Fluid vesicles with viscous membranes in shear flow

Probing wrapping dynamics of spherical nanoparticles by 3D vesicles using force-based simulations

Nanoparticles present in various environments can interact with living organisms, potentially leading to deleterious effects. Understanding how these nanoparticles interact with cell membranes is crucial for rational assessment of their impact on diverse biological processes. While previous research has explored particle-membrane interactions, the dynamic processes of particle wrapping by fluid vesicles remain incompletely understood. In this study, we introduce a force-based, continuum-scale model utilizing triangulated mesh representation and discrete differential geometry to investigate particle-vesicle interaction dynamics. Our model captures the transformation of vesicle shape and nanoparticle wrapping by calculating the forces arising from membrane bending energy and particle adhesion energy. Inspired by cell phagocytosis of large particles, we focus on establishing a quantitative understanding of large-scale vesicle deformation induced by the interaction with particles of comparable sizes. We first examine the interactions between spherical vesicles and individual nanospheres, both externally and internally, and quantify energy landscapes across different wrapping fractions of the nanoparticles. Furthermore, we explore multiple particle interactions with biologically relevant fluid vesicles with nonspherical shapes. Our study reveals that initial particle positions and interaction sequences are critical in determining the final equilibrium shapes of the vesicle-particle complexes in these interactions. These findings emphasize the importance of nanoparticle positioning and wrapping fractions in the dynamics of particle-vesicle interactions, providing crucial insights for future research in the field.

Excluded volume effects on tangentially driven active ring polymers

The conformational and dynamical properties of active ring polymers are studied by numerical simulations. The two-dimensionally confined polymer is modeled as a closed bead-spring chain, driven by tangential forces, put in contact with a heat bath described by the Brownian multiparticle collision dynamics. Both phantom polymers and chains comprising excluded volume interactions are considered for different bending rigidities. The size and shape are found to be dependent on persistence length, driving force, and bead mutual exclusion. The lack of excluded volume interactions is responsible for a shrinkage of active rings when increasing driving force in the flexible limit, while the presence induces a moderate swelling of chains. The internal dynamics of flexible phantom active rings shows activity-enhanced diffusive behavior at large activity values while, in the case of self-avoiding active chains, it is characterized by active ballistic motion not depending on stiffness. The long-time dynamics of active rings is marked by rotational motion whose period scales as the inverse of the applied tangential force, irrespective of persistence length and beads' self-exclusion.

Mesoscale simulation of biomembranes with FreeDTS

We present FreeDTS software for performing computational research on biomembranes at the mesoscale. In this software, a membrane is represented by a dynamically triangulated surface equipped with vertex-based inclusions to integrate the effects of integral and peripheral membrane proteins. Several algorithms are included in the software to simulate complex membranes at different conditions such as framed membranes with constant tension, vesicles and high-genus membranes with various fixed volumes or constant pressure differences and applying external forces to membrane regions. Furthermore, the software allows the user to turn off the shape evolution of the membrane and focus solely on the organization of proteins. As a result, we can take realistic membrane shapes obtained from, for example, cryo-electron tomography and backmap them into a finer simulation model. In addition to many biomembrane applications, this software brings us a step closer to simulating realistic biomembranes with molecular resolution. Here we provide several interesting showcases of the power of the software but leave a wide range of potential applications for interested users.

Modelling how curved active proteins and shear flow pattern cellular shape and motility

Cell spreading and motility on an adhesive substrate are driven by the active physical forces generated by the actin cytoskeleton. We have recently shown that coupling curved membrane complexes to protrusive forces, exerted by the actin polymerization that they recruit, provides a mechanism that can give rise to spontaneous membrane shapes and patterns. In the presence of an adhesive substrate, this model was shown to give rise to an emergent motile phenotype, resembling a motile cell. Here, we utilize this "minimal-cell" model to explore the impact of external shear flow on the cell shape and migration on a uniform adhesive flat substrate. We find that in the presence of shear the motile cell reorients such that its leading edge, where the curved active proteins aggregate, faces the shear flow. The flow-facing configuration is found to minimize the adhesion energy by allowing the cell to spread more efficiently over the substrate. For the non-motile vesicle shapes, we find that they mostly slide and roll with the shear flow. We compare these theoretical results with experimental observations, and suggest that the tendency of many cell types to move against the flow may arise from the very general, and non-cell-type-specific mechanism predicted by our model.

Stability of Branched Tubular Membrane Structures

We study the energetics and stability of branched tubular membrane structures by computer simulations of a triangulated network model. We find that triple (Y) junctions can be created and stabilized by applying mechanical forces, if the angle between branches is 120°. The same holds for tetrahedral junctions with tetraeder angles. If the wrong angles are enforced, the branches coalesce to a linear structure, a pure tube. After releasing the mechanical force, Y-branched structures remain metastable if one constrains the enclosed volume and the average curvature (the area difference) to a fixed value; tetrahedral junctions however split up into two Y junctions. Somewhat counterintuitively, the energy cost of adding a Y branch is negative in structures with fixed surface area and tube diameter, even if one accounts for the positive contribution of the additional branch end. For fixed average curvature, however, adding a branch also enforces a thinning of tubes, therefore the overall curvature energy cost is positive. Possible implications for the stability of branched networks structures in cells are discussed.

Three-dimensional numerical study on wrinkling of vesicles in elongation flow based on the immersed boundary method

We study the wrinkling dynamics of three-dimensional vesicles in a time-dependent elongation flow by utilizing an immersed boundary method. For a quasispherical vesicle, our numerical results well match the predictions of perturbation analysis, where similar exponential relationships between wrinkles' characteristic wavelength and the flow strength are observed. Using the same parameters as in the experiments by Kantsler et al. [V. Kantsler et al., Phys. Rev. Lett. 99, 178102 (2007)0031-900710.1103/PhysRevLett.99.178102], our simulations of an elongated vesicle are in good agreement with their results. In addition, we get rich three-dimensional morphological details, which are favorable to comprehend the two-dimensional snapshots. This morphological information helps identify wrinkle patterns. We analyze the morphological evolution of wrinkles using spherical harmonics. We find discrepancies in elongated vesicle dynamics between simulations and perturbation analysis, highlighting the importance of the nonlinear effects. Finally, we investigate the unevenly distributed local surface tension, which largely determines the position of wrinkles excited on the vesicle membrane.

