A streamlined molecular-dynamics workflow for computing solubilities of molecular and ionic crystals.

Computing the solubility of crystals in a solvent using atomistic simulations is notoriously challenging due to the complexities and convergence issues associated with free-energy methods, as well as the slow equilibration in direct-coexistence simulations. This paper introduces a molecular-dynamics workflow that simplifies and robustly computes the solubility of molecular or ionic crystals. This method is considerably more straightforward than the state-of-the-art, as we have streamlined and optimised each step of the process. Specifically, we calculate the chemical potential of the crystal using the gas-phase molecule as a reference state, and employ the S0 method to determine the concentration dependence of the chemical potential of the solute. We use this workflow to predict the solubilities of sodium chloride in water, urea polymorphs in water, and paracetamol polymorphs in both water and ethanol. Our findings indicate that the predicted solubility is sensitive to the chosen potential energy surface. Furthermore, we note that the harmonic approximation often fails for both molecular crystals and gas molecules at or above room temperature, and that the assumption of an ideal solution becomes less valid for highly soluble substances.

Quantitative real-time in-cell imaging reveals heterogeneous clusters of proteins prior to condensation.

Our current understanding of biomolecular condensate formation is largely based on observing the final near-equilibrium condensate state. Despite expectations from classical nucleation theory, pre-critical protein clusters were recently shown to form under subsaturation conditions in vitro; if similar long-lived clusters comprising more than a few molecules are also present in cells, our understanding of the physical basis of biological phase separation may fundamentally change. Here, we combine fluorescence microscopy with photobleaching analysis to quantify the formation of clusters of NELF proteins in living, stressed cells. We categorise small and large clusters based on their dynamics and their response to p38 kinase inhibition. We find a broad distribution of pre-condensate cluster sizes and show that NELF protein cluster formation can be explained as non-classical nucleation with a surprisingly flat free-energy landscape for a wide range of sizes and an inhibition of condensation in unstressed cells.

Aromatic and arginine content drives multiphasic condensation of protein-RNA mixtures.

Multiphasic architectures are found ubiquitously in biomolecular condensates and are thought to have important implications for the organization of multiple chemical reactions within the same compartment. Many of these multiphasic condensates contain RNA in addition to proteins. Here, we investigate the importance of different interactions in multiphasic condensates comprising two different proteins and RNA using computer simulations with a residue-resolution coarse-grained model of proteins and RNA. We find that in multilayered condensates containing RNA in both phases, protein-RNA interactions dominate, with aromatic residues and arginine forming the key stabilizing interactions. The total aromatic and arginine content of the two proteins must be appreciably different for distinct phases to form, and we show that this difference increases as the system is driven toward greater multiphasicity. Using the trends observed in the different interaction energies of this system, we demonstrate that we can also construct multilayered condensates with RNA preferentially concentrated in one phase. The "rules" identified can thus enable the design of synthetic multiphasic condensates to facilitate further study of their organization and function.

Thermodynamic origins of two-component multiphase condensates of proteins.

Intracellular condensates are highly multi-component systems in which complex phase behaviour can ensue, including the formation of architectures comprising multiple immiscible condensed phases. Relying solely on physical intuition to manipulate such condensates is difficult because of the complexity of their composition, and systematically learning the underlying rules experimentally would be extremely costly. We address this challenge by developing a computational approach to design pairs of protein sequences that result in well-separated multilayered condensates and elucidate the molecular origins of these compartments. Our method couples a genetic algorithm to a residue-resolution coarse-grained protein model. We demonstrate that we can design protein partners to form multiphase condensates containing naturally occurring proteins, such as the low-complexity domain of hnRNPA1 and its mutants, and show how homo- and heterotypic interactions must differ between proteins to result in multiphasicity. We also show that in some cases the specific pattern of amino-acid residues plays an important role. Our findings have wide-ranging implications for understanding and controlling the organisation, functions and material properties of biomolecular condensates.

Phase diagrams-Why they matter and how to predict them.

Understanding the thermodynamic stability and metastability of materials can help us to, for example, gauge whether crystalline polymorphs in pharmaceutical formulations are likely to be durable. It can also help us to design experimental routes to novel phases with potentially interesting properties. In this Perspective, we provide an overview of how thermodynamic phase behavior can be quantified both in computer simulations and machine-learning approaches to determine phase diagrams, as well as combinations of the two. We review the basic workflow of free-energy computations for condensed phases, including some practical implementation advice, ranging from the Frenkel-Ladd approach to thermodynamic integration and to direct-coexistence simulations. We illustrate the applications of such methods on a range of systems from materials chemistry to biological phase separation. Finally, we outline some challenges, questions, and practical applications of phase-diagram determination which we believe are likely to be possible to address in the near future using such state-of-the-art free-energy calculations, which may provide fundamental insight into separation processes using multicomponent solvents.

