Coarse-grained Model Research Articles

A coarse-grained bacterial cell model for resource-aware analysis and design of synthetic gene circuits

Within a cell, synthetic and native genes compete for expression machinery, influencing cellular process dynamics through resource couplings. Models that simplify competitive resource binding kinetics can guide the design of strategies for countering these couplings. However, in bacteria resource availability and cell growth rate are interlinked, which complicates resource-aware biocircuit design. Capturing this interdependence requires coarse-grained bacterial cell models that balance accurate representation of metabolic regulation against simplicity and interpretability. We propose a coarse-grained E. coli cell model that combines the ease of simplified resource coupling analysis with appreciation of bacterial growth regulation mechanisms and the processes relevant for biocircuit design. Reliably capturing known growth phenomena, it provides a unifying explanation to disparate empirical relations between growth and synthetic gene expression. Considering a biomolecular controller that makes cell-wide ribosome availability robust to perturbations, we showcase our model’s usefulness in numerically prototyping biocircuits and deriving analytical relations for design guidance.

Understanding Self-Assembly and Molecular Packing in Methylcellulose Aqueous Solutions Using Multiscale Modeling and Simulations.

We present a multiscale molecular dynamics (MD) simulation study on self-assembly in methylcellulose (MC) aqueous solutions. First, using MD simulations with a new coarse-grained (CG) model of MC chains in implicit water, we establish how the MC chains self-assemble to form fibrils and fibrillar networks and elucidate the MC chains' packing within the assembled fibrils. The CG model for MC is extended from a previously developed model for unsubstituted cellulose and captures the directionality of H-bonding interactions between the -OH groups. The choice and placement of the CG beads within each monomer facilitates explicit modeling of the exact degree and position of methoxy substitutions in the monomers along the MC chain. CG MD simulations show that with increasing hydrophobic effect and/or increasing H-bonding strength, the commercial MC chains (with degree of methoxy substitution, DS, ∼1.8) assemble from a random dispersed configuration into fibrils. The assembled fibrils exhibit consistent fibril diameters regardless of the molecular weight and concentration of MC chains, in agreement with past experiments. Most MC chains' axes are aligned with the fibril axis, and some MC chains exhibit twisted conformations in the fibril. To understand the molecular driving force for the twist, we conduct atomistic simulations of MC chains preassembled in fibrils (without any chain twists) in explicit water at 300 and 348 K. These atomistic simulations also show that at DS = 1.8, MC chains adopt twisted conformations, with these twists being more prominent at higher temperatures, likely as a result of shielding of hydrophobic methyl groups from water. For MC chains with varying DS, at 348 K, atomistic simulations show a nonmonotonic effect of DS on water-monomer contacts. For 0.0 < DS < 0.6, the MC monomers have more water contacts than at DS = 0.0 or DS > 0.6, suggesting that with few methoxy substitutions, the MC chains are effectively hydrophilic, letting the water molecules diffuse into the fibril to participate in H-bonds with the MC chains' remaining -OH groups. At DS > 0.6, the MC monomers become increasingly hydrophobic, as seen by decreasing water contacts around each monomer. We conclude based on the atomistic observations that MC chains with lower degrees of substitutions (DS ≤ 0.6) should exhibit solubility in water over broader temperature ranges than DS ∼ 1.8 chains.

Constant-pH Simulations of a Coarse-Grained Model of Polyfunctional Weak Charged Biopolymers

A coarse-grained model of linear polyfunctional weak charged biopolymers was implemented, formed of different proportions of acid-base groups resembling the composition of humic substances. These substances are mainly present in dissolved organic matter in natural water. The influence of electrostatic interactions computing methods, factors concerning the structure of the chain, different functional groups, and the ionic strength on polyelectrolytes were studied. Langevin dynamics with constant pH simulations were performed using the ESPResSO package and the Python-based Molecule Builder for ESPResSo (pyMBE) library. The coverage was fitted to a polyfunctional Frumkin isotherm, with a mean-field interaction between charged beads. The composition of the chain affects the charge while ionic strength affects both the charge and the radius of gyration. Additionally, the parameters intrinsic to the polyelectrolyte model were well reproduced by fitting the polyfunctional Frumkin isotherm. In contrast, the non-intrinsic parameters depended on the ionic strength. The method developed and applied to a polyfunctional polypeptide model, that resembles a humic acid, will be very useful for characterizing biopolymers with several acid-base functional groups, where their structure, the composition of the different functional groups, and the determination of the main intrinsic proton binding constants and their proportion are not exactly known.