Non-equilibrium shapes and dynamics of active vesicles

Active vesicles, constructed through the confinement of self-propelled particles (SPPs) inside a lipid membrane shell, exhibit a large variety of non-equilibrium shapes, ranging from the formation of local tethers and dendritic conformations, to prolate and bola-like structures. To better understand the behavior of active vesicles, we perform simulations of membranes modelled as dynamically triangulated surfaces enclosing active Brownian particles. A systematic analysis of membrane deformations and SPP clustering, as a function of SPP activity and volume fraction inside the vesicle is carried out. Distributions of membrane local curvature, and the clustering and mobility of SPPs obtained from simulations of active vesicles are analysed. There exists a feedback mechanism between the enhancement of membrane curvature, the formation of clusters of active particles, and local or global changes in vesicle shape. The emergence of active tension due to the activity of SPPs can well be captured by the Young-Laplace equation. Furthermore, a simple numerical method for tether detection is presented and used to determine correlations between the number of tethers, their length, and local curvature. We also provide several geometrical arguments to explain different tether characteristics for various conditions. These results contribute to the future development of steerable active vesicles or soft micro-robots whose behaviour can be controlled and used for potential applications.

Circulating cell clusters aggravate the hemorheological abnormalities in COVID-19

Microthrombi and circulating cell clusters are common microscopic findings in patients with coronavirus disease 2019 (COVID-19) at different stages in the disease course, implying that they may function as the primary drivers in disease progression. Inspired by a recent flow imaging cytometry study of the blood samples from patients with COVID-19, we perform computational simulations to investigate the dynamics of different types of circulating cell clusters, namely white blood cell (WBC) clusters, platelet clusters, and red blood cell clusters, over a range of shear flows and quantify their impact on the viscosity of the blood. Our simulation results indicate that the increased level of fibrinogen in patients with COVID-19 can promote the formation of red blood cell clusters at relatively low shear rates, thereby elevating the blood viscosity, a mechanism that also leads to an increase in viscosity in other blood diseases, such as sickle cell disease and type 2 diabetes mellitus. We further discover that the presence of WBC clusters could also aggravate the abnormalities of local blood rheology. In particular, the extent of elevation of the local blood viscosity is enlarged as the size of the WBC clusters grows. On the other hand, the impact of platelet clusters on the local rheology is found to be negligible, which is likely due to the smaller size of the platelets. The difference in the impact of WBC and platelet clusters on local hemorheology provides a compelling explanation for the clinical finding that the number of WBC clusters is significantly correlated with thrombotic events in COVID-19 whereas platelet clusters are not. Overall, our study demonstrates that our computational models based on dissipative particle dynamics can serve as a powerful tool to conduct quantitative investigation of the mechanism causing the pathological alterations of hemorheology and explore their connections to the clinical manifestations in COVID-19.

The many faces of membrane tension: Challenges across systems and scales

Our understanding of the role of membrane tension in the field of membrane biophysics is rapidly evolving from a passive construct to an active player in a variety of cellular phenomena. Membrane tension has been shown to be a key regulator of many cellular processes ranging including trafficking, ion channel activation, and the invasion of red blood cells by malaria parasites. Recent experimental advances in cells, including the development of a fluorescent tension reporter, have shown that membrane tension is heterogeneous. In this mini-review, I summarize the recent advances in membrane tension measurements and discuss the contributions from different cellular constituents such as the cortical cytoskeleton. Then, I will explore how these different complexities can be considered in biophysical models of different scales. Finally, I will elaborate on the need for iterations between models and experiments as technologies in both fields advance to enable us to obtain critical insights into the physiological role of membrane tension as a critical component of mechanotransduction.

Dynamics of active polar ring polymers

The conformational and dynamical properties of isolated semiflexible active polar ring polymers are investigated analytically. A ring is modeled as a continuous Gaussian polymer exposed to tangential active forces. The analytical solution of the linear non-Hermitian equation of motion in terms of an eigenfunction expansion shows that ring conformations are independent of activity. In contrast, activity strongly affects the internal ring dynamics and yields characteristic time regimes, which are absent in passive rings. On intermediate timescales, flexible rings show an activity-enhanced diffusive regime, while semiflexible rings exhibit ballistic motion. Moreover, a second active time regime emerges on longer timescales, where rings display a snake-like motion, which is reminiscent to a tank-treading rotational dynamics in shear flow, dominated by the mode with the longest relaxation time.

Evaluation of hydromechanical and functional properties of diversion-type microcapsule-suspension bioreactor for bioartificial liver

AIM

To evaluate the performance of a diversion-type microcapsulesuspension fluidized bed bioreactor and a choanoid fluidized bed bioreactor as bioartificial liver support systems.

MATERIALS AND METHODS

We evaluated the performance between the modified fluidized bed bioreactor based on diversion-type microcapsule suspension (DMFBB) and choanoid fluidized bed bioreactor (CFBB). The fluidization performance, fluidized height, bed expansion, and the mechanical stability and strength of microcapsule were determined. The viability, synthetic, metabolism, and apoptosis of microcapsulated HepLi5 cells were evaluated. Finally, samples were collected for measurement of alanine aminotransferase, total bilirubin, direct bilirubin, and albumin concentrations.

RESULTS

Uniform fluidization was established in both DMFBB and CFBB. The bed expansion, shear force, retention rate, swelling rate, and breakage rate of microcapsules differed significantly between two bioreactors over 3 days. The viability of microencapsulated HepLi5 cells and the activities of cytochrome P450 1A2 and 3A4 increased on each day in DMFBB compared to the control. The albumin and urea concentrations in the DMFBB displayed obvious improvements compared to the control. Caspase3/7 activities in the DMFBB decreased compared to those in the CFBB. At 24 h, the alanine aminotransferase concentration in the DMFBB declined significantly compared to the control. The total and direct bilirubin concentrations within plasma perfusion were decreased and albumin was increased in the DMFBB at 24 h than in the CFBB.

CONCLUSION

The DMFBB shows a promising alternative bioreactor for use in bioartificial liver support systems for application of clinical practice.

