The MUMO (minimal under-restraining minimal over-restraining) method for the determination of native state ensembles of proteins

Richter, Barbara; Gsponer, Joerg; Várnai, Péter; Salvatella, Xavier; Vendruscolo, Michele

doi:10.1007/s10858-006-9117-7

Cited by 128 publications

(243 citation statements)

References 63 publications

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“…This is consistent with previous applications of restrainedensemble simulations. [4][5][6][7][8][9][10][11]14 The implication, particularly if the basic molecular system considered is large, is that the approach can be computationally onerous. On the other hand, high dimensional stochastic optimization problems such as in Eq.…”

Section: Discussionmentioning

confidence: 99%

“…[4][5][6][7][8][9][10][11]14 It consists of carrying out parallel MD simulations of N replicas of the basic system in the presence of a biasing potential that restrains the ensemble-averaged property toward the experimental value Q. In practice, the restraining potential U RE could have the form,…”

Section: B Restrained-ensemble Schemementioning

confidence: 99%

“…3 With the notion of "restrained-ensemble" MD simulations, Vendruscolo and co-workers extended these ideas to incorporate information from dynamically-averaged NMR scalar couplings, 4 and to explore structure and fluctuations of proteins in solution, 5,6 and implemented them into the MUMO (minimal under-restraining minimal over-restraining) scheme. 7 More recently, Im and co-workers extended the usage of restrained-ensemble MD simulations to deduce the structure of membrane-bound peptides on the basis of solid state NMR observables such as dipolar coupling, deuterium quadrupolar splitting, and chemical shift anisotropy; [8][9][10] for a review see Ref. 11.…”

Section: Introductionmentioning

confidence: 99%

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On the statistical equivalence of restrained-ensemble simulations with the maximum entropy method

Roux

Weare

2013

The Journal of Chemical Physics

191

270

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An issue of general interest in computer simulations is to incorporate information from experiments into a structural model. An important caveat in pursuing this goal is to avoid corrupting the resulting model with spurious and arbitrary biases. While the problem of biasing thermodynamic ensembles can be formulated rigorously using the maximum entropy method introduced by Jaynes, the approach can be cumbersome in practical applications with the need to determine multiple unknown coefficients iteratively. A popular alternative strategy to incorporate the information from experiments is to rely on restrained-ensemble molecular dynamics simulations. However, the fundamental validity of this computational strategy remains in question. Here, it is demonstrated that the statistical distribution produced by restrained-ensemble simulations is formally consistent with the maximum entropy method of Jaynes. This clarifies the underlying conditions under which restrained-ensemble simulations will yield results that are consistent with the maximum entropy method.

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Section: Discussionmentioning

confidence: 99%

Section: B Restrained-ensemble Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the statistical equivalence of restrained-ensemble simulations with the maximum entropy method

Roux

Weare

2013

The Journal of Chemical Physics

191

270

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“…A powerful strategy for characterizing the structure and dynamics of proteins in solution is emerging from methodsthatcombineNMRandmoleculardynamicssimulations. 5,8,12,13 Although these methods have offered encouraging results, it is still unclear whether they can provide ensembles of structures with the correct equilibrium statistical weights.Here we address this fundamental question by showing that molecular dynamics simulations with ensemble-averaged restraints 5,12,13 serve as a very accurate tool for calculating the free energies associated with the equilibrium ensembles corresponding to the native states of proteins. We adopted an approach in which AMBER 14 molecular dynamics simulations of ubiquitin are used to generate a collection of structures, forming a reference ensemble representing the state of the protein in solution.…”

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confidence: 99%

Toward an Accurate Determination of Free Energy Landscapes in Solution States of Proteins

