Competing Protein-RNA Interaction Networks Control Multiphase Intracellular Organization.

Liquid-liquid phase separation (LLPS) mediates formation of membraneless condensates such as those associated with RNA processing, but the rules that dictate their assembly, substructure, and coexistence with other liquid-like compartments remain elusive. Here, we address the biophysical mechanism of this multiphase organization using quantitative reconstitution of cytoplasmic stress granules (SGs) with attached P-bodies in human cells. Protein-interaction networks can be viewed as interconnected complexes (nodes) of RNA-binding domains (RBDs), whose integrated RNA-binding capacity determines whether LLPS occurs upon RNA influx. Surprisingly, both RBD-RNA specificity and disordered segments of key proteins are non-essential, but modulate multiphase condensation. Instead, stoichiometry-dependent competition between protein networks for connecting nodes determines SG and P-body composition and miscibility, while competitive binding of unconnected proteins disengages networks and prevents LLPS. Inspired by patchy colloid theory, we propose a general framework by which competing networks give rise to compositionally specific and tunable condensates, while relative linkage between nodes underlies multiphase organization.

Accessibility of the Shine-Dalgarno Sequence Dictates N-Terminal Codon Bias in E. coli.

Despite considerable efforts, no physical mechanism has been shown to explain N-terminal codon bias in prokaryotic genomes. Using a systematic study of synonymous substitutions in two endogenous E. coli genes, we show that interactions between the coding region and the upstream Shine-Dalgarno (SD) sequence modulate the efficiency of translation initiation, affecting both intracellular mRNA and protein levels due to the inherent coupling of transcription and translation in E. coli. We further demonstrate that far-downstream mutations can also modulate mRNA levels by occluding the SD sequence through the formation of non-equilibrium secondary structures. By contrast, a non-endogenous RNA polymerase that decouples transcription and translation largely alleviates the effects of synonymous substitutions on mRNA levels. Finally, a complementary statistical analysis of the E. coli genome specifically implicates avoidance of intra-molecular base pairing with the SD sequence. Our results provide general physical insights into the coding-level features that optimize protein expression in prokaryotes.

Self-assembly of photonic crystals by controlling the nucleation and growth of DNA-coated colloids.

DNA-coated colloids can self-assemble into an incredible diversity of crystal structures, but their applications have been limited by poor understanding and control over the crystallization dynamics. To address this challenge, we use microfluidics to quantify the kinetics of DNA-programmed self-assembly along the entire crystallization pathway, from thermally activated nucleation through reaction-limited and diffusion-limited phases of crystal growth. Our detailed measurements of the temperature and concentration dependence of the kinetics at all stages of crystallization provide a stringent test of classical theories of nucleation and growth. After accounting for the finite rolling and sliding rates of micrometer-sized DNA-coated colloids, we show that modified versions of these classical theories predict the absolute nucleation and growth rates with quantitative accuracy. We conclude by applying our model to design and demonstrate protocols for assembling large single crystals with pronounced structural coloration, an essential step in creating next-generation optical metamaterials from colloids.

Competition between Self-Assembly and Phase Separation Governs High-Temperature Condensation of a DNA Liquid.

In many biopolymer solutions, attractive interactions that stabilize finite-sized clusters at low concentrations also promote phase separation at high concentrations. Here we study a model biopolymer system that exhibits the opposite behavior, whereby self-assembly of DNA oligonucleotides into finite-sized, stoichiometric clusters tends to inhibit phase separation. We first use microfluidics-based experiments to map a novel phase transition in which the oligonucleotides condense as the temperature increases at high concentrations of divalent cations. We then show that a theoretical model of competition between self-assembly and phase separation quantitatively predicts changes in experimental phase diagrams arising from DNA sequence perturbations. Our results point to a general mechanism by which self-assembly shapes phase boundaries in complex biopolymer solutions.

Tuning Nucleation Kinetics via Nonequilibrium Chemical Reactions.

Unlike fluids at thermal equilibrium, biomolecular mixtures in living systems can sustain nonequilibrium steady states, in which active processes modify the conformational states of the constituent molecules. Despite qualitative similarities between liquid-liquid phase separation in these systems, the extent to which the phase-separation kinetics differ remains unclear. Here we show that inhomogeneous chemical reactions can alter the nucleation kinetics of liquid-liquid phase separation in a manner that is consistent with classical nucleation theory, but can only be rationalized by introducing a nonequilibrium interfacial tension. We identify conditions under which nucleation can be accelerated without changing the energetics or supersaturation, thus breaking the correlation between fast nucleation and strong driving forces that is typical of phase separation and self-assembly at thermal equilibrium.

Nonequilibrium interfacial properties of chemically driven fluids.