Nonlinear Transient and Steady State Stretching of Deflated Vesicles in Flow

Membrane-bound vesicles and organelles exhibit a wide array of nonspherical shapes at equilibrium, including biconcave and tubular morphologies. Despite recent progress, the stretching dynamics of deflated vesicles is not fully understood, particularly far from equilibrium where complex nonspherical shapes undergo large deformations in flow. Here, we directly observe the transient and steady-state nonlinear stretching dynamics of deflated vesicles in extensional flow using a Stokes trap. Automated flow control is used to observe vesicle dynamics over a wide range of flow rates, shape anisotropy, and viscosity contrast. Our results show that deflated vesicle membranes stretch into highly deformed shapes in flow above a critical capillary number Ca_c1. We further identify a second critical capillary number Ca_c2, above which vesicle stretch diverges in flow. Vesicles are robust to multiple nonlinear stretch-relax cycles, evidenced by relaxation of dumbbell-shaped vesicles containing thin lipid tethers following flow cessation. An analytical model is developed for vesicle deformation in flow, which enables comparison of nonlinear steady-state stretching results with theories for different reduced volumes. Our results show that the model captures the steady-state stretching of moderately deflated vesicles; however, it underpredicts the steady-state nonlinear stretching of highly deflated vesicles. Overall, these results provide a new understanding of the nonlinear stretching dynamics and membrane mechanics of deflated vesicles in flow.

Curvature-driven feedback on aggregation-diffusion of proteins in lipid bilayers

Membrane bending is an extensively studied problem from both modeling and experimental perspectives because of the wide implications of curvature generation in cell biology. Many of the curvature generating aspects in membranes can be attributed to interactions between proteins and membranes. These interactions include protein diffusion and formation of aggregates due to protein-protein interactions in the plane of the membrane. Recently, we developed a model that couples the in-plane flow of lipids and diffusion of proteins with the out-of-plane bending of the membrane. Building on this work, here, we focus on the role of explicit aggregation of proteins on the surface of the membrane in the presence of membrane bending and diffusion. We develop a comprehensive framework that includes lipid flow, membrane bending, the entropy of protein distribution, along with an explicit aggregation potential and derive the governing equations for the coupled system. We compare this framework to the Cahn-Hillard formalism to predict the regimes in which the proteins form patterns on the membrane. We demonstrate the utility of this model using numerical simulations to predict how aggregation and diffusion, when coupled with curvature generation, can alter the landscape of membrane-protein interactions.

Hydrodynamic coupling for particle-based solvent-free membrane models

The great challenge with biological membrane systems is the wide range of scales involved, from nanometers and picoseconds for individual lipids to the micrometers and beyond millisecond for cellular signaling processes. While solvent-free coarse-grained membrane models are convenient for large-scale simulations and promising to provide insight into slow processes involving membranes, these models usually have unrealistic kinetics. One major obstacle is the lack of an equally convenient way of introducing hydrodynamic coupling without significantly increasing the computational cost of the model. To address this, we introduce a framework based on anisotropic Langevin dynamics, for which major in-plane and out-of-plane hydrodynamic effects are modeled via friction and diffusion tensors from analytical or semi-analytical solutions to Stokes hydrodynamic equations. Using this framework, in conjunction with our recently developed membrane model, we obtain accurate dispersion relations for planar membrane patches, both free-standing and in the vicinity of a wall. We briefly discuss how non-equilibrium dynamics is affected by hydrodynamic interactions. We also measure the surface viscosity of the model membrane and discuss the affecting dissipative mechanisms.

Hydrodynamics of immiscible binary fluids with viscosity contrast: a multiparticle collision dynamics approach

We present a multiparticle collision dynamics (MPC) implementation of layered immiscible fluids A and B of different shear viscosities separated by planar interfaces. The simulated flow profile for imposed steady shear motion and the time-dependent shear stress functions are in excellent agreement with our continuum hydrodynamics results for the composite fluid. The wave-vector dependent transverse velocity auto-correlation functions (TVAF) in the bulk-fluid regions of the layers decay exponentially, and agree with those of single-phase isotropic MPC fluids. In addition, we determine the hydrodynamic mobilities of an embedded colloidal sphere moving steadily parallel or transverse to a fluid-fluid interface, as functions of the distance from the interface. The obtained mobilities are in good agreement with hydrodynamic force multipoles calculations, for a no-slip sphere moving under creeping flow conditions near a clean, ideally flat interface. The proposed MPC fluid-layer model can be straightforwardly implemented, and it is computationally very efficient. Yet, owing to the spatial discretization inherent to the MPC method, the model can not reproduce all hydrodynamic features of an ideally flat interface between immiscible fluids.

Impact of simulated lung fluid components on the solubility of inhaled drugs and predicted in vivo performance

Orally inhaled products (OIPs) are gaining increased attention, as pulmonary delivery is a preferred route for the treatment of various diseases. Yet, the field of inhalation biopharmaceutics is still in development phase. For a successful correlation between various in vitro data obtained during formulation characterization and in vivo performance, it is necessary to understand the impact of parameters such as solubility and dissolution of drugs. In this work, we used in vitro-in silico feedback-feedforward approach to gain a better insight into the biopharmaceutics behavior of inhaled Salbutamol Sulphate (SS) and Budesonide (BUD). The thorough characterization of the in vitro test media and the impact of different in vitro fluid components such as lipids and protein on the solubility of aforementioned drugs was studied. These results were subsequently used as an input into the developed in silico models to investigate potential PK parameter changes in vivo. Results revealed that media comprising lipids and albumin were the most biorelevant and impacted the solubility of BUD the most. On the contrary, no notable impact was seen in case of SS. The use of simple media such as phosphate buffer saline (PBS) might be sufficient to use in solubility studies of the highly soluble and permeable drugs. However, its use for the poorly soluble drugs is limited due to the greater potential for interactions within in vivo environment. The use of in silico tools showed that the model response varies, depending on the used media. Therefore, this work highlights the relevance of carefully selecting the media composition when investigating solubility and dissolution behavior, especially in the early phases of drug development and of poorly soluble drugs.

Multiparticle collision dynamics simulations of a squirmer in a nematic fluid

We study the dynamics of a squirmer in a nematic liquid crystal using the multiparticle collision dynamics (MPCD) method. A recently developed nematic MPCD method [Phys. Rev. E 99, 063319 (2019)] which employs a tensor order parameter to describe the spatial and temporal variations of the nematic order is used to simulate the suspending anisotropic fluid. Considering both nematodynamic effects (anisotropic viscosity and elasticity) and thermal fluctuations, in the present study, we couple the nematic MPCD algorithm with a molecular dynamics (MD) scheme for the squirmer. A unique feature of the proposed method is that the nematic order, the fluid, and the squirmer are all represented in a particle-based framework. To test the applicability of this nematic MPCD-MD method, we simulate the dynamics of a spherical squirmer with homeotropic surface anchoring conditions in a bulk domain. The importance of anisotropic viscosity and elasticity on the squirmer's speed and orientation is studied for different values of self-propulsion strength and squirmer type (pusher, puller or neutral). In sharp contrast to Newtonian fluids, the speed of the squirmer in a nematic fluid depends on the squirmer type. Interestingly, the speed of a strong pusher is smaller in the nematic fluid than for the Newtonian case. The orientational dynamics of the squirmer in the nematic fluid also shows a non-trivial dependence on the squirmer type. Our results compare well with existing experimental and numerical data. The full particle-based framework could be easily extended to model the dynamics of multiple squirmers in anisotropic fluids.

