the dotted arrows. Each arrow starts at the metastable state in the higher-order network which contains the majority of microstates in the state the arrow points to. For example, the microstates of state in the network are split into three substates, , , and , in the network.

Image of FIG. 5.
FIG. 5.

Scheme illustrating the concept of a kinetic trap in the absence of a predefined native structure. The free energy minimum is understood as the “native state” while high-energy minima that are separated from the main basins with high barriers are defined as “kinetic traps.” Such traps are rarely visited, but when they are visited, they are stable for a long time.

Image of FIG. 6.
FIG. 6.

(Color) Hierarchical transition network analysis for the peptide for two, three, and five metastable sets containing hydrogen-bonding pattern microstates. See caption of Fig. 4 for a complete description.

Image of FIG. 7.
FIG. 7.

Scheme illustrating why some definitions of microstates are better than others. Set (a) is optimal because, when partitioned into metastable sets, it splits exactly on the transition state. Set (b) is inappropriate because one of its states reaches across the transition state, including states from both basins. Such a definition yields shorter lifetimes because transitions into and out of cell are frequent from both basins, even though no actual transition may occur. This also produces apparent connections between metastable states that are actually not directly connected.

Image of FIG. 8.
FIG. 8.

(Color) Hierarchical transition network analysis for the peptide for two, six, and eight metastable sets containing backbone torsion rotamer microstates. See caption of Fig. 4 for a complete description.

Image of FIG. 9.
FIG. 9.

(Color) Hierarchical transition network analysis for the peptide for two, three, and five metastable sets containing hydrogen-bonding pattern microstates. See caption of Fig. 4 for a complete description.

Image of FIG. 10.
FIG. 10.

(Color) Transition network analysis for the peptide for 20 metastable sets containing backbone torsion rotamer microstates. See caption of Fig. 4 for a complete description.

Image of FIG. 11.
FIG. 11.

(Color) The implied time scales of the processes associated with individual eigenvectors, depending on the lag time , computed as , where is the eigenvalue. The implied time scales for the torsion rotamer states, [(a) and (c)] become flat for most processes at around , which is much below the lifetimes of the metastable states, indicating that the interstate transitions are Markovian. This is not the case for the hydrogen-bond state definition, [(b) and (d)] whose time scales do not converge within .

Image of FIG. 12.
FIG. 12.

(Color) Comparison of the molecular dynamics simulations (solid lines) with the Markov dynamics produced by the model network of metastable states (dashed lines) for a lag time of . The Markov dynamics is run times, each time initializing a single state with population 1 and the remaining states with population 0. The dashed curves show the relaxation of the selected state’s population towards equilibrium. For comparison, the relaxation dynamics directly computed from the molecular dynamics trajectory are shown. While the Markov model based H-bond definition fails for all states, the one based on the torsion rotamer definition fits well for many states and for all states in the high population range. The mismatch for some states in the low-population range [states and for and states , , , and for are likely to be due to insufficient sampling in the molecular dynamics trajectory (see number of long-time visits in Tables I and III]. The mismatch for states and which are frequently visited indicates the presence of considerable internal energy barriers, which means that these states should be further decomposed into substates (by increasing the number of metastable states in the model) in order to improve the fit.

Tables

Generic image for table
Table I.

Quantitative description of the metastable states found by grouping backbone torsion rotamers in . , name of the state. , free energy difference with respect to the free energy minimum (state ) and its error. , potential energy difference. is the free energy difference due to entropy. All energies are given in kcal/mol, the uncertainties in are negligible, and thus the uncertainties in are identical to the uncertainties . , mean lifetime of the state in picoseconds and its error. , number of visits in the state long enough to be useful to fitting the mean lifetime. The lines are ordered so as to indicate how states split when going to a network with more metastable states, e.g., state for splits up into , , and for .

Generic image for table
Table II.

Quantitative description of the metastable states found by grouping hydrogen-bonding patterns in . See Table I for a complete description.

Generic image for table
Table III.

Quantitative description of the metastable states found by grouping torsion rotamers in . See Table I for a complete description.

Generic image for table
Table IV.

Quantitative description of the metastable states found by grouping hydrogen-bonding patterns in . See Table I for a complete description.

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/content/aip/journal/jcp/126/15/10.1063/1.2714539
2007-04-19
2014-04-19
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Hierarchical analysis of conformational dynamics in biomolecules: Transition networks of metastable states
http://aip.metastore.ingenta.com/content/aip/journal/jcp/126/15/10.1063/1.2714539
10.1063/1.2714539
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