A unified theory of free energy functionals and applications to diffusion.

SignificanceThe free energy functional is a central component of continuum dynamical models used to describe phase transitions, microstructural evolution, and pattern formation. However, despite the success of these models in many areas of physics, chemistry, and biology, the standard free energy frameworks are frequently characterized by physically opaque parameters and incorporate assumptions that are difficult to assess. Here, we introduce a mathematical formalism that provides a unifying umbrella for constructing free energy functionals. We show that Ginzburg-Landau framework is a special case of this umbrella and derive a generalization of the widely employed Cahn-Hilliard equation. More broadly, we expect the framework will also be useful for generalizing higher-order theories, establishing formal connections to microscopic physics, and coarse graining.

Thrombin spatial distribution determines protein C activation during hemostasis and thrombosis.

Rebalancing the hemostatic system by targeting endogenous anticoagulant pathways, like the protein C (PC) system, is being tested as a means of improving hemostasis in patients with hemophilia. Recent intravital studies of hemostasis demonstrated that, in some vascular contexts, thrombin activity is sequestered in the extravascular compartment. These findings raise important questions about the context-dependent contribution of activated PC (APC) to the hemostatic response, because PC activation occurs on the surface of endothelial cells. We used a combination of pharmacologic, genetic, imaging, and computational approaches to examine the relationships among thrombin spatial distribution, PC activation, and APC anticoagulant function. We found that inhibition of APC activity, in mice either harboring the factor V Leiden mutation or infused with an APC-blocking antibody, significantly enhanced fibrin formation and platelet activation in a microvascular injury model, consistent with the role of APC as an anticoagulant. In contrast, inhibition of APC activity had no effect on hemostasis after penetrating injury of the mouse jugular vein. Computational studies showed that differences in blood velocity, injury size, and vessel geometry determine the localization of thrombin generation and, consequently, the extent of PC activation. Computational predictions were tested in vivo and showed that when thrombin generation occurred intravascularly, without penetration of the vessel wall, inhibition of APC significantly increased fibrin formation in the jugular vein. Together, these studies show the importance of thrombin spatial distribution in determining PC activation during hemostasis and thrombosis.

Multiparticle collision dynamics simulations of hydrodynamic interactions in colloidal suspensions: How well does the discrete particle approach do at short range?

The multiparticle collision dynamics (MPCD) simulation method is an attractive technique for studying the effects of hydrodynamic interactions in colloidal suspensions because of its flexibility, computational efficiency, and ease of implementation. Here, we analyze an extension of the basic MPCD method in which colloidal particles are discretized with a surface mesh of sensor nodes/particles that interact with solvent particles (MPCD + Discrete Particle or MPCD + DP). We use several situations that have been described analytically to probe the impact of colloidal particle mesh resolution on the ability of the MPCD + DP method to resolve short-ranged hydrodynamic interactions, which are important in crowded suspensions and especially in self-assembling systems that create high volume fraction phases. Specifically, we consider (A) hard-sphere diffusion near a wall, (B) two-particle diffusion, (C) hard-sphere diffusion in crowded suspensions, and (D) the dynamics of aggregation in an attractive colloidal suspension. We show that in each case, the density of sensor nodes plays a significant role in the accuracy of the simulation and that a surprisingly high number of surface nodes are needed to fully capture hydrodynamic interactions.

A three-dimensional multiscale model for the prediction of thrombus growth under flow with single-platelet resolution.