Switchable positioning of plate-like inclusions in lipid membranes: Elastically mediated interactions of planar colloids in 2D fluids

We demonstrate how manipulating curvature in an elastic fluid lamella enables the reversible relative positioning of flat, rigid, plate-like micrometer-scale inclusions, with spacings from about a micrometer to tens of micrometers. In an experimental model comprising giant unilamellar vesicles containing solid domain pairs coexisting in a fluid membrane, we adjusted vesicle inflation to manipulate membrane curvature and mapped the interdomain separation. A two-dimensional model of the pair potential predicts the salient experimental observations and reveals both attractions and repulsions, producing a potential minimum entirely a result of the solid domain rigidity and bending energy in the fluid membrane. The impact of vesicle inflation on domain separation in vesicles containing two solid domains was qualitatively consistent with observations in vesicles containing many domains. The behavior differs qualitatively from the pure repulsions between fluid membrane domains or interactions between nanoscopic inclusions whose repulsive or attractive character is not switchable.

Role of combined cell membrane and wall mechanical properties regulated by polarity signals in cell budding

Budding yeast, Saccharomyces cerevisiae, serves as a prime biological model to study mechanisms underlying asymmetric growth. Previous studies have shown that prior to bud emergence, polarization of a conserved small GTPase Cdc42 must be established on the cell membrane of a budding yeast. Additionally, such polarization contributes to the delivery of cell wall remodeling enzymes and hydrolase from cytosol through the membrane, to change the mechanical properties of the cell wall. This leads to the hypothesis that Cdc42 and its associated proteins at least indirectly regulate cell surface mechanical properties. However, how the surface mechanical properties in the emerging bud are changed and whether such change is important are not well understood. To test several hypothesised mechanisms, a novel three-dimensional coarse-grained particle-based model has been developed which describes inhomogeneous mechanical properties of the cell surface. Model simulations predict alternation of the levels of stretching and bending stiffness of the cell surface in the bud region by the polarized Cdc42 signals is essential for initiating bud formation. Model simulations also suggest that bud shape depends strongly on the distribution of the polarized signaling molecules while the neck width of the emerging bud is strongly impacted by the mechanical properties of the chitin and septin rings. Moreover, the temporal change of the bud mechanical properties is shown to affect the symmetry of the bud shape. The 3D model of asymmetric cell growth can also be used for studying viral budding and other vegetative reproduction processes performed via budding, as well as detailed studies of cell growth.

Quantitative Synaptic Biology: A Perspective on Techniques, Numbers and Expectations

Synapses play a central role for the processing of information in the brain and have been analyzed in countless biochemical, electrophysiological, imaging, and computational studies. The functionality and plasticity of synapses are nevertheless still difficult to predict, and conflicting hypotheses have been proposed for many synaptic processes. In this review, we argue that the cause of these problems is a lack of understanding of the spatiotemporal dynamics of key synaptic components. Fortunately, a number of emerging imaging approaches, going beyond super-resolution, should be able to provide required protein positions in space at different points in time. Mathematical models can then integrate the resulting information to allow the prediction of the spatiotemporal dynamics. We argue that these models, to deal with the complexity of synaptic processes, need to be designed in a sufficiently abstract way. Taken together, we suggest that a well-designed combination of imaging and modelling approaches will result in a far more complete understanding of synaptic function than currently possible.

Active particles induce large shape deformations in giant lipid vesicles

Biological cells generate intricate structures by sculpting their membrane from within to actively sense and respond to external stimuli or to explore their environment^1-4. Several pathogenic bacteria also provide examples of how localized forces strongly deform cell membranes from inside, leading to the invasion of neighbouring healthy mammalian cells⁵. Giant unilamellar vesicles have been successfully used as a minimal model system with which to mimic biological cells^6-11, but the realization of a minimal system with localized active internal forces that can strongly deform lipid membranes from within and lead to dramatic shape changes remains challenging. Here we present a combined experimental and simulation study that demonstrates how self-propelled particles enclosed in giant unilamellar vesicles can induce a plethora of non-equilibrium shapes and active membrane fluctuations. Using confocal microscopy, in the experiments we explore the membrane response to local forces exerted by self-phoretic Janus microswimmers. To quantify dynamic membrane changes, we perform Langevin dynamics simulations of active Brownian particles enclosed in thin membrane shells modelled by dynamically triangulated surfaces. The most pronounced shape changes are observed at low and moderate particle loadings, with the formation of tether-like protrusions and highly branched, dendritic structures, whereas at high volume fractions globally deformed vesicle shapes are observed. The resulting state diagram predicts the conditions under which local internal forces generate various membrane shapes. A controlled realization of such distorted vesicle morphologies could improve the design of artificial systems such as small-scale soft robots and synthetic cells.

Clusterization of self-propelled particles in a two-component system

We consider a mixture of active solute molecules in a suspension of passive solvent particles comprising a thermal bath. The solute molecules are considered to be extended objects with two chemically distinct heads, one head of which having chemical affinity towards the solvent particles. The coupled Langevin equations for the solvent particles along with the equations governing the dynamics of active molecules are numerically simulated to show how the active molecules self-assemble to form clusters which remain in dynamic equilibrium with the free solute molecules. We observe an interesting crossover at an intermediate time in the variation of the order parameter with time when the temperature of the bath is changed signifying the differential behavior of clusterization below and above the crossover time associated with a transition between a thermodynamic and a quasithermodynamic regime. Enthalpy-entropy compensation in the formation of clusters below the crossover is demonstrated.

