Jeremy Copperman

An analytical coarse-graining method which preserves the free energy, structural correlations, and thermodynamic state of polymer melts from the atomistic to the mesoscale

Structural and thermodynamic consistency of coarse-graining models across multiple length scales is essential for the predictive role of multi-scale modeling and molecular dynamic simulations that use mesoscale descriptions. Our approach is a coarse-grained model based on integral equation theory, which can represent polymer chains at variable levels of chemical details. The model is analytical and depends on molecular and thermodynamic parameters of the system under study, as well as on the direct correlation function in the k → 0 limit, c0. A numerical solution to the PRISM integral equations is used to determine c0, by adjusting the value of the effective hard sphere diameter, dHS, to agree with the predicted equation of state. This single quantity parameterizes the coarse-grained potential, which is used to perform mesoscale simulations that are directly compared with atomistic-level simulations of the same system. We test our coarse-graining formalism by comparing structural correlations, isothermal compressibility, equation of state, Helmholtz and Gibbs free energies, and potential energy and entropy using both united atom and coarse-grained descriptions. We find quantitative agreement between the analytical formalism for the thermodynamic properties, and the results of Molecular Dynamics simulations, independent of the chosen level of representation. In the mesoscale description, the potential energy of the soft-particle interaction becomes a free energy in the coarse-grained coordinates which preserves the excess free energy from an ideal gas across all levels of description. The structural consistency between the united-atom and mesoscale descriptions means the relative entropy between descriptions has been minimized without any variational optimization parameters. The approach is general and applicable to any polymeric system in different thermodynamic conditions.

Modifications to toxic CUG RNAs induce structural stability, rescue mis-splicing in a myotonic dystrophy cell model and reduce toxicity in a myotonic dystrophy zebrafish model.

CUG repeat expansions in the 3′ UTR of dystrophia myotonica protein kinase (DMPK) cause myotonic dystrophy type 1 (DM1). As RNA, these repeats elicit toxicity by sequestering splicing proteins, such as MBNL1, into protein–RNA aggregates. Structural studies demonstrate that CUG repeats can form A-form helices, suggesting that repeat secondary structure could be important in pathogenicity. To evaluate this hypothesis, we utilized structure-stabilizing RNA modifications pseudouridine (Ψ) and 2′-O-methylation to determine if stabilization of CUG helical conformations affected toxicity. CUG repeats modified with Ψ or 2′-O-methyl groups exhibited enhanced structural stability and reduced affinity for MBNL1. Molecular dynamics and X-ray crystallography suggest a potential water-bridging mechanism for Ψ-mediated CUG repeat stabilization. Ψ modification of CUG repeats rescued mis-splicing in a DM1 cell model and prevented CUG repeat toxicity in zebrafish embryos. This study indicates that the structure of toxic RNAs has a significant role in controlling the onset of neuromuscular diseases.

Pseudouridine Modification Inhibits Muscleblind-like 1 (MBNL1) Binding to CCUG Repeats and Minimally Structured RNA through Reduced RNA Flexibility

Myotonic dystrophy type 2 is a genetic neuromuscular disease caused by the expression of expanded CCUG repeat RNAs from the non-coding region of the CCHC-type zinc finger nucleic acid-binding protein (CNBP) gene. These CCUG repeats bind and sequester a family of RNA-binding proteins known as Muscleblind-like 1, 2, and 3 (MBNL1, MBNL2, and MBNL3), and sequestration plays a significant role in pathogenicity. MBNL proteins are alternative splicing regulators that bind to the consensus RNA sequence YGCY (Y = pyrimidine). This consensus sequence is found in the toxic RNAs (CCUG repeats) and in cellular RNA substrates that MBNL proteins have been shown to bind. Replacing the uridine in CCUG repeats with pseudouridine (Ψ) resulted in a modest reduction of MBNL1 binding. Interestingly, Ψ modification of a minimally structured RNA containing YGCY motifs resulted in more robust inhibition of MBNL1 binding. The different levels of inhibition between CCUG repeat and minimally structured RNA binding appear to be due to the ability to modify both pyrimidines in the YGCY motif, which is not possible in the CCUG repeats. Molecular dynamic studies of unmodified and pseudouridylated minimally structured RNAs suggest that reducing the flexibility of the minimally structured RNA leads to reduced binding by MBNL1.

Coarse-Grained Langevin Equation for Protein Dynamics: Global Anisotropy and a Mode Approach to Local Complexity.

We utilize a multiscale approach where molecular dynamic simulations are performed to obtain quantitative structural averages used as input to a coarse-grained Langevin equation for protein dynamics, which can be solved analytically. The approach describes proteins as fundamentally semiflexible objects collapsed into the free energy well representing the folded state. The normal-mode analytical solution to this Langevin equation naturally separates into global modes describing the fully anisotropic tumbling of the macromolecule as a whole and internal modes which describe local fluctuations about the folded structure. Complexity in the configurational free-energy landscape of the macromolecule leads to a renormalization of the internal modes, while the global modes provide a basis set in which the dipolar orientation and global anisotropy can be accounted for when comparing to experiments. This simple approach predicts the dynamics of both global rotational diffusion and internal motion from the picosecond to the nanosecond regime and is quantitative when compared to time correlation functions calculated from molecular dynamic simulations and in good agreement with nuclear magnetic resonance relaxation experiments. Fundamental to this approach is the inclusion of internal dissipation, which is absent in any rigid-body hydrodynamical modeling scheme.

