Hydro-osmotic instabilities in active membrane tubes Sami C. Al-Izzi,1, 2, 3, 4 George Rowlands,2 Pierre Sens,3, 4 and Matthew S. Turner2, 5 1Department of Mathematics, University of Warwick, Coventry CV4 7AL, UK 2Department of Physics, University of Warwick, Coventry CV4 7AL, UK 3Institut Curie, PSL Research University, CNRS, Physical Chemistry Curie, F-75005, Paris, France 4Sorbonne Universit´es,UPMC Univ Paris 06, CNRS, UMR 168, F-75005, Paris, France 5Centre for Complexity Science, University of Warwick, Coventry CV4 7AL, UK We study a membrane tube with unidirectional ion pumps driving an osmotic pressure difference. A pressure driven peristaltic instability is identified, qualitatively distinct from similar tension- driven Rayleigh type instabilities on membrane tubes. We discuss how this instability could be related to the function and biogenesis of membrane bound organelles, in particular the contractile vacuole complex. The unusually long natural wavelength of this instability is in agreement with that observed in cells. The \blueprint" for internal structures in living cells Plasma (a) membrane (b) is genetically encoded but their spatio-temporal organi- sation ultimately rely on physical mechanisms. Pore u(z) A key contemporary challenge in cellular biophysics R is to understand the physical self-organization and reg- ulation of organelles [1, 2]. Eukaryotic organelles bound Tubular arm by lipid membranes perform a variety of mechanical and chemical functions inside the cell, and range in size, con- struction, and complexity [3]. A quantitative under- standing of how such membrane bound organelles func- tion have applications in bioengineering, synthetic biol- ogy and medicine. Most models of the shape regulation t of membrane bound organelles invoke local driving forces, e.g. membrane proteins that alter the morphology (of- FIG. 1. (a) Diagram of the contractile vacuole complex. The ten curvature) [4{6]. However other mechanisms, such as tube is shown connected to the main body of the CV (left). As ions are pumped in, increasing the osmotic pressure, the osmotic pressure, could play an important role [7]. tube undergoes a swelling instability and undulations develop Membrane tubes are ubiquitous in cells, being found in with some wavelength λ. This phenomena is observed in the organelles such as the Golgi and endoplasmic reticulum contractile vacuoles of, e.g. paramecium multimicronucleatum [3] and elsewhere. Models for their formation typically [9, 10]. (b) Schematic of a membrane tube with ion pumps involve the spontaneous curvature of membrane proteins and surface undulations. A cartoon of a representative ion pump is shown in the top right. [5] or forces arising from molecular motors, attached to the membrane, that pull tubular tethers as they move along microtubules [8]. Many of these tubules may con- tain trans-membrane proteins that can alter the osmotic environment, thereby regulating cell volume [9, 14]. Wa- pressure by active transport of ions. Most work on the ter influx into the CVC is due to an osmotic gradient gen- biogenesis of cellular organelles has focused on their static erated by ATP-hydrolysing proton pumps in the mem- morphology and generally not on their non-equilibrium brane that move protons into the CVC [12, 15{17]. In dynamics. In what follows we consider an example in many organisms such as Paramecium multimicronuclea- which the out-of-equilibrium dynamics drives the mor- tum, the CVC includes several membrane tubular arms arXiv:1709.02703v3 [physics.bio-ph] 30 Jan 2018 phology, Fig. 1. Our study is inspired by the biophysics connected to the main vesicles, which are thought to be of an organelle called the Contractile Vacuole Complex associated with the primary sites of proton pumping and but additionally reveals a new class of instabilities not water influx activity [18]. The tubular arms do not swell previously studied that are of broad, perhaps even uni- homogeneously in response to water influx, but rather versal, physiological relevance. show large undulatory bulges with a size comparable to The Contractile Vacuole Complex (CVC) is an or- the size of the main CV, leading us to speculate that this ganelle found in most freshwater protists and algae that might even play a role in CV formation de novo. These regulates osmotic pressure by expelling excess water [10{ tubular arms appear to be undergoing a process simi- 14]. Its primary features is a main vesicle (CV) that lar to the Pearling or Rayleigh instability of a membrane is inflated by osmosis and periodically expels its contents tube under high tension [19{26] or an axon under osmotic through the opening of a large pore - probably in