A mathematical model of fibrinogen-mediated erythrocyte-erythrocyte adhesion.

Erythrocytes are deformable cells that undergo progressive biophysical and biochemical changes affecting the normal blood flow. Fibrinogen, one of the most abundant plasma proteins, is a primary determinant for changes in haemorheological properties, and a major independent risk factor for cardiovascular diseases. In this study, the adhesion between human erythrocytes is measured by atomic force microscopy (AFM) and its effect observed by micropipette aspiration technique, in the absence and presence of fibrinogen. These experimental data are then used in the development of a mathematical model to examine the biomedical relevant interaction between two erythrocytes. Our designed mathematical model is able to explore the erythrocyte-erythrocyte adhesion forces and changes in erythrocyte morphology. AFM erythrocyte-erythrocyte adhesion data show that the work and detachment force necessary to overcome the adhesion between two erythrocytes increase in the presence of fibrinogen. The changes in erythrocyte morphology, the strong cell-cell adhesion and the slow separation of the two cells are successfully followed in the mathematical simulation. Erythrocyte-erythrocyte adhesion forces and energies are quantified and matched with experimental data. The changes observed on erythrocyte-erythrocyte interactions may give important insights about the pathophysiological relevance of fibrinogen and erythrocyte aggregation in hindering microcirculatory blood flow.

The dynamics of shapes of vesicle membranes with time dependent spontaneous curvature.

We study the time evolution of the shape of a vesicle membrane under time-dependent spontaneous curvature by means of phase-field model. We introduce the variation in time of the spontaneous curvature via a second field which represents the concentration of a substance that anchors with the lipid bilayer thus changing the local curvature and producing constriction. This constriction is mediated by the action on the membrane of an structure resembling the role of a Z ring. Our phase-field model is able to reproduce a number of different shapes that have been experimentally observed. Different shapes are associated with different constraints imposed upon the model regarding conservation of membrane area. In particular, we show that if area is conserved our model reproduces the so-called L-form shape. By contrast, if the area of the membrane is allowed to grow, our model reproduces the formation of a septum in the vicinity of the constriction. Furthermore, we propose a new term in the free energy which allows the membrane to evolve towards eventual pinching.

Front microrheology of the non-Newtonian behaviour of blood: scaling theory of erythrocyte aggregation by aging.

We introduce a new framework to study the non-Newtonian behaviour of fluids at the microscale based on the analysis of front advancement. We apply this methodology to study the non-linear rheology of blood in microchannels. We carry out experiments in which the non-linear viscosity of blood samples is quantified at different haematocrits and ages. Under these conditions, blood exhibits a power-law dependence on the shear rate. In order to analyse our experimental data, we put forward a scaling theory which allows us to define an adhesion scaling number. This theory yields a scaling behaviour of the viscosity expressed as a function of the adhesion capillary number. By applying this scaling theory to samples of different ages, we are able to quantify how the characteristic adhesion energy varies as time progresses. This connection between microscopic and mesoscopic properties allows us to estimate quantitatively the change in the cell-cell adhesion energies as the sample ages.

Phase-field modelling of the dynamics of Z-ring formation in liposomes: Onset of constriction and coarsening.

We propose a model for the dynamics of the formation of rings of FtsZ on tubular liposomes which produce constriction on the corresponding membrane. Our phase-field model is based on a simple bending energy that captures the dynamics of the interplay between the protein and the membrane. The short-time regime is analyzed by a linear dispersion relation, with which we are able to predict the number of rings per unit length on a tubular liposome. We study numerically the long-time dynamics of the system in the non-linear regime where we observe coarsening of Z-rings on tubular liposomes. In particular, our numerical results show that, during the coarsening process, the number of Z-rings decreases as the radius of tubular liposome increases. This is consistent with the experimental observation that the separation between rings is proportional to the radius of the liposome. Our model predicts that the mechanism for the increased rate of coarsening in liposomes of larger radius is a consequence of the increased interface energy.

Theory of wetting-induced fluid entrainment by advancing contact lines on dry surfaces.

We report on the onset of fluid entrainment when a contact line is forced to advance over a dry solid of arbitrary wettability. We show that entrainment occurs at a critical advancing speed beyond which the balance between capillary, viscous, and contact-line forces sustaining the shape of the interface is no longer satisfied. Wetting couples to the hydrodynamics by setting both the morphology of the interface at small scales and the viscous friction of the front. We find that the critical deformation that the interface can sustain is controlled by the friction at the contact line and the viscosity contrast between the displacing and displaced fluids, leading to a rich variety of wetting-entrainment regimes. We discuss the potential use of our theory to measure contact-line forces using atomic force microscopy and to study entrainment under microfluidic conditions exploiting colloid-polymer fluids of ultralow surface tension.

Phase-field model for the morphology of monolayer lipid domains.