Chemically driven fluids can demix to form condensed droplets that exhibit phase behaviors not observed at equilibrium. In particular, nonequilibrium interfacial properties can emerge when the chemical reactions are driven differentially between the interior and exterior of the phase-separated droplets. Here, we use a minimal model to study changes in the interfacial tension between coexisting phases away from equilibrium. Simulations of both droplet nucleation and interface roughness indicate that the nonequilibrium interfacial tension can either be increased or decreased relative to its equilibrium value, depending on whether the driven chemical reactions are accelerated or decelerated within the droplets. Finally, we show that these observations can be understood using a predictive theory based on an effective thermodynamic equilibrium.

Programmable phase behavior in fluids with designable interactions.

We introduce a method for solving the "inverse" phase equilibria problem: How should the interactions among a collection of molecular species be designed in order to achieve a target phase diagram? Using techniques from convex optimization theory, we show how to solve this problem for phase diagrams containing a large number of components and many coexisting phases with prescribed compositions. We apply our approach to commonly used mean-field models of multicomponent fluids and then use molecular simulations to verify that the designed interactions result in the target phase diagrams. Our approach enables the rational design of "programmable" fluids, such as biopolymer and colloidal mixtures, with complex phase behavior.

Cotranslational folding allows misfolding-prone proteins to circumvent deep kinetic traps.

Many large proteins suffer from slow or inefficient folding in vitro. It has long been known that this problem can be alleviated in vivo if proteins start folding cotranslationally. However, the molecular mechanisms underlying this improvement have not been well established. To address this question, we use an all-atom simulation-based algorithm to compute the folding properties of various large protein domains as a function of nascent chain length. We find that for certain proteins, there exists a narrow window of lengths that confers both thermodynamic stability and fast folding kinetics. Beyond these lengths, folding is drastically slowed by nonnative interactions involving C-terminal residues. Thus, cotranslational folding is predicted to be beneficial because it allows proteins to take advantage of this optimal window of lengths and thus avoid kinetic traps. Interestingly, many of these proteins' sequences contain conserved rare codons that may slow down synthesis at this optimal window, suggesting that synthesis rates may be evolutionarily tuned to optimize folding. Using kinetic modeling, we show that under certain conditions, such a slowdown indeed improves cotranslational folding efficiency by giving these nascent chains more time to fold. In contrast, other proteins are predicted not to benefit from cotranslational folding due to a lack of significant nonnative interactions, and indeed these proteins' sequences lack conserved C-terminal rare codons. Together, these results shed light on the factors that promote proper protein folding in the cell and how biomolecular self-assembly may be optimized evolutionarily.

Self-Assembly of Biomolecular Condensates with Shared Components.

Biomolecular condensates self-assemble when proteins and nucleic acids spontaneously demix to form droplets within the crowded intracellular milieu. This simple mechanism underlies the formation of a wide variety of membraneless compartments in living cells. To understand how multiple condensates with distinct compositions can self-assemble in such a heterogeneous system, which may not be at thermodynamic equilibrium, we study a minimal model in which we can "program" the pairwise interactions among hundreds of species. We show that the number of distinct condensates that can be reliably assembled grows superlinearly with the number of species in the mixture when the condensates share components. Furthermore, we show that we can predict the maximum number of distinct condensates in a mixture without knowing the details of the pairwise interactions. Simulations of condensate growth confirm these predictions and suggest that the physical rules governing the achievable complexity of condensate-mediated spatial organization are broadly applicable to biomolecular mixtures.

Validation of DBFOLD: An efficient algorithm for computing folding pathways of complex proteins.

Atomistic simulations can provide valuable, experimentally-verifiable insights into protein folding mechanisms, but existing ab initio simulation methods are restricted to only the smallest proteins due to severe computational speed limits. The folding of larger proteins has been studied using native-centric potential functions, but such models omit the potentially crucial role of non-native interactions. Here, we present an algorithm, entitled DBFOLD, which can predict folding pathways for a wide range of proteins while accounting for the effects of non-native contacts. In addition, DBFOLD can predict the relative rates of different transitions within a protein's folding pathway. To accomplish this, rather than directly simulating folding, our method combines equilibrium Monte-Carlo simulations, which deploy enhanced sampling, with unfolding simulations at high temperatures. We show that under certain conditions, trajectories from these two types of simulations can be jointly analyzed to compute unknown folding rates from detailed balance. This requires inferring free energies from the equilibrium simulations, and extrapolating transition rates from the unfolding simulations to lower, physiologically-reasonable temperatures at which the native state is marginally stable. As a proof of principle, we show that our method can accurately predict folding pathways and Monte-Carlo rates for the well-characterized Streptococcal protein G. We then show that our method significantly reduces the amount of computation time required to compute the folding pathways of large, misfolding-prone proteins that lie beyond the reach of existing direct simulation. Our algorithm, which is available online, can generate detailed atomistic models of protein folding mechanisms while shedding light on the role of non-native intermediates which may crucially affect organismal fitness and are frequently implicated in disease.