Patterning and polarization of cells by intracellular flows

Beginning with Turing’s seminal work [1], decades of research have demonstrated the fundamental ability of biochemical networks to generate and sustain the formation of patterns. However, it is increasingly appreciated that biochemical networks both shape and are shaped by physical and mechanical processes [2, 3, 4]. One such process is fluid flow. In many respects, the cytoplasm, membrane and actin cortex all function as fluids, and as they flow, they drive bulk transport of molecules throughout the cell. By coupling biochemical activity to long range molecular transport, flows can shape the distributions of molecules in space. Here we review the various types of flows that exist in cells, with the aim of highlighting recent advances in our understanding of how flows are generated and how they contribute to intracellular patterning processes, such as the establishment of cell polarity.

Hydrodynamic correlations of viscoelastic fluids by multiparticle collision dynamics simulations

The emergent fluctuating hydrodynamics of a viscoelastic fluid modeled by the multiparticle collision dynamics (MPC) approach is studied. The fluid is composed of flexible, Gaussian phantom polymers that interact by local momentum-conserving stochastic MPCs. For comparison, the analytical solution of the linearized Navier-Stokes equation is calculated, where viscoelasticity is taken into account by a time-dependent shear relaxation modulus. The fluid properties are characterized by the transverse velocity autocorrelation function in Fourier space as well as in real space. Various polymer lengths are considered-from dumbbells to (near-)continuous polymers. Viscoelasticity affects the fluid properties and leads to strong correlations, which overall decay exponentially in Fourier space. In real space, the center-of-mass velocity autocorrelation function of individual polymers exhibits a long-time tail, independent of the polymer length, which decays as t^-3/2, similar to a Newtonian fluid, in the asymptotic limit t → ∞. Moreover, for long polymers, an additional power-law decay appears at time scales shorter than the longest polymer relaxation time with the same time dependence, but negative correlations, and the polymer length dependence L^-1/2. Good agreement is found between the analytical and simulation results.

A theoretical study on the dynamics of a compound vesicle in shear flow

The dynamics of nucleate cells in shear flow is of great relevance in cancer cells and circulatory tumor cells where they determine the flow properties of blood. Buoyed by the success of giant unilamellar vesicles in explaining the dynamics of anucleate cells such as red blood cells, compound vesicles have been suggested as a simple model for nucleate cells. A compound vesicle consists of two concentric unilamellar vesicles with the inner, annular and outer regions filled with aqueous Newtonian solvents. In this work, a theoretical model is presented to study the deformation and dynamics of a compound vesicle in linear shear flow using small deformation theory and spherical harmonics with higher order approximation to the membrane forces. A coupling of viscous and membrane stresses at the membrane interface of the two vesicles results in highly nonlinear shape evolution equations for the inner and the outer vesicles which are solved numerically. The results indicate that the size of the inner vesicle (χ) does not affect the tank-treading dynamics of the outer vesicle. The inner vesicle admits a greater inclination angle than the outer vesicle. However, the transition to trembling/swinging and tumbling is significantly affected. The inner and outer vesicles exhibit identical dynamics in the parameter space defined by the nondimensional rotational (Λ_an) and extensional (S) strength of the general shear flow. At moderate χ, a swinging mode is observed for the inner vesicle while the outer vesicle exhibits tumbling. The inner vesicle also exhibits modification of the TU mode to IUS (intermediate tumbling swinging) mode. Moreover, synchronization of the two vesicles at higher χ and a Capillary number sensitive motion at lower χ is observed in the tumbling regime. These results are in accordance with the few experimental observations reported by Levant and Steinberg. A reduction in the inclination angle is observed with an increase in χ when the inner vesicle is replaced by a solid inclusion. Additionally, a very elaborate phase diagram is presented in the Λ_an-S parameter space, which could be tested in future experiments or numerical simulations.

Oblate to prolate transition of a vesicle in shear flow

Vesicles are micrometric soft particles whose membrane is a two-dimensional incompressible fluid governed by bending resistance leading to a zoology of shapes. The dynamics of deflated vesicles in shear flow with a bottom wall, a first minimal configuration to consider confined vesicles, is investigated using numerical simulations. Coexistence under flow of oblate (metastable) and prolate (stable) shapes is studied in details. In particular, we discuss the boundaries of the region of coexistence in the (v, Ca -plane where v is the reduced volume of the vesicle and Ca the Capillary number. We characterize the transition from oblate to prolate and analyse the divergence of the transition time near the critical capillary number. We then analyse the lift dynamics of an oblate vesicle in the weak flow regime.

Multi-particle collision dynamics for a coarse-grained model of soft colloids

The growing interest in the dynamical properties of colloidal suspensions, both in equilibrium and under an external drive such as shear or pressure flow, requires the development of accurate methods to correctly include hydrodynamic effects due to the suspension in a solvent. In the present work, we generalize Multiparticle Collision Dynamics (MPCD) to be able to deal with soft, polymeric colloids. Our methods build on the knowledge of the monomer density profile that can be obtained from monomer-resolved simulations without hydrodynamics or from theoretical arguments. We hereby propose two different approaches. The first one simply extends the MPCD method by including in the simulations effective monomers with a given density profile, thus neglecting monomer-monomer interactions. The second one considers the macromolecule as a single penetrable soft colloid (PSC), which is permeated by an inhomogeneous distribution of solvent particles. By defining an appropriate set of rules to control the collision events between the solvent and the soft colloid, both linear and angular momenta are exchanged. We apply these methods to the case of linear chains and star polymers for varying monomer lengths and arm number, respectively, and compare the results for the dynamical properties with those obtained within monomer-resolved simulations. We find that the effective monomer method works well for linear chains, while the PSC method provides very good results for stars. These methods pave the way to extend MPCD treatments to complex macromolecular objects such as microgels or dendrimers and to work with soft colloids at finite concentrations.

State diagram for wall adhesion of red blood cells in shear flow: from crawling to flipping

Red blood cells in shear flow show a variety of different shapes due to the complex interplay between hydrodynamics and membrane elasticity. Malaria-infected red blood cells become generally adhesive and less deformable. Adhesion to a substrate leads to a reduction in shape variability and to a flipping motion of the non-spherical shapes during the mid-stage of infection. Here, we present a complete state diagram for wall adhesion of red blood cells in shear flow obtained by simulations, using a particle-based mesoscale hydrodynamics approach, multiparticle collision dynamics. We find that cell flipping at a substrate is replaced by crawling beyond a critical shear rate, which increases with both membrane stiffness and viscosity contrast between the cytosol and suspending medium. This change in cell dynamics resembles the transition between tumbling and tank-treading for red blood cells in free shear flow. In the context of malaria infections, the flipping-crawling transition would strongly increase the adhesive interactions with the vascular endothelium, but might be suppressed by the combined effect of increased elasticity and viscosity contrast.

