The role of cell volume in the dynamics of seizure, spreading depression, and anoxic depolarization. Rep. 24, 386–434. The PLM stabilizes cells against changes in extracellular osmolarity. Regulation of cellular volume. In addition, I will assume that the composition of the solution in the extracellular space is fixed, which is reasonable since the volume of the extracellular space if far larger than that of a single cell. Figure A1 shows that the form of the C−p curve is maintained even with this nonlinear pump mechanism. Boron, W. F., and Boulpaep, E. L. (2016). The polar heads electrostatically attract polarized water molecules while the nonpolar tails lie between the layers, shielded from water and creating a dry middle layer. Biosci. x is a heterogeneous collection of molecules, yet both it and z can be calculated. Burton, R. F. (1983a). Rev. As mentioned above the Na+ pump hyperpolarizes the cell. doi: 10.1111/j.1749-6632.1959.tb44168.x, Lee, J., and Kim, S. J. Figure 1. Physiol. An Integrated Theoretical Approach. This works by means of special channels called pumps, which are present in the cell plasma membrane, and that use up energy when they move molecules across the membrane. 100, 1–63. doi: 10.1007/978-3-642-81008-4. Perspect. It is worth noting that if a cell has no impermeant ions, which is clearly impossible, the system is stable in the absence of an NKA. Neuroimage 51, 713–722. Mol. This is shown in Figure 7 for two different values of the pump current. To see how this arises I will consider the effect of impermeant ions on the osmotic balance of cells. Moreover, as my simulations show (Figure 2), it may take a long time for ion gradients to dissipate after the application of a pump blocker like ouabain. It allows movement across its barrier by diffusion, osmosis, or active transport. This equation was derived by Boyle and Conway (1941), and I follow Fraser and Huang (2004) in calling it the “constraint equation.”. J. Bacteriol. p in (C mm−2 s−1). It endows them with an osmoregulatory mechanism that does not require rigid cells and it gives them a negative resting potential, which turned out to be very convenient for evolving excitable cells (Jakobsson, 1980; Armstrong, 2015). Lett. I haven't quite figured out what it has to do with selective permeability, any ideas? Figure 6. The % of the minimum volume is plotted as function of p on the lowest panel. To break the expressions down into their components, the term gNa[V-RTFln(NaoNa)] is Ohm's law, which combines the chemical and electrical potentials, and RTTln(NaoNa) is often referred to as the “Nernst potential.” While, the NKA is represented by a constant pump rate p with n and q being the number of Na+ and K+ ions, respectively transported per cycle. Clin. Wood, J. M. (2015). If the cells were larger by an order of magnitude the average diffusion time across a cell would rise by two orders of magnitude. Biol. Where w is the cell volume. At equilibrium, diffusion continues, but the net flow balances except for random fluctuations. This can be seen by plotting these variables as a function of z with a fixed number of moles of x (Figure 4). As p is increased, although Na+ and K+ are being pumped, it is Cl− and X that first respond to this applied force. J. Gen. Physiol. 15, 637–654. A mathematical model of the volume, pH, and ion content regulation in reticulocytes. J. Gen. Physiol. The PLM also liberates cells from the straitjacket of cell walls allowing the evolution of contractile mechanisms. doi: 10.1073/pnas.0931278100, Armstrong, C. M. (2015). Acta 25, 118–128. Osmolarity is the term used to describe the concentration of solute particles per liter. membranes enable organisms to maintain homeostasis by regulating the materials that may enter or ... • semi-permeability (selective permeability) • transport protein . Transport by osmosis is affected by the concentration of solute (the number of particles) in the water. Moreover, if the sodium conductance is zero, the system of equations is unstable, with the Na+ concentration becoming negative, which is clearly impossible. Stein, W. D. (1990). In addition it is possible that other pumps may take over and sustain the volume. (2011). The effect of z on ion concentrations, voltage and cell radius. 349, 263–269. I thank James Keener (Univ. 25, 127–135. It is this PLM that I will discuss in the rest of this article. So I think the fluid mosaic model has to do with selective permeability because they both have to do with the cell's membrane. The fluid mosaic model is a model that shows where the substances such as proteins and lipids within the cell membrane are arranged. Allen, R. D., and Naitoh, Y. 1: Transport. Cellular homeostasis involves maintaining a balance of several factors that make a cell healthy. Sunderland, MA: Sinauer Associates, Inc. Ostby, I., Oyehaug, L., Einevoll, G. T., Nagelhus, E. A., Plahte, E., Zeuthen, T., et al. Astrocytic mechanisms explaining neural-activity-induced shrinkage of extraneuronal space. doi: 10.1006/jtbi.1994.1129, Kager, H., Wadman, W. J., and Somjen, G. G. (2002). Rev. PLoS Comput. Hernandez, J., and Cristina, E. (1998). A solute is the substance dissolved in the solvent. Ion Transport through Biological Membranes. The outer surface of each layer is made up of tightly packed hydrophilic (or water-loving) polar heads. (1980). doi: 10.1152/advan.00039.2014, Keener, J., and Sneyd, J. Front. Berlin; Heidelberg; New York: Springer Verlag. To model the Na+/H+ exchanger I assume that it can be described by the following chemical equilibrium: Where H and H0 are the inside and outside proton concentrations respectively. For the case where z = −1, pmin can be found by differentiating Equation (A12) with respect to p, setting this derivative equal to zero and solving for p: Equation (9) encapsulates how the energy required to maintain cell volume depends on the systems parameters. At steady state each of the current flows is zero and the system of Equations (1, 2, 4–6) can be solved exactly for the five unknowns Na, K, Cl, w, and V: I follow Keener and