Predicting protein dynamics from structural ensembles

The biological properties of proteins are uniquely determined by their structure and dynamics. A protein in solution populates a structural ensemble of metastable configurations around the global fold. From overall rotation to local fluctuations, the dynamics of proteins can cover several orders of magnitude in time scales. We propose a simulation-free coarse-grained approach which utilizes knowledge of the important metastable folded states of the protein to predict the protein dynamics. This approach is based upon the Langevin Equation for Protein Dynamics (LE4PD), a Langevin formalism in the coordinates of the protein backbone. The linear modes of this Langevin formalism organize the fluctuations of the protein, so that more extended dynamical cooperativity relates to increasing energy barriers to mode diffusion. The accuracy of the LE4PD is verified by analyzing the predicted dynamics across a set of seven different proteins for which both relaxation data and NMR solution structures are available. Using experimental NMR conformers as the input structural ensembles, LE4PD predicts quantitatively accurate results, with correlation coefficient ρ = 0.93 to NMR backbone relaxation measurements for the seven proteins. The NMR solution structure derived ensemble and predicted dynamical relaxation is compared with molecular dynamics simulation-derived structural ensembles and LE4PD predictions and is consistent in the time scale of the simulations. The use of the experimental NMR conformers frees the approach from computationally demanding simulations.

Mode localization in the cooperative dynamics of protein recognition

The biological function of proteins is encoded in their structure and expressed through the mediation of their dynamics. This paper presents a study on the correlation between local fluctuations, binding, and biological function for two sample proteins, starting from the Langevin Equation for Protein Dynamics (LE4PD). The LE4PD is a microscopic and residue-specific coarse-grained approach to protein dynamics, which starts from the static structural ensemble of a protein and predicts the dynamics analytically. It has been shown to be accurate in its prediction of NMR relaxation experiments and Debye-Waller factors. The LE4PD is solved in a set of diffusive modes which span a vast range of time scales of the protein dynamics, and provides a detailed picture of the mode-dependent localization of the fluctuation as a function of the primary structure of the protein. To investigate the dynamics of protein complexes, the theory is implemented here to treat the coarse-grained dynamics of interacting macromolecules. As an example, calculations of the dynamics of monomeric and dimerized HIV protease and the free Insulin Growth Factor II Receptor (IGF2R) domain 11 and its IGF2R:IGF2 complex are presented. Either simulation-derived or experimentally measured NMR conformers are used as input structural ensembles to the theory. The picture that emerges suggests a dynamical heterogeneous protein where biologically active regions provide energetically comparable conformational states that are trapped by a reacting partner in agreement with the conformation-selection mechanism of binding.

Universality and Specificity in Protein Fluctuation Dynamics

We investigate the universal scaling of protein fluctuation dynamics with a site-specific diffusive model of protein motion, which predicts an initial subdiffusive regime in the configurational relaxation. The long-time dynamics of proteins is controlled by an activated regime. We argue that the hierarchical free energy barriers set the time scales of biological processes and establish an upper limit to the size of single protein domains. We find it compelling that the scaling behavior for the protein dynamics is in close agreement with the Kardar-Parisi-Zhang scaling exponents.

A Coarse-Grained Langevin Equation for Protein Dynamics: Global Anisotropy and a Mode Approach to Local Complexity

We utilize a multi-scale approach where molecular dynamic simulations are performed to obtain quantitative structural averages used as input to a coarse-grained Langevin Equation for Protein Dynamics, which can be solved analytically. The approach describes proteins as fundamentally semiflexible objects collapsed into the free energy well representing the folded state. The normal mode analytical solution to this Langevin equation naturally separates into global modes describing the fully anisotropic tumbling of the macromolecule as a whole, and internal modes which describe local fluctuations about the folded structure. Complexity in the configurational free energy landscape around the folded state of the macromolecule leads to a renormalization of the internal modes, while the global modes provide a basis set in which the dipolar orientation and global anisotropy can be accounted for when comparing to experiments. Fundamental to this approach is the inclusion of internal dissipation which is absent in any rigid-body hydrodynamical modeling scheme. This simple approach predicts the dynamics of both global rotational diffusion and internal motion from the picosecond to the nanosecond regime, and is quantitative when compared to time correlation functions calculated from molecular dynamic simulations and in good agreement with Nuclear Magnetic Resonance relaxation experiments. Results for several well-characterized globular proteins are presented, suggesting our method describes the relevant dynamics around the global minimum well. Use of non-equilibrium simulation techniques such as metadynamics to sample the full free-energy landscape of the protein, and extension of the theoretical treatment to describe the dynamics into the biologically interesting microsecond to millisecond regime, will be discussed.