A finite difference method with subsampling for immersed boundary simulations of the capsule dynamics with viscoelastic membranes

The membrane or interfacial viscosity is an important property in many multiphase and biofluidic situations, such as the red blood cell dynamics and emulsion stability. The immersed boundary method (IBM), which incorporates the dynamic flow-membrane interaction via force distribution and velocity interpolation, has been extensively employed in simulations of such systems. Unfortunately, direct implementation of membrane viscosity in IBM suffers severe numerical instability, which causes an IBM calculation to break down before generating any useful results. Few attempts have been recently reported; however, several concerns exist in these attempts, such as the inconsistency to the classical definition of membrane viscosity, the inability to model the shear and dilatational viscosities separately, the unjustified mathematical formulations, and the complicated algorithms and computation. To overcome these concerns, in this paper, we propose a finite difference approach for implementing membrane viscosity in immersed boundary simulations. The viscous stress is obtained via finite difference approximations to the differential strain-stress relationship, with the help of a subsampling scheme to reduce the numerical noise in the calculated strain rates. This simple method has also avoided the complicated matrix calculations in previous attempts, and hence, a better computational efficiency is expected. Detailed mathematical description of the method and key steps for its implementation in immersed boundary programs are provided. Validation and illustration calculations are performed, and our results are compared with analytical solutions and previous publications with satisfactory agreement. The influences of membrane mesh resolution and simulation time step are also examined; and the results show no indication that our finite difference method has downgraded the general IBM accuracy. Based on these simulations and analysis, we believe that our method would be a better choice for future IBM simulations of capsule dynamics with viscoelastic membranes.

High-Throughput Microfluidic Characterization of Erythrocyte Shapes and Mechanical Variability

The motion of red blood cells (RBCs) in microchannels is important for microvascular blood flow and biomedical applications such as blood analysis in microfluidics. The current understanding of the complexity of RBC shapes and dynamics in microchannels is mainly based on several simulation studies, but there are a few systematic experimental investigations. Here, we present a combined study that systematically characterizes RBC behavior for a wide range of flow rates and channel sizes. Even though simulations and experiments generally show good agreement, experimental observations demonstrate that there is no single well-defined RBC state for fixed flow conditions but rather a broad distribution of states. This result can be attributed to the inherent variability in RBC mechanical properties, which is confirmed by a model that takes the variation in RBC shear elasticity into account. This represents a significant step toward a quantitative connection between RBC behavior in microfluidic devices and their mechanical properties, which is essential for a high-throughput characterization of diseased cells.

Modeling Membrane Curvature Generation due to Membrane⁻Protein Interactions

To alter and adjust the shape of the plasma membrane, cells harness various mechanisms of curvature generation. Many of these curvature generation mechanisms rely on the interactions between peripheral membrane proteins, integral membrane proteins, and lipids in the bilayer membrane. Mathematical and computational modeling of membrane curvature generation has provided great insights into the physics underlying these processes. However, one of the challenges in modeling these processes is identifying the suitable constitutive relationships that describe the membrane free energy including protein distribution and curvature generation capability. Here, we review some of the commonly used continuum elastic membrane models that have been developed for this purpose and discuss their applications. Finally, we address some fundamental challenges that future theoretical methods need to overcome to push the boundaries of current model applications.

Steady state sedimentation of ultrasoft colloids

The structural and dynamical properties of ultra-soft colloids-star polymers-exposed to a uniform external force field are analyzed by applying the multiparticle collision dynamics technique, a hybrid coarse-grain mesoscale simulation approach, which captures thermal fluctuations and long-range hydrodynamic interactions. In the weak-field limit, the structure of the star polymer is nearly unchanged; however, in an intermediate regime, the radius of gyration decreases, in particular transverse to the sedimentation direction. In the limit of a strong field, the radius of gyration increases with field strength. Correspondingly, the sedimentation coefficient increases with increasing field strength, passes through a maximum, and decreases again at high field strengths. The maximum value depends on the functionality of the star polymer. High field strengths lead to symmetry breaking with trailing, strongly stretched polymer arms and a compact star-polymer body. In the weak-field-linear response regime, the sedimentation coefficient follows the scaling relation of a star polymer in terms of functionality and arm length.

Elastic particle deformation in rectangular channel flow as a measure of particle stiffness

In this study, we experimentally observed and characterized soft elastic particle deformation in confined flow in a microchannel with a rectangular cross-section. Hydrogel microparticles of pNIPAM were produced using two different concentrations of crosslinker. This resulted in particles with two different shear moduli of 13.3 ± 5.5 Pa and 32.5 ± 15.7 Pa and compressive moduli of 66 ± 10 Pa and 79 ± 15 Pa, respectively, as measured by capillary micromechanics. Under flow, the particle shapes transitioned from circular to egg, triangular, arrowhead, and ultimately parachute shaped with increasing shear rate. The shape changes were reversible, and deformed particles relaxed back to circular/spherical in the absence of flow. The thresholds for each shape transition were quantified using a non-dimensional radius of curvature at the tip, particle deformation, circularity, and the depth of the concave dimple at the trailing edge. Several of the observed shapes were distinct from those previously reported in the literature for vesicles and capsules; the elastic particles had a narrower leading tip and a lower circularity. Due to variations in the shear moduli between particles within a batch of particles, each flow rate corresponded to a small but finite range of capillary number (Ca) and resulted in a series of shapes. By arranging the images on a plot of Ca versus circularity, a direct correlation was developed between shape and Ca and thus between particle deformation and shear modulus. As the shape was very sensitive to differences in shear modulus, particle deformation in confined flow may allow for better differentiation of microparticle shear modulus than other methods.

Self-assembly and rheology of dipolar colloids in simple shear studied using multi-particle collision dynamics

Magnetic nanoparticles in a colloidal solution self-assemble in various aligned structures, which has a profound influence on the flow behavior. However, the precise role of the microstructure in the development of the rheological response has not been reliably quantified. We investigate the self-assembly of dipolar colloids in simple shear using hybrid molecular dynamics and multi-particle collision dynamics simulations with explicit coarse-grained hydrodynamics, conduct simulated rheometric studies and apply micromechanical models to produce master curves, showing evidence of the universality of the structural behavior governed by the competition between the bonding (dipolar) and erosive (thermal and/or hydrodynamic) stresses. The simulations display viscosity changes across several orders of magnitude in fair quantitative agreement with various literature sources, substantiating the universality of the approach, which seems to apply generally across vastly different length scales and a broad range of physical systems.

