Fluctuating hydrodynamics and the Rayleigh-Plateau instability.

The Rayleigh-Plateau instability occurs when surface tension makes a fluid column become unstable to small perturbations. At nanometer scales, thermal fluctuations are comparable to interfacial energy densities. Consequently, at these scales, thermal fluctuations play a significant role in the dynamics of the instability. These microscopic effects have previously been investigated numerically using particle-based simulations, such as molecular dynamics (MD), and stochastic partial differential equation-based hydrodynamic models, such as stochastic lubrication theory. In this paper, we present an incompressible fluctuating hydrodynamics model with a diffuse-interface formulation for binary fluid mixtures designed for the study of stochastic interfacial phenomena. An efficient numerical algorithm is outlined and validated in numerical simulations of stable equilibrium interfaces. We present results from simulations of the Rayleigh-Plateau instability for long cylinders pinching into droplets for Ohnesorge numbers of Oh = 0.5 and 5.0. Both stochastic and perturbed deterministic simulations are analyzed and ensemble results show significant differences in the temporal evolution of the minimum radius near pinching. Short cylinders, with lengths less than their circumference, were also investigated. As previously observed in MD simulations, we find that thermal fluctuations cause these to pinch in cases where a perturbed cylinder would be stable deterministically. Finally, we show that the fluctuating hydrodynamics model can be applied to study a broader range of surface tension-driven phenomena.

Surface coverage dynamics for reversible dissociative adsorption on finite linear lattices.

Dissociative adsorption onto a surface introduces dynamic correlations between neighboring sites not found in non-dissociative absorption. We study surface coverage dynamics where reversible dissociative adsorption of dimers occurs on a finite linear lattice. We derive analytic expressions for the equilibrium surface coverage as a function of the number of reactive sites, N, and the ratio of the adsorption and desorption rates. Using these results, we characterize the finite size effect on the equilibrium surface coverage. For comparable N's, the finite size effect is significantly larger when N is even than when N is odd. Moreover, as N increases, the size effect decays more slowly in the even case than in the odd case. The finite-size effect becomes significant when adsorption and desorption rates are considerably different. These finite-size effects are related to the number of accessible configurations in a finite system where the odd-even dependence arises from the limited number of accessible configurations in the even case. We confirm our analytical results with kinetic Monte Carlo simulations. We also analyze the surface-diffusion case where adsorbed atoms can hop into neighboring sites. As expected, the odd-even dependence disappears because more configurations are accessible in the even case due to surface diffusion.

Fluctuation-enhanced electric conductivity in electrolyte solutions.

We analyze the effects of an externally applied electric field on thermal fluctuations for a binary electrolyte fluid. We show that the fluctuating Poisson-Nernst-Planck (PNP) equations for charged multispecies diffusion coupled with the fluctuating fluid momentum equation result in enhanced charge transport via a mechanism distinct from the well-known enhancement of mass transport that accompanies giant fluctuations. Although the mass and charge transport occurs by advection by thermal velocity fluctuations, it can macroscopically be represented as electrodiffusion with renormalized electric conductivity and a nonzero cation-anion diffusion coefficient. Specifically, we predict a nonzero cation-anion Maxwell-Stefan coefficient proportional to the square root of the salt concentration, a prediction that agrees quantitatively with experimental measurements. The renormalized or effective macroscopic equations are different from the starting PNP equations, which contain no cross-diffusion terms, even for rather dilute binary electrolytes. At the same time, for infinitely dilute solutions the renormalized electric conductivity and renormalized diffusion coefficients are consistent and the classical PNP equations with renormalized coefficients are recovered, demonstrating the self-consistency of the fluctuating hydrodynamics equations. Our calculations show that the fluctuating hydrodynamics approach recovers the electrophoretic and relaxation corrections obtained by Debye-Huckel-Onsager theory, while elucidating the physical origins of these corrections and generalizing straightforwardly to more complex multispecies electrolytes. Finally, we show that strong applied electric fields result in anisotropically enhanced "giant" velocity fluctuations and reduced fluctuations of salt concentration.

Fluctuating hydrodynamics of reactive liquid mixtures.

