Chemical Potential of a Flexible Polymer Liquid in a Coarse-Grained Representation.

While the excess chemical potential is the key quantity in determining phase diagrams, its direct computation for high-density liquids of long polymer chains has posed a significant challenge. Computationally, the excess chemical potential is calculated using the Widom insertion method, which involves monitoring the change in internal energy as one incrementally introduces individual molecules into the liquid. However, when dealing with dense polymer liquids, inserting long chains requires generating trial configurations with a bias that favors those at low energy on a unit-by-unit basis: a procedure that becomes more challenging as the number of units increases. Thus, calculating the excess chemical potential of dense polymer liquids using this method becomes computationally intractable as the chain length exceeds N ≥ 30. Here, we adopt a coarse-grained model derived from the integral equation theory for which inserting long polymer chains becomes feasible. The integral equation theory of coarse graining (IECG) represents a polymer as a sphere or a collection of blobs interacting through a soft potential. We employ the IECG approach to compute the excess chemical potential using Widom's method for polymer chains of increasing lengths, extending up to N = 720 monomers, and at densities reaching up to ρ = 0.767 g/cm3. From a fundamental perspective, we demonstrate that the excess chemical potentials remain nearly constant across various levels of coarse graining, offering valuable insights into the consistency of this type of procedure. Ultimately, we argue that current Monte Carlo algorithms, originally designed for atomistic simulations, such as configurational bias Monte Carlo (CBMC) methods, can significantly benefit from the integration of the IECG approach, thereby enhancing their performance in the study of phase diagrams of polymer liquids.

Comparison between slow anisotropic LE4PD fluctuations and the principal component analysis modes of ubiquitin.

The biological function and folding mechanisms of proteins are often guided by large-scale slow motions, which involve crossing high energy barriers. In a simulation trajectory, these slow fluctuations are commonly identified using a principal component analysis (PCA). Despite the popularity of this method, a complete analysis of its predictions based on the physics of protein motion has been so far limited. This study formally connects the PCA to a Langevin model of protein dynamics and analyzes the contributions of energy barriers and hydrodynamic interactions to the slow PCA modes of motion. To do so, we introduce an anisotropic extension of the Langevin equation for protein dynamics, called the LE4PD-XYZ, which formally connects to the PCA "essential dynamics." The LE4PD-XYZ is an accurate coarse-grained diffusive method to model protein motion, which describes anisotropic fluctuations in the alpha carbons of the protein. The LE4PD accounts for hydrodynamic effects and mode-dependent free-energy barriers. This study compares large-scale anisotropic fluctuations identified by the LE4PD-XYZ to the mode-dependent PCA predictions, starting from a microsecond-long alpha carbon molecular dynamics atomistic trajectory of the protein ubiquitin. We observe that the inclusion of free-energy barriers and hydrodynamic interactions has important effects on the identification and timescales of ubiquitin's slow modes.

Identifying the leading dynamics of ubiquitin: A comparison between the tICA and the LE4PD slow fluctuations in amino acids' position.

Molecular Dynamics (MD) simulations of proteins implicitly contain the information connecting the atomistic molecular structure and proteins' biologically relevant motion, where large-scale fluctuations are deemed to guide folding and function. In the complex multiscale processes described by MD trajectories, it is difficult to identify, separate, and study those large-scale fluctuations. This problem can be formulated as the need to identify a small number of collective variables that guide the slow kinetic processes. The most promising method among the ones used to study the slow leading processes in proteins' dynamics is the time-structure based on time-lagged independent component analysis (tICA), which identifies the dominant components in a noisy signal. Recently, we developed an anisotropic Langevin approach for the dynamics of proteins, called the anisotropic Langevin Equation for Protein Dynamics or LE4PD-XYZ. This approach partitions the protein's MD dynamics into mostly uncorrelated, wavelength-dependent, diffusive modes. It associates with each mode a free-energy map, where one measures the spatial extension and the time evolution of the mode-dependent, slow dynamical fluctuations. Here, we compare the tICA modes' predictions with the collective LE4PD-XYZ modes. We observe that the two methods consistently identify the nature and extension of the slowest fluctuation processes. The tICA separates the leading processes in a smaller number of slow modes than the LE4PD does. The LE4PD provides time-dependent information at short times and a formal connection to the physics of the kinetic processes that are missing in the pure statistical analysis of tICA.

