Dynamic force matching: Construction of dynamic coarse-grained models with realistic short time dynamics and accurate long time dynamics

Resolving the dynamic properties of entangled linear polymers in non-equilibrium coarse grain simulation with a priori scaling factors

The molecular weight of polymers can influence the material properties, but the molecular weight at the experiment level sometimes can be a huge burden for property prediction with full-atomic simulations. The traditional bottom-up coarse grain (CG) simulation can reduce the computation cost. However, the dynamic properties predicted by the CG simulation can deviate from the full-atomic simulation result. Usually, in CG simulations, the diffusion is faster and the viscosity and modulus are much lower. The fast dynamics in CG are usually solved by a posteriori scaling on time, temperature, or potential modifications, which usually have poor transferability to other non-fitted physical properties because of a lack of fundamental physics. In this work, a priori scaling factors were calculated by the loss of degrees of freedom and implemented in the iterative Boltzmann inversion. According to the simulation results on 3 different CG levels at different temperatures and loading rates, such a priori scaling factors can help in reproducing some dynamic properties of polycaprolactone in CG simulation more accurately, such as heat capacity, Young's modulus, and viscosity, while maintaining the accuracy in the structural distribution prediction. The transferability of entropy-enthalpy compensation and a dissipative particle dynamics thermostat is also presented for comparison. The proposed method reveals the huge potential for developing customized CG thermostats and offers a simple way to rebuild multiphysics CG models for polymers with good transferability.

Understanding dynamics in coarse-grained models. III. Roles of rotational motion and translation-rotation coupling in coarse-grained dynamics

This paper series aims to establish a complete correspondence between fine-grained (FG) and coarse-grained (CG) dynamics by way of excess entropy scaling (introduced in Paper I). While Paper II successfully captured translational motions in CG systems using a hard sphere mapping, the absence of rotational motions in single-site CG models introduces differences between FG and CG dynamics. In this third paper, our objective is to faithfully recover atomistic diffusion coefficients from CG dynamics by incorporating rotational dynamics. By extracting FG rotational diffusion, we unravel, for the first time reported to our knowledge, a universality in excess entropy scaling between the rotational and translational diffusion. Once the missing rotational dynamics are integrated into the CG translational dynamics, an effective translation-rotation coupling becomes essential. We propose two different approaches for estimating this coupling parameter: the rough hard sphere theory with acentric factor (temperature-independent) or the rough Lennard-Jones model with CG attractions (temperature-dependent). Altogether, we demonstrate that FG diffusion coefficients can be recovered from CG diffusion coefficients by (1) incorporating "entropy-free" rotational diffusion with translation-rotation coupling and (2) recapturing the missing entropy. Our findings shed light on the fundamental relationship between FG and CG dynamics in molecular fluids.

The SIRAH force field: A suite for simulations of complex biological systems at the coarse-grained and multiscale levels

The different combinations of molecular dynamics simulations with coarse-grained representations have acquired considerable popularity among the scientific community. Especially in biocomputing, the significant speedup granted by simplified molecular models opened the possibility of increasing the diversity and complexity of macromolecular systems, providing realistic insights on large assemblies for more extended time windows. However, a holistic view of biological ensembles' structural and dynamic features requires a self-consistent force field, namely, a set of equations and parameters that describe the intra and intermolecular interactions among moieties of diverse chemical nature (i.e., nucleic and amino acids, lipids, solvent, ions, etc.). Nevertheless, examples of such force fields are scarce in the literature at the fully atomistic and coarse-grained levels. Moreover, the number of force fields capable of handling simultaneously different scales is restricted to a handful. Among those, the SIRAH force field, developed in our group, furnishes a set of topologies and tools that facilitate the setting up and running of molecular dynamics simulations at the coarse-grained and multiscale levels. SIRAH uses the same classical pairwise Hamiltonian function implemented in the most popular molecular dynamics software. In particular, it runs natively in AMBER and Gromacs engines, and porting it to other simulation packages is straightforward. This review describes the underlying philosophy behind the development of SIRAH over the years and across families of biological molecules, discussing current limitations and future implementations.