Fluctuating hydrodynamics (FHD) provides a framework for modeling microscopic fluctuations in a manner consistent with statistical mechanics and nonequilibrium thermodynamics. This paper presents an FHD formulation for isothermal reactive incompressible liquid mixtures with stochastic chemistry. Fluctuating multispecies mass diffusion is formulated using a Maxwell-Stefan description without assuming a dilute solution, and momentum dynamics is described by a stochastic Navier-Stokes equation for the fluid velocity. We consider a thermodynamically consistent generalization for the law of mass action for non-dilute mixtures and use it in the chemical master equation (CME) to model reactions as a Poisson process. The FHD approach provides remarkable computational efficiency over traditional reaction-diffusion master equation methods when the number of reactive molecules is large, while also retaining accuracy even when there are as few as ten reactive molecules per hydrodynamic cell. We present a numerical algorithm to solve the coupled FHD and CME equations and validate it on both equilibrium and nonequilibrium problems. We simulate a diffusively driven gravitational instability in the presence of an acid-base neutralization reaction, starting from a perfectly flat interface. We demonstrate that the coupling between velocity and concentration fluctuations dominates the initial growth of the instability.

Stochastic simulation of reaction-diffusion systems: A fluctuating-hydrodynamics approach.

We develop numerical methods for stochastic reaction-diffusion systems based on approaches used for fluctuatinghydrodynamics (FHD). For hydrodynamicsystems, the FHD formulation is formally described by stochastic partial differential equations (SPDEs). In the reaction-diffusion systems we consider, our model becomes similar to the reaction-diffusion master equation (RDME) description when our SPDEs are spatially discretized and reactions are modeled as a source term having Poissonfluctuations. However, unlike the RDME, which becomes prohibitively expensive for an increasing number of molecules, our FHD-based description naturally extends from the regime where fluctuations are strong, i.e., each mesoscopic cell has few (reactive) molecules, to regimes with moderate or weak fluctuations, and ultimately to the deterministic limit. By treating diffusion implicitly, we avoid the severe restriction on time step size that limits all methods based on explicit treatments of diffusion and construct numerical methods that are more efficient than RDME methods, without compromising accuracy. Guided by an analysis of the accuracy of the distribution of steady-state fluctuations for the linearized reaction-diffusion model, we construct several two-stage (predictor-corrector) schemes, where diffusion is treated using a stochastic Crank-Nicolson method, and reactions are handled by the stochastic simulation algorithm of Gillespie or a weakly second-order tau leaping method. We find that an implicit midpoint tau leaping scheme attains second-order weak accuracy in the linearized setting and gives an accurate and stable structure factor for a time step size of an order of magnitude larger than the hopping time scale of diffusing molecules. We study the numerical accuracy of our methods for the Schlögl reaction-diffusion model both in and out of thermodynamic equilibrium. We demonstrate and quantify the importance of thermodynamicfluctuations to the formation of a two-dimensional Turing-like pattern and examine the effect of fluctuations on three-dimensional chemical front propagation. By comparing stochastic simulations to deterministic reaction-diffusion simulations, we show that fluctuations accelerate pattern formation in spatially homogeneous systems and lead to a qualitatively different disordered pattern behind a traveling wave.

Fluctuating hydrodynamics of multi-species reactive mixtures.

We formulate and study computationally the fluctuating compressible Navier-Stokes equations for reactive multi-species fluid mixtures. We contrast two different expressions for the covariance of the stochastic chemical production rate in the Langevin formulation of stochastic chemistry, and compare both of them to predictions of the chemical master equation for homogeneous well-mixed systems close to and far from thermodynamic equilibrium. We develop a numerical scheme for inhomogeneous reactive flows, based on our previous methods for non-reactive mixtures [Balakrishnan , Phys. Rev. E 89, 013017 (2014)]. We study the suppression of non-equilibrium long-ranged correlations of concentration fluctuations by chemical reactions, as well as the enhancement of pattern formation by spontaneous fluctuations. Good agreement with available theory demonstrates that the formulation is robust and a useful tool in the study of fluctuations in reactive multi-species fluids. At the same time, several problems with Langevin formulations of stochastic chemistry are identified, suggesting that future work should examine combining Langevin and master equation descriptions of hydrodynamic and chemical fluctuations.