Molecular insights into nanoplastics-peptides binding and their interactions with the lipid membrane

Micro- and nanoplastics have become a significant concern, due to their ubiquitous presence in the environment. These particles can be internalized by the human body through ingestion, inhalation, or dermal contact, and then they can interact with environmental or biological molecules, such as proteins, resulting in the formation of the protein corona. However, information on the role of protein corona in the human body is still missing. Coarse-grain models of the nanoplastics and pentapeptides were created and simulated at the microscale to study the role of protein corona. Additionally, a lipid bilayer coarse-grain model was reproduced to investigate the behavior of the coronated nanoplastics in proximity of a lipid bilayer. Hydrophobic and aromatic amino acids have a high tendency to create stable bonds with all nanoplastics. Moreover, polystyrene and polypropylene establish bonds with polar and charged amino acids. When the coronated nanoplastics are close to a lipid bilayer, different behaviors can be observed. Polyethylene creates a single polymeric chain, while polypropylene tends to break down into its single chains. Polystyrene can both separate into its individual chains and remain aggregated. The protein corona plays an important role when interacting with the nanoplastics and the lipid membrane. More studies are needed to validate the results and to enhance the complexity of the systems.

Cellulose-Callose Hydrogels: Computational Exploration of Their Nanostructure and Mechanical Properties.

Polysaccharides play a crucial role in virtually all living systems. They also represent the biocompatible and fully sustainable component of a variety of nanoparticles, which are of increasing interest in biomedicine, food processing, cosmetics, and structural reinforcement of polymeric materials. The computational modeling of complex polysaccharide phases will assist in understanding the properties and behavior of all these systems. In this paper, structural, bonding, and mechanical properties of 10 wt % cellulose-callose hydrogels (β-glucans coexisting in plant cell walls) were investigated by atomistic simulations. Systems of this kind have recently been introduced in experiments revealing unexpected interactions between the polysaccharides. Starting from initial configurations inspired by X-ray diffraction data, atomistic models made of ∼1.6 × 106 atoms provide a qualitatively consistent view of these hydrogels, displaying stability, homogeneity, connectivity, and elastic properties beyond those of a liquid suspension. The simulation shows that the relatively homogeneous distribution of saccharide nanofibers and chains in water is not due to the solubility of cellulose and callose, but to the formation of a number of cross-links among the various sample components. The broad distribution of strength and elasticity among the links implies a degree of anharmonicity and irreversible deformation already evident at low external load. Besides the qualitative agreement with experimental observations, the simulation results display also quantitative disagreements in the estimation of elastic coefficients, such as the Young's modulus, that require further investigation. Complementary simulations of dense cellulose-callose mixtures (no hydrogels) highlight the role of callose in smoothing the contact surface of different nanofibers forming larger bundles. Cellulose-callose structures in these systems displayed an enhanced water uptake and delayed dye release when compared to cellulose alone, highlighting potential new applications as drug delivery scaffolds. The simulation trajectories provide a tuning and testing ground for the development of coarse-grained models that are required for the large scale investigation of mechanical properties of cellulose and callose mixtures in a watery environment.

Dissipation and recovery in collagen fibrils under cyclic loading: A molecular dynamics study.