Systems biology of cellular membranes: a convergence with biophysics

Systems biology and systems medicine have played an important role in the last two decades in shaping our understanding of biological processes. While systems biology is synonymous with network maps and '-omics' approaches, it is not often associated with mechanical processes. Here, we make the case for considering the mechanical and geometrical aspects of biological membranes as a key step in pushing the frontiers of systems biology of cellular membranes forward. We begin by introducing the basic components of cellular membranes, and highlight their dynamical aspects. We then survey the functions of the plasma membrane and the endomembrane system in signaling, and discuss the role and origin of membrane curvature in these diverse cellular processes. We further give an overview of the experimental and modeling approaches to study membrane phenomena. We close with a perspective on the converging futures of systems biology and membrane biophysics, invoking the need to include physical variables such as location and geometry in the study of cellular membranes. WIREs Syst Biol Med 2017, 9:e1386. doi: 10.1002/wsbm.1386 For further resources related to this article, please visit the WIREs website.

Toward Hydrodynamics with Solvent Free Lipid Models: STRD Martini

Solvent hydrodynamics are incorporated into simulations of the solvent-free Dry Martini model. The solvent hydrodynamics are modeled with the stochastic rotation dynamics (SRD) algorithm, a particle-based method for resolving fluid hydrodynamics. SRD does not require calculation of particle-particle distances in the solvent, and so is scalable to arbitrary volumes of solvent with minimal additional computational overhead. The viscosity of the solvent is easily tuned via parameters of the algorithm to span an order of magnitude in viscosity around the viscosity of water at room temperature. The combination "Stochastic Thermostatted Rotation Dynamics (STRD) with Martini" was implemented in Gromacs v.5.01. Simulations of an SRD/palmitoyloleoylphosphatidylcholine membrane demonstrate that the solvent may be included without reparametrizing the lipid model, with minimal perturbation to the thermodynamics. A recent generalization of Saffman-Delbruck theory to periodic geometries by Camley and Brown indicates that lipid dynamics are contaminated by a finite-size effect in typical molecular dynamics (MD) simulations, and that very large systems are required for quantitative simulation of dynamics. Analysis of lipid translational diffusion in this work shows good agreement with the theory, and with explicitly solvated simulations. This indicates that STRD Martini is a viable approach for quantitative simulation of membrane dynamics and does not require massive computational overhead to model the solvent.

Role of molecular turnover in dynamic deformation of a three-dimensional cellular membrane

In cells, the molecular constituents of membranes are dynamically turned over by transportation from one membrane to another. This molecular turnover causes the membrane to shrink or expand by sensing the stress state within the cell, changing its morphology. At present, little is known as to how this turnover regulates the dynamic deformation of cellular membranes. In this study, we propose a new physical model by which molecular turnover is coupled with three-dimensional membrane deformation to explore mechanosensing roles of turnover in cellular membrane deformations. In particular, as an example of microscopic machinery, based on a coarse-graining description, we suppose that molecular turnover depends on the local membrane strain. Using the proposed model, we demonstrate computational simulations of a single vesicle. The results show that molecular turnover adaptively facilitates vesicle deformation, owing to its stress dependence; while the vesicle drastically expands in the case with low bending rigidity, it shrinks in that with high bending rigidity. Moreover, localized active tension on the membrane causes cellular migration by driving the directional transport of molecules within the cell. These results illustrate the use of the proposed model as well as the role of turnover in the dynamic deformations of cellular membranes.

Diffusion in systems crowded by active force-dipole molecules

Experimental studies of systems containing active proteins that undergo conformational changes driven by catalytic chemical reactions have shown that the diffusion coefficients of passive tracer particles and active molecules are larger than the corresponding values when chemical activity is absent. Various mechanisms have been proposed for such behavior, including, among others, force dipole interactions of molecular motors moving on filaments and collective hydrodynamic effects arising from active proteins. Simulations of a multi-component system containing active dumbbell molecules that cycle between open and closed states, a passive tracer particle and solvent molecules are carried out. Consistent with experiments, it is shown that the diffusion coefficients of both passive particles and the dumbbells themselves are enhanced when the dumbbells are active. The dependence of the diffusion enhancement on the volume fraction of dumbbells is determined, and the effects of crowding by active dumbbell molecules are shown to differ from those due to inactive molecules.

Theory and algorithms to compute Helfrich bending forces: a review

Cell membranes are vital to shield a cell's interior from the environment. At the same time they determine to a large extent the cell's mechanical resistance to external forces. In recent years there has been considerable interest in the accurate computational modeling of such membranes, driven mainly by the amazing variety of shapes that red blood cells and model systems such as vesicles can assume in external flows. Given that the typical height of a membrane is only a few nanometers while the surface of the cell extends over many micrometers, physical modeling approaches mostly consider the interface as a two-dimensional elastic continuum. Here we review recent modeling efforts focusing on one of the computationally most intricate components, namely the membrane's bending resistance. We start with a short background on the most widely used bending model due to Helfrich. While the Helfrich bending energy by itself is an extremely simple model equation, the computation of the resulting forces is far from trivial. At the heart of these difficulties lies the fact that the forces involve second order derivatives of the local surface curvature which by itself is the second derivative of the membrane geometry. We systematically derive and compare the different routes to obtain bending forces from the Helfrich energy, namely the variational approach and the thin-shell theory. While both routes lead to mathematically identical expressions, so-called linear bending models are shown to reproduce only the leading order term while higher orders differ. The main part of the review contains a description of various computational strategies which we classify into three categories: the force, the strong and the weak formulation. We finally give some examples for the application of these strategies in actual simulations.

Shear-sensitive nanocapsule drug release for site-specific inhibition of occlusive thrombus formation

Essentials Vessel stenosis due to large thrombus formation increases local shear 1-2 orders of magnitude. High shear at stenotic sites was exploited to trigger eptifibatide release from nanocapsules. Local delivery of eptifibatide prevented vessel occlusion without increased tail bleeding times. Local nanocapsule delivery of eptifibatide may be safer than systemic antiplatelet therapies.