Modeling thrombus growth in pathological flows allows evaluation of risk under patient-specific pharmacological, hematological, and hemodynamical conditions. We have developed a 3D multiscale framework for the prediction of thrombus growth under flow on a spatially resolved surface presenting collagen and tissue factor (TF). The multiscale framework is composed of four coupled modules: a Neural Network (NN) that accounts for platelet signaling, a Lattice Kinetic Monte Carlo (LKMC) simulation for tracking platelet positions, a Finite Volume Method (FVM) simulator for solving convection-diffusion-reaction equations describing agonist release and transport, and a Lattice Boltzmann (LB) flow solver for computing the blood flow field over the growing thrombus. A reduced model of the coagulation cascade was embedded into the framework to account for TF-driven thrombin production. The 3D model was first tested against in vitro microfluidics experiments of whole blood perfusion with various antiplatelet agents targeting COX-1, P2Y1, or the IP receptor. The model was able to accurately capture the evolution and morphology of the growing thrombus. Certain problems of 2D models for thrombus growth (artifactual dendritic growth) were naturally avoided with realistic trajectories of platelets in 3D flow. The generalizability of the 3D multiscale solver enabled simulations of important clinical situations, such as cylindrical blood vessels and acute flow narrowing (stenosis). Enhanced platelet-platelet bonding at pathologically high shear rates (e.g., von Willebrand factor unfolding) was required for accurately describing thrombus growth in stenotic flows. Overall, the approach allows consideration of patient-specific platelet signaling and vascular geometry for the prediction of thrombotic episodes.

Hydrodynamic and frictional modulation of deformations in switchable colloidal crystallites.

Displacive transformations in colloidal crystals may offer a pathway for increasing the diversity of accessible configurations without the need to engineer particle shape or interaction complexity. To date, binary crystals composed of spherically symmetric particles at specific size ratios have been formed that exhibit floppiness and facile routes for transformation into more rigid structures that are otherwise not accessible by direct nucleation and growth. There is evidence that such transformations, at least at the micrometer scale, are kinetically influenced by concomitant solvent motion that effectively induces hydrodynamic correlations between particles. Here, we study quantitatively the impact of such interactions on the transformation of binary bcc-CsCl analog crystals into close-packed configurations. We first employ principal-component analysis to stratify the explorations of a bcc-CsCl crystallite into orthogonal directions according to displacement. We then compute diffusion coefficients along the different directions using several dynamical models and find that hydrodynamic correlations, depending on their range, can either enhance or dampen collective particle motions. These two distinct effects work synergistically to bias crystallite deformations toward a subset of the available outcomes.

Shear-driven rolling of DNA-adhesive microspheres.

Many biologically important cell binding processes, such as the rolling of leukocytes in the vasculature, are multivalent, being mediated by large numbers of weak binding ligands. Quantitative agreement between experiments and models of rolling has been elusive and often limited by the poor understanding of the binding and unbinding kinetics of the ligands involved. Here, we present a cell-free experimental model for such rolling, consisting of polymer microspheres whose adhesion to a glass surface is mediated by ligands with well-understood force-dependent binding free energy-short complementary DNA strands. We observe robust rolling activity for certain values of the shear rate and the grafted DNA strands' binding free energy and force sensitivity. The simulation framework developed to model leukocyte rolling, adhesive dynamics, quantitatively captures the mean rolling velocity and lateral diffusivity of the experimental particles using known values of the experimental parameters. Moreover, our model captures the velocity variations seen within the trajectories of single particles. Particle-to-particle variations can be attributed to small, plausible differences in particle characteristics. Overall, our findings confirm that state-of-the-art adhesive dynamics simulations are able to capture the complex physics of particle rolling, boding well for their extension to modeling more complex systems of rolling cells.

Coagulopathy implications using a multiscale model of traumatic bleeding matching macro- and microcirculation.