Thermodynamics of high-pressure ice phases explored with atomistic simulations.

Most experimentally known high-pressure ice phases have a body-centred cubic (bcc) oxygen lattice. Our large-scale molecular-dynamics simulations with a machine-learning potential indicate that, amongst these bcc ice phases, ices VII, VII' and X are the same thermodynamic phase under different conditions, whereas superionic ice VII″ has a first-order phase boundary with ice VII'. Moreover, at about 300 GPa, the transformation between ice X and the Pbcm phase has a sharp structural change but no apparent activation barrier, whilst at higher pressures the barrier gradually increases. Our study thus clarifies the phase behaviour of the high-pressure ices and reveals peculiar solid-solid transition mechanisms not known in other systems.

Simulations of DNA-Origami Self-Assembly Reveal Design-Dependent Nucleation Barriers.

Nucleation is the rate-determining step in the kinetics of many self-assembly processes. However, the importance of nucleation in the kinetics of DNA-origami self-assembly, which involves both the binding of staple strands and the folding of the scaffold strand, is unclear. Here, using Monte Carlo simulations of a lattice model of DNA origami, we find that some, but not all, designs can have a nucleation barrier and that this barrier disappears at lower temperatures, rationalizing the success of isothermal assembly. We show that the height of the nucleation barrier depends primarily on the coaxial stacking of staples that are adjacent on the same helix, a parameter that can be modified with staple design. Creating a nucleation barrier to DNA-origami assembly could be useful in optimizing assembly times and yields, while eliminating the barrier may allow for fast molecular sensors that can assemble/disassemble without hysteresis in response to changes in the environment.

Targeted modulation of protein liquid-liquid phase separation by evolution of amino-acid sequence.

Rationally and efficiently modifying the amino-acid sequence of proteins to control their ability to undergo liquid-liquid phase separation (LLPS) on demand is not only highly desirable, but can also help to elucidate which protein features are important for LLPS. Here, we propose a computational method that couples a genetic algorithm to a sequence-dependent coarse-grained protein model to evolve the amino-acid sequences of phase-separating intrinsically disordered protein regions (IDRs), and purposely enhance or inhibit their capacity to phase-separate. We validate the predicted critical solution temperatures of the mutated sequences with ABSINTH, a more accurate all-atom model. We apply the algorithm to the phase-separating IDRs of three naturally occurring proteins, namely FUS, hnRNPA1 and LAF1, as prototypes of regions that exist in cells and undergo homotypic LLPS driven by different types of intermolecular interaction, and we find that the evolution of amino-acid sequences towards enhanced LLPS is driven in these three cases, among other factors, by an increase in the average size of the amino acids. However, the direction of change in the molecular driving forces that enhance LLPS (such as hydrophobicity, aromaticity and charge) depends on the initial amino-acid sequence. Finally, we show that the evolution of amino-acid sequences to modulate LLPS is strongly coupled to the make-up of the medium (e.g. the presence or absence of RNA), which may have significant implications for our understanding of phase separation within the many-component mixtures of biological systems.

Quantum-mechanical exploration of the phase diagram of water.

The set of known stable phases of water may not be complete, and some of the phase boundaries between them are fuzzy. Starting from liquid water and a comprehensive set of 50 ice structures, we compute the phase diagram at three hybrid density-functional-theory levels of approximation, accounting for thermal and nuclear fluctuations as well as proton disorder. Such calculations are only made tractable because we combine machine-learning methods and advanced free-energy techniques. The computed phase diagram is in qualitative agreement with experiment, particularly at pressures ≲ 8000 bar, and the discrepancy in chemical potential is comparable with the subtle uncertainties introduced by proton disorder and the spread between the three hybrid functionals. None of the hypothetical ice phases considered is thermodynamically stable in our calculations, suggesting the completeness of the experimental water phase diagram in the region considered. Our work demonstrates the feasibility of predicting the phase diagram of a polymorphic system from first principles and provides a thermodynamic way of testing the limits of quantum-mechanical calculations.

Physics-driven coarse-grained model for biomolecular phase separation with near-quantitative accuracy.

Various physics- and data-driven sequence-dependent protein coarse-grained models have been developed to study biomolecular phase separation and elucidate the dominant physicochemical driving forces. Here, we present Mpipi, a multiscale coarse-grained model that describes almost quantitatively the change in protein critical temperatures as a function of amino-acid sequence. The model is parameterised from both atomistic simulations and bioinformatics data and accounts for the dominant role of π-π and hybrid cation-π/π-π interactions and the much stronger attractive contacts established by arginines than lysines. We provide a comprehensive set of benchmarks for Mpipi and seven other residue-level coarse-grained models against experimental radii of gyration and quantitative in-vitro phase diagrams; Mpipi predictions agree well with experiment on both fronts. Moreover, it can account for protein-RNA interactions, correctly predicts the multiphase behaviour of a charge-matched poly-arginine/poly-lysine/RNA system, and recapitulates experimental LLPS trends for sequence mutations on FUS, DDX4 and LAF-1 proteins.

