Modelling the dynamics of cellular membranes

[eng] Membranes are present in all cells, in some viruses, and are involved in all kinds of biological functions. The goal of this thesis is to expand our knowledge of this element, in hope that this -on top of all the other knowledge of biology and physics- can help someday improve people's life. With this aim, what I tried to do was to understand how cells react when things happen to them. This is the drive behind the two different research paths written in this thesis: membranes inside a fluid flow, and membranes during topological transitions and fluctuations. For the first research path -on membranes inside a flow- while this is not a new topic we wanted to start by making it more approachable. That has been achieved by introducing a new methodology to couple membranes and flows by using the stream function and the vorticity to solve the Navier Stokes equation. This approach creates a model derived straight from the hydrodynamic equations and grounded on the physics of the system rather than other more complex approaches. With this model we tried to study the effects of confinement for membranes inside a Poiseuille flow. We mainly tried to replicate red blood cell shapes as it is a very researched case and there is plenty of experimental data on them. First starting with cells inside channels slightly bigger than their diameter, which is known to give a set of shapes named parachutes and slippers. We use this knowledge to prove the validity of our model. For very wide channels, the low confinement Poiseuille flows have shown a different meta-stable shape which we named anti-parachute. Moreover, tumbling can be produced by introducing a different viscosity for the cell fluid, higher than the surrounding fluid. In very narrow super-confined channels we have a Poiseuille flow where the cell is much bigger than the channel and gives very different shapes. However, the model is capable of studying other flows rather than Poiseuille. Couette flow has been studied, where one can see a lift perpendicular to the flow that depends on the reduced volume of the cell as well as the viscosity contrast. The most important thing has been leaving behind a methodology ready for expansion to time-dependent flows, inertial flows, or to generalize to 3 dimensions. For the second research path --on topological transitions-- we have implemented the Gaussian curvature energy term to the membrane model, to allow study of fission and fusion. With this methodology we study fission of tubes with the use of the spontaneous curvature, which deforms a membrane tube into a pearled tube. This pearled tube formed by an array of spheres connected through membrane tethers undergoes fission if the Gaussian rigidity is negative and high enough. A phase diagram of what happens depending on the values of Gaussian and bending rigidity is obtained. Then we expand to study geometries less helpful for fission, such as a flat planar membrane. It will not matter how big is the spontaneous curvature of the Gaussian rigidity, as a perfectly flat membrane is a meta-stable shape. This is due the fact that to start the fission process we need an area with enough curvature so that the spontaneous curvature can kick-off the membrane budding process. To solve this, we added a white noise to mimic temperature. This noise makes each point of the membrane position to fluctuate. There is a phase transition between a flat membrane that is not undergoing fission and one that does.[cat] Les membranes estan presents a totes les cèl·lules, en alguns virus, i estan implicades en tot tipus de funcions biològiques. L'objectiu d'aquesta tesi és ampliar el nostre coneixement d'aquest element, amb l'esperança que això –junt amb tots els altres coneixements de biologia i física– pugui ajudar algun dia a millorar la vida de les persones. Amb aquest objectiu, el que vaig intentar va ser entendre com reaccionen les cèl·lules quan els passen coses. Aquest és el objectiu de les dues parts en les que està dividida la recerca en aquesta tesi: membranes dins d'un flux de fluids i membranes durant transicions topològiques. Per a la primera part de la recerca –sobre membranes dins d'un flux–, tot i que no és un tema extremadament nou, hem volgut començar per fer-lo més accessible. Això s'ha aconseguit introduint una nova metodologia per acoblar membranes i fluxos. Amb aquest model hem intentat estudiar els efectes del confinament de les membranes dins d'un flux. Tanmateix, el més important ha estat deixar enrere una metodologia preparada per a l'expansió a fluxos dependents del temps, fluxos inercials o per expandir-se a 3 dimensions. Per a la segona part de la recerca –sobre transicions topològiques– hem implementat el terme d'energia de curvatura gaussiana al model de membrana, per permetre l'estudi de la fissió i la fusió. Amb això s'estudia la fissió de tubs amb l'ús de la curvatura espontània. A partir d'això ens expandim per estudiar geometries menys tolerants per a la fissió, com ara una membrana plana plana. Afegint una temperatura a les simulacions s'estudia com en funció de la temperatura es pot promoure fins i tot una membrana amb geometria difícil per generar vesícules