Quantifying the relationship between vascular injury and the dynamic bleeding rate requires a multiscale model that accounts for changing and coupled hemodynamics between the global and microvascular levels. A lumped, global hemodynamic model of the human cardiovascular system with baroreflex control was coupled to a local 24-level bifurcating vascular network that spanned diameters from the muscular artery scale (0.1-1.3 mm) to capillaries (5-10 µm) via conservation of momentum and conservation of mass boundary conditions. For defined injuries of severing all vessels at each nth-level, the changing pressures and flowrates were calculated using prescribed shear-dependent hemostatic clot growth rates (normal or coagulopathic). Key results were as follows: 1) the upstream vascular network rapidly depressurizes to reduce blood loss; 2) wall shear rates at the hemorrhaging wound exit are sufficiently high (~10,000 s-1) to drive von Willebrand factor unfolding; 3) full coagulopathy results in >2-liter blood loss in 2 h for severing all vessels of 0.13- to 0.005-mm diameter within the bifurcating network, whereas full hemostasis limits blood loss to <100 ml within 2 min; and 4) hemodilution from transcapillary refill increases blood loss and could be implicated in trauma-induced coagulopathy. A sensitivity analysis on length-to-diameter ratio and branching exponent demonstrated that bleeding was strongly dependent on these tissue-dependent network parameters. This is the first bleeding model that prescribes the geometry of the injury to calculate the rate of pressure-driven blood loss and local wall shear rate in the presence or absence of coagulopathic blood. NEW & NOTEWORTHY We developed a multiscale model that couples a lumped, global hemodynamic model of a patient to resolved, single-vessel wounds ranging from the small artery to capillary scale. The model is able to quantify wall shear rates, seal rates, and blood loss rates in the presence and absence of baroreflex control and hemodilution.

Carbon solubility in liquid silicon: A computational analysis across empirical potentials.

The nucleation and growth of SiC precipitates in liquid silicon is important in the crystallization of silicon used for the photovoltaic industry. These processes depend strongly on the carbon concentration as well as the equilibrium solubility relative to the precipitate phase. Here, using a suite of statistical thermodynamic techniques, we calculate the solubility of carbon atoms in liquid silicon relative to the ß-SiC phase. We employ several available empirical potentials to assess whether these potentials may reasonably be used to computationally analyze SiC precipitation. We find that some of the Tersoff-type potentials provide an excellent picture for carbon solubility in liquid silicon but, because of their severe silicon melting point overestimation, are limited to high temperatures where the carbon solubility is several percent, a value that is irrelevant for typical solidification conditions. Based on chemical potential calculations for pure silicon, we suggest that this well-known issue is confined to the description of the liquid phase and demonstrate that some recent potential models for silicon might address this weakness while preserving the excellent description of the carbon-silicon interaction found in the existing models.

Dynamics of Thrombin Generation and Flux from Clots during Whole Human Blood Flow over Collagen/Tissue Factor Surfaces.

Coagulation kinetics are well established for purified blood proteases or human plasma clotting isotropically. However, less is known about thrombin generation kinetics and transport within blood clots formed under hemodynamic flow. Using microfluidic perfusion (wall shear rate, 200 s-1) of corn trypsin inhibitor-treated whole blood over a 250-µm long patch of type I fibrillar collagen/lipidated tissue factor (TF; ∼1 TF molecule/µm2), we measured thrombin released from clots using thrombin-antithrombin immunoassay. The majority (>85%) of generated thrombin was captured by intrathrombus fibrin as thrombin-antithrombin was largely undetectable in the effluent unless Gly-Pro-Arg-Pro (GPRP) was added to block fibrin polymerization. With GPRP present, the flux of thrombin increased to ∼0.5 × 10-12 nmol/µm2-s over the first 500 s of perfusion and then further increased by ∼2-3-fold over the next 300 s. The increased thrombin flux after 500 s was blocked by anti-FXIa antibody (O1A6), consistent with thrombin-feedback activation of FXI. Over the first 500 s, ∼92,000 molecules of thrombin were generated per surface TF molecule for the 250-µm-long coating. A single layer of platelets (obtained with αIIbß3 antagonism preventing continued platelet deposition) was largely sufficient for thrombin production. Also, the overall thrombin-generating potential of a 1000-µm-long coating became less efficient on a per µm2 basis, likely due to distal boundary layer depletion of platelets. Overall, thrombin is robustly generated within clots by the extrinsic pathway followed by late-stage FXIa contributions, with fibrin localizing thrombin via its antithrombin-I activity as a potentially self-limiting hemostatic mechanism.