et al. 2009

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In the 50 years since the first determination of the structures of proteins, our understanding of the states that they adopt in solution has enormously improved.1 It is now well-established that proteins populate a wide variety of different states in solution, many of which are conformationally highly heterogeneous. [1][2][3][4][5][6][7][8] Even in their native states, proteins constantly undergo structural fluctuations on time scales ranging from picoseconds to seconds and beyond. [2][3][4][5][6][7][8] It is therefore indispensable to achieve an accurate description of the inherent protein dynamics in order to account for biologically relevant processes, including enzymatic activity 6,9,10 and the formation of biomolecular complexes. 8,11 This goal is challenging, especially with respect to the treatment of dynamical regions of proteins, such as loop or unstructured sections, which often play important biological roles. A powerful strategy for characterizing the structure and dynamics of proteins in solution is emerging from methodsthatcombineNMRandmoleculardynamicssimulations. 5,8,12,13 Although these methods have offered encouraging results, it is still unclear whether they can provide ensembles of structures with the correct equilibrium statistical weights.Here we address this fundamental question by showing that molecular dynamics simulations with ensemble-averaged restraints 5,12,13 serve as a very accurate tool for calculating the free energies associated with the equilibrium ensembles corresponding to the native states of proteins. We adopted an approach in which AMBER 14 molecular dynamics simulations of ubiquitin are used to generate a collection of structures, forming a reference ensemble representing the state of the protein in solution. NMR observables are then back-calculated from this ensemble and used as restraints in CHARMM 5,12 molecular dynamics simulations aimed at reconstructing the distribution of structures of the reference ensemble. Approaches using reference ensembles have proved to be very powerful in structural biology 12,15 since they allow for an objective cross-validation analysis in which the atomic coordinates of the conformations to be reconstructed are known exactly; thus, both the average structure and the structural heterogeneity obtained from the restrained simulations can be compared with great accuracy to those of the reference ensemble. To define a reference representation of the ubiquitin solution ensemble, we employed the AMBER99SB force field, 16 which has been shown to accurately reproduce the native-state dynamics of ubiquitin; 14 a comparison of the referenceensemble-calculated and experimentally measured S 2 order parameters and residual dipolar couplings (RDCs) is presented in Figure S1 in the Supporting Information. As structural restraints, we employed RDCs, 17 which are particularly suitable for probing protein structure and dynamics with sensitivity up to the millisecond time scale. 8,13,18,19 We extracted 36 reference RDC data sets from the reference ensemble...

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“…21,22 Richter et al provided a solution to solve overfitting and underfitting problems when calculating ensemble of structures with NMR constraints. 23 To generate explicit conformations of the TSE, Vendruscolo et al used information from experimental φ values. 24,25 The φ value at individual residue position is defined as the ratio of stability change to the transition state upon mutation versus stability change of the native folded state upon the same mutation.…”

Section: Introductionmentioning

confidence: 99%

Constrained proper sampling of conformations of transition state ensemble of protein folding

Lin

Jian

et al. 2011

The Journal of Chemical Physics

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Characterizing the conformations of protein in the transition state ensemble (TSE) is important for studying protein folding. A promising approach pioneered by Vendruscolo et al. [Nature (London) 409, 641 (2001) (2007)] to generate TSE conformations of acylphosphatase of 98 residues that satisfy the φ-value constraints, as well as the criterion that each conformation has a folding probability of 0.5 by Monte Carlo simulations. We adopt a two stage process and first generate 5000 contact maps satisfying the φ-value constraints. Each contact map is then used to generate 1000 properly weighted conformations. After clustering similar conformations, we obtain a set of properly weighted samples of 4185 candidate clusters. Representative conformation of each of these cluster is then selected and 50 runs of Markov chain Monte Carlo (MCMC) simulation are carried using a regrowth move set. We then select a subset of 1501 conformations that have equal probabilities to fold and to unfold as the set of TSE. These 1501 samples characterize well the distribution of transition state ensemble conformations of acylphosphatase. Compared with previous studies, our approach can access much wider conformational space and can objectively generate conformations that satisfy the φ-value constraints and the criterion of 0.5 folding probability without bias. In contrast to previous studies, our results show that transition state conformations are very diverse and are far from nativelike when measured in cartesian root-mean-square deviation (cRMSD): the average cRMSD between TSE conformations and the native structure is 9.4 Å for this short protein, instead of 6 Å reported in previous studies. In addition, we found that the average fraction of native contacts in the TSE is 0.37, with enrichment in native-like β-sheets and a shortage of long range contacts, suggesting such contacts form at a later stage of folding. We further calculate the first passage time of folding of TSE conformations through calculation of physical time associated with the regrowth moves in MCMC simulation through mapping such moves to a Markovian state model, whose transition time was obtained by Langevin dynamics simulations. Our results indicate that despite the large structural diversity of the TSE, they are characterized by similar folding time. Our approach is general and can be used to study TSE in other macromolecules.

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The MUMO (minimal under-restraining minimal over-restraining) method for the determination of native state ensembles of proteins

Cited by 128 publications

References 63 publications

On the statistical equivalence of restrained-ensemble simulations with the maximum entropy method

On the statistical equivalence of restrained-ensemble simulations with the maximum entropy method

Toward an Accurate Determination of Free Energy Landscapes in Solution States of Proteins

Constrained proper sampling of conformations of transition state ensemble of protein folding

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