Staggered scheme for the compressible fluctuating hydrodynamics of multispecies fluid mixtures.

We present a numerical formulation for the solution of nonisothermal, compressible Navier-Stokes equations with thermal fluctuations to describe mesoscale transport phenomena in multispecies fluid mixtures. The novelty of our numerical method is the use of staggered grid momenta along with a finite volume discretization of the thermodynamic variables to solve the resulting stochastic partial differential equations. The key advantages of the numerical scheme are that it significantly simplifies the discretization of diffusive and stochastic momentum fluxes into a more compact form, and it provides an unambiguous prescription of boundary conditions involving pressure. The staggered grid scheme more accurately reproduces the equilibrium static structure factor of hydrodynamic fluctuations in gas mixtures compared to a collocated scheme described previously by Balakrishnan et al. [Phys. Rev. E 89, 013017 (2014)1539-375510.1103/PhysRevE.89.013017]. The numerical method is tested for ideal noble gases mixtures under various nonequilibrium conditions, such as applied thermal and concentration gradients, to assess the role of cross-diffusion effects, such as Soret and Dufour, on the long-ranged correlations of hydrodynamic fluctuations, which are also more accurately reproduced compared to the collocated scheme. We numerically study giant nonequilibrium fluctuations driven by concentration gradients and fluctuation-driven Rayleigh-Taylor instability in gas mixtures. Wherever applicable, excellent agreement is observed with theory and measurements from the direct simulation Monte Carlo method.

Diffusive transport by thermal velocity fluctuations.

We study the contribution of advection by thermal velocity fluctuations to the effective diffusion coefficient in a mixture of two identical fluids. We find good agreement between a simple fluctuating hydrodynamics theory and particle and finite-volume simulations. The enhancement of the diffusive transport depends on the system size L and grows as ln(L/L0) in quasi-two-dimensional systems, while in three dimensions it scales as L0⁻¹ - L⁻¹, where L0 is a reference length. Our results demonstrate that fluctuations play an important role in the hydrodynamics of small-scale systems.

Analyzing and tracking burning structures in lean premixed hydrogen flames.

This paper presents topology-based methods to robustly extract, analyze, and track features defined as subsets of isosurfaces. First, we demonstrate how features identified by thresholding isosurfaces can be defined in terms of the Morse complex. Second, we present a specialized hierarchy that encodes the feature segmentation independent of the threshold while still providing a flexible multiresolution representation. Third, for a given parameter selection, we create detailed tracking graphs representing the complete evolution of all features in a combustion simulation over several hundred time steps. Finally, we discuss a user interface that correlates the tracking information with interactive rendering of the segmented isosurfaces enabling an in-depth analysis of the temporal behavior. We demonstrate our approach by analyzing three numerical simulations of lean hydrogen flames subject to different levels of turbulence. Due to their unstable nature, lean flames burn in cells separated by locally extinguished regions. The number, area, and evolution over time of these cells provide important insights into the impact of turbulence on the combustion process. Utilizing the hierarchy, we can perform an extensive parameter study without reprocessing the data for each set of parameters. The resulting statistics enable scientists to select appropriate parameters and provide insight into the sensitivity of the results with respect to the choice of parameters. Our method allows for the first time to quantitatively correlate the turbulence of the burning process with the distribution of burning regions, properly segmented and selected. In particular, our analysis shows that counterintuitively stronger turbulence leads to larger cell structures, which burn more intensely than expected. This behavior suggests that flames could be stabilized under much leaner conditions than previously anticipated.

Antagonizing scalloped with a novel vestigial construct reveals an important role for scalloped in Drosophila melanogaster leg, eye and optic lobe development.