Accuracy, Transferability, and Efficiency of Coarse-Grained Models of Molecular Liquids.

Coarse-graining (CG) approaches are becoming essential tools in the study of complex systems because they can considerably speed up computer simulations, with the promise of determining properties in a range of length scales and time scales never before possible. While much progress in this field has been achieved in recent years, application of CG methods is still inhibited by the limited understanding of a number of conceptual points that need to be resolved to open up the field of CG to a wide range of applications in material science and biology. In this paper, we present some of the key findings that emerged from the development of the integral equation theory of coarse-graining (IECG), which addresses some of the fundamental questions in coarse-graining. Although the IECG method pertains to the CG of polymer liquids, and specifically homopolymer melts are illustrated here, many of the results that emerge from the study of the IECG approach are general and apply to the CG of any molecular liquid. Through this method, we developed a formal relation between the statistical mechanics of CG and a number of predicted physical properties. On the basis of the theory of liquids, the IECG affords the analytical solution of the intermolecular potential for macromolecules represented by a Markov chain of CG sites, thus providing a transparent tool for analysis of the properties in coarse-graining. We identify three key requirements that render a CG model useful: accuracy, transferability, and computational efficiency. When these three requirements are fulfilled, the CG model becomes widely applicable and useful for studying regions in the phase space that are not covered by atomistic simulations. In the process, the IECG answers formally a number of relevant questions on how structural, thermodynamic, and dynamical properties are modified during coarse-graining. It sheds light upon how the level of CG affects the shape of the CG potential and how, in turn, the shape of the potential affects the physical properties. It tests the validity of selecting the potential-of-mean force as the effective pairwise CG potential and the role of higher-order many-body corrections to the pairwise potential to recover structural and thermodynamic consistency of the CG model. Because the IECG theory can be analytically formalized, it does not suffer from the problem of transferability and, in the canonical ensemble, leads to consistent pair distribution functions, pressure, isothermal compressibility, and excess free energy at variable levels of CG from the atomistic to the ultra-CG model, where macromolecules are represented as interpenetrable soft spheres.

GRadient Adaptive Decomposition (GRAD) Method: Optimized Refinement Along Macrostate Borders in Markov State Models.

Markov state models (MSM) are used to model the kinetics of processes sampled by molecular dynamics (MD) simulations. MSM reduce the high dimensionality inherent to MD simulations as they partition the free energy landscape into discrete states, generating a kinetic model as a series of uncorrelated jumps between states. Here, we detail a new method, called GRadient Adaptive Decomposition, which optimizes coarse-grained MSM by refining borders with respect to the gradient along the free energy surface. The proposed method requires only a small number of initial microstates because it corrects for errors produced by limited sampling. Whereas many methods rely on fuzzy partitions for proper statistics, GRAD retains a crisp decomposition. Two test studies are presented to illustrate the method and assess its accuracy: the first analyzes MSM of idealized model potentials, while the second is a study of the dynamics of unstacking of the deoxyribose adenosine monophosphate dinucleotide.

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.

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.

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.

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.

Localization of chain dynamics in entangled polymer melts.

The dynamics of polymer melts in both the unentangled and entangled regimes is described by a Langevin equation for the correlated motion of a group of chains, interacting through both intra- and inter-molecular potentials. Entanglements are represented by an intermolecular monomer-monomer confining potential that has no effect on short chains, while interpolymer interactions, responsible for correlated motion and subdiffusive center-of-mass dynamics, are represented by an intermolecular center-of-mass potential derived from the Ornstein-Zernike equation. This potential ensures that the liquid of phantom chains reproduces the compressibility and free energy of the real samples. For polyethylene melts the calculated dynamic structure factor is found to be in quantitative agreement with neutron spin echo experiments of polyethylene melts with chain lengths that span both the unentangled and the entangled regimes. The theory shows a progressive localization of the cooperative chain dynamics at the crossover from the unentangled to the entangled regime, in the spirit of the reptation model.