Interaction Heterogeneity can Favorably Impact Colloidal Crystal Nucleation.

Colloidal particles with short-ranged attractions, e.g., micron-scale spheres functionalized with single-stranded DNA oligomers, are susceptible to becoming trapped in disordered configurations even when a crystalline arrangement is the ground state. Moreover, for reasons that are not well understood, seemingly minor variations in the particle formulation can lead to dramatic changes in the crystallization outcome. We demonstrate, using a combination of equilibrium and nonequilibrium computer simulations, that interaction heterogeneity-variations in the energetic interactions among different particle pairs in the population-may favorably impact crystal nucleation. Specifically, interaction heterogeneity is found to lower the free energy barrier to nucleation via the formation of clusters comprised preferentially of strong-binding particle pairs. Moreover, gelation is inhibited by "spreading out over time" the nucleation process, resulting in a reduced density of stable nuclei, allowing each to grow unhindered and larger. Our results suggest a simple and robust approach for enhancing colloidal crystallization near the "sticky sphere" limit, and support the notion that differing extents of interaction heterogeneity arising from various particle functionalization protocols may contribute to the otherwise unexplained variations in crystallization outcomes reported in the literature.

Self-assembly with colloidal clusters: facile crystal design using connectivity landscape analysis.

Recent experimental and theoretical studies demonstrate that prefabricated micron-scale colloidal clusters functionalized with DNA oligomers offer a practical way for introducing anisotropic interactions, significantly extending the scope of DNA-mediated colloidal assembly, and enabling the formation of interesting crystalline superstructures that are otherwise inaccessible with short-ranged, spherically symmetric interactions. However, it is apparent that the high-dimensional parameter space that defines the geometric and interaction properties of such systems poses an obstacle to assembly design and optimization. Here, we present a geometrical analysis that generates connectivity landscapes for target superstructures, greatly reducing the space over which subsequent experimental trials must search. We focus on several superstructures that are assembled from binary systems comprised of 'merged' or 'sintered' tetrahedral clusters and single spheres. We also validate and extend the analytical constraint approach with direct MD simulations of superstructure nucleation and growth.

Hydrodynamics selects the pathway for displacive transformations in DNA-linked colloidal crystallites.

The degree to which DNA-linked particle crystals, particularly those composed of micrometer-scale colloids, are able to dynamically evolve or whether they are kinetically arrested after formation remains poorly understood. Here, we study a recently observed displacive transformation in colloidal binary superlattice crystals, whereby a body-centered cubic to face-centered cubic transformation is found to proceed spontaneously under some annealing conditions. Using a comprehensive suite of computer simulation tools, we develop a framework for analyzing the many displacive transformation pathways corresponding to distinct, but energetically degenerate, random hexagonal close-packed end states. Due to the short-ranged, spherically symmetric nature of the particle interactions the pathways are all barrierless, suggesting that all end states should be equally likely. Instead, we find that hydrodynamic correlations between particles result in anisotropic mobility along the various possible displacive pathways, strongly selecting for pathways that lead to the fcc-CuAu-I configuration, explaining recent experimental observations. This finding may provide clues for discovering new approaches for controlling structure in this emerging class of materials.

A systems approach to hemostasis: 2. Computational analysis of molecular transport in the thrombus microenvironment.

Hemostatic thrombi formed after a penetrating injury have a heterogeneous architecture in which a core of highly activated, densely packed platelets is covered by a shell of less-activated, loosely packed platelets. In the first manuscript in this series, we show that regional differences in intrathrombus protein transport rates emerge early in the hemostatic response and are preserved as the thrombus develops. Here, we use a theoretical approach to investigate this process and its impact on agonist distribution. The results suggest that hindered diffusion, rather than convection, is the dominant mechanism responsible for molecular movement within the thrombus. The analysis also suggests that the thrombus core, as compared with the shell, provides an environment for retaining soluble agonists such as thrombin, affecting the extent of platelet activation by establishing agonist-specific concentration gradients radiating from the site of injury. This analysis accounts for the observed weaker activation and relative instability of platelets in the shell and predicts that a failure to form a tightly packed thrombus core will limit thrombin accumulation, a prediction tested by analysis of data from mice with a defect in clot retraction.