SUMMARY

Background Myocardial infarction and stroke remain the leading causes of mortality and morbidity. The major limitation of current antiplatelet therapy is that the effective concentrations are limited because of bleeding complications. Targeted delivery of antiplatelet drug to sites of thrombosis would overcome these limitations. Objectives Here, we have exploited a key biomechanical feature specific to thrombosis, i.e. significantly increased blood shear stress resulting from a reduction in the lumen of the vessel, to achieve site-directed delivery of the clinically used antiplatelet agent eptifibatide by using shear-sensitive phosphatidylcholine (PC)-based nanocapsules. Methods PC-based nanocapsules (2.8 × 1012 ) with high-dose encapsulated eptifibatide were introduced into microfluidic blood perfusion assays and into in vivo models of thrombosis and tail bleeding. Results Shear-triggered nanocapsule delivery of eptifibatide inhibited in vitro thrombus formation selectively under stenotic and high shear flow conditions above a shear rate of 1000 s-1 while leaving thrombus formation under physiologic shear rates unaffected. Thrombosis was effectively prevented in in vivo models of vessel wall damage. Importantly, mice infused with shear-sensitive antiplatelet nanocapsules did not show prolonged bleeding times. Conclusions Targeted delivery of eptifibatide by shear-sensitive nanocapsules offers site-specific antiplatelet potential, and may form a basis for developing more potent and safer antiplatelet drugs.

Hydrodynamic coupling of particle inclusions embedded in curved lipid bilayer membranes

We develop theory and computational methods to investigate particle inclusions embedded within curved lipid bilayer membranes. We consider the case of spherical lipid vesicles where inclusion particles are coupled through (i) intramembrane hydrodynamics, (ii) traction stresses with the external and trapped solvent fluid, and (iii) intermonolayer slip between the two leaflets of the bilayer. We investigate relative to flat membranes how the membrane curvature and topology augment hydrodynamic responses. We show how both the translational and rotational mobility of protein inclusions are effected by the membrane curvature, ratio of intramembrane viscosity to solvent viscosity, and intermonolayer slip. For general investigations of many-particle dynamics, we also discuss how our approaches can be used to treat the collective diffusion and hydrodynamic coupling within spherical bilayers.

Dynamic and rheological properties of soft biological cell suspensions

Quantifying dynamic and rheological properties of suspensions of soft biological particles such as vesicles, capsules, and red blood cells (RBCs) is fundamentally important in computational biology and biomedical engineering. In this review, recent studies on dynamic and rheological behavior of soft biological cell suspensions by computer simulations are presented, considering both unbounded and confined shear flow. Furthermore, the hemodynamic and hemorheological characteristics of RBCs in diseases such as malaria and sickle cell anemia are highlighted.

Viscoelastic properties of marginal networks in a solvent

Polymer networks at the margins of mechanical stability are known to be highly sensitive to applied forces and fields and to exhibit an anomalously large resistance to deformation. In this paper, we study the effects of hydrodynamic interactions on the behavior of marginal networks using a hybrid molecular dynamics and multiparticle collision dynamics simulation technique. We examine how the filament and solvent properties affect the response of marginal networks to shear. We find that the stiffening of the network shows a stronger dependence on the shear frequency when hydrodynamic interactions are present than when they are not. The network shear modulus scales as G'∼ω(α(c)), with a critical stiffening exponent α(c) that can be controlled by varying the relative concentrations of the network and the solvent. Our results show that this arises due to the solvent aiding the relaxation of the network and suppressing the network nonaffinity, with the system deforming more affinely when hydrodynamic interactions are maximized.

Understanding particle margination in blood flow - A step toward optimized drug delivery systems

Targeted delivery of drugs and imaging agents is very promising to develop new strategies for the treatment of various diseases such as cancer. For an efficient targeted adhesion, the particles have to migrate toward the walls in blood flow - a process referred to as margination. Due to a huge diversity of available carriers, a good understanding of their margination properties in blood flow depending on various flow conditions and particle properties is required. We employ a particle-based mesoscopic hydrodynamic simulation approach to investigate the margination of different carriers for a wide range of hematocrits (volume fraction of red blood cells) and flow rates. Our results show that margination strongly depends on the thickness of the available free space close to the wall, the so-called red blood cell-free layer (RBC-FL), in comparison to the carrier size. The carriers with a few micrometers in size are comparable with the RBC-FL thickness and marginate better than their sub-micrometer counterparts. Deformable carriers, in general, show worse margination properties than rigid particles. Particle margination is also found to be most pronounced in small channels with a characteristic size comparable to blood capillaries. Finally, different margination mechanisms are discussed.

Selective flow-induced vesicle rupture to sort by membrane mechanical properties

Vesicle and cell rupture caused by large viscous stresses in ultrasonication is central to biomedical and bioprocessing applications. The flow-induced opening of lipid membranes can be exploited to deliver drugs into cells, or to recover products from cells, provided that it can be obtained in a controlled fashion. Here we demonstrate that differences in lipid membrane and vesicle properties can enable selective flow-induced vesicle break-up. We obtained vesicle populations with different membrane properties by using different lipids (SOPC, DOPC, or POPC) and lipid:cholesterol mixtures (SOPC:chol and DOPC:chol). We subjected vesicles to large deformations in the acoustic microstreaming flow generated by ultrasound-driven microbubbles. By simultaneously deforming vesicles with different properties in the same flow, we determined the conditions in which rupture is selective with respect to the membrane stretching elasticity. We also investigated the effect of vesicle radius and excess area on the threshold for rupture, and identified conditions for robust selectivity based solely on the mechanical properties of the membrane. Our work should enable new sorting mechanisms based on the difference in membrane composition and mechanical properties between different vesicles, capsules, or cells.

Monte Carlo simulations of fluid vesicles

Lipid vesicles are closed two dimensional fluid surfaces that are studied extensively as model systems for understanding the physical properties of biological membranes. Here we review the recent developments in the Monte Carlo techniques for simulating fluid vesicles and discuss some of their applications. The technique, which treats the membrane as an elastic sheet, is most suitable for the study of large scale conformations of membranes. The model can be used to study vesicles with fixed and varying topologies. Here we focus on the case of multi-component membranes with the local lipid and protein composition coupled to the membrane curvature leading to a variety of shapes. The phase diagram is more intriguing in the case of fluid vesicles having an in-plane orientational order that induce anisotropic directional curvatures. Methods to explore the steady state morphological structures due to active flux of materials have also been described in the context of Monte Carlo simulations.