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.

Effective potentials for representing polymers in melts as chains of interacting soft particles.

This paper outlines the derivation of an analytical pair potential in a coarse grained description of polymer melts where each chain is represented as a collection of soft spheres. Each particle is located at the center of mass of a polymer subchain, while the polymer is divided into an arbitrary number of identical chain subsections, each comprised of a large number of monomers. It is demonstrated that the soft effective pair potentials acting between these center-of-mass sites is described by a soft repulsive region at separation distances less than the average size of each coarse grained unit and a long repulsive tail, with a small attractive component. The attractive component is located at a length scale beyond the size of the coarse grained unit and its form varies with the level of interpenetration between the coarse-grained units. Consistent with numerically derived potentials, it is found that the short range features of the potential dominate the liquid structure, while the long-tail features dominate the virial-route thermodynamics of the system. It follows that the accurate determination of the effective potential in both short and large separation distances is relevant for ensuring structural and thermodynamic consistency in the coarse-grained description of the macromolecular liquid. It is further shown that due to the sensitivity of thermodynamic properties to the large-scale features of the potential, which are irrelevant to the reproducibility of structural correlations, the determination of thermodynamically accurate potentials by numerical optimization of structure alone is not a reliable strategy in the high-density regime for high levels of coarse-graining.

Theoretical reconstruction of realistic dynamics of highly coarse-grained cis-1,4-polybutadiene melts.

The theory to reconstruct the atomistic-level chain diffusion from the accelerated dynamics that is measured in mesoscale simulations of the coarse-grained system, is applied here to the dynamics of cis-1,4-polybutadiene melts where each chain is described as a soft interacting colloidal particle. The rescaling formalism accounts for the corrections in the dynamics due to the change in entropy and the change in friction that are a consequence of the coarse-graining procedure. By including these two corrections the dynamics is rescaled to reproduce the realistic dynamics of the system described at the atomistic level. The rescaled diffusion coefficient obtained from mesoscale simulations of coarse-grained cis-1,4-polybutadiene melts shows good agreement with data from united atom simulations performed by Tsolou et al. [Macromolecules 38, 1478 (2005)]. The derived monomer friction coefficient is used as an input to the theory for cooperative dynamics that describes the internal dynamics of a polymer moving in a transient regions of slow cooperative motion in a liquid of macromolecules. Theoretically predicted time correlation functions show good agreement with simulations in the whole range of length and time scales in which data are available.

Thermodynamic consistency in variable-level coarse graining of polymeric liquids.

Numerically optimized reduced descriptions of macromolecular liquids often present thermodynamic inconsistency with atomistic level descriptions even if the total correlation function, i.e. the structure, appears to be in agreement. An analytical expression for the effective potential between a pair of coarse-grained units is derived starting from the first-principles Ornstein-Zernike equation, for a polymer liquid where each chain is represented as a collection of interpenetrating blobs, with a variable number of blobs, n(b), of size N(b). The potential is characterized by a long tail, slowly decaying with characteristic scaling exponent of N(b)(1/4). This general result applies to any coarse-grained model of polymer melts with units larger than the persistence length, highlighting the importance of the long, repulsive, potential tail for the model to correctly predict both structural and thermodynamic properties of the macromolecular liquid.

First-principle approach to rescale the dynamics of simulated coarse-grained macromolecular liquids.