Memory Unlocks the Future of Biomolecular Dynamics: Transformative Tools to Uncover Physical Insights Accurately and Efficiently

Conformational changes underpin function and encode complex biomolecular mechanisms. Gaining atomic-level detail of how such changes occur has the potential to reveal these mechanisms and is of critical importance in identifying drug targets, facilitating rational drug design, and enabling bioengineering applications. While the past two decades have brought Markov state model techniques to the point where practitioners can regularly use them to glimpse the long-time dynamics of slow conformations in complex systems, many systems are still beyond their reach. In this Perspective, we discuss how including memory (i.e., non-Markovian effects) can reduce the computational cost to predict the long-time dynamics in these complex systems by orders of magnitude and with greater accuracy and resolution than state-of-the-art Markov state models. We illustrate how memory lies at the heart of successful and promising techniques, ranging from the Fokker-Planck and generalized Langevin equations to deep-learning recurrent neural networks and generalized master equations. We delineate how these techniques work, identify insights that they can offer in biomolecular systems, and discuss their advantages and disadvantages in practical settings. We show how generalized master equations can enable the investigation of, for example, the gate-opening process in RNA polymerase II and demonstrate how our recent advances tame the deleterious influence of statistical underconvergence of the molecular dynamics simulations used to parameterize these techniques. This represents a significant leap forward that will enable our memory-based techniques to interrogate systems that are currently beyond the reach of even the best Markov state models. We conclude by discussing some current challenges and future prospects for how exploiting memory will open the door to many exciting opportunities.

Flow-Matching: Efficient Coarse-Graining of Molecular Dynamics without Forces

Coarse-grained (CG) molecular simulations have become a standard tool to study molecular processes on time and length scales inaccessible to all-atom simulations. Parametrizing CG force fields to match all-atom simulations has mainly relied on force-matching or relative entropy minimization, which require many samples from costly simulations with all-atom or CG resolutions, respectively. Here we present flow-matching, a new training method for CG force fields that combines the advantages of both methods by leveraging normalizing flows, a generative deep learning method. Flow-matching first trains a normalizing flow to represent the CG probability density, which is equivalent to minimizing the relative entropy without requiring iterative CG simulations. Subsequently, the flow generates samples and forces according to the learned distribution in order to train the desired CG free energy model via force-matching. Even without requiring forces from the all-atom simulations, flow-matching outperforms classical force-matching by an order of magnitude in terms of data efficiency and produces CG models that can capture the folding and unfolding transitions of small proteins.

Bottom-Up Informed and Iteratively Optimized Coarse-Grained Non-Markovian Water Models with Accurate Dynamics

Molecular dynamics (MD) simulations based on coarse-grained (CG) particle models of molecular liquids generally predict accelerated dynamics and misrepresent the time scales for molecular vibrations and diffusive motions. The parametrization of Generalized Langevin Equation (GLE) thermostats based on the microscopic dynamics of the fine-grained model provides a promising route to address this issue, in conjunction with the conservative interactions of the CG model obtained with standard coarse graining methods, such as iterative Boltzmann inversion, force matching, or relative entropy minimization. We report the application of a recently introduced bottom-up dynamic coarse graining method, based on the Mori-Zwanzig formalism, which provides accurate estimates of isotropic GLE memory kernels for several CG models of liquid water. We demonstrate that, with an additional iterative optimization of the memory kernels (IOMK) for the CG water models based on a practical iterative optimization technique, the velocity autocorrelation function of liquid water can be represented very accurately within a few iterations. By considering the distinct Van Hove function, we demonstrate that, with the presented methods, an accurate representation of structural relaxation can be achieved. We consider several distinct CG potentials to study how the choice of the CG potential affects the performance of bottom-up informed and iteratively optimized models.

Local Molecular Field Theory for Coulomb Interactions in Aqueous Solutions

Coulomb interactions play a crucial role in a wide array of processes in aqueous solutions but present conceptual and computational challenges to both theory and simulations. We review recent developments in an approach addressing these challenges─local molecular field (LMF) theory. LMF theory exploits an exact and physically suggestive separation of intermolecular Coulomb interactions into strong short-range and uniformly slowly varying long-range components. This allows us to accurately determine the averaged effects of the long-range components on the short-range structure using effective single particle fields and analytical corrections, greatly reducing the need for complex lattice summation techniques used in most standard approaches. The simplest use of these ideas in aqueous solutions leads to the short solvent (SS) model, where both solvent-solvent and solute-solvent Coulomb interactions have only short-range components. Here we use the SS model to give a simple description of pairing of nucleobases and biologically relevant ions in water.