Sneyd in defining the following non-dimensional variables: Keener and Sneyd demonstrated that for nNaogNa>qKogK there is a range of p for which the system has a finite positive cell volume, with the following solutions: In the Keener-Sneyd equations the NKA is represented by as constant term (Equations 4 and 5). Application to the pathophysiology of sickle cell dehydration. A useful way of thinking about the PLM is that it creates a negative potential and Cl− follows its equilibrium making space to accommodate x. Cell size regulation is a fundamental aspect of both single and multi-cellular organisms. The Biological Chemistry of the Elements: The Inorganic Chemistry of Life. A Cp plot of a PLM model incorporating a Na/H exchanger, a proton pump and conductance (see Appendix A7). Mathematical properties of pump-leak models of cell volume control and electrolyte balance. Cation-chloride cotransporters in neuronal development, plasticity and disease. I will argue that this is so because, for the most part, the understanding of electrical current flow is not considered a necessary part of a cell biologist's intellectual toolbox. All cells have a problem that stems from their need for an inventory of impermeant molecules (metabolites, proteins, nucleic acids, etc., Burton, 1983b) that sets up an unstable osmotic condition, which could rupture the plasma membrane if left unattended (Stein, 2002; Armstrong, 2003; Dawson and Liu, 2008). Physiol. In passive transport across a selectively permeable membrane, molecules move across the membrane without the cell having to expend any extra energy. New York, NY: Garland Science. Plants and prokaryotes solve the problem by building cell walls that can resist turgor pressure (Haswell and Verslues, 2015; Wood, 2015). Lond. Sunderland, MA: Sinauer Associates, Inc. Sperelakis, N. (2012). Microbiol. doi: 10.1172/JCI114958. If p is increased above this, the volume then increases very slowly. 1. Water is a polar molecule that will not pass through the lipid bilayer; however, it’s small enough to move through the pores — formed by protein molecules — of most cell membranes. Modelling and Simulation in Medicine and the Life Sciences. I thought it useful to present it in part as a tutorial, since some of the information is collected in rather specialized publications, often directed at excitable cells. 87, 100–112. doi: 10.1007/978-0-387-21571-6, Hutchison, C. A. III, Chuang, R. Y., Noskov, V. N., Assad-Garcia, N., Deerinck, T. J., Ellisman, M. H., et al. 109, 173–177. The assumption of constant area is adopted to reflect the fact that over a short time scale the number of channels and transporters is unlikely to change much. Berg, H. C. (1993). Rungta, R. L., Choi, H. B., Tyson, J. R., Malik, A., Dissing-Olesen, L., Lin, P. J., et al. Physiol. This constitutes something of a problem for evaluating the PLM experimentally; even with very low pump rates the volume may be stable. These carrier or transport proteins interact with the passenger molecules and use the ATP-supplied energy to move them against the gradient. Stein, W. D. (1995). The Na+ and K+ gradients also serve as energy reservoirs for transporting other molecules against their concentration gradients and in establishing a negative resting membrane potential. (2009). Cell Physiol. This means that they move across the cell membrane from an area of high concentration to an area of low concentration. (2015). doi: 10.1007/s00285-011-0483-8. Cellular swelling occurs after cerebral trauma, stroke, spreading depression and epilepsy (Rungta et al., 2015; Stokum et al., 2016). J. Theor. 559, 459–478. When one plots the concentration of intracellular ions as a function of pump rate the ion concentrations form a braid (symmetrical for the case of z = −1 because of the osmotic and charge constraints (Figure 5, 2nd panel from the top and see Appendix A4). Cytol. J. Gen. Physiol. 9, 330–343. Although the link between active and passive ion fluxes and cell size has been known for over 50 years, it seems not to have had much impact in cell biology. New York, NY: Springer. The diffusion rate depends on the mass and temperature of the molecule; lighter and warmer molecules move faster. Water is removed or added respectively to the default extracellular solution. In addition, there are secondary active transport processes that are similar to diffusion but instead use imbalances in electrostatic forces to move molecules across the membrane. However, it only determines the sum of Na+ and K+. This mechanism is incapable of generating a stable volume and generates a strong depolarization. Mori (2012) has demonstrated that the PLEs have an asymptotically stable steady state so long as the pump current is not too large. Biochim. Physiol. A hypertonic solution has more solute and lower water potential than inside the cell. Physiol. At first glance in the huge array of cellular factors, these may seem somewhat remote but they are tied together through the osmotic movement of water across the plasma membrane. Cell biology. Both cytoplasm and the matrix, the material in which cells lie, are primarily water. Physiol. They move at high speeds, colliding with one another, changing directions, and moving away from areas of greater concentration to areas of lower concentration. Maintaining water balance in the body Kidneys. Whereas in the absence of impermeant molecules the “cell” is stable if the solution within the cell is identical to that outside. To determine the individual concentrations, I have to consider the forces acting on all the ions. Molecular pathophysiology of cerebral edema. Keener and Sneyd's solutions (KSS) to the PLEs are given in Appendix A1. To improve the flow of the equations I have adopted the following conventions: capitalized solutes represent concentrations, and lower cases represent amounts in moles. Rev. From the definition of capacitance, the voltage is given by the following equation (Varghese and Sell, 1997): I shall refer to the system of Equations (1), (2), (4), (5), (6), and (8) as the “pump-leak equations” (PLE).