Effects of macromolecular crowding on the folding of a polymer chain: A Wang-Landau simulation study

Computational Approaches to Predict Protein-Protein Interactions in Crowded Cellular Environments

Investigating protein-protein interactions is crucial for understanding cellular biological processes because proteins often function within molecular complexes rather than in isolation. While experimental and computational methods have provided valuable insights into these interactions, they often overlook a critical factor: the crowded cellular environment. This environment significantly impacts protein behavior, including structural stability, diffusion, and ultimately the nature of binding. In this review, we discuss theoretical and computational approaches that allow the modeling of biological systems to guide and complement experiments and can thus significantly advance the investigation, and possibly the predictions, of protein-protein interactions in the crowded environment of cell cytoplasm. We explore topics such as statistical mechanics for lattice simulations, hydrodynamic interactions, diffusion processes in high-viscosity environments, and several methods based on molecular dynamics simulations. By synergistically leveraging methods from biophysics and computational biology, we review the state of the art of computational methods to study the impact of molecular crowding on protein-protein interactions and discuss its potential revolutionizing effects on the characterization of the human interactome.

Molecular Crowding: The History and Development of a Scientific Paradigm

It is now generally accepted that macromolecules do not act in isolation but "live" in a crowded environment, that is, an environment populated by numerous different molecules. The field of molecular crowding has its origins in the far 80s but became accepted only by the end of the 90s. In the present issue, we discuss various aspects that are influenced by crowding and need to consider its effects. This Review is meant as an introduction to the theme and an analysis of the evolution of the crowding concept through time from colloidal and polymer physics to a more biological perspective. We introduce themes that will be more thoroughly treated in other Reviews of the present issue. In our intentions, each Review may stand by itself, but the complete collection has the aspiration to provide different but complementary perspectives to propose a more holistic view of molecular crowding.

Predicting Polymer Brush Behavior in Solvents Using the Steepest-Entropy-Ascent Quantum Thermodynamic Framework

The steepest-entropy-ascent quantum thermodynamic (SEAQT) framework is utilized to study the effects of temperature on polymer brushes. The brushes are represented by a discrete energy spectrum, and energy degeneracies obtained through the replica-exchange Wang-Landau algorithm. The SEAQT equation of motion is applied to the density of states to establish a unique kinetic path from an initial thermodynamic state to a stable equilibrium state. The kinetic path describes the brush's evolution in state space, as it interacts with a thermal reservoir. The predicted occupation probabilities along the kinetic path are used to determine the expected thermodynamic and structural properties. The polymer density profile of a polystyrene brush in cyclohexane solvent is predicted using the equation of motion, and it agrees qualitatively with the experimental density profiles. The Flory-Huggins parameter chosen to describe brush-solvent interactions affects the solvent distribution in the brush but has a minimal impact on the polymer density profile. Three types of nonequilibrium kinetic paths with differing amounts of entropy production are considered: a heating path, a cooling path, and a heating-cooling path. Properties such as tortuosity, radius of gyration, brush density, solvent density, and brush chain conformations are calculated for each path.

The conformational phase diagram of charged polymers in the presence of attractive bridging crowders

Using extensive molecular dynamics simulations, we obtain the conformational phase diagram of a charged polymer in the presence of oppositely charged counterions and neutral attractive crowders for monovalent, divalent, and trivalent counterion valencies. We demonstrate that the charged polymer can exist in three phases: (1) an extended phase for low charge densities and weak polymer-crowder attractive interactions [Charged Extended (CE)]; (2) a collapsed phase for high charge densities and weak polymer-crowder attractive interactions, primarily driven by counterion condensation [Charged Collapsed due to Intra-polymer interactions [(CCI)]; and (3) a collapsed phase for strong polymer-crowder attractive interactions, irrespective of the charge density, driven by crowders acting as bridges or cross-links [Charged Collapsed due to Bridging interactions [(CCB)]. Importantly, simulations reveal that the interaction with crowders can induce collapse, despite the presence of strong repulsive electrostatic interactions, and can replace condensed counterions to facilitate a direct transition from the CCI and CE phases to the CCB phase.

Coil-globule transition in two-dimensional polymer chains in an explicit solvent

The structure of two-dimensional polymer chains in a solvent at different temperatures is still far from being fully understood. Computer simulations of high-density macromolecular systems require the use of appropriate algorithms, and therefore the simulations were carried out using the Cooperative Motion Algorithm. The polymer model studied was exactly two-dimensional, coarse-grained and based on a triangular lattice. The theta temperature and temperature of coil-to-globule transition, and critical exponents were determined. The differences between the structure of such a disk and that of a chain in a dense polymer liquid were shown.

