Mesoscopic coarse-grained representations of fluids rigorously derived from atomistic models

A Gauss-Newton method for iterative optimization of memory kernels for generalized Langevin thermostats in coarse-grained molecular dynamics simulations

In molecular dynamics simulations, dynamically consistent coarse-grained (CG) models commonly use stochastic thermostats to model friction and fluctuations that are lost in a CG description. While Markovian, i.e., time-local, formulations of such thermostats allow for an accurate representation of diffusivities/long-time dynamics, a correct description of the dynamics on all time scales generally requires non-Markovian, i.e., non-time-local, thermostats. These thermostats typically take the form of a Generalized Langevin Equation (GLE) determined by a memory kernel. In this work, we use a Markovian embedded formulation of a position-independent GLE thermostat acting independently on each CG degree of freedom. Extracting the memory kernel of this CG model from atomistic reference data requires several approximations. Therefore, this task is best understood as an inverse problem. While our recently proposed approximate Newton scheme allows for the iterative optimization of memory kernels (IOMK), Markovian embedding remained potentially error-prone and computationally expensive. In this work, we present an IOMK-Gauss-Newton scheme (IOMK-GN) based on IOMK that allows for the direct parameterization of a Markovian embedded model.

Chemically Specific Systematic Coarse-Grained Polymer Model with Both Consistently Structural and Dynamical Properties

The coarse-grained (CG) model serves as a powerful tool for the simulation of polymer systems; its reliability depends on the accurate representation of both structural and dynamical properties. However, strong correlations between structural and dynamical properties on different scales and also a strong memory effect, enforced by chain connectivity between monomers in polymer systems, render developing a chemically specific systematic CG model a formidable task. In this study, we report a systematic CG approach that combines the iterative Boltzmann inversion (IBI) method and the generalized Langevin equation (GLE) dynamics. Structural properties are ensured by using conservative CG potentials derived from the IBI method. To retrieve the correct dynamical properties in the system, we demonstrate that using a combination of a Rouse-type delta function and a time-dependent short-time kernel in the GLE simulation is practically efficient. The former can be used to adjust the long-time diffusion dynamics, and the latter can be reconstructed from an iterative procedure according to the velocity autocorrelation function (ACF) from all-atomistic (AA) simulations. Taking the polystyrene as an example, we show that not only structural properties of radial distribution function, intramolecular bond, and angle distributions can be reproduced but also dynamical properties of mean-square displacement, velocity ACF, and force ACF resulted from our CG model have quantitative agreement with the reference AA model. In addition, reasonable agreements are observed in other collective properties between our GLE-CG model and the AA simulations as well.

Maximum Entropy Theory of Multiscale Coarse-Graining via Matching Thermodynamic Forces: Application to a Molecular Crystal (TATB)

The MSCG/FM (multiscale coarse-graining via force-matching) approach is an efficient supervised machine learning method to develop microscopically informed coarse-grained (CG) models. We present a theory based on the principle of maximum entropy (PME) enveloping the existing MSCG/FM approaches. This theory views the MSCG/FM method as a special case of matching the thermodynamic forces from the extended ensemble described by the set of thermodynamic (relevant) system coordinates. This set may include CG coordinates, the stress tensor, applied external fields, and so forth, and may be characterized by nonequilibrium conditions. Following the presentation of the theory, we discuss the consistent matching of both bonded and nonbonded interactions. The proposed PME formulation is used as a starting point to extend the MSCG/FM method to the constant strain ensemble, which together with the explicit matching of the bonded forces is better suited for coarse-graining anisotropic media at a submolecular resolution. The theory is demonstrated by performing the fine coarse-graining of crystalline 1,3,5-triamino-2,4,6-trinitrobenzene (TATB), a well-known insensitive molecular energetic material, which exhibits highly anisotropic mechanical properties.