Scalloped (SD), a TEA/ATTS-domain-containing protein, is required for the proper development of Drosophila melanogaster. Despite being expressed in a variety of tissues, most of the work on SD has been restricted to understanding its role and function in patterning the adult wing. To gain a better understanding of its role in development, we generated sd(47M) flip-in mitotic clones. The mitotic clones had developmental defects in the leg and eye. Further, by removing the VG domains involved in activation, we created a reagent (VGDeltaACT) that disrupts the ability of SD to form a functional transcription factor complex and produced similar phenotypes to the flip-in mitotic clones. The VGDeltaACT construct also disrupted adult CNS development. Expression of the VGDeltaACT construct in the wing alters the cellular localization of VG and produces a mutant phenotype, indicating that the construct is able to antagonize the normal function of the SD/VG complex. Expression of the protein:protein interaction portion of SD is also able to elicit similar phenotypes, suggesting that SD interacts with other cofactors in the leg, eye, and adult CNS. Furthermore, antagonizing SD in larval tissues results in cell death, indicating that SD may also have a role in cell survival.

Numerical methods for the stochastic Landau-Lifshitz Navier-Stokes equations.

The Landau-Lifshitz Navier-Stokes (LLNS) equations incorporate thermal fluctuations into macroscopic hydrodynamics by using stochastic fluxes. This paper examines explicit Eulerian discretizations of the full LLNS equations. Several computational fluid dynamics approaches are considered (including MacCormack's two-step Lax-Wendroff scheme and the piecewise parabolic method) and are found to give good results for the variance of momentum fluctuations. However, neither of these schemes accurately reproduces the fluctuations in energy or density. We introduce a conservative centered scheme with a third-order Runge-Kutta temporal integrator that does accurately produce fluctuations in density, energy, and momentum. A variety of numerical tests, including the random walk of a standing shock wave, are considered and results from the stochastic LLNS solver are compared with theory, when available, and with molecular simulations using a direct simulation Monte Carlo algorithm.

Transvection at the vestigial locus of Drosophila melanogaster.

Transvection is a phenomenon wherein gene expression is effected by the interaction of alleles in trans and often results in partial complementation between mutant alleles. Transvection is dependent upon somatic pairing between homologous chromosome regions and is a form of interallelic complementation that does not occur at the polypeptide level. In this study we demonstrated that transvection could occur at the vestigial (vg) locus by revealing that partial complementation between two vg mutant alleles could be disrupted by changing the genomic location of the alleles through chromosome rearrangement. If chromosome rearrangements affect transvection by disrupting somatic pairing, then combining chromosome rearrangements that restore somatic pairing should restore transvection. We were able to restore partial complementation in numerous rearrangement trans-heterozygotes, thus providing substantial evidence that the observed complementation at vg results from a transvection effect. Cytological analyses revealed this transvection effect to have a large proximal critical region, a feature common to other transvection effects. In the Drosophila interphase nucleus, paired chromosome arms are separated into distinct, nonoverlapping domains. We propose that if the relative position of each arm in the nucleus is determined by the centromere as a relic of chromosome positions after the last mitotic division, then a locus will be displaced to a different territory of the interphase nucleus relative to its nonrearranged homolog by any rearrangement that links that locus to a different centromere. This physical displacement in the nucleus hinders transvection by disrupting the somatic pairing of homologous chromosomes and gives rise to proximal critical regions.

An in vivo analysis of the vestigial gene in Drosophila melanogaster defines the domains required for Vg function.

Considerable evidence indicates an obligate partnership of the Drosophila melanogaster Vestigial (VG) and Scalloped (SD) proteins within the context of wing development. These two proteins interact physically and a 56-amino-acid motif within VG is necessary and sufficient for this binding. While the importance of this SD-binding domain has been clearly demonstrated both in vitro and in vivo, the remaining portions of VG have not been examined for in vivo function. Herein, additional regions within VG were tested for possible in vivo functions. The results identify two additional domains that must be present for optimal VG function as measured by the loss of ability to rescue vg mutants, to induce ectopic sd expression, and to perform other normal VG functions when they are deleted. An in vivo study such as this one is fundamentally important because it identifies domains of VG that are necessary in the cellular context in which wing development actually occurs. The results also indicate that an additional large portion of VG, outside of these two domains and the SD-binding domain, is dispensable in the execution of these normal VG functions.

Molecular and functional analysis of scalloped recessive lethal alleles in Drosophila melanogaster.

