Phase-field theories for mathematical modeling of biological membranesView Publication
Biological membranes are complex structures whose mechanics are usually described at a mesoscopic level, such as the Helfrich bending theory. In this article, we present the phase-field methods, a useful tool for studying complex membrane problems which can be applied to very different phenomena. We start with an overview of the general theory of elasticity, paying special attention to its derivation from a molecular scale. We then study the particular case of membrane elasticity, explicitly obtaining the Helfrich bending energy. Within the framework of this theory, we derive a phase-field model for biological membranes and explore its physical basis and interpretation in terms of membrane elasticity. We finally explain three examples of applications of these methods to membrane related problems. First, the case of vesicle pearling and tubulation, when lipidic vesicles are exposed to the presence of hydrophobic polymers that anchor to the membrane, inducing a shape instability. Finally, we study the behavior of red blood cells while flowing in narrow microchannels, focusing on the importance of membrane elasticity to the cell flow capabilities.
Dynamics of Z-ring formation in liposomesView Publication
We propose a model for the dynamics of the formation of rings of FtsZ on tubular liposomes which produce constriction on the corresponding membrane. The main characteristic of our model is the energy of interaction between the FtsZ protein and the membrane. This coupling is achieved through the protein increasing the spontaneous curvature energy. Our analysis is based on a linear dispersion relation, valid in general in the linear, short-time regime, which allows us to estimate the minimum concentration of FtsZ protein in order to induce constriction. Using this dispersion relation we are also able to predict the effect of increasing membrane rigidity on the constriction process. Our model allows to study the non-linear, long- term dynamics of the system where we observe coarsening of Z-rings on tubular liposomes.
Rheology of red blood cells under flow in higly confined microchannels: II. Effect of focusing and confinementView Publication
We study the focusing of red blood cells and vesicles in pressure-driven flows in highly confined microchannels (10–30 ?m), identifying the control parameters that dictate the cell distribution along the channel. Our results show that an increase in the flow velocity leads to a sharper cell distribution in a lateral position of the channel. This position depends on the channel width, with cells flowing at outer (closer to the walls) positions in thicker channels. We also study the relevance of the object shape, exploring the different behaviour of red blood cells and different vesicles. We also analyze the implications of these phenomena in the cell suspension rheology, highlighting the crucial role of the wall confinement in the rheological properties of the suspension.
Rheology of red blood cells under flow in higly confined microchannels: I. Effect of elasticityView Publication
We analyze the rheology of dilute red blood cell suspensions in pressure driven flows at low Reynolds number, in terms of the morphologies and elasticity of the cells. We focus on narrow channels of width similar to the cell diameter, when the interactions with the walls dominate the cell dynamics. The suspension presents a shear-thinning behaviour, with a Newtonian-behaviour at low shear rates, an intermediate region of strong decay of the suspension viscosity, and an asymptotic regime at high shear rates in which the effective viscosity converges to that of the solvent. We identify the relevant aspects of cell elasticity that contribute to the rheological response of blood at high confinement. In a second paper, we will explore the focusing of red blood cells while flowing at high shear rates and how this effect is controlled by the geometry of the channel.
Elastic energies and morphologies of the first stages of the discoechinocyte transitionView Publication
Red blood cells are highly sensitive to changes in the relative areas of the two lipid leaflets of the cell membrane. Expansion of the outer leaflet forces the membrane to bend, leading to the deformation of the biconcave discocyte into increasingly spiculated shapes, in a well-defined series of cell shapes known as the discoechinocyte transition. We explore the first stages of this transition by means of an elastic membrane energy model that accounts for the bilayer and cytoskeleton contributions. The morphological evolution is explained in terms of the elastic response of these membrane components. Our results highlight the importance of the cytoskeleton as a stabilizing component and how it determines the strong sequential character of the development of different morphologies. In general, cells develop undulations around the cell contour prior to the growth of out-of-plane bumps; this was found to be due to the high energetic penalty relative to a limited area-difference benefit.
Theory of wetting-induced fluid entrainment by advancing contact lines on dry surfacesView Publication
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.
AFM measurements and lipid rearrangements: evidence from red blood cell shape changesView Publication
Application of force to echinocytes during atomic force microscopy measurements was shown to be able to convert the cells to stable discocyte shapes. The echinocyte shape is associated with a relative excess of the area of the outer leaflet of the cell membrane; the AFM measurements are therefore associated with a change in the relative areas of the inner and outer membrane leaflets. It was hypothesized that localised damage in the lipid bilayer that is caused by an AFM tip can permit the lipids to flip-flop between the two membrane leaflets, thus changing their relative areas. The conditions in which AFM measurements on cells could induce shape changes were investigated both experimentally and by modelling. The relative area change of the membrane leaflets, attributed here to lipid movement, was characterised in terms of the membrane energy levels; membrane energy was calculated using a version of the area-difference-elasticity model that was applied to predetermined shapes, rather than being used to generate shapes as solutions found at the energy minima. Shapes were generated by rotation of Cassini ovals with a superimposed undulation in order to generate spikes similar to those of the echinocytes. The membrane energy was considered as a function of the membrane curvature, the area difference between the two membrane leaflets, and the deformation of the cytoskeleton. This led to the conclusions that the minimisation of the membrane energy causes the lipid translocation, with the relaxation of the cytoskeleton being a significant driving force.
Phase-field model for the morphology of monolayer lipid domainsView Publication
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.
Controlling viscoelastic flow in microchannels with slipView Publication
Growth saturation of unstable thin films on transversed-striped hydrophilic-hydrophobic micropatterns;View Publication
Curvature multiphase-field model for phase separation on a membraneView Publication
Tumor angiogenesis and vascular patterning: A mathematica model;View Publication
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 microchanels;View Publication
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;View Publication
The controlled formation of micrometre-sized drops is of great importance to many technological applications 1, 2, 3, 4, 5. 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 solids6. 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.
Flower development as an interplay between dynamical physical field and genetic networks;View Publication
Dynamics of gravity driven three dimensional thin films on hydrophilic-hydrophobic patterned substrates;View Publication
We investigate numerically the dynamics of unstable gravity driven three-dimensional thin liquid films on hydrophilic-hydrophobic patterned substrates of longitudinal stripes and checkerboard arrangements. The thin film can be guided preferentially on hydrophilic longitudinal stripes, while fingers develop on adjacent hydrophobic stripes if their width is large enough. On checkerboard patterns, the film fingering occurs on hydrophobic domains, while lateral spreading is favoured on 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 towards potential techniques that control thin film growth in experimental setups.