Representability and Dynamical Consistency in Coarse-Grained Models

We address the challenge of representativity and dynamical consistency when unbonded fine-grained particles are collected together into coarse-grained particles. We implement a hybrid procedure for identifying and tracking the underlying fine-grained particles─e.g., atoms or molecules─by exchanging them between the coarse-grained particles periodically at a characteristic time. The exchange involves a back-mapping of the coarse-grained particles into fine-grained particles and a subsequent reassignment to coarse-grained particles conserving total mass and momentum. We find that an appropriate choice of the characteristic exchange time can lead to the correct effective diffusion rate of the fine-grained particles when simulated in hybrid coarse-grained dynamics. In the compressed (supercritical) fluid regime, without the exchange term, fine-grained particles remain associated with a given coarse-grained particle, leading to substantially lower diffusion rates than seen in all-atom molecular dynamics of the fine-grained particles. Thus, this work confirms the need for addressing the representativity of fine-grained particles within coarse-grained particles and offers a simple exchange mechanism so as to retain dynamical consistency between the fine- and coarse-grained scales.

Formalizing Coarse-Grained Representations of Anisotropic Interactions at Multimeric Protein Interfaces Using Virtual Sites

Molecular simulations of biomacromolecules that assemble into multimeric complexes remain a challenge due to computationally inaccessible length and time scales. Low-resolution and implicit-solvent coarse-grained modeling approaches using traditional nonbonded interactions (both pairwise and spherically isotropic) have been able to partially address this gap. However, these models may fail to capture the complex anisotropic interactions present at macromolecular interfaces unless higher-order interaction potentials are incorporated at the expense of the computational cost. In this work, we introduce an alternate and systematic approach to represent directional interactions at protein-protein interfaces by using virtual sites restricted to pairwise interactions. We show that virtual site interaction parameters can be optimized within a relative entropy minimization framework by using only information from known statistics between coarse-grained sites. We compare our virtual site models to traditional coarse-grained models using two case studies of multimeric protein assemblies and find that the virtual site models predict pairwise correlations with higher fidelity and, more importantly, assembly behavior that is morphologically consistent with experiments. Our study underscores the importance of anisotropic interaction representations and paves the way for more accurate yet computationally efficient coarse-grained simulations of macromolecular assembly in future research.

Phase behaviour of coarse-grained fluids

Soft condensed matter structures often challenge us with complex many-body phenomena governed by collective modes spanning wide spatial and temporal domains. In order to successfully tackle such problems, mesoscopic coarse-grained (CG) statistical models are being developed, providing a dramatic reduction in computational complexity. CG models provide an intermediate step in the complex statistical framework of linking the thermodynamics of condensed phases with the properties of their constituent atoms and molecules. These allow us to offload part of the problem to the CG model itself and reformulate the remainder in terms of reduced CG phase space. However, such exchange of pawns to chess pieces, or 'Hamiltonian renormalization', is a radical step and the thermodynamics of the primary atomic and CG models could be quite distinct. Here, we present a comprehensive study of the phase diagram including binodal and interfacial properties of a dissipative particle dynamics (DPD) model, extended to include finite-range attraction to support the liquid-gas equilibrium. Despite the similarities with the atomic model potentials, its phase envelope is markedly different featuring several anomalies such as an unusually broad liquid range, change in concavity of the liquid coexistence branch with variation of the model parameters, volume contraction on fusion, temperature of maximum density in the liquid phase and negative thermal expansion in the solid phase. These results provide new insight into the connection between simple potential models and complex emergent condensed matter phenomena.

Hierarchical Machine Learning of Low-Resolution Coarse-Grained Free Energy Potentials