Numerical calculation of free-energy barriers for entangled polymer nucleation.

The crystallization of entangled polymers from their melt is investigated using computer simulation with a coarse-grained model. Using hybrid Monte Carlo simulations enables us to probe the behavior of long polymer chains. We identify solid-like beads with a centrosymmetry local order parameter and compute the nucleation free-energy barrier at relatively high supercooling with adaptive-bias windowed umbrella sampling. Our results demonstrate that the critical nucleus sizes and the heights of free-energy barriers do not significantly depend on the molecular weight of the polymer; however, the nucleation rate decreases with the increase in molecular weight. Moreover, an analysis of the composition of the critical nucleus suggests that intra-molecular growth of the nucleated cluster does not contribute significantly to crystallization for this system.

Predicting the phase diagram of titanium dioxide with random search and pattern recognition.

Predicting phase stabilities of crystal polymorphs is central to computational materials science and chemistry. Such predictions are challenging because they first require searching for potential energy minima and then performing arduous free-energy calculations to account for entropic effects at finite temperatures. Here, we develop a framework that facilitates such predictions by exploiting all the information obtained from random searches of crystal structures. This framework combines automated clustering, classification and visualisation of crystal structures with machine-learning estimation of their enthalpy and entropy. We demonstrate the framework on the technologically important system of TiO2, which has many polymorphs, without relying on prior knowledge of known phases. We find a number of new phases and predict the phase diagram and metastabilities of crystal polymorphs at 1600 K, benchmarking the results against full free-energy calculations.

Programming the Nucleation of DNA Brick Self-Assembly with a Seeding Strand.

Recently, the DNA brick strategy has provided a highly modular and scalable approach for the construction of complex structures, which can be used as nanoscale pegboards for the precise organization of molecules and nanoparticles for many applications. Despite the dramatic increase of structural complexity provided by the DNA brick method, the assembly pathways are still poorly understood. Herein, we introduce a "seed" strand to control the crucial nucleation and assembly pathway in DNA brick assembly. Through experimental studies and computer simulations, we successfully demonstrate that the regulation of the assembly pathways through seeded growth can accelerate the assembly kinetics and increase the optimal temperature by circa 4-7 °C for isothermal assembly. By improving our understanding of the assembly pathways, we provide new guidelines for the design of programmable pathways to improve the self-assembly of DNA nanostructures.

Microscopic analysis of thermo-orientation in systems of off-centre Lennard-Jones particles.

When fluids of anisotropic molecules are placed in temperature gradients, the molecules may align themselves along the gradient: this is called thermo-orientation. We discuss the theory of this effect in a fluid of particles that interact by a spherically symmetric potential, where the particles' centres of mass do not coincide with their interaction centres. Starting from the equations of motion of the molecules, we show how a simple assumption of local equipartition of energy can be used to predict the thermo-orientation effect, recovering the result of Wirnsberger et al. [Phys. Rev. Lett. 120, 226001 (2018)]. Within this approach, we show that for particles with a single interaction centre, the thermal centre of the molecule must coincide with the interaction centre. The theory also explains the coupling between orientation and kinetic energy that is associated with this non-Boltzmann distribution. We discuss deviations from this local equipartition assumption, showing that these can occur in linear response to a temperature gradient. We also present numerical simulations showing significant deviations from the local equipartition predictions, which increase as the centre of mass of the molecule is displaced further from its interaction centre.

Lattice models and Monte Carlo methods for simulating DNA origami self-assembly.

The optimal design of DNA origami systems that assemble rapidly and robustly is hampered by the lack of a model for self-assembly that is sufficiently detailed yet computationally tractable. Here, we propose a model for DNA origami that strikes a balance between these two criteria by representing these systems on a lattice at the level of binding domains. The free energy of hybridization between individual binding domains is estimated with a nearest-neighbour model. Double helical segments are treated as being rigid, but we allow flexibility at points where the backbone of one of the strands is interrupted, which provides a reasonably realistic representation of partially and fully assembled states. Particular attention is paid to the constraints imposed by the double helical twist, as they determine where strand crossovers between adjacent helices can occur. To improve the efficiency of sampling configuration space, we develop Monte Carlo methods for sampling scaffold conformations in near-assembled states, and we carry out simulations in the grand canonical ensemble, enabling us to avoid considering states with unbound staples. We demonstrate that our model can quickly sample assembled configurations of a small origami design previously studied with the oxDNA model, as well as a design with staples that span longer segments of the scaffold. The sampling ability of our method should allow for good statistics to be obtained when studying the assembly pathways and is suited to investigating, in particular, the effects of design and assembly conditions on these pathways and their resulting final assembled structures.

