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Combination of Markov state models and kinetic networks for the analysis of molecular dynamics simulations of peptide folding.

Radford IH, Fersht AR, Settanni G - J Phys Chem B (2011)

Bottom Line: The trajectories have been analyzed using a Markov state model defined on the projections along two significant observables and a kinetic network approach.The kinetic network analysis served to extract the main transition state for folding of the peptide and to validate the results from the Markov state analysis.The transition state for the main folding reaction is similar to the intermediate state, although a more native like side-chain packing is observed.

View Article: PubMed Central - PubMed

Affiliation: MRC-Centre for Protein Engineering, Cambridge, UK.

ABSTRACT
Atomistic molecular dynamics simulations of the TZ1 beta-hairpin peptide have been carried out using an implicit model for the solvent. The trajectories have been analyzed using a Markov state model defined on the projections along two significant observables and a kinetic network approach. The Markov state model allowed for an unbiased identification of the metastable states of the system, and provided the basis for commitment probability calculations performed on the kinetic network. The kinetic network analysis served to extract the main transition state for folding of the peptide and to validate the results from the Markov state analysis. The combination of the two techniques allowed for a consistent and concise characterization of the dynamics of the peptide. The slowest relaxation process identified is the exchange between variably folded and denatured species, and the second slowest process is the exchange between two different subsets of the denatured state which could not be otherwise identified by simple inspection of the projected trajectory. The third slowest process is the exchange between a fully native and a partially folded intermediate state characterized by a native turn with a proximal backbone H-bond, and frayed side-chain packing and termini. The transition state for the main folding reaction is similar to the intermediate state, although a more native like side-chain packing is observed.

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(a) Equilibrium distribution of the microstates of the Markov state model fitting simulation data at 300 K (left), 330 K (middle), and 360 K (right). The simulation data binned into the microstates provide very similar (undistinguishable) distributions. The variable width of the microstates in the HB/rmsd space is obtained via an adaptive binning algorithm (see “Methods”) to limit the presence of poorly populated microstates. (b) Relaxation times (inverse eigenvalues) of the Markov state model as a function of lag time for the simulations at 330 K (error bars at 99% confidence interval). Convergence is reached for lag times equal or larger than 2.5 ns. Similar plots are obtained at 300 and 360 K (not shown). First (left), second (middle), and third (right) average non- left eigenvectors of the rate matrices at (c) 300 K, (d) 330 K, and (e) 360 K. The color scale ranges from black for the smallest (most negative) components to white for the largest (most positive) components, with the midpoint of the scale fixed at 50% gray for the zero components.
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fig2: (a) Equilibrium distribution of the microstates of the Markov state model fitting simulation data at 300 K (left), 330 K (middle), and 360 K (right). The simulation data binned into the microstates provide very similar (undistinguishable) distributions. The variable width of the microstates in the HB/rmsd space is obtained via an adaptive binning algorithm (see “Methods”) to limit the presence of poorly populated microstates. (b) Relaxation times (inverse eigenvalues) of the Markov state model as a function of lag time for the simulations at 330 K (error bars at 99% confidence interval). Convergence is reached for lag times equal or larger than 2.5 ns. Similar plots are obtained at 300 and 360 K (not shown). First (left), second (middle), and third (right) average non- left eigenvectors of the rate matrices at (c) 300 K, (d) 330 K, and (e) 360 K. The color scale ranges from black for the smallest (most negative) components to white for the largest (most positive) components, with the midpoint of the scale fixed at 50% gray for the zero components.

Mentions: The stable states of the peptide were identified by optimizing a MSM using Bayesian analysis, as described in “Methods”. Initial calculations carried out using only the backbone H-bonds for the definition of the microstates did not show Markovianity, as reported by the eigenvalue spectrum which did not converge at any lag time (data not shown). Simulations were then projected along both HB and rmsd, and the data were binned into a discrete number of two-dimensional microstates defined by both a range of HB and rmsd values (Figure 2a). Thus, 55, 53, and 41 microstates were obtained at 300, 330, and 360 K, respectively. The variable width of the microstates/bins, obtained using the adaptive binning procedure described in the “Methods” section, reduces the presence of scarcely populated bins which would introduce large uncertainties on the elements of the transition matrix. The MSM was then built using those microstates. The ensemble of rate matrices compatible with the simulation data was obtained using the Bayesian analysis and the likelihood function L (eq 11). The rate matrices were then diagonalized. At all three temperatures (300, 330, 360 K), the slowest non- eigenvalue converged for lag times not less than 2.56 ns, and a well-defined gap was observed between the slowest and second-slowest non- eigenvalues (the eigenvector corresponding to the eigenvalue is the equilibrium state) and between the third and fourth non- eigenvalues (Figure 2b). The relaxation times for the three slowest processes at the three temperatures are reported in Table 1. Although the slowest relaxation process at 360 K is on average slightly faster than the lag time, the subsequent analysis of the corresponding macrostates provided results in line with those obtained at the lower temperatures, so we decided to present the 360 K data as well.