A force-matching-based method for supervised machine learning (ML) of coarse-grained (CG) free energy (FE) potentials─known as multiscale coarse-graining via force-matching (MSCG/FM)─is an efficient method to develop microscopically informed CG models that are thermodynamically and statistically equivalent to the reference microscopic models. For low-resolution models, when the coarse-graining is at supramolecular scales, objective-oriented clustering of nonbonded particles is required and the reduced description becomes a function of the clustering algorithm. In the present work, we explore the dependence of the ML of the CG Helmholtz FE potential on the clustering algorithm. We consider coarse-graining based on partitional (k-means, leading to Voronoi diagram) and hierarchical agglomerative (bottom-up) clustering algorithms common in unsupervised ML and develop theory connecting the MSCG/FM learned CG Helmholtz potential and the clustering statistics. By combining the agglomerative clustering and the MSCG/FM learning in a recursive manner, we propose an efficient ML methodology to develop the fine-to-low resolution hierarchies of the CG models. The methodology does not suffer from degrading accuracy or increased computational cost to construct larger hierarchies and as such does not impose an upper size limitation of the CG particles resulting from the extended hierarchies. The utility of the methodology is demonstrated by obtaining the bottom-up agglomerative hierarchy for liquid nitromethane from all-atom molecular dynamics (MD) simulations. For agglomerative hierarchies, we prove the existence of renormalization group transformations that indicate self-similarity and allow for learning the low-resolution MSCG/FM potentials at low computational cost by rescaling and renormalizing the certain finer-resolution members of the hierarchy. The hierarchies of the CG models can be used to carry out simulations under constant-pressure conditions.

Statistical Mechanical Design Principles for Coarse-Grained Interactions across Different Conformational Free Energy Surfaces

Systematic bottom-up coarse-graining (CG) of molecular systems provides a means to explore different coupled length and time scales while treating the molecular-scale physics at a reduced level. However, the configuration dependence of CG interactions often results in CG models with limited applicability for exploring the parametrized configurations. We propose a statistical mechanical theory to design CG interactions across different configurations and conditions. In order to span wide ranges of conformational space, distinct classical CG free energy surfaces for characteristic configurations are identified using molecular collective variables. The coupling interaction between different CG free energy surfaces can then be systematically determined by analogy to quantum mechanical approaches describing coupled states. The present theory can accurately capture the underlying many-body potentials of mean force in the CG variables for various order parameters applied to liquids, interfaces, and in principle proteins, uncovering the complex nature underlying the coupling interaction and imparting a new protocol for the design of predictive multiscale models.

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.

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.

Conservative Potentials for a Lattice-Mapped, Coarse-Grain Scheme with Fuzzy Switching Functions

We extend our previous work (Luo, S.; Thachuk, M. J. Phys. Chem. A 2021, 125, 64866497) on determining conservative potentials for lattice-like, coarse-grain (CG) mapping schemes to the case where the boundaries between different spatial regions are not sharply defined but are fuzzy. In other words, the system is divided into interpenetrating "subcells" such that atomistic particles continuously change their memberships as they move through space. This is done by using fuzzy switching functions to define overlapping regions between subcells with fractional particle occupations. In this case, a full mass matrix is required to describe the system, and its off-diagonal elements are nonzero and contribute to the CG potential. As the overlapping region increases in size, we observe the mass distribution transitions from a discrete spectrum, through an intermediate state, and finally to a continuous Gaussian-like function. We interpret this as a quantitative measure for signaling when a continuum-theory description of the system is appropriate. Nonzero correlations among all CG variables are calculated and are found to depend strongly on the degree of overlap. In particular, those for the diagonal mass elements decrease in magnitude, and there exists a specific value of the overlap for which the correlations are zero. Other correlations are strong only when the overlap is quite large, so there is a trade-off between the complexity of the interactions in the system and the degree of fuzziness between the subcells. However, if the number of particles in a subcell is large enough and the overlap is moderate, then the CG potential is found to be well-approximated by a generalized quadratic function. These results demonstrate the transition between atomistic and continuum resolutions in a system and have implications for designing CG schemes with mixed atomistic and continuum character.

Conservative Potentials for a Lattice-Mapped Coarse-Grained Scheme

The conservative potential, arising from a coarse-grain (CG) mapping scheme for nonbonded atomistic particles, is studied. This is a bottom-up approach from first-principles that maps atomistic particles to fluid element-like subcells whose centers lie on a regular, cubic lattice. Unlike standard CG mapping schemes, the current one uses dynamic labeling which on-the-fly changes the CG labels of the particles. The subcells can also be different sizes and shapes, in principle. Equilibrium atomistic molecular dynamics trajectories for different Lennard-Jones fluids are calculated and converted to CG ones, from which CG probability distribution functions are calculated. Correlation studies show position and mass CG variables are uncoupled in a given subcell, as are different vector components of position. Furthermore, the strongest coupling occurs with neighboring cells in specific directions, and the resulting distribution is well described by a multivariate Gaussian. This implies the CG potential has a generalized quadratic form, whose derivative can be determined analytically. A microscopic rationalization is provided for the signs and relative magnitudes of different correlation coefficients, and in some cases, a connection is made with bulk properties of the fluid. We argue the generalized quadratic form should be robust to changes in the particulars of the CG scheme, as well as the nature of the atomistic intermolecular potential. Only a few potential parameters need to be calculated from the underlying atomistic system. This is significant because it indicates the transferability of this form to other, more complex systems. This transferability will be tested in future work, where mapping schemes with fuzzy boundaries will be considered.