Direct observation and rational design of nucleation behavior in addressable self-assembly.

To optimize a self-assembly reaction, it is essential to understand the factors that govern its pathway. Here, we examine the influence of nucleation pathways in a model system for addressable, multicomponent self-assembly based on a prototypical "DNA-brick" structure. By combining temperature-dependent dynamic light scattering and atomic force microscopy with coarse-grained simulations, we show how subtle changes in the nucleation pathway profoundly affect the yield of the correctly formed structures. In particular, we can increase the range of conditions over which self-assembly occurs by using stable multisubunit clusters that lower the nucleation barrier for assembling subunits in the interior of the structure. Consequently, modifying only a small portion of a structure is sufficient to optimize its assembly. Due to the generality of our coarse-grained model and the excellent agreement that we find with our experimental results, the design principles reported here are likely to apply generically to addressable, multicomponent self-assembly.

Investigating the role of boundary bricks in DNA brick self-assembly.

In the standard DNA brick set-up, distinct 32-nucleotide strands of single-stranded DNA are each designed to bind specifically to four other such molecules. Experimentally, it has been demonstrated that the overall yield is increased if certain bricks which occur on the outer faces of target structures are merged with adjacent bricks. However, it is not well understood by what mechanism such 'boundary bricks' increase the yield, as they likely influence both the nucleation process and the final stability of the target structure. Here, we use Monte Carlo simulations with a patchy particle model of DNA bricks to investigate the role of boundary bricks in the self-assembly of complex multicomponent target structures. We demonstrate that boundary bricks lower the free-energy barrier to nucleation and that boundary bricks on edges stabilize the final structure. However, boundary bricks are also more prone to aggregation, as they can stabilize partially assembled intermediates. We explore some design strategies that permit us to benefit from the stabilizing role of boundary bricks whilst minimizing their ability to hinder assembly; in particular, we show that maximizing the total number of boundary bricks is not an optimal strategy.

Self-assembly of two-dimensional binary quasicrystals: a possible route to a DNA quasicrystal.

We use Monte Carlo simulations and free-energy techniques to show that binary solutions of penta- and hexavalent two-dimensional patchy particles can form thermodynamically stable quasicrystals even at very narrow patch widths, provided their patch interactions are chosen in an appropriate way. Such patchy particles can be thought of as a coarse-grained representation of DNA multi-arm 'star' motifs, which can be chosen to bond with one another very specifically by tuning the DNA sequences of the protruding arms. We explore several possible design strategies and conclude that DNA star tiles that are designed to interact with one another in a specific but not overly constrained way could potentially be used to construct soft quasicrystals in experiment. We verify that such star tiles can form stable dodecagonal motifs using oxDNA, a realistic coarse-grained model of DNA.

DNA brick self-assembly with an off-lattice potential.

We report Monte Carlo simulations of a simple off-lattice patchy-particle model for DNA 'bricks'. We relate the parameters that characterise this model with the binding free energy of pairs of single-stranded DNA molecules. We verify that an off-lattice potential parameterised in this way reproduces much of the behaviour seen with a simpler lattice model we introduced previously, although the relaxation of the geometric constraints leads to a more error-prone self-assembly pathway. We investigate the self-assembly process as a function of the strength of the non-specific interactions. We show that our off-lattice model for DNA bricks results in robust self-assembly into a variety of target structures.

Effects of co-ordination number on the nucleation behaviour in many-component self-assembly.

We report canonical and grand-canonical lattice Monte Carlo simulations of the self-assembly of addressable structures comprising hundreds of distinct component types. The nucleation behaviour, in the form of free-energy barriers to nucleation, changes significantly as the co-ordination number of the building blocks is changed from 4 to 8 to 12. Unlike tetrahedral structures - which roughly correspond to DNA bricks that have been studied in experiments - the shapes of the free-energy barriers of higher co-ordination structures depend strongly on the supersaturation, and such structures require a very significant driving force for structure growth before nucleation becomes thermally accessible. Although growth at high supersaturation results in more defects during self-assembly, we show that high co-ordination number structures can still be assembled successfully in computer simulations and that they exhibit self-assembly behaviour analogous to DNA bricks. In particular, the self-assembly remains modular, enabling in principle a wide variety of nanostructures to be assembled, with a greater spatial resolution than is possible in low co-ordination structures.