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.

Effect of Protein Structure on Evolution of Cotranslational Folding.

Cotranslational folding depends on the folding speed and stability of the nascent protein. It remains difficult, however, to predict which proteins cotranslationally fold. Here, we simulate evolution of model proteins to investigate how native structure influences evolution of cotranslational folding. We developed a model that connects protein folding during and after translation to cellular fitness. Model proteins evolved improved folding speed and stability, with proteins adopting one of two strategies for folding quickly. Low contact order proteins evolve to fold cotranslationally. Such proteins adopt native conformations early on during the translation process, with each subsequently translated residue establishing additional native contacts. On the other hand, high contact order proteins tend not to be stable in their native conformations until the full chain is nearly extruded. We also simulated evolution of slowly translating codons, finding that slower translation speeds at certain positions enhances cotranslational folding. Finally, we investigated real protein structures using a previously published data set that identified evolutionarily conserved rare codons in Escherichia coli genes and associated such codons with cotranslational folding intermediates. We found that protein substructures preceding conserved rare codons tend to have lower contact orders, in line with our finding that lower contact order proteins are more likely to fold cotranslationally. Our work shows how evolutionary selection pressure can cause proteins with local contact topologies to evolve cotranslational folding.

Evidence of evolutionary selection for cotranslational folding.

Recent experiments and simulations have demonstrated that proteins can fold on the ribosome. However, the extent and generality of fitness effects resulting from cotranslational folding remain open questions. Here we report a genome-wide analysis that uncovers evidence of evolutionary selection for cotranslational folding. We describe a robust statistical approach to identify loci within genes that are both significantly enriched in slowly translated codons and evolutionarily conserved. Surprisingly, we find that domain boundaries can explain only a small fraction of these conserved loci. Instead, we propose that regions enriched in slowly translated codons are associated with cotranslational folding intermediates, which may be smaller than a single domain. We show that the intermediates predicted by a native-centric model of cotranslational folding account for the majority of these loci across more than 500 Escherichia coli proteins. By making a direct connection to protein folding, this analysis provides strong evidence that many synonymous substitutions have been selected to optimize translation rates at specific locations within genes. More generally, our results indicate that kinetics, and not just thermodynamics, can significantly alter the efficiency of self-assembly in a biological context.

Rational design of self-assembly pathways for complex multicomponent structures.

The field of complex self-assembly is moving toward the design of multiparticle structures consisting of thousands of distinct building blocks. To exploit the potential benefits of structures with such "addressable complexity," we need to understand the factors that optimize the yield and the kinetics of self-assembly. Here we use a simple theoretical method to explain the key features responsible for the unexpected success of DNA-brick experiments, which are currently the only demonstration of reliable self-assembly with such a large number of components. Simulations confirm that our theory accurately predicts the narrow temperature window in which error-free assembly can occur. Even more strikingly, our theory predicts that correct assembly of the complete structure may require a time-dependent experimental protocol. Furthermore, we predict that low coordination numbers result in nonclassical nucleation behavior, which we find to be essential for achieving optimal nucleation kinetics under mild growth conditions. We also show that, rather surprisingly, the use of heterogeneous bond energies improves the nucleation kinetics and in fact appears to be necessary for assembling certain intricate 3D structures. This observation makes it possible to sculpt nucleation pathways by tuning the distribution of interaction strengths. These insights not only suggest how to improve the design of structures based on DNA bricks, but also point the way toward the creation of a much wider class of chemical or colloidal structures with addressable complexity.

Phase Transitions in Biological Systems with Many Components.

Biological mixtures such as the cytosol may consist of thousands of distinct components. There is now a substantial body of evidence showing that, under physiological conditions, intracellular mixtures can phase separate into spatially distinct regions with differing compositions. In this article we present numerical evidence indicating that such spontaneous compartmentalization exploits general features of the phase diagram of a multicomponent biomolecular mixture. In particular, we show that demixed domains are likely to segregate when the variance in the intermolecular interaction strengths exceeds a well-defined threshold. Multiple distinct phases are likely to become stable under very similar conditions, which can then be tuned to achieve multiphase coexistence. As a result, only minor adjustments to the composition of the cytosol or the strengths of the intermolecular interactions are needed to regulate the formation of different domains with specific compositions, implying that phase separation is a robust mechanism for creating spatial organization. We further predict that this functionality is only weakly affected by increasing the number of components in the system. Our model therefore suggests that, for purely physico-chemical reasons, biological mixtures are naturally poised to undergo a small number of demixing phase transitions.