Liquid structure of bistable responsive macromolecules using mean-field density-functional theory

Macromolecular crowding typically applies to biomolecular and polymer-based systems in which the individual particles often feature a two-state folded/unfolded or coil-to-globule transition, such as found for proteins and peptides, DNA and RNA, or supramolecular polymers. Here, we employ a mean-field density functional theory (DFT) of a model of soft and bistable responsive colloids (RCs) in which the size of the macromolecule is explicitly resolved as a degree of freedom living in a bimodal 'Landau' energy landscape (exhibiting big and small states), thus directly responding to the crowding environment. Using this RC-DFT we study the effects of self-crowding on the liquid bulk structure and thermodynamics for different energy barriers and softnesses of the bimodal energy landscape, in conditions close to the coil-to-globule transition. We find substantial crowding effects on the internal distributions, a complex polydispersity behavior, and quasi-universal compression curves for increasing (generalized) packing fractions. Moreover, we uncover distinct signatures of bimodal versus unimodal behavior in the particle compression. Finally, the analysis of the pair structure - derived from the test particle route - reveals that the microstructure of the liquid is quite inhomogeneous due to local depletion effects, tuneable by particle softness.

The conformational phase diagram of neutral polymers in the presence of attractive crowders

Extensive coarse-grained molecular dynamics simulations are performed to investigate the conformational phase diagram of a neutral polymer in the presence of attractive crowders. We show that, for low crowder densities, the polymer predominantly shows three phases as a function of both intra-polymer and polymer-crowder interactions: (1) weak intra-polymer and weak polymer-crowder attractive interactions induce extended or coil polymer conformations (phase E), (2) strong intra-polymer and relatively weak polymer-crowder attractive interactions induce collapsed or globular conformations (phase CI), and (3) strong polymer-crowder attractive interactions, regardless of intra-polymer interactions, induce a second collapsed or globular conformation that encloses bridging crowders (phase CB). The detailed phase diagram is obtained by determining the phase boundaries delineating the different phases based on an analysis of the radius of gyration as well as bridging crowders. The dependence of the phase diagram on strength of crowder-crowder attractive interactions and crowder density is clarified. We also show that when the crowder density is increased, a third collapsed phase of the polymer emerges for weak intra-polymer attractive interactions. This crowder density-induced compaction is shown to be enhanced by stronger crowder-crowder attraction and is different from the depletion-induced collapse mechanism, which is primarily driven by repulsive interactions. We also provide a unified explanation of the observed re-entrant swollen/extended conformations of the earlier simulations of weak and strongly self-interacting polymers in terms of crowder-crowder attractive interactions.

Predicting non-equilibrium folding behavior of polymer chains using the steepest-entropy-ascent quantum thermodynamic framework

The steepest-entropy-ascent quantum thermodynamic (SEAQT) framework is used to explore the influence of heating and cooling on polymer chain folding kinetics. The framework predicts how a chain moves from an initial non-equilibrium state to stable equilibrium along a unique thermodynamic path. The thermodynamic state is expressed by occupation probabilities corresponding to the levels of a discrete energy landscape. The landscape is generated using the Replica Exchange Wang-Landau method applied to a polymer chain represented by a sequence of hydrophobic and polar monomers with a simple hydrophobic-polar amino acid model. The chain conformation evolves as energy shifts among the levels of the energy landscape according to the principle of steepest entropy ascent. This principle is implemented via the SEAQT equation of motion. The SEAQT framework has the benefit of providing insight into structural properties under non-equilibrium conditions. Chain conformations during heating and cooling change continuously without sharp transitions in morphology. The changes are more drastic along non-equilibrium paths than along quasi-equilibrium paths. The SEAQT-predicted kinetics are fitted to rates associated with the experimental intensity profiles of cytochrome c protein folding with Rouse dynamics.