The hysteretic behavior exhibited by collagen fibrils, when subjected to cyclic loading, is known to result in both dissipation as well as accumulation of residual strain. On subsequent relaxation, partial recovery has also been reported. Cross-links have been considered to play a key role in overall mechanical properties. Here, we modify an existing coarse-grained molecular dynamics model for collagen fibril with initially cross-linked collagen molecules, which is known to reproduce the response to uniaxial strain, by incorporating reformation of cross-links to allow for possible recovery of the fibril. Using molecular dynamics simulations, we show that our model successfully replicates the key features observed in experimental data, including the movement of hysteresis loops, the time evolution of residual strains and energy dissipation, as well as the recovery observed during relaxation. We also show that the characteristic cycle number, describing the approach toward steady state, has a value similar to that in experiments. We also emphasize the vital role of the degree of cross-linking on the key features of the macroscopic response to cyclic loading.

Ion-Induced PIP2 Clustering with Martini3: Modification of Phosphate-Ion Interactions and Comparison with CHARMM36.

Phosphatidylinositol 4,5-bisphosphate (PIP2) is a critical lipid for cellular signaling. The specific phosphorylation of the inositol ring controls protein binding as well as clustering behavior. Two popular models to describe ion-mediated clustering of PIP2 are Martini3 (M3) and CHARMM36 (C36). Molecular dynamics simulations of PIP2-containing bilayers in solutions of potassium chloride, sodium chloride, and calcium chloride, and at two different resolutions are performed to understand the aggregation and the model parameters that drive it. The average M3 clusters of PIP2 in bilayers of 1-palmitoyl-2-oleoyl-glycero-3-phosphocholine and PIP2 bilayers in the presence of K+, Na+, or Ca2+ contained 2.2, 2.6, and 6.4 times more PIP2 than C36 clusters, respectively. Indeed, the Ca2+-containing systems often formed a single large aggregate. Reparametrization of the M3 ion-phosphate Lennard-Jones interaction energies to reproduce experimental osmotic pressure of sodium dimethyl phosphate (DMP), K[DMP], and Ca[DMP]2 solutions, the same experimental target as C36, yielded comparably sized PIP2 clusters for the two models. Furthermore, C36 and the modified M3 predict similar saturation of the phosphate groups with increasing Ca2+, although the coarse-grained model does not capture the cooperativity between K+ and Ca2+. This characterization of the M3 behavior in the presence of monovalent and divalent ions lays a foundation to study cation/protein/PIP2 clustering.

Ion Conductivity in Salt-Doped Polymers: Combined Effects of Temperature and Salt Concentration.

We construct a coarse-grained molecular dynamics model based on poly(ethylene oxide) and lithium bis(trifluoromethane)sulfonimide salt to examine the combined effects of temperature and salt concentration on the transport properties. Salt doping notably slows the dynamics of polymer chains and reduces ion diffusivity, resulting in a glass transition temperature increase proportional to the salt concentration. The polymer diffusion is shown to be well represented by a modified Vogel-Fulcher-Tamman (M-VFT) equation that accounts for both the temperature and salt concentration dependence. Furthermore, we find that, at any temperature, the concentration dependence of the conductivity is well described by the product of its infinite dilution value and a correction factor accounting for the reduced segmental mobility with increasing salt concentration. These results highlight the important role of polymer segmental mobility in the salt concentration dependence of ion conductivity for temperatures near and above the glass transition.

Transfer operators on graphs: spectral clustering and beyond

Graphs and networks play an important role in modeling and analyzing complex interconnected systems such as transportation networks, integrated circuits, power grids, citation graphs, and biological and artificial neural networks. Graph clustering algorithms can be used to detect groups of strongly connected vertices and to derive coarse-grained models. We define transfer operators such as the Koopman operator and the Perron–Frobenius operator on graphs, study their spectral properties, introduce Galerkin projections of these operators, and illustrate how reduced representations can be estimated from data. In particular, we show that spectral clustering of undirected graphs can be interpreted in terms of eigenfunctions of the Koopman operator and propose novel clustering algorithms for directed graphs based on generalized transfer operators. We demonstrate the efficacy of the resulting algorithms on several benchmark problems and provide different interpretations of clusters.