Understanding dynamics in coarse-grained models. II. Coarse-grained diffusion modeled using hard sphere theory

The first paper of this series [J. Chem. Phys. 158, 034103 (2023)] demonstrated that excess entropy scaling holds for both fine-grained and corresponding coarse-grained (CG) systems. Despite its universality, a more exact determination of the scaling relationship was not possible due to the semi-empirical nature. In this second paper, an analytical excess entropy scaling relation is derived for bottom-up CG systems. At the single-site CG resolution, effective hard sphere systems are constructed that yield near-identical dynamical properties as the target CG systems by taking advantage of how hard sphere dynamics and excess entropy can be analytically expressed in terms of the liquid packing fraction. Inspired by classical equilibrium perturbation theories and recent advances in constructing hard sphere models for predicting activated dynamics of supercooled liquids, we propose a new approach for understanding the diffusion of molecular liquids in the normal regime using hard sphere reference fluids. The proposed "fluctuation matching" is designed to have the same amplitude of long wavelength density fluctuations (dimensionless compressibility) as the CG system. Utilizing the Enskog theory to derive an expression for hard sphere diffusion coefficients, a bridge between the CG dynamics and excess entropy is then established. The CG diffusion coefficient can be roughly estimated using various equations of the state, and an accurate prediction of accelerated CG dynamics at different temperatures is also possible in advance of running any CG simulation. By introducing another layer of coarsening, these findings provide a more rigorous method to assess excess entropy scaling and understand the accelerated CG dynamics of molecular fluids.

Understanding dynamics in coarse-grained models. I. Universal excess entropy scaling relationship

Coarse-grained (CG) models facilitate an efficient exploration of complex systems by reducing the unnecessary degrees of freedom of the fine-grained (FG) system while recapitulating major structural correlations. Unlike structural properties, assessing dynamic properties in CG modeling is often unfeasible due to the accelerated dynamics of the CG models, which allows for more efficient structural sampling. Therefore, the ultimate goal of the present series of articles is to establish a better correspondence between the FG and CG dynamics. To assess and compare dynamical properties in the FG and the corresponding CG models, we utilize the excess entropy scaling relationship. For Paper I of this series, we provide evidence that the FG and the corresponding CG counterpart follow the same universal scaling relationship. By carefully reviewing and examining the literature, we develop a new theory to calculate excess entropies for the FG and CG systems while accounting for entropy representability. We demonstrate that the excess entropy scaling idea can be readily applied to liquid water and methanol systems at both the FG and CG resolutions. For both liquids, we reveal that the scaling exponents remain unchanged from the coarse-graining process, indicating that the scaling behavior is universal for the same underlying molecular systems. Combining this finding with the concept of mapping entropy in CG models, we show that the missing entropy plays an important role in accelerating the CG dynamics.

Bottom-up Coarse-Graining: Principles and Perspectives

Large-scale computational molecular models provide scientists a means to investigate the effect of microscopic details on emergent mesoscopic behavior. Elucidating the relationship between variations on the molecular scale and macroscopic observable properties facilitates an understanding of the molecular interactions driving the properties of real world materials and complex systems (e.g., those found in biology, chemistry, and materials science). As a result, discovering an explicit, systematic connection between microscopic nature and emergent mesoscopic behavior is a fundamental goal for this type of investigation. The molecular forces critical to driving the behavior of complex heterogeneous systems are often unclear. More problematically, simulations of representative model systems are often prohibitively expensive from both spatial and temporal perspectives, impeding straightforward investigations over possible hypotheses characterizing molecular behavior. While the reduction in resolution of a study, such as moving from an atomistic simulation to that of the resolution of large coarse-grained (CG) groups of atoms, can partially ameliorate the cost of individual simulations, the relationship between the proposed microscopic details and this intermediate resolution is nontrivial and presents new obstacles to study. Small portions of these complex systems can be realistically simulated. Alone, these smaller simulations likely do not provide insight into collectively emergent behavior. However, by proposing that the driving forces in both smaller and larger systems (containing many related copies of the smaller system) have an explicit connection, systematic bottom-up CG techniques can be used to transfer CG hypotheses discovered using a smaller scale system to a larger system of primary interest. The proposed connection between different CG systems is prescribed by (i) the CG representation (mapping) and (ii) the functional form and parameters used to represent the CG energetics, which approximate potentials of mean force (PMFs). As a result, the design of CG methods that facilitate a variety of physically relevant representations, approximations, and force fields is critical to moving the frontier of systematic CG forward. Crucially, the proposed connection between the system used for parametrization and the system of interest is orthogonal to the optimization used to approximate the potential of mean force present in all systematic CG methods. The empirical efficacy of machine learning techniques on a variety of tasks provides strong motivation to consider these approaches for approximating the PMF and analyzing these approximations.

