A sampling problem in molecular dynamics simulations of macromolecules.

Abstract

Correlations in low-frequency atomic displacements predicted by molecular dynamics simulations on the order of 1 ns are undersampled for the time scales currently accessible by the technique. This is shown with three different representations of the fluctuations in a macromolecule: the reciprocal space of crystallography using diffuse x-ray scattering data, real three-dimensional Cartesian space using covariance matrices of the atomic displacements, and the 3N-dimensional configuration space of the protein using dimensionally reduced projections to visualize the extent to which phase space is sampled. Experimental Tests of Molecular DynamicsClassical molecular dynamics is a computational technique to simulate the behaviors, both thermodynamic and dynamic, of an atomic model. In the long time limit, the sampled trajectory yields detailed information about the approximate model Hamiltonian. Shorter trajectories yield incomplete information and confound comparison of the model with experiment. Relatively short trajectories of biological macromolecules, on the order of 1 ns, have been used with reasonable success for a number of experimental comparisons-for example, crystallographic B values (1) from x-ray scattering, incoherent structure factors from neutron scattering (2, 3), and order parameters from NMR spectroscopy (4). Because these experimental techniques probe the movement of single atoms we can conclude that the amplitude and frequency of displacement for particular individual atoms in the protein are being simulated accurately.However, on the basis of such local results we cannot also conclude that these same molecular dynamics calculations accurately simulate, or converge on, the correlation between two or more atoms in a protein (e.g., whether one part of a protein tends to act concertedly with another). The state of our present models, then, is not completely tested by our experimental methods. In this article we consider a particular test set to measure the convergence of two-body properties in protein simulations.An experimental technique that is sensitive to statistics between pairs of atoms, and thus to collective behavior in macromolecules, is diffuse x-ray scattering (5, 6). All x-ray intensity on a detector besides the sharp Bragg peaks is typically ignored in structural studies. However, this remaining intensity, Diffuse scattering experiments on several proteins and nucleic acids (7-10) demonstrate that correlations in displacements of neighboring atoms fall off roughly exponentially with the distance between the atoms, with a characteristic length of 4-8 A. That is, (8iA8) = e(Wx(J)exp(-Jr| rjl/y), where the correlation length y is less than the dimension of the protein molecule. Thus, in analogy with liquid structures, where the correlations in atomicpositions decay approximately exponentially with pair separation, the motion in proteins is often called liquid-like, in the sense that correlations in atomic displacements fall off rapidly with distance.As a test of whether the pair corr...

Algorithmic dimensionality reduction for molecular structure analysis

“…32 In addition to filtering high-frequency motions, dimensionality reduction of molecular simulations can be utilized to identify discrete conformational substates 24,33,34 and for understanding the extent of configuration space sampling and the topography of the system's energy hypersurface. 35 As reviewed by Altis et al, 36 dimensionality reduction can be utilized to obtain representations for reaction coordinates and free energy landscapes as well as the transition matrix between metastable conformational states.…”

Efficient Characterization of Collective Motions and Interresidue Correlations in Proteins by Low-Resolution Simulations

“…A typical protein MD simulation samples only a limited portion of the overall conformational space, as evidenced by the projection of the trajectory onto the subspace spanned by the three dominant eigenvectors of the displacement covariance matrix (2). Another difficulty with MD simulations is that cross-correlations between the displacements of different atoms in a given protein cannot be precisely captured (2). Such limitations motivate the quest for simplified models and more efficient computational tools.…”

“…Atomic models used in such simulations necessitate the adoption of time steps of the order of femtoseconds, which do not permit attaining time scales longer than nanoseconds. A typical protein MD simulation samples only a limited portion of the overall conformational space, as evidenced by the projection of the trajectory onto the subspace spanned by the three dominant eigenvectors of the displacement covariance matrix (2). Another difficulty with MD simulations is that cross-correlations between the displacements of different atoms in a given protein cannot be precisely captured (2).…”

“…Likewise, Garcia and Harman (11) applied this same method to analyze the MD trajectories of the cyclic 3′,5′-adenosine monophosphate (cAMP) receptor protein (CRP) complexed with (cAMP) 2 to demonstrate the occurrence of a subunit transition from the open to the closed conformation in noncrystalline environments.…”

Entropic Folding Pathway of Human Epidermal Growth Factor Explored by Disulfide Scrambling and Amplified Collective Motion Simulations,

“…However, the standard MD technique faces a major limitation because of the so-called sampling problem (18). On the typical nanosecond to microsecond time scales one can only sample a limited number of functional states of the protein, leaving many other states to be poorly sampled.…”