Phase-separated domains exist in multicomponent lipid monolayers and bilayers. We present here a phase-field model that takes into account the competition between lipid dipole-dipole interactions and line tension to define the domain morphology. A dynamic equation for the phase-field is solved numerically showing stationary non-circular shapes like starfish shapes. This phase-field model could be applied to study the dynamic properties of complex problems like phase segregation in pulmonary surfactant membranes and films.

Tumor angiogenesis and vascular patterning: a mathematical model.

Understanding tumor induced angiogenesis is a challenging problem with important consequences for diagnosis and treatment of cancer. Recently, strong evidences suggest the dual role of endothelial cells on the migrating tips and on the proliferating body of blood vessels, in consonance with further events behind lumen formation and vascular patterning. In this paper we present a multi-scale phase-field model that combines the benefits of continuum physics description and the capability of tracking individual cells. The model allows us to discuss the role of the endothelial cells' chemotactic response and proliferation rate as key factors that tailor the neovascular network. Importantly, we also test the predictions of our theoretical model against relevant experimental approaches in mice that displayed distinctive vascular patterns. The model reproduces the in vivo patterns of newly formed vascular networks, providing quantitative and qualitative results for branch density and vessel diameter on the order of the ones measured experimentally in mouse retinas. Our results highlight the ability of mathematical models to suggest relevant hypotheses with respect to the role of different parameters in this process, hence underlining the necessary collaboration between mathematical modeling, in vivo imaging and molecular biology techniques to improve current diagnostic and therapeutic tools.

Pinning and avalanches in hydrophobic microchannels.

Rare events appear in a wide variety of phenomena such as rainfall, floods, earthquakes, and risk. We demonstrate that the stochastic behavior induced by the natural roughening present in standard microchannels is so important that the dynamics for the advancement of a water front displacing air has plenty of rare events. We observe that for low pressure differences the hydrophobic interactions of the water front with the walls of the microchannel put the front close to the pinning point. This causes a burstlike dynamics, characterized by series of pinning and avalanches, that leads to an extreme-value Gumbel distribution for the velocity fluctuations and a nonclassical time exponent for the advancement of the mean front position as low as 0.38.

Controlled drop emission by wetting properties in driven liquid filaments.

The controlled formation of micrometre-sized drops is of great importance to many technological applications. Here we present a wetting-based destabilization mechanism of forced microfilaments on either hydrophilic or hydrophobic stripes that leads to the periodic emission of droplets. The drop emission mechanism is triggered above the maximum critical forcing at which wetting, capillarity, viscous friction and gravity can balance to sustain a stable driven contact line. The corresponding critical filament velocity is predicted as a function of the static wetting angle, which can be tuned through the substrate behaviour, and shows a strong dependence on the filament size. This sensitivity explains the qualitative difference in the critical velocity between hydrophilic and hydrophobic stripes, and accounts for previous experimental results of splashing solids. We demonstrate that this mechanism can be used to control independently the drop size and emission period, opening the possibility of highly monodisperse and flexible drop production techniques in open microfluidic geometries.

Controlling viscoelastic flow in microchannels with slip.

We show that viscoelastic flow in a microchannel under a dynamic pressure gradient dramatically changes with the value of the apparent slip. We demonstrate this by using classical hydrodynamics and the Navier boundary condition for the apparent slip. At certain driving frequencies, the flow is orders of magnitude different for systems with and without slip, implying that controlling the degree of hydrophobicity of a microchannel can lead to the control of the magnitude of the flow. We verify this for viscoelastic fluids with very different constitutive equations. Moreover, we demonstrate that flow, given a value of the apparent slip, is a non-monotonic function of the driving frequency and can be increased or reduced by orders of magnitude by slightly changing the frequency of the driving pressure gradient. Finally, we show that, for dynamic situations, slip causes and effectively thicker channel whose effective thickness depends on frequency. We have calculated relevant quantities for blood and a polymeric fluid in order to motivate experimental studies.

Curvature multiphase field model for phase separation on a membrane.

We propose a model to describe the physical mechanisms by which chemical substances separate in the vicinity of a membrane. We assume that the adsorption of the different components of a complex liquid on a membrane is governed by interactions that couple them to the spontaneous curvature of the membrane. This problem is relevant to many fields in science, as cell constriction and division, micelles with cosurfactants, holometamorphosis, and morphogenesis in general.

Frequency-induced stratification in viscoelastic microfluidics.

We present a mechanism in the field of microfluidics by which the stratification of a viscoelastic fluid can be induced in a channel on the microscale by applying a dynamic pressure gradient at frequencies within the range of sound. Stratification is obtained with identical layers, parallel to the channel walls, whose number can be tailored. These layers are separated by 2D zero-velocity planes. This would allow different tracer particles with small diffusion coefficients to be confined in different fluid layers within the same microchannel. We obtain analytical results that allow us to make theoretical predictions regarding the possible experimental realization of stratification in a microchannel using a biofluid. We find a relation among the diffusion coefficient, fluid properties, and microchannel thickness that establishes a condition for the confinement of tracer particles to a layer. This mechanism has potential use in micrototal analysis systems and MEMS-containing viscoelastic fluids.

