The dynamics of association between diffusing and reacting molecular species are routinely quantified using simple rate-equation kinetics that assume both well-mixed concentrations of species and a single rate constant for parameterizing the binding rate. In two-dimensions (2D), however, even when systems are well-mixed, the assumption of a single characteristic rate constant for describing association is not generally accurate, due to the properties of diffusional searching in dimensions d ≤ 2. Establishing rigorous bounds for discriminating between 2D reactive systems that will be accurately described by rate equations with a single rate constant, and those that will not, is critical for both modeling and experimentally parameterizing binding reactions restricted to surfaces such as cellular membranes. We show here that in regimes of intrinsic reaction rate (k a ) and diffusion (D) parameters k a /D > 0.05, a single rate constant cannot be fit to the dynamics of concentrations of associating species independently of the initial conditions. Instead, a more sophisticated multiparametric description than rate-equations is necessary to robustly characterize bimolecular reactions from experiment. Our quantitative bounds derive from our new analysis of 2D rate-behavior predicted from Smoluchowski theory. Using a recently developed single particle reaction-diffusion algorithm we extend here to 2D, we are able to test and validate the predictions of Smoluchowski theory and several other theories of reversible reaction dynamics in 2D for the first time. Finally, our results also mean that simulations of reactive systems in 2D using rate equations must be undertaken with caution when reactions have k a /D > 0.05, regardless of the simulation volume. We introduce here a simple formula for an adaptive concentration dependent rate constant for these chemical kinetics simulations which improves on existing formulas to better capture non-equilibrium reaction dynamics from dilute to dense systems. C 2015 AIP Publishing LLC. [http://dx

The physical underpinnings of fibrosarcoma cell dissemination from a tumor in a surrounding collagen-rich matrix are poorly understood. Here we show that a tumor spheroid embedded in a 3D collagen matrix exerts large contractile forces on the matrix before invasion. Cell invasion is accompanied by complex spatially and temporally dependent patterns of cell migration within and at the surface of the spheroids that are fundamentally different from migratory patterns of individual fibrosarcoma cells homogeneously distributed in the same type of matrix. Cells display a continuous transition from a round morphology at the spheroid core, to highly aligned elongated morphology at the spheroid periphery, which depends on both β1-integrin-based cell-matrix adhesion and myosin II/ROCK-based cell contractility. This isotropic-to-anisotropic transition corresponds to a shift in migration, from a slow and unpolarized movement at the core, to a fast, polarized and persistent one at the periphery. Our results also show that the ensuing collective invasion of fibrosarcoma cells is induced by anisotropic contractile stresses exerted on the surrounding matrix.

Cell division, endocytosis, and viral budding would not function without the localization and assembly of protein complexes on membranes. What is poorly appreciated, however, is that by localizing to membranes, proteins search in a reduced space that effectively drives up concentration. Here we derive an accurate and practical analytical theory to quantify the significance of this dimensionality reduction in regulating protein assembly on membranes. We define a simple metric, an effective equilibrium constant, that allows for quantitative comparison of protein-protein interactions with and without membrane present. To test the importance of membrane localization for driving protein assembly, we collected the protein-protein and protein-lipid affinities, protein and lipid concentrations, and volume-to-surface-area ratios for 46 interactions between 37 membrane-targeting proteins in human and yeast cells. We find that many of the protein-protein interactions between pairs of proteins involved in clathrin-mediated endocytosis in human and yeast cells can experience enormous increases in effective protein-protein affinity (10–1000 fold) due to membrane localization. Localization of binding partners thus triggers robust protein complexation, suggesting that it can play an important role in controlling the timing of endocytic protein coat formation. Our analysis shows that several other proteins involved in membrane remodeling at various organelles have similar potential to exploit localization. The theory highlights the master role of phosphoinositide lipid concentration, the volume-to-surface-area ratio, and the ratio of 3D to 2D equilibrium constants in triggering (or preventing) constitutive assembly on membranes. Our simple model provides a novel quantitative framework for interpreting or designing in vitro experiments of protein complexation influenced by membrane binding.

Conserved residues in protein-protein interfaces correlate with residue hot-spots. To obtain insight into their roles, we have studied their mobility. We have performed 39 explicit solvent simulations of 15 complexes and their monomers, with the interfaces varying in size, shape, and function. The dynamic behavior of conserved residues in unbound monomers illustrates significantly lower flexibility as compared to their environment, suggesting that already before binding they are constrained in a boundlike configuration. To understand this behavior, we have analyzed the inter- and intrachain hydrogen-bond residence-time in the interfaces. We find that conserved residues are not involved significantly in hydrogen bonds across the interface as compared to nonconserved. However, the monomer simulations reveal that conserved residues contribute dominantly to hydrogen-bond formation before binding. Packing of conserved residues across the trajectories is significantly higher before and after the binding, rationalizing their lower mobility. Backbone torsional angle distributions show that conserved residues assume restricted regions of space and the most visited conformations in the bound and unbound trajectories are similar, suggesting that conserved residues are preorganized. Combined with previous studies, we conclude that conserved residues, hot spots, anchor, and interface-buried residues may be similar residues, fulfilling similar roles.

Currently, a significant barrier to building predictive models of cellular self-assembly processes is that molecular models cannot capture minutes-long dynamics that couple distinct components with active processes, whereas reaction-diffusion models cannot capture structures of molecular assembly. Here, we introduce the nonequilibrium reaction-diffusion self-assembly simulator (NERDSS), which addresses this spatiotemporal resolution gap. NERDSS integrates efficient reactiondiffusion algorithms into generalized software that operates on user-defined molecules through diffusion, binding and orientation, unbinding, chemical transformations, and spatial localization. By connecting the fast processes of binding with the slow timescales of large-scale assembly, NERDSS integrates molecular resolution with reversible formation of ordered, multisubunit complexes. NERDSS encodes models using rule-based formatting languages to facilitate model portability, usability, and reproducibility. Applying NERDSS to steps in clathrin-mediated endocytosis, we design multicomponent systems that can form lattices in solution or on the membrane, and we predict how stochastic but localized dephosphorylation of membrane lipids can drive lattice disassembly. The NERDSS simulations reveal the spatial constraints on lattice growth and the role of membrane localization and cooperativity in nucleating assembly. By modeling viral lattice assembly and recapitulating oscillations in protein expression levels for a circadian clock model, we illustrate the adaptability of NERDSS. NERDSS simulates user-defined assembly models that were previously inaccessible to existing software tools, with broad applications to predicting self-assembly in vivo and designing high-yield assemblies in vitro.