Oligomers of Heat-Shock Proteins: Structures That Don't Imply Function.

Most proteins must remain soluble in the cytosol in order to perform their biological functions. To protect against undesired protein aggregation, living cells maintain a population of molecular chaperones that ensure the solubility of the proteome. Here we report simulations of a lattice model of interacting proteins to understand how low concentrations of passive molecular chaperones, such as small heat-shock proteins, suppress thermodynamic instabilities in protein solutions. Given fixed concentrations of chaperones and client proteins, the solubility of the proteome can be increased by tuning the chaperone-client binding strength. Surprisingly, we find that the binding strength that optimizes solubility while preventing irreversible chaperone binding also promotes the formation of weakly bound chaperone oligomers, although the presence of these oligomers does not significantly affect the thermodynamic stability of the solution. Such oligomers are commonly observed in experiments on small heat-shock proteins, but their connection to the biological function of these chaperones has remained unclear. Our simulations suggest that this clustering may not have any essential biological function, but rather emerges as a natural side-effect of optimizing the thermodynamic stability of the proteome.

Structure-Based Prediction of Protein-Folding Transition Paths.

We propose a general theory to describe the distribution of protein-folding transition paths. We show that transition paths follow a predictable sequence of high-free-energy transient states that are separated by free-energy barriers. Each transient state corresponds to the assembly of one or more discrete, cooperative units, which are determined directly from the native structure. We show that the transition state on a folding pathway is reached when a small number of critical contacts are formed between a specific set of substructures, after which folding proceeds downhill in free energy. This approach suggests a natural resolution for distinguishing parallel folding pathways and provides a simple means to predict the rate-limiting step in a folding reaction. Our theory identifies a common folding mechanism for proteins with diverse native structures and establishes general principles for the self-assembly of polymers with specific interactions.

Self-Assembly of Structures with Addressable Complexity.

The self-assembly of structures with "addressable complexity", where every component is distinct and is programmed to occupy a specific location within a target structure, is a promising route to engineering materials with precisely defined morphologies. Because systems with many components are inherently complicated, one might assume that the chances of successful self-assembly are extraordinarily small. Yet recent advances suggest otherwise: addressable structures with hundreds of distinct building blocks have been designed and assembled with nanometer precision. Despite this remarkable success, it is often challenging to optimize a self-assembly reaction to ensure that the intended structure is kinetically accessible. In this Perspective, we focus on the prediction of kinetic pathways for self-assembly and implications for the design of robust experimental protocols. The development of general principles to predict these pathways will enable the engineering of complex materials using a much wider range of building blocks than is currently possible.

Self-assembly protocol design for periodic multicomponent structures.

Assembling molecular superstructures with many distinct components will allow unprecedented control over morphology at the nanoscale. Recently, this approach has been used to assemble periodic structures with precisely defined features, such as repeating arrays of pores and channels, using a large number of building blocks. Here we propose a predictive tool that allows us to optimize the nucleation and growth of unbounded, ordered structures. In what follows, we call these structures 'crystals', even though they may only be periodic in one or two dimensions. We find that the nucleation barriers and growth pathways for crystals consisting of many components exhibit generic features that are very different from those of simple crystals. To illustrate the very non-classical nature of the nucleation and growth of such structures, we study the formation of one and two-dimensional crystals with multicomponent unit cells. We find that, whilst the boundaries in the non-periodic dimensions significantly affect the stabilities of these crystals, the nucleation barriers are largely determined by the local connectivity of the associated bulk crystal and are independent of the number of distinct components in the unit cell. We predict that the self-assembly of crystals with complex morphologies can be made to follow specific pathways toward the target structure that successively incorporate key features of the three-dimensional target structure. In contrast with simple crystals, it is possible to tune the kinetics of nucleation and growth separately, thus minimizing defect formation. We show how control over self-assembly pathways can be used to optimize the kinetics of formation of extended structures with arbitrary nanoscale patterns.

Communication: theoretical prediction of free-energy landscapes for complex self-assembly.

We present a technique for calculating free-energy profiles for the nucleation of multicomponent structures that contain as many species as building blocks. We find that a key factor is the topology of the graph describing the connectivity of the target assembly. By considering the designed interactions separately from weaker, incidental interactions, our approach yields predictions for the equilibrium yield and nucleation barriers. These predictions are in good agreement with corresponding Monte Carlo simulations. We show that a few fundamental properties of the connectivity graph determine the most prominent features of the assembly thermodynamics. Surprisingly, we find that polydispersity in the strengths of the designed interactions stabilizes intermediate structures and can be used to sculpt the free-energy landscape for self-assembly. Finally, we demonstrate that weak incidental interactions can preclude assembly at equilibrium due to the combinatorial possibilities for incorrect association.