Unveiling the three-dimensional network and deformation mechanism of foamed polyurethane by coarse-grained and graph theory

Non-aqueous reactive foamed polyurethane (PU) has gained widespread applications in engineering trenchless rehabilitation, and the three-dimensional (3D) network formed by crosslinking is the key to their applications through strength and durability. However, there remains a major gap between the general understanding of the 3D network in foamed PU and how it affects the strength of closed cell. Here classical molecular dynamics simulations combined with coarse-grained modeling, X-ray photoelectron spectroscopy and graph theory were employed to characterize the key components and deformations of the 3D network, also the resulting mechanical properties of foamed PU. The results quantified the number and deformations of simple cycles in 3D networks and indicate that the simple cycles are crucial structure that reflect the connectivity and deformation of 3D network. A stronger connectivity results in a higher quantity of shorter simple cycles in the 3D network. These shorter simple cycles impose more limitations on the deformation of the 3D network, manifested as smaller degree of flipping and stretching, resulting from the extension of linear chain segments within soft segments and the flipping of hard segments. Consequently, the stronger connectivity contributes to more uniform deformation and greater tensile strength of PU closed cells. The results can provide a blueprint for future characterization and design of this materials.

Impact of Glycosylation on Protein-Protein Self-Interactions of Monoclonal Antibodies.

Protein self-interactions measured via second osmotic virial coefficients (B22) and dynamic light scattering interaction parameter values (kD) are often used as metrics for assessing the favorability of protein candidates and different formulations during monoclonal antibody (MAb) product development. Model predictions of B22 or kD typically do not account for glycans, though glycosylation can potentially impact experimental MAb self-interactions. To the best of our knowledge, the impact of MAb glycosylation on the experimentally measured B22 and kD values has not yet been reported. B22 and kD values of two fully deglycosylated MAbs and their native (i.e., fully glycosylated) counterparts were measured by light scattering over a range of pH and ionic strength conditions. Significant differences between B22 and kD of the native and deglycosylated forms were observed at a range of low to high ionic strengths used to modulate the effect of electrostatic contributions. Differences were most pronounced at low ionic strength, indicating that electrostatic interactions are a contributing factor. Though B22 and kD values were statistically equivalent at high ionic strengths where electrostatics were fully screened, we observed protein-dependent qualitative differences, which indicate that steric interactions may also play a role in the observed B22 and kD differences. A domain-level coarse-grained molecular model accounting for charge differences was considered to potentially provide additional insight but was not fully predictive of the behavior across all of the solution conditions investigated. This highlights that both the level of modeling and lack of inclusion of glycans may limit existing models in making quantitatively accurate predictions of self-interactions.

Development of a bottom-up coarse-grained model for interactions of lipids with TiO nanoparticles.

Understanding interactions of inorganic nanoparticles with biomolecules is important in many biotechnology, nanomedicine, and toxicological research, however, the size of typical nanoparticles makes their direct modeling by atomistic simulations unfeasible. Here, we present a bottom-up coarse-graining approach for modeling titanium dioxide (TiO ) nanomaterials in contact with phospholipids that uses the inverse Monte Carlo method to optimize the effective interactions from the structural data obtained in small-scale all-atom simulations of TiO surfaces with lipids in aqueous solution. The resulting coarse-grained models are able to accurately reproduce the structural details of lipid adsorption on different titania surfaces without the use of an explicit solvent, enabling significant computational resource savings and favorable scaling. Our coarse-grained simulations show that small spherical TiO nanoparticles ( nm) can only be partially wrapped by a lipid bilayer with phosphoethanolamine headgroups, however, the lipid adsorption increases with the radius of the nanoparticle. The current approach can be used to study the effect of the size and shape of TiO nanoparticles on their interactions with cell membrane lipids, which can be a determining factor in membrane wrapping as well as the recently discovered phenomenon of nanoquarantining, which involves the formation of layered nanomaterial-lipid structures.

Simulations of Functional Motions of Super Large Biomolecules with a Mixed-Resolution Model.