Introducing Memory in Coarse-Grained Molecular Simulations

Preserving the correct dynamics at the coarse-grained (CG) level is a pressing problem in the development of systematic CG models in soft matter simulation. Starting from the seminal idea of simple time-scale mapping, there have been many efforts over the years toward establishing a meticulous connection between the CG and fine-grained (FG) dynamics based on fundamental statistical mechanics approaches. One of the most successful attempts in this context has been the development of CG models based on the Mori-Zwanzig (MZ) theory, where the resulting equation of motion has the form of a generalized Langevin equation (GLE) and closely preserves the underlying FG dynamics. In this Review, we describe some of the recent studies in this regard. We focus on the construction and simulation of dynamically consistent systematic CG models based on the GLE, both in the simple Markovian limit and the non-Markovian case. Some recent studies of physical effects of memory are also discussed. The Review is aimed at summarizing recent developments in the field while highlighting the major challenges and possible future directions.

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.

A new one-site coarse-grained model for water: Bottom-up many-body projected water (BUMPer). I. General theory and model

Water is undoubtedly one of the most important molecules for a variety of chemical and physical systems, and constructing precise yet effective coarse-grained (CG) water models has been a high priority for computer simulations. To recapitulate important local correlations in the CG water model, explicit higher-order interactions are often included. However, the advantages of coarse-graining may then be offset by the larger computational cost in the model parameterization and simulation execution. To leverage both the computational efficiency of the CG simulation and the inclusion of higher-order interactions, we propose a new statistical mechanical theory that effectively projects many-body interactions onto pairwise basis sets. The many-body projection theory presented in this work shares similar physics from liquid state theory, providing an efficient approach to account for higher-order interactions within the reduced model. We apply this theory to project the widely used Stillinger-Weber three-body interaction onto a pairwise (two-body) interaction for water. Based on the projected interaction with the correct long-range behavior, we denote the new CG water model as the Bottom-Up Many-Body Projected Water (BUMPer) model, where the resultant CG interaction corresponds to a prior model, the iteratively force-matched model. Unlike other pairwise CG models, BUMPer provides high-fidelity recapitulation of pair correlation functions and three-body distributions, as well as N-body correlation functions. BUMPer extensively improves upon the existing bottom-up CG water models by extending the accuracy and applicability of such models while maintaining a reduced computational cost.

Scaling-Up Simulations of Diffusion in Microporous Materials

We introduce and demonstrate the coarse-graining of static and dynamical properties of host-guest systems constituted by methane in two different microporous materials. The reference systems are mapped to occupancy-based pore-scale lattice models. Each coarse-grained model is equipped with an appropriate coarse-grained potential and a local dynamical operator, which represents the probability of interpore molecular jumps between different cages. Coarse-grained thermodynamics and dynamics are both defined based on small-scale atomistic simulations of the reference systems. We considered two host materials: the widely studied ITQ-29 zeolite and the LTA-zeolite-templated carbon, which was recently theorized. Our method allows for representing with satisfactory accuracy and a considerably reduced computational effort the reference systems while providing new interesting physical insights in terms of static and diffusive properties.