Combination of Markov state models and kinetic networks for the analysis of molecular dynamics simulations of peptide folding.

Radford IH, Fersht AR, Settanni G - J Phys Chem B (2011)

(a) Equilibrium distribution of the microstates of the Markov state model fitting simulation data at 300 K (left), 330 K (middle), and 360 K (right). The simulation data binned into the microstates provide very similar (undistinguishable) distributions. The variable width of the microstates in the HB/rmsd space is obtained via an adaptive binning algorithm (see “Methods”) to limit the presence of poorly populated microstates. (b) Relaxation times (inverse eigenvalues) of the Markov state model as a function of lag time for the simulations at 330 K (error bars at 99% confidence interval). Convergence is reached for lag times equal or larger than 2.5 ns. Similar plots are obtained at 300 and 360 K (not shown). First (left), second (middle), and third (right) average non- left eigenvectors of the rate matrices at (c) 300 K, (d) 330 K, and (e) 360 K. The color scale ranges from black for the smallest (most negative) components to white for the largest (most positive) components, with the midpoint of the scale fixed at 50% gray for the zero components.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3106446&req=5

fig2: (a) Equilibrium distribution of the microstates of the Markov state model fitting simulation data at 300 K (left), 330 K (middle), and 360 K (right). The simulation data binned into the microstates provide very similar (undistinguishable) distributions. The variable width of the microstates in the HB/rmsd space is obtained via an adaptive binning algorithm (see “Methods”) to limit the presence of poorly populated microstates. (b) Relaxation times (inverse eigenvalues) of the Markov state model as a function of lag time for the simulations at 330 K (error bars at 99% confidence interval). Convergence is reached for lag times equal or larger than 2.5 ns. Similar plots are obtained at 300 and 360 K (not shown). First (left), second (middle), and third (right) average non- left eigenvectors of the rate matrices at (c) 300 K, (d) 330 K, and (e) 360 K. The color scale ranges from black for the smallest (most negative) components to white for the largest (most positive) components, with the midpoint of the scale fixed at 50% gray for the zero components.
Mentions: The stable states of the peptide were identified by optimizing a MSM using Bayesian analysis, as described in “Methods”. Initial calculations carried out using only the backbone H-bonds for the definition of the microstates did not show Markovianity, as reported by the eigenvalue spectrum which did not converge at any lag time (data not shown). Simulations were then projected along both HB and rmsd, and the data were binned into a discrete number of two-dimensional microstates defined by both a range of HB and rmsd values (Figure 2a). Thus, 55, 53, and 41 microstates were obtained at 300, 330, and 360 K, respectively. The variable width of the microstates/bins, obtained using the adaptive binning procedure described in the “Methods” section, reduces the presence of scarcely populated bins which would introduce large uncertainties on the elements of the transition matrix. The MSM was then built using those microstates. The ensemble of rate matrices compatible with the simulation data was obtained using the Bayesian analysis and the likelihood function L (eq 11). The rate matrices were then diagonalized. At all three temperatures (300, 330, 360 K), the slowest non- eigenvalue converged for lag times not less than 2.56 ns, and a well-defined gap was observed between the slowest and second-slowest non- eigenvalues (the eigenvector corresponding to the eigenvalue is the equilibrium state) and between the third and fourth non- eigenvalues (Figure 2b). The relaxation times for the three slowest processes at the three temperatures are reported in Table 1. Although the slowest relaxation process at 360 K is on average slightly faster than the lag time, the subsequent analysis of the corresponding macrostates provided results in line with those obtained at the lower temperatures, so we decided to present the 360 K data as well.

Bottom Line: The trajectories have been analyzed using a Markov state model defined on the projections along two significant observables and a kinetic network approach.The kinetic network analysis served to extract the main transition state for folding of the peptide and to validate the results from the Markov state analysis.The transition state for the main folding reaction is similar to the intermediate state, although a more native like side-chain packing is observed.

View Article: PubMed Central - PubMed

Affiliation: MRC-Centre for Protein Engineering, Cambridge, UK.

ABSTRACT
Atomistic molecular dynamics simulations of the TZ1 beta-hairpin peptide have been carried out using an implicit model for the solvent. The trajectories have been analyzed using a Markov state model defined on the projections along two significant observables and a kinetic network approach. The Markov state model allowed for an unbiased identification of the metastable states of the system, and provided the basis for commitment probability calculations performed on the kinetic network. The kinetic network analysis served to extract the main transition state for folding of the peptide and to validate the results from the Markov state analysis. The combination of the two techniques allowed for a consistent and concise characterization of the dynamics of the peptide. The slowest relaxation process identified is the exchange between variably folded and denatured species, and the second slowest process is the exchange between two different subsets of the denatured state which could not be otherwise identified by simple inspection of the projected trajectory. The third slowest process is the exchange between a fully native and a partially folded intermediate state characterized by a native turn with a proximal backbone H-bond, and frayed side-chain packing and termini. The transition state for the main folding reaction is similar to the intermediate state, although a more native like side-chain packing is observed.

Show MeSH