We present a detailed derivation and testing of our approach to rescale the dynamics of mesoscale simulations of coarse-grained polymer melts (I. Y. Lyubimov, J. McCarty, A. Clark, and M. G. Guenza, J. Chem. Phys. 132, 224903 (2010)). Starting from the first-principle Liouville equation and applying the Mori-Zwanzig projection operator technique, we derive the generalized Langevin equations (GLEs) for the coarse-grained representations of the liquid. The chosen slow variables in the projection operators define the length scale of coarse graining. Each polymer is represented at two levels of coarse graining: monomeric as a bead-and-spring model and molecular as a soft colloid. In the long-time regime where the center-of-mass follows Brownian motion and the internal dynamics is completely relaxed, the two descriptions must be equivalent. By enforcing this formal relation we derive from the GLEs the analytical rescaling factors to be applied to dynamical data in the coarse-grained representation to recover the monomeric description. Change in entropy and change in friction are the two corrections to be accounted for to compensate the effects of coarse graining on the polymer dynamics. The solution of the memory functions in the coarse-grained representations provides the dynamical rescaling of the friction coefficient. The calculation of the internal degrees of freedom provides the correction of the change in entropy due to coarse graining. The resulting rescaling formalism is a function of the coarse-grained model and thermodynamic parameters of the system simulated. The rescaled dynamics obtained from mesoscale simulations of polyethylene, represented as soft-colloidal particles, by applying our rescaling approach shows a good agreement with data of translational diffusion measured experimentally and from simulations. The proposed method is used to predict self-diffusion coefficients of new polyethylene samples.

Multiscale modeling of binary polymer mixtures: Scale bridging in the athermal and thermal regime.

Obtaining a rigorous and reliable method for linking computer simulations of polymer blends and composites at different length scales of interest is a highly desirable goal in soft matter physics. In this paper a multiscale modeling procedure is presented for the efficient calculation of the static structural properties of binary homopolymer blends. The procedure combines computer simulations of polymer chains on two different length scales, using a united atom representation for the finer structure and a highly coarse-grained approach on the mesoscale, where chains are represented as soft colloidal particles interacting through an effective potential. A method for combining the structural information by inverse mapping is discussed, allowing for the efficient calculation of partial correlation functions, which are compared with results from full united atom simulations. The structure of several polymer mixtures is obtained in an efficient manner for several mixtures in the homogeneous region of the phase diagram. The method is then extended to incorporate thermal fluctuations through an effective χ parameter. Since the approach is analytical, it is fully transferable to numerous systems.

Analytical rescaling of polymer dynamics from mesoscale simulations.

We present a theoretical approach to scale the artificially fast dynamics of simulated coarse-grained polymer liquids down to its realistic value. As coarse graining affects entropy and dissipation, two factors enter the rescaling: inclusion of intramolecular vibrational degrees of freedom and rescaling of the friction coefficient. Because our approach is analytical, it is general and transferable. Translational and rotational diffusion of unentangled and entangled polyethylene melts, predicted from mesoscale simulations of coarse-grained polymer melts using our rescaling procedure, are in quantitative agreement with united-atom simulations and with experiments.

Mapping of polymer melts onto liquids of soft-colloidal chains.

Microscopic computer simulations of fluids of long polymers are greatly restricted by the limits of current computational power, and so course-grained descriptions, accurate on molecular length scales, are essential to extending the range of accessible systems. For some phenomena, particularly dynamical entanglement, descriptions that eliminate all internal degrees of freedom from the polymers are too drastic, as intermediate wavelength degrees of freedom are essential to the effect. Employing first-principles liquid-state theory, we have developed a course-grained model for the intermolecular structure of melts of long homopolymer chains that maps each chain of hard-sphere monomers onto a chain of connected soft colloids. All dependence on system parameters is analytically expressed so the results may be immediately applied to melts with different polymer and thermodynamic properties to calculate effective potentials between the soft colloids on the chains, which can then be used to perform molecular dynamics simulations. These simulations will be able to capture the large wavelength structure of the system at greatly reduced computational cost, while still retaining enough internal degrees of freedom explicitly to describe the phenomena that occur on length scales much larger than the monomeric units that comprise the chain, but shorter than the size of the molecule.

Multiscale modeling of coarse-grained macromolecular liquids.

A first-principle multiscale modeling approach is presented, which is derived from the solution of the Ornstein-Zernike equation for the coarse-grained representation of polymer liquids. The approach is analytical, and for this reason is transferable. It is here applied to determine the structure of several polymeric systems, which have different parameter values, such as molecular length, monomeric structure, local flexibility, and thermodynamic conditions. When the pair distribution function obtained from this procedure is compared with the results from a full atomistic simulation, it shows quantitative agreement. Moreover, the multiscale procedure accurately captures both large and local scale properties while remaining computationally advantageous.