Confinement free energy for a polymer chain: Corrections to scaling

Spatial confinement of a polymer chain results in a reduction of conformational entropy. For confinement of a flexible N-mer chain in a planar slit or cylindrical pore (confining dimension D), a blob model analysis predicts the asymptotic scaling behavior ΔF/N ∼ D^-γ with γ ≈ 1.70, where ΔF is the free energy increase due to confinement. Here, we extend this scaling analysis to include the variation of local monomer density upon confinement giving ΔF/N ∼ D^-γ(1 - h(N, D)), where the correction-to-scaling term has the form h ∼ D^y/N^Δ with exponents y = 3 - γ ≈ 1.30 and Δ = 3/γ - 1 ≈ 0.76. To test these scaling predictions, we carry out Wang-Landau simulations of confined and unconfined tangent-hard-sphere chains (bead diameter σ) in the presence of a square-well trapping potential. The fully trapped chain provides a common reference state, allowing for an absolute determination of the confinement free energy. Our simulation results for 32 ≤ N ≤ 1024 and 3 ≤ D/σ ≤ 14 are well-described by the extended scaling relation giving exponents of γ = 1.69(1), y = 1.25(2), and Δ = 0.75(6).

Quantification of Entropic Excluded Volume Effects Driving Crowding-Induced Collapse and Folding of a Disordered Protein

We investigate the conformational properties of the intrinsically disordered DNA-binding domain of CytR in the presence of the polymeric crowder polyethylene glycol (PEG). Integrating circular dichroism, nuclear magnetic resonance, and single-molecule Förster resonance energy transfer measurements, we demonstrate that disordered CytR populates a well-folded minor conformation in its native ensemble, while the unfolded ensemble collapses and folds with an increase in crowder density independent of the crowder size. Employing a statistical-mechanical model, the effective reduction in the accessible conformational space of a residue in the unfolded state is estimated to be 10% at 300 mg/mL PEG8000, relative to dilute conditions. The experimentally consistent PEG-temperature phase diagram thus constructed reveals that entropic effects can stabilize disordered CytR by 10 kJ mol^-1, driving the equilibrium toward folded conformations under physiological conditions. Our work highlights the malleable conformational landscape of CytR, the presence of a folded conformation in the disordered ensemble, and proposes a scaling relation for quantifying excluded volume effects on protein stability.

Collapse transition of a heterogeneous polymer in a crowded medium

Long chain molecules can be entropically compacted in a crowded medium. We study the compaction transition of a heterogeneous polymer with ring topology by crowding effects in a free or confined space. For this, we use molecular dynamics simulations in which the effects of crowders are taken into account through effective interactions between chain segments. Our parameter choices are inspired by the Escherichia coli chromosome. The polymer consists of small and big monomers; the big monomers dispersed along the backbone are to mimic the binding of RNA polymerases. Our results show that the compaction transition is a two-step process: initial compaction induced by the association (clustering) of big monomers followed by a gradual overall compaction. They also indicate that cylindrical confinement makes the initial transition more effective; for representative parameter choices, the initial compaction accounts for about 60% reduction in the chain size. Our simulation results support the view that crowding promotes clustering of active transcription units into transcription factories.

Transforming eukaryotic cell culture with macromolecular crowding

In multicellular organisms, the intracellular and extracellular spaces are considerably packed with a diverse range of macromolecular species. Yet, standard eukaryotic cell culture is performed in dilute, and deprived of macromolecules culture media, that barely imitate the density and complex macromolecular composition of tissues. Essentially, we drown cells in a sea of media and then expect them to perform physiologically. Herein, we argue the use of macromolecular crowding (MMC) in eukaryotic cell culture for regenerative medicine and drug discovery purposes.

Stabilizing or Destabilizing: Simulations of Chymotrypsin Inhibitor 2 under Crowding Reveal Existence of a Crossover Temperature

The effect of macromolecular crowding on the stability of proteins can change with temperature. This dependence might reveal a delicate balance between two factors: the entropic excluded volume and the stability-modulating quinary interactions. Here we computationally investigate the thermal stability of the native state of chymotrypsin inhibitor 2 (CI2), which was previously shown by experiments to be destabilized by protein crowders at room temperature. Mimicking experimental conditions, our enhanced-sampling atomistic simulations of CI2 surrounded by lysozyme and bovine serum albumin reproduce this destabilization but also provide evidence of a crossover temperature above which lysozyme is found to become stabilizing, as previously predicted by analysis of thermodynamic data. We relate this crossover to the different CI2-crowder interactions and the local packing experienced by CI2. In fact, we clearly show that the pronounced stabilization induced by lysozyme at high temperatures stems from the tight local packing created around CI2 by this smaller crowder.