Many large protein machines function through an interplay between large-scale movements and intricate conformational changes. Understanding functional motions of these proteins through simulations becomes challenging for both all-atom and coarse-grained (CG) modeling techniques because neither approach alone can readily capture the full details of these motions. In this study, we develop a multiscale model by employing the popular MARTINI CG model to represent a heterogeneous environment and structurally stable proteins and using the united-atom (UA) model PACE to describe proteins undergoing subtle conformational changes. PACE was previously developed to be compatible with the MARTINI solvent and membrane. Here, we couple the protein descriptions of the two models by directly mixing UA and CG interaction parameters to greatly simplify parameter determination. Through extensive validations with diverse protein systems in solution or membrane, we demonstrate that only additional parameter rescaling is needed to enable the resulting model to recover the stability of native structures of proteins under mixed representation. Moreover, we identify the optimal scaling factors that can be applied to various protein systems, rendering the model potentially transferable. To further demonstrate its applicability for realistic systems, we apply the model to a mechanosensitive ion channel Piezo1 that has peripheral arms for sensing membrane tension and a central pore for ion conductance. The model can reproduce the coupling between Piezo1's large-scale arm movement and subtle pore opening in response to membrane stress while consuming much less computational costs than all-atom models. Therefore, our model shows promise for studying functional motions of large protein machines.

Diffusive dynamics of a model protein chain in solution.

A Markov state model is a powerful tool that can be used to track the evolution of populations of configurations in an atomistic representation of a protein. For a coarse-grained linear chain model with discontinuous interactions, the transition rates among states that appear in the Markov model when the monomer dynamics is diffusive can be determined by computing the relative entropy of states and their mean first passage times, quantities that are unchanged by the specification of the energies of the relevant states. In this paper, we verify the folding dynamics described by a diffusive linear chain model of the crambin protein in three distinct solvent systems, each differing in complexity: a hard-sphere solvent, a solvent undergoing multi-particle collision dynamics, and an implicit solvent model. The predicted transition rates among configurations agree quantitatively with those observed in explicit molecular dynamics simulations for all three solvent models. These results suggest that the local monomer-monomer interactions provide sufficient friction for the monomer dynamics to be diffusive on timescales relevant to changes in conformation. Factors such as structural ordering and dynamic hydrodynamic effects appear to have minimal influence on transition rates within the studied solvent densities.

Mobility, response and transport in non-equilibrium coarse-grained models

We investigate two different types of non-Markovian coarse-grained models extracted from a linear, non-equilibrium microscopic system, featuring a tagged particle coupled to underdamped oscillators. The first model is obtained by analytically ‘integrating out’ the oscillators and the second is based on a derivation using projection operator techniques. We observe that these two models behave very differently when the tagged particle is exposed to external harmonic potentials or pulling forces. Most importantly, we find that the analytic model has a well defined friction kernel and can be used to extract work, consistent with the microscopic system, while the projection model corresponds to an effective equilibrium model, which cannot be used to extract work. We apply the analysis to two popular non-equilibrium systems, time-delay feedback control and the active Ornstein–Uhlenbeck process. Finally, we highlight that our study could have important consequences for dynamic coarse-graining of non-equilibrium systems going far beyond the linear systems investigated in this manuscript.

Closing the Gap between Experiment and Simulation─A Holistic Study on the Complexation of Small Interfering RNAs with Polyethylenimine.

Rational design is pivotal in the modern development of nucleic acid nanocarrier systems. With the rising prominence of polymeric materials as alternatives to lipid-based carriers, understanding their structure-function relationships becomes paramount. Here, we introduce a newly developed coarse-grained model of polyethylenimine (PEI) based on the Martini 3 force field. This model facilitates molecular dynamics simulations of true-sized PEI molecules, exemplified by molecules with molecular weights of 1.3, 5, 10, and 25 kDa, with degrees of branching between 50.0 and 61.5%. We employed this model to investigate the thermodynamics of small interfering RNA (siRNA) complexation with PEI. Our simulations underscore the pivotal role of electrostatic interactions in the complexation process. Thermodynamic analyses revealed a stronger binding affinity with increased protonation, notably in acidic (endosomal) pH, compared to neutral conditions. Furthermore, the molecular weight of PEI was found to be a critical determinant of binding dynamics: smaller PEI molecules closely enveloped the siRNA, whereas larger ones extended outward, facilitating the formation of complexes with multiple RNA molecules. Experimental validations, encompassing isothermal titration calorimetry and single-molecule fluorescence spectroscopy, aligned well with our computational predictions. Our findings not only validate the fidelity of our PEI model but also accentuate the importance of in silico data in the rational design of polymeric drug carriers. The synergy between computational predictions and experimental validations, as showcased here, signals a refined and precise approach to drug carrier design.