Analysis of the lattice kinetic Monte Carlo method in systems with external fields.

The lattice kinetic Monte Carlo (LKMC) method is studied in the context of Brownian particles subjected to drift forces, here principally represented by external fluid flow. LKMC rate expressions for particle hopping are derived that satisfy detailed balance at equilibrium while also providing correct dynamical trajectories in advective-diffusive situations. Error analyses are performed for systems in which collections of particles undergo Brownian motion while also being advected by plug and parabolic flows. We demonstrate how the flow intensity, and its associated drift force, as well as its gradient, each impact the accuracy of the method in relation to reference analytical solutions and Brownian dynamics simulations. Finally, we show how a non-uniform grid that everywhere retains full microscopic detail may be employed to increase the computational efficiency of lattice kinetic Monte Carlo simulations of particles subjected to drift forces arising from the presence of external fields.

Interactions and stress relaxation in monolayers of soft nanoparticles at fluid-fluid interfaces.

Nanoparticles with grafted layers of ligand molecules behave as soft colloids when they adsorb at fluid-fluid interfaces. The ligand brush can deform and reconfigure, adopting a lens-shaped configuration at the interface. This behavior strongly affects the interactions between soft nanoparticles at fluid-fluid interfaces, which have proven challenging to probe experimentally. We measure the surface pressure for a stable 2D interfacial suspension of nanoparticles grafted with ligands, and extract the interaction potential from these data by comparison to Brownian dynamics simulations. A soft repulsive potential with an exponential form accurately reproduces the measured surface pressure data. A more realistic interaction potential model is also fitted to the data to provide insights into the ligand configuration at the interface. The stress of the 2D interfacial suspension upon step compression exhibits a single relaxation time scale, which is also attributable to ligand reconfiguration.

Interaction potentials from arbitrary multi-particle trajectory data.

Understanding the complex physics of particle-based systems at the nanoscale and mesoscale increasingly relies on simulation methods, empowered by exponential advances in computing speed. A major impediment to progress lies in reliably obtaining the interaction potential functions that control system behavior - which are key inputs for any simulation approach - and which are often difficult or impossible to obtain directly using traditional experimental methods. Here, we present a straightforward methodology for generating pair potential functions from large multi-particle trajectory datasets, with no operational constraints regarding their state of equilibration, degree of damping or presence of hydrodynamic interactions. Using simulated datasets, we demonstrate that the method is highly robust against trajectory perturbations from Brownian motion and common errors introduced by particle tracking algorithms. Given the recent rapid pace of advancement in high-speed and three-dimensional microscopy and associated particle tracking algorithms, we anticipate a near future experimental regime where easily collected high-dimensional trajectory sets can be rapidly converted to the detailed interaction and hydrodynamic force fields required to replicate the system's physics in simulation.

On coarse projective integration for atomic deposition in amorphous systems.

Direct molecular dynamics simulation of atomic deposition under realistic conditions is notoriously challenging because of the wide range of time scales that must be captured. Numerous simulation approaches have been proposed to address the problem, often requiring a compromise between model fidelity, algorithmic complexity, and computational efficiency. Coarse projective integration, an example application of the "equation-free" framework, offers an attractive balance between these constraints. Here, periodically applied, short atomistic simulations are employed to compute time derivatives of slowly evolving coarse variables that are then used to numerically integrate differential equations over relatively large time intervals. A key obstacle to the application of this technique in realistic settings is the "lifting" operation in which a valid atomistic configuration is recreated from knowledge of the coarse variables. Using Ge deposition on amorphous SiO2 substrates as an example application, we present a scheme for lifting realistic atomistic configurations comprised of collections of Ge islands on amorphous SiO2 using only a few measures of the island size distribution. The approach is shown to provide accurate initial configurations to restart molecular dynamics simulations at arbitrary points in time, enabling the application of coarse projective integration for this morphologically complex system.