Constructing many-body dissipative particle dynamics models of fluids from bottom-up coarse-graining

Since their emergence in the 1990s, mesoscopic models of fluids have been widely used to study complex organization and transport phenomena beyond the molecular scale. Even though these models are designed based on results from physics at the meso- and macroscale, such as fluid mechanics and statistical field theory, the underlying microscopic foundation of these models is not as well defined. This paper aims to build such a systematic connection using bottom-up coarse-graining methods. From the recently developed dynamic coarse-graining scheme, we introduce a statistical inference framework of explicit many-body conservative interaction that quantitatively recapitulates the mesoscopic structure of the underlying fluid. To further consider the dissipative and fluctuation forces, we design a novel algorithm that parameterizes these forces. By utilizing this algorithm, we derive pairwise decomposable friction kernels under both non-Markovian and Markovian limits where both short- and long-time features of the coarse-grained dynamics are reproduced. Finally, through these new developments, the many-body dissipative particle dynamics type of equations of motion are successfully derived. The methodologies developed in this work thus open a new avenue for the construction of direct bottom-up mesoscopic models that naturally bridge the meso- and macroscopic physics.

Machine Learning for Molecular Simulation

Machine learning (ML) is transforming all areas of science. The complex and time-consuming calculations in molecular simulations are particularly suitable for an ML revolution and have already been profoundly affected by the application of existing ML methods. Here we review recent ML methods for molecular simulation, with particular focus on (deep) neural networks for the prediction of quantum-mechanical energies and forces, on coarse-grained molecular dynamics, on the extraction of free energy surfaces and kinetics, and on generative network approaches to sample molecular equilibrium structures and compute thermodynamics. To explain these methods and illustrate open methodological problems, we review some important principles of molecular physics and describe how they can be incorporated into ML structures. Finally, we identify and describe a list of open challenges for the interface between ML and molecular simulation.

Coarse-graining molecular dynamics: stochastic models with non-Gaussian force distributions

Incorporating atomistic and molecular information into models of cellular behaviour is challenging because of a vast separation of spatial and temporal scales between processes happening at the atomic and cellular levels. Multiscale or multi-resolution methodologies address this difficulty by using molecular dynamics (MD) and coarse-grained models in different parts of the cell. Their applicability depends on the accuracy and properties of the coarse-grained model which approximates the detailed MD description. A family of stochastic coarse-grained (SCG) models, written as relatively low-dimensional systems of nonlinear stochastic differential equations, is presented. The nonlinear SCG model incorporates the non-Gaussian force distribution which is observed in MD simulations and which cannot be described by linear models. It is shown that the nonlinearities can be chosen in such a way that they do not complicate parametrization of the SCG description by detailed MD simulations. The solution of the SCG model is found in terms of gamma functions.

Recent Progress towards Chemically-Specific Coarse-Grained Simulation Models with Consistent Dynamical Properties

Coarse-grained (CG) models can provide computationally efficient and conceptually simple characterizations of soft matter systems. While generic models probe the underlying physics governing an entire family of free-energy landscapes, bottom-up CG models are systematically constructed from a higher-resolution model to retain a high level of chemical specificity. The removal of degrees of freedom from the system modifies the relationship between the relative time scales of distinct dynamical processes through both a loss of friction and a “smoothing” of the free-energy landscape. While these effects typically result in faster dynamics, decreasing the computational expense of the model, they also obscure the connection to the true dynamics of the system. The lack of consistent dynamics is a serious limitation for CG models, which not only prevents quantitatively accurate predictions of dynamical observables but can also lead to qualitatively incorrect descriptions of the characteristic dynamical processes. With many methods available for optimizing the structural and thermodynamic properties of chemically-specific CG models, recent years have seen a stark increase in investigations addressing the accurate description of dynamical properties generated from CG simulations. In this review, we present an overview of these efforts, ranging from bottom-up parameterizations of generalized Langevin equations to refinements of the CG force field based on a Markov state modeling framework. We aim to make connections between seemingly disparate approaches, while laying out some of the major challenges as well as potential directions for future efforts.