Constricting force of filamentary protein rings evaluated from experimental results.

We present a model of Z -ring constriction in bacteria based on different experimental in vitro results. The forces produced by the Z ring due to lateral attraction of its constituent parts, estimated in previous studies that were based on FtsZ filaments observed by atomic force microscopy, are in good agreement with an estimation of the force required for recently found deformations in liposomes caused by FtsZ. These forces are calculated within the usual Helfrich energy formalism. In this context, we also explain the apparent attraction of multiple Z rings in the liposomes initially separated by small distances, as well as the stable distribution of rings separated by distances greater than approximately twice the diameter of the cylindrical liposomes. We adapted the model to the in vivo conditions imposed by the bacterial cell wall, concluding that the proposed mechanism gives a qualitative explanation for the force generation during bacterial division.

Activity statistics, avalanche kinetics, and velocity correlations in surface growth.

We investigate the complex spatiotemporal dynamics in avalanche driven surface growth by means of scaling theory. We study local activity statistics, avalanche kinetics, and temporal correlations in the global interface velocity, obtaining different scaling relationships among the involved critical exponents depending on how far from or close to a critical point the system is. Our scaling arguments are very general and connect local and global magnitudes through several scaling relationships. We expect our results to be applicable in a wide range of systems exhibiting interface kinetic roughening driven by avalanches of local activity, either critical or not. As an example we apply the scaling theory to analyze avalanches and roughening of forced-flow imbibition fronts in excellent agreement with phase-field numerical integrations.

Dynamics of gravity driven three-dimensional thin films on hydrophilic-hydrophobic patterned substrates.

We investigate numerically the dynamics of unstable gravity driven three-dimensional thin liquid films on hydrophilic-hydrophobic patterned substrates. We explore longitudinally striped and checkerboard arrangements. Simulations show that for longitudinal stripes, the thin film can be guided preferentially on the hydrophilic stripes, while fingers develop on adjacent hydrophobic stripes if the width of the stripes is large enough. On checkerboard patterns, the film develops as a finger on hydrophobic domains, while it spreads laterally to cover the hydrophilic domains, providing a mechanism to tune the growth rate of the film. By means of kinematical arguments, we quantitatively predict the growth rate of the contact line on checkerboard arrangements, providing a first step toward potential techniques that control thin film growth in experimental setups.

Single-phase-field model of stepped surfaces.

We formulate a phase-field description of step dynamics on vicinal surfaces that makes use of a single dynamical field, at variance with previous analogous works in which two coupled fields are employed, namely, a phase-field proper plus the physical adatom concentration. Within an asymptotic sharp interface limit, our formulation is shown to retrieve the standard Burton-Cabrera-Frank model in the general case of asymmetric attachment coefficients (Ehrlich-Schwoebel effect). We confirm our analytical results by means of numerical simulations of our phase-field model. Our present formulation seems particularly well adapted to generalization when additional physical fields are required.

Avalanche dynamics in fluid imbibition near the depinning transition.

We study avalanche dynamics and local activity of forced-flow imbibition fronts in disordered media. We focus on the front dynamics as the mean velocity of the interface v is decreased and the pinning state is approached. Scaling arguments allow us to obtain the statistics of avalanche sizes and durations, which become power-law distributed due to the existence of a critical point at v=0 . Results are compared with phase-field numerical simulations.

Dynamic characterization of permeabilities and flows in microchannels.

We make an analytical study of the nonsteady flow of Newtonian fluids in microchannels. We consider the slip boundary condition at the solid walls with Navier hypothesis and calculate the dynamic permeability, which gives the system's response to dynamic pressure gradients. We find a scaling relation in the absence of slip that is broken in its presence. We discuss how this might be useful to experimentally determine--by means of microparticle image velocimetry technology--whether slip exists or not in a system, the value of the slip length, and the validity of Navier hypothesis in dynamic situations.

Influence of disorder strength on phase-field models of interfacial growth.

We study the influence of disorder strength on the interface roughening process in a phase-field model with locally conserved dynamics. We consider two cases where the mobility coefficient multiplying the locally conserved current is either constant throughout the system (the two-sided model) or becomes zero in the phase into which the interface advances (one-sided model). In the limit of weak disorder, both models are completely equivalent and can reproduce the physical process of a fluid diffusively invading a porous media, where super-rough scaling of the interface fluctuations occurs. On the other hand, increasing disorder causes the scaling properties to change to intrinsic anomalous scaling. In the limit of strong disorder this behavior prevails for the one-sided model, whereas for the two-sided case, nucleation of domains in front of the invading front are observed.