response shock [27], but with a much longer natural wavelength: to membrane tension - connecting it to the extracellular Rayleigh instabilities have a natural wave length λ ∼ R 2 where R is the tube radius. Here we derive the dynamical Identifying the static pressure difference ∆P with the 3 evolution of a membrane tube driven out-of-equilibrium Laplace pressure PL = −κ/(2R ) + γ=R, the point at by osmotic pumping. which the q = 0 mode goes unstable can be identified:p In the CVC, the tubular arms are surrounded by a the membrane tube is unstable for tube radii R > 3Req membrane structure resembling a bicontinuous phase q κ where Req = is the equilibrium radius of a tube with made up of a labyrinth tubular network called the smooth 2γ ∆P = 0. This criterion for the onset of the instability is spongiome (SS). We assume this to represent a reservoir the same as the Rayleigh instability on a membrane tube of membrane keeping membrane tension constant and [24], however the instability is now driven by pressure uniform during tube inflation. It is possible to imple- not surface tension. This is a crucial difference. It leads ment more realistic area-tension relations [26], however to a qualitatively different evolution of the instability, this is beyond the scope of the present work. as we now show. In what follows we are interested in The CVC is comprised of a phospholipid bilayer mem- the dynamics of the growth of unstable modes after the brane. This bilayer behaves in an elastic manner [28, 29]. p cylinder has reached radius 3R . Our initial condition At physiological temperatures these lipids are in the fluid eq is a tube under zero net pressure, although the choice phase [3, 29]. For simplicity we will treat the bilayer as of initial condition is not crucial. We assume that the a purely elastic, fluid membrane in the constant tension number of proton pumps moving ions from the cytosol regime, neglecting the separate dynamics of each leaflet. into the tubular arm depends only on the initial surface The membrane free energy involves the mean curvature area, i.e. it is fixed as the tube volume (and surface) H and surface tension γ [28, 30, 31] varies. Z κ Z F = dA (2H)2 + γ − ∆P dV , (1) We denote the number of ions per unit length in the S 2 tube as n and write an equation for the growth of n as where dA and dV are the area and volume elements on ( dn 0; t 2 (−∞; 0) S, κ is the bending rigidity, and ∆P is the pressure dif- = (6) ference between the fluid inside and outside the tube. dt 2πβReq; t 2 [0; 1) Assuming radial symmetry and integrating over the vol- ume of the tube we obtain where β is a constant equal to the pumping rate of a sin- gle pump multiplied by the initial area density of pumps. 2 2 Z 1 ! The density of ions, ρI , can be obtained by solving κ 1 @zzr 1 F = 2π dz 4 r q 2 − Eq. (6) and dividing by volume per unit length, v(t), −∞ 2 2 1 + (@zr) r 1 + (@zr) n(t) n 2πβR t q 1 ρ = = 0 + eq . (7) +γr 1 + (@ r)2 − r2∆P (2) I v(t) v(t) v(t) z 2 where r(z; t) is the radial distance of the axisymmetric The growth of the tube radius is driven by a differ- membrane from the cylindrical symmetry axis and z mea- ence between osmotic and Laplace pressure [32]. This sures the coordinate along that axis, see S.I. for details. means the rate equation for the increase in volume can be We use Eq. (2) as a model for the free energy of a radial written in terms of the membrane permeability to water. arm of the CVC. Ion pumps create an osmotic pressure Assuming that the water permeability (number of wa- difference that drive a flux of water to permeate through ter permeable pores) is constant during tube inflation, we write the volume permeability per unit tube length the membrane. We calculate the dominant mode of the 0 hydro-osmotic instability resulting from the volume in- µ = 2πReqµ, where µ is the (initial) permeability of the crease of the tube lumen. We write the radius of the tube membrane. Thus as r(z; t) = R + u(z; t), with u assumed small, and make dv 0 P {qz = µ (kBT (ρI − ρI (t = 0)) − P ) (8) use of the Fourier representation u(z; t) = q u¯qe . dt Absorbing the q = 0 mode into R = R(t) allows us to write R u dz = 0. The free-energy per unit length can be where the osmotic pressure is approximated by an ideal written at leading order as gas law. This can be transformed into an equation for R(t) on the time interval t 2 [0; 1). We identify P with π X F = F (0) + α(q)ju¯ j2 (3) the Laplace pressure. This leads to R q q dR~ τ 1 t~ γ~ 1 where = pump + 1 + − 1 (9) dt~ τµ R~ R~2 R~ R~2 κ (qR)2 α(q) = (qR)4 − + 1 + γ(qR)2 − ∆PR (4) 2 γ Req ρI (t=0) t R 2 whereγ ~ = , τpump = , t~ = , kB TReq ρI (t=0) 2β τpump ~ R Req and R = and τµ = 0 .