Implicit-solvent coarse-grained modeling for polymer solutions via Mori-Zwanzig formalism

We present a bottom-up coarse-graining (CG) method to establish implicit-solvent CG modeling for polymers in solution, which conserves the dynamic properties of the reference microscopic system. In particular, tens to hundreds of bonded polymer atoms (or Lennard-Jones beads) are coarse-grained as one CG particle, and the solvent degrees of freedom are eliminated. The dynamics of the CG system is governed by the generalized Langevin equation (GLE) derived via the Mori-Zwanzig formalism, by which the CG variables can be directly and rigorously linked to the microscopic dynamics generated by molecular dynamics (MD) simulations. The solvent-mediated dynamics of polymers is modeled by the non-Markovian stochastic dynamics in GLE, where the memory kernel can be computed from the MD trajectories. To circumvent the difficulty in direct evaluation of the memory term and generation of colored noise, we exploit the equivalence between the non-Markovian dynamics and Markovian dynamics in an extended space. To this end, the CG system is supplemented with auxiliary variables that are coupled linearly to the momentum and among themselves, subject to uncorrelated Gaussian white noise. A high-order time-integration scheme is used to solve the extended dynamics to further accelerate the CG simulations. To assess, validate, and demonstrate the established implicit-solvent CG modeling, we have applied it to study four different types of polymers in solution. The dynamic properties of polymers characterized by the velocity autocorrelation function, diffusion coefficient, and mean square displacement as functions of time are evaluated in both CG and MD simulations. Results show that the extended dynamics with auxiliary variables can construct arbitrarily high-order CG models to reproduce dynamic properties of the reference microscopic system and to characterize long-time dynamics of polymers in solution.

Adversarial-residual-coarse-graining: Applying machine learning theory to systematic molecular coarse-graining

We utilize connections between molecular coarse-graining (CG) approaches and implicit generative models in machine learning to describe a new framework for systematic molecular CG. Focus is placed on the formalism encompassing generative adversarial networks. The resulting method enables a variety of model parameterization strategies, some of which show similarity to previous CG methods. We demonstrate that the resulting framework can rigorously parameterize CG models containing CG sites with no prescribed connection to the reference atomistic system (termed virtual sites); however, this advantage is offset by the lack of a closed-form expression for the CG Hamiltonian at the resolution obtained after integration over the virtual CG sites. Computational examples are provided for cases in which these methods ideally return identical parameters as relative entropy minimization CG but where traditional relative entropy minimization CG optimization equations are not applicable.

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.

SWINGER: a clustering algorithm for concurrent coupling of atomistic and supramolecular liquids

In this contribution, we review recent developments and applications of a dynamic clustering algorithm SWINGER tailored for the multiscale molecular simulations of biomolecular systems. The algorithm on-the-fly redistributes solvent molecules among supramolecular clusters. In particular, we focus on its applications in combination with the adaptive resolution scheme, which concurrently couples atomistic and coarse-grained molecular representations. We showcase the versatility of our multiscale approach on a few applications to biomolecular systems coupling atomistic and supramolecular water models such as the well-established MARTINI and dissipative particle dynamics models and provide an outlook for future work.

Relative entropy indicates an ideal concentration for structure-based coarse graining of binary mixtures

Many methodological approaches have been proposed to improve systematic or bottom-up coarse-graining techniques to enhance the representability and transferability of the derived interaction potentials. Transferability describes the ability of a coarse-grained (CG) model to be predictive, i.e., to describe a system at state points different from those chosen for parametrization. Whereas the representability characterizes the accuracy of a CG model to reproduce target properties of the underlying reference or fine-grained model at a given state point. In this article, we shift the focus away from methodological aspects and rather raise the question whether we can overcome the disadvantages of a given method in terms of representability and transferability by systematically selecting the state point at which the CG model gets parametrized. We answer this question by applying the inverse Monte Carlo (IMC) approach-a structure-based coarse-graining method-to derive effective interactions for binary mixtures of simple Lennard-Jones (LJ) particles, which are different in size. For such simple systems we indeed can identify a concentration where the derived potentials show the best performance in terms of structural representability and transferability. This specific concentration is identified by computing the relative entropy which quantifies the information loss between different IMC models and the reference LJ model at varying mixture compositions. Further, we show that an IMC model for mixtures of n-hexane and n-perfluorohexane shows the same trend in transferability as the IMC models for the LJ system. All derived models are more transferable in the direction of increasing concentration of the larger-sized compound.