Multi-resolution dimer models in heat baths with short-range and long-range interactions

This work investigates multi-resolution methodologies for simulating dimer models. The solvent particles which make up the heat bath interact with the monomers of the dimer either through direct collisions (short-range) or through harmonic springs (long-range). Two types of multi-resolution methodologies are considered in detail: (a) describing parts of the solvent far away from the dimer by a coarser approach; (b) describing each monomer of the dimer by using a model with different level of resolution. These methodologies are then used to investigate the effect of a shared heat bath versus two uncoupled heat baths, one for each monomer. Furthermore, the validity of the multi-resolution methods is discussed by comparison to dynamics of macroscopic Langevin equations.

Encoding and selecting coarse-grain mapping operators with hierarchical graphs

Coarse-grained (CG) molecular dynamics (MD) can simulate systems inaccessible to fine-grained (FG) MD simulations. A CG simulation decreases the degrees of freedom by mapping atoms from an FG representation into agglomerate CG particles. The FG to CG mapping is not unique. Research into systematic selection of these mappings is challenging due to their combinatorial growth with respect to the number of atoms in a molecule. Here we present a method of reducing the total count of mappings by imposing molecular topology and symmetry constraints. The count reduction is illustrated by considering all mappings for nearly 50 000 molecules. The resulting number of mapping operators is still large, so we introduce a novel hierarchical graphical approach which encodes multiple CG mapping operators. The encoding method is demonstrated for methanol and a 14-mer peptide. With the test cases, we show how the encoding can be used for automated selection of reasonable CG mapping operators.

Coarse-graining simulation approaches for polymer melts: the effect of potential range on computational efficiency

The integral equation coarse-graining (IECG) approach is a promising high-level coarse-graining (CG) method for polymer melts, with variable resolution from soft spheres to multi CG sites, which preserves the structural and thermodynamical consistencies with the related atomistic simulations. When compared to the atomistic description, the procedure of coarse-graining results in smoother free energy surfaces, longer-ranged potentials, a decrease in the number of interaction sites for a given polymer, and more. Because these changes have competing effects on the computational efficiency of the CG model, care needs to be taken when studying the effect of coarse-graining on the computational speed-up in CG molecular dynamics simulations. For instance, treatment of long-range CG interactions requires the selection of cutoff distances that include the attractive part of the effective CG potential and force. In particular, we show how the complex nature of the range and curvature of the effective CG potential, the selection of a suitable CG timestep, the choice of the cutoff distance, the molecular dynamics algorithms, and the smoothness of the CG free energy surface affect the efficiency of IECG simulations. By direct comparison with the atomistic simulations of relatively short chain polymer melts, we find that the overall computational efficiency is highest for the highest level of CG (soft spheres), with an overall improvement of the computational efficiency being about 106-108 for various CG levels/resolutions. Therefore, the IECG method can have important applications in molecular dynamics simulations of polymeric systems. Finally, making use of the standard spatial decomposition algorithm, the parallel scalability of the IECG simulations for various levels of CG is presented. Optimal parallel scaling is observed for a reasonably large number of processors. Although this study is performed using the IECG approach, its results on the relation between the level of CG and the computational efficiency are general and apply to any properly-constructed CG model.

Single molecule translocation in smectics illustrates the challenge for time-mapping in simulations on multiple scales

Understanding the connections between the characteristic dynamical time scales associated with a coarse-grained (CG) and a detailed representation is central to the applicability of the coarse-graining methods to understand molecular processes. The process of coarse graining leads to an accelerated dynamics, owing to the smoothening of the underlying free-energy landscapes. Often a single time-mapping factor is used to relate the time scales associated with the two representations. We critically examine this idea using a model system ideally suited for this purpose. Single molecular transport properties are studied via molecular dynamics simulations of the CG and atomistic representations of a liquid crystalline, azobenzene containing mesogen, simulated in the smectic and the isotropic phases. The out-of-plane dynamics in the smectic phase occurs via molecular hops from one smectic layer to the next. Hopping can occur via two mechanisms, with and without significant reorientation. The out-of-plane transport can be understood as a superposition of two (one associated with each mode of transport) independent continuous time random walks for which a single time-mapping factor would be rather inadequate. A comparison of the free-energy surfaces, relevant to the out-of-plane transport, qualitatively supports the above observations. Thus, this work underlines the need for building CG models that exhibit both structural and dynamical consistency to the underlying atomistic model.