The Drosophila melanogaster scalloped (sd) gene is a homolog of the human TEF-1 gene and is a member of the TEA/ATTS domain-containing family of transcription factors. In Drosophila, sd is involved in wing development as well as neural development. Herein, data are presented from a molecular analysis of five recessive lethal sd alleles. Only one of these alleles complements a viable allele associated with an sd mutant wing phenotype, suggesting that functions important for wing development are compromised by the noncomplementing alleles. Two of the wing noncomplementing alleles have mutations that help to define a VG-binding domain for the SD protein in vivo, and another noncomplementing allele has a lesion within the TEA DNA-binding domain. The VG-binding domain overlaps with a domain important for viability of the fly, since two of the sd lethal lesions are located there. The fifth lethal affects a yet undefined motif lying just outside the VG-binding domain in the C-terminal direction that affects both wing phenotype and viability. This is the first example linking mutations affecting specific amino acids in the SD protein with phenotypic consequences for the organism.

Further developmental roles of the Vestigial/Scalloped transcription complex during wing development in Drosophila melanogaster.

The Drosophila homologue of the human TEF-1 gene, scalloped (sd), is required for wing development. The SD protein forms part of a transcriptional activation complex with the protein encoded by vestigial (vg) that, in turn, activates target genes important for wing formation. One sd function involves a regulatory feedback loop with vg and wingless (wg) that is essential in this process. The dorsal-ventral (D/V) margin-specific expression of wg is lost in sd mutant wing discs while the hinge-specific expression appears normal. In the context of wing development, a VG::sdTEA domain fusion produces a protein that mimics the wild-type SD/VG complex and restores the D/V boundary-specific expression of wg in a sd mutant background. Further, targeted expression of wg at the D/V boundary in the wing disc was able to partially rescue the sd mutant phenotype. This infers that sd could function in either the maintenance or induction of wg at the D/V border. Another functional role for sd is the establishment of sensory organ precursors (SOP) of the peripheral nervous system at the wing margin. Thus, the relationship between sd and senseless (sens) in the development of these cells is also examined, and it appears that sd must be functional for proper sens expression, and ultimately, for sensory organ precursor development.

Modeling multiphase flow using fluctuating hydrodynamics.

Fluctuating hydrodynamics provides a model for fluids at mesoscopic scales where thermal fluctuations can have a significant impact on the behavior of the system. Here we investigate a model for fluctuating hydrodynamics of a single-component, multiphase flow in the neighborhood of the critical point. The system is modeled using a compressible flow formulation with a van der Waals equation of state, incorporating a Korteweg stress term to treat interfacial tension. We present a numerical algorithm for modeling this system based on an extension of algorithms developed for fluctuating hydrodynamics for ideal fluids. The scheme is validated by comparison of measured structure factors and capillary wave spectra with equilibrium theory. We also present several nonequilibrium examples to illustrate the capability of the algorithm to model multiphase fluid phenomena in a neighborhood of the critical point. These examples include a study of the impact of fluctuations on the spinodal decomposition following a rapid quench, as well as the piston effect in a cavity with supercooled walls. The conclusion in both cases is that thermal fluctuations affect the size and growth of the domains in off-critical quenches.

Fluctuating hydrodynamics of multispecies nonreactive mixtures.

In this paper we discuss the formulation of the fluctuating Navier-Stokes equations for multispecies, nonreactive fluids. In particular, we establish a form suitable for numerical solution of the resulting stochastic partial differential equations. An accurate and efficient numerical scheme, based on our previous methods for single species and binary mixtures, is presented and tested at equilibrium as well as for a variety of nonequilibrium problems. These include the study of giant nonequilibrium concentration fluctuations in a ternary mixture in the presence of a diffusion barrier, the triggering of a Rayleigh-Taylor instability by diffusion in a four-species mixture, as well as reverse diffusion in a ternary mixture. Good agreement with theory and experiment demonstrates that the formulation is robust and can serve as a useful tool in the study of thermal fluctuations for multispecies fluids.

Identification of a classical bipartite nuclear localization signal in the Drosophila TEA/ATTS protein scalloped.