Evolution of cooperation in deme-structured populations on graphs.

Understanding how cooperation can evolve in populations despite its cost to individual cooperators is an important challenge. Models of spatially structured populations with one individual per node of a graph have shown that cooperation, modeled via the prisoner's dilemma, can be favored by natural selection. These results depend on microscopic update rules, which determine how birth, death, and migration on the graph are coupled. Recently, we developed coarse-grained models of spatially structured populations on graphs, where each node comprises a well-mixed deme, and where migration is independent from division and death, thus bypassing the need for update rules. Here, we study the evolution of cooperation in these models in the rare-migration regime, within the prisoner's dilemma. We find that cooperation is not favored by natural selection in these coarse-grained models on graphs where overall deme fitness does not directly impact migration from a deme. This is due to a separation of scales, whereby cooperation occurs at a local level within demes, while spatial structure matters between demes.

On the Validity of Constant pH Simulations.

Constant pH (cpH) simulations are now a standard tool for investigating charge regulation in coarse-grained models of polyelectrolytes and colloidal systems. Originally developed for studying solutions with implicit ions, extending this method to systems with explicit ions or solvents presents several challenges. Ensuring proper charge neutrality within the simulation cell requires performing titration moves in sync with the insertion or deletion of ions, a crucial aspect often overlooked in the literature. Contrary to the prevailing views, cpH simulations are inherently grand-canonical, meaning that the controlled pH is that of the reservoir. The presence of the Donnan potential between the implicit reservoir and the simulation cell introduces significant differences between titration curves calculated for open and closed systems; the pH of an isolated (closed) system is different from the pH of the reservoir for the same protonation state of the polyelectrolyte. To underscore this point, in this paper, we will compare the titration curves calculated using the usual cpH algorithm with those from the exact canonical simulation algorithm. In the latter case, titration moves adhere to the correct detailed balance condition, and pH is calculated using the recently introduced surface Widom insertion algorithm. Our findings reveal a very significant difference between the titration isotherms obtained using the standard cpH algorithm and the canonical titration algorithm, emphasizing the importance of using the correct simulation approach when studying charge regulation of polyelectrolytes, proteins, and colloidal particles.

Coarse-Grained Model of Glycosaminoglycans for Biomolecular Simulations.

Proteoglycans contain glycosaminoglycans (GAGs) which are negatively charged linear polymers made of repeating disaccharide units of uronic acid and hexosamine units. They play vital roles in numerous physiological and pathological processes, particularly in governing cellular communication and attachment. Depending on their sulfonation state, acetylation, and glycosidic linkages, GAGs belong to different families. The high molecular weight, heterogeneity, and flexibility of GAGs hamper their characterization at atomic resolution, but this may be circumvented via coarse-grained (CG) approaches. In this work, we report a CG model for a library of common GAG types in their isolated or proteoglycan-linked states compatible with version 2.2 (v2.2) of the widely popular CG Martini force field. The model reproduces conformational and thermodynamic properties for a wide variety of GAGs, as well as matching structural and binding data for selected proteoglycan test systems. The parameters developed here may thus be employed to study a range of GAG-containing biomolecular systems, thereby benefiting from the efficiency and broad applicability of the Martini framework.

Development of anisotropic force fields for homopolymer melts at the mesoscale.

With the aim of producing realistic coarse-grained models of homopolymers, we introduce a tabulated backbone-oriented anisotropic potential. The parameters of the model are optimized using statistical trajectory matching. The impact of grain anisotropy is evaluated at different coarse-graining levels using cis-polybutadiene as a test case. We show that, at the same time, tuning the aspect ratio of the grains can lead to a better density and structure and may reduce the unphysical bond crossings by up to 90%, without increasing the computation time too much and thereby jeopardizing the main advantage of coarse-grained models.