Multiscale prediction of patient-specific platelet function under flow.

During thrombotic or hemostatic episodes, platelets bind collagen and release ADP and thromboxane A(2), recruiting additional platelets to a growing deposit that distorts the flow field. Prediction of clotting function under hemodynamic conditions for a patient's platelet phenotype remains a challenge. A platelet signaling phenotype was obtained for 3 healthy donors using pairwise agonist scanning, in which calcium dye-loaded platelets were exposed to pairwise combinations of ADP, U46619, and convulxin to activate the P2Y(1)/P2Y(12), TP, and GPVI receptors, respectively, with and without the prostacyclin receptor agonist iloprost. A neural network model was trained on each donor's pairwise agonist scanning experiment and then embedded into a multiscale Monte Carlo simulation of donor-specific platelet deposition under flow. The simulations were compared directly with microfluidic experiments of whole blood flowing over collagen at 200 and 1000/s wall shear rate. The simulations predicted the ranked order of drug sensitivity for indomethacin, aspirin, MRS-2179 (a P2Y(1) inhibitor), and iloprost. Consistent with measurement and simulation, one donor displayed larger clots and another presented with indomethacin resistance (revealing a novel heterozygote TP-V241G mutation). In silico representations of a subject's platelet phenotype allowed prediction of blood function under flow, essential for identifying patient-specific risks, drug responses, and novel genotypes.

A general method for spatially coarse-graining Metropolis Monte Carlo simulations onto a lattice.

A recently introduced method for coarse-graining standard continuous Metropolis Monte Carlo simulations of atomic or molecular fluids onto a rigid lattice of variable scale [X. Liu, W. D. Seider, and T. Sinno, Phys. Rev. E 86, 026708 (2012)] is further analyzed and extended. The coarse-grained Metropolis Monte Carlo technique is demonstrated to be highly consistent with the underlying full-resolution problem using a series of detailed comparisons, including vapor-liquid equilibrium phase envelopes and spatial density distributions for the Lennard-Jones argon and simple point charge water models. In addition, the principal computational bottleneck associated with computing a coarse-grained interaction function for evolving particle positions on the discretized domain is addressed by the introduction of new closure approximations. In particular, it is shown that the coarse-grained potential, which is generally a function of temperature and coarse-graining level, can be computed at multiple temperatures and scales using a single set of free energy calculations. The computational performance of the method relative to standard Monte Carlo simulation is also discussed.

Coarse-grained Monte Carlo simulations of non-equilibrium systems.

We extend the scope of a recent method for generating coarse-grained lattice Metropolis Monte Carlo simulations [X. Liu, W. D. Seider, and T. Sinno, Phys. Rev. E 86, 026708 (2012); and J. Chem. Phys. 138, 114104 (2013)] from continuous interaction potentials to non-equilibrium situations. The original method has been shown to satisfy detailed balance at the coarse scale and to provide a good representation of various equilibrium properties in both atomic and molecular systems. However, we show here that the original method is inconsistent with non-equilibrium trajectories generated by full-resolution Monte Carlo simulations, which, under certain conditions, have been shown to correspond to Langevin dynamics. The modified coarse-grained method is generated by simultaneously biasing the forward and backward transition probability for every possible move, thereby preserving the detailed balance of the original method. The resulting coarse-grained Monte Carlo simulations are shown to provide trajectories that are consistent with overdamped Langevin (Smoluchowski) dynamics using a sequence of simple non-equilibrium examples. We first consider the purely diffusional spreading of a Gaussian pulse of ideal-gas particles and then include an external potential to study the influence of drift. Finally, we validate the method using a more general situation in which the particles interact via a Lennard-Jones interparticle potential.