Drosophila melanogaster wing development has been shown to rely on the activity of a complex of two proteins, Scalloped (Sd) and Vestigial (Vg). Within this complex, Sd is known to provide DNA binding though its TEA/ATTS domain, while Vg modulates this binding and provides transcriptional activation through N- and C-terminal activation domains. There is also evidence that Sd is required for the nuclear translocation of Vg. Indeed, a candidate sequence which shows consensus to the bipartite family of nuclear localization signals (NLSs) has been identified within Sd previously, though it is not known if it is functional, or if additional unpredicted signals that mediate nuclear transport exist within the protein. By expressing various enhanced green fluorescent protein (eGFP) tagged constructs within Drosophila S2 cells, we demonstrate that this NLS is indeed functional and necessary for the proper nuclear localization of Sd. Additionally, the region containing the NLS is critical for the wildtype function of ectopically expressed Sd, in the context of wing development. Using site-directed mutagenesis, we have identified a group of five amino acids within this NLS which is critical for its function, as well as another group of two which is of lesser importance. Together with data that suggests that this sequence mediates interactions with Importin-α3, we conclude that the identified NLS is likely a classical bipartite signal. Further dissection of Sd has also revealed that a large portion of the C-terminal domain of the protein is required its proper nuclear localization. Finally, a Leptomycin B (LB) sensitive signal which appears to facilitate nuclear export is identified, raising the possibility that Sd also contains a nuclear export signal (NES).

Interactive Exploration and Analysis of Large-Scale Simulations Using Topology-Based Data Segmentation.

Large-scale simulations are increasingly being used to study complex scientific and engineering phenomena. As a result, advanced visualization and data analysis are also becoming an integral part of the scientific process. Often, a key step in extracting insight from these large simulations involves the definition, extraction, and evaluation of features in the space and time coordinates of the solution. However, in many applications, these features involve a range of parameters and decisions that will affect the quality and direction of the analysis. Examples include particular level sets of a specific scalar field, or local inequalities between derived quantities. A critical step in the analysis is to understand how these arbitrary parameters/decisions impact the statistical properties of the features, since such a characterization will help to evaluate the conclusions of the analysis as a whole. We present a new topological framework that in a single-pass extracts and encodes entire families of possible features definitions as well as their statistical properties. For each time step we construct a hierarchical merge tree a highly compact, yet flexible feature representation. While this data structure is more than two orders of magnitude smaller than the raw simulation data it allows us to extract a set of features for any given parameter selection in a postprocessing step. Furthermore, we augment the trees with additional attributes making it possible to gather a large number of useful global, local, as well as conditional statistic that would otherwise be extremely difficult to compile. We also use this representation to create tracking graphs that describe the temporal evolution of the features over time. Our system provides a linked-view interface to explore the time-evolution of the graph interactively alongside the segmentation, thus making it possible to perform extensive data analysis in a very efficient manner. We demonstrate our framework by extracting and analyzing burning cells from a large-scale turbulent combustion simulation. In particular, we show how the statistical analysis enabled by our techniques provides new insight into the combustion process.

Vestigial is required during late-stage muscle differentiation in Drosophila melanogaster embryos.

The somatic muscles of Drosophila develop in a complex pattern that is repeated in each embryonic hemi-segment. During early development, progenitor cells fuse to form a syncytial muscle, which further differentiates via expression of muscle-specific factors that induce specific responses to external signals to regulate late-stage processes such as migration and attachment. Initial communication between somatic muscles and the epidermal tendon cells is critical for both of these processes. However, later establishment of attachments between longitudinal muscles at the segmental borders is largely independent of the muscle-epidermal attachment signals, and relatively little is known about how this event is regulated. Using a combination of null mutations and a truncated version of Sd that binds Vg but not DNA, we show that Vestigial (Vg) is required in ventral longitudinal muscles to induce formation of stable intermuscular attachments. In several muscles, this activity may be independent of Sd. Furthermore, the cell-specific differentiation events induced by Vg in two cells fated to form attachments are coordinated by Drosophila epidermal growth factor signaling. Thus, Vg is a key factor to induce specific changes in ventral longitudinal muscles 1-4 identity and is required for these cells to be competent to form stable intermuscular attachments with each other.