Multi-scale simulations of biological systems using the OPEP coarse-grained model

Biomolecules are complex machines that are optimized by evolution to properly fulfill or contribute to a variety of biochemical tasks in the cellular environment. Computer simulations based on quantum mechanics and atomistic force fields have been proven to be a powerful microscope for obtaining valuable insights into many biological, physical, and chemical processes. Many interesting phenomena involve, however, a time scale and a number of degrees of freedom, notably if crowding is considered, that cannot be explored at an atomistic resolution. To bridge the gap between reality and simulation, many different advanced computational techniques and coarse-grained (CG) models have been developed. Here, we report some applications of the CG OPEP protein model to amyloid fibril formation, the response of catch-bond proteins to two types of fluid flow, and interactive simulations to fold peptides with well-defined 3D structures or with intrinsic disorder.

Mesoscopic Adaptive Resolution Scheme toward Understanding of Interactions between Sickle Cell Fibers

Understanding of intracellular polymerization of sickle hemoglobin (HbS) and subsequent interaction with the membrane of a red blood cell (RBC) is important to predict the altered morphologies and mechanical properties of sickle RBCs in sickle cell anemia. However, modeling the integrated processes of HbS nucleation, polymerization, HbS fiber interaction, and subsequent distortion of RBCs is challenging as they occur at multispatial scales, ranging from nanometers to micrometers. To make progress toward simulating the integrated processes, we propose a hybrid HbS fiber model, which couples fine-grained and coarse-grained HbS fiber models through a mesoscopic adaptive resolution scheme (MARS). To this end, we apply a microscopic model to capture the dynamic process of polymerization of HbS fibers, while maintaining the mechanical properties of polymerized HbS fibers by the mesoscopic model, thus providing a means of bridging the subcellular and cellular phenomena in sickle cell disease. At the subcellular level, this model can simulate HbS polymerization with preexisting HbS nuclei. At the cellular level, if combined with RBC models, the generated HbS fibers could be applied to study the morphologies and membrane stiffening of sickle RBCs. One important feature of the MARS is that it can be easily employed in other particle-based multiscale simulations where a dynamic coarse-graining and force-blending method is required. As demonstrations, we first apply the hybrid HbS fiber model to simulate the interactions of two growing fibers and find that their final configurations depend on the orientation and interaction distance between two fibers, in good agreement with experimental observations. We also model the formation of fiber bundles and domains so that we explore the mechanism that causes fiber branching.

Energy-Renormalization for Achieving Temperature Transferable Coarse-Graining of Polymer Dynamics

The bottom-up prediction of the properties of polymeric materials based on molecular dynamics simulation is a major challenge in soft matter physics. Coarse-grained (CG) models are often employed to access greater spatiotemporal scales required for many applications, but these models normally experience significantly altered thermodynamics and highly accelerated dynamics due to the reduced number of degrees of freedom upon coarse-graining. While CG models can be calibrated to meet certain properties at particular state points, there is unfortunately no temperature transferable and chemically specific coarse-graining method that allows for modeling of polymer dynamics over a wide temperature range. Here, we pragmatically address this problem by "correcting" for deviations in activation free energies that occur upon coarse-graining the dynamics of a model polymeric material (polystyrene). In particular, we propose a new strategy based on concepts drawn from the Adam-Gibbs (AG) theory of glass formation. Namely we renormalize the cohesive interaction strength and effective interaction length-scale parameters to modify the activation free energy. We show that this energy-renormalization method for CG modeling allows accurate prediction of atomistic dynamics over the Arrhenius regime, the non-Arrhenius regime of glass formation, and even the non-equilibrium glassy regime, thus allowing for the predictive modeling of dynamic properties of polymer over the entire range of glass formation. Our work provides a practical scheme for establishing temperature transferable coarse-grained models for predicting and designing the properties of polymeric materials.