Mechanokinetic model of cell membrane: theoretical analysis of plasmalemma homeostasis, growth and division

A theoretical model dealing with endocytosis, exocytosis and caveolae invagination, describing plasmalemma homeostasis during cell growth and division, is proposed. It considers transmembrane pressure, membrane tension and mechanosensitivity of membrane processes. Membrane hydraulic conductivity and the flux of transmembrane nonvesicular transport are taken into account. The developed mathematical analysis operates with a formulated set of constitutive equations describing the mechanical state and kinetics of changes in an open dynamic membrane system. The standard version of a model with adjusted parameters was implemented, and predictions including a discussion on the effect of possible parameter modifications were presented. Computer simulations indicate big changes in the magnitude of membrane tension and elasticity, and in the number of membrane buddings in young cells and during mitosis. They also show the extent of cell growth inhibition resulting from a decrease in transmembrane transport or an increase in the exerted difference in osmotic pressure. Moreover, the simulations reveal that exocytosis regulated during mitosis may not be as important for cell growth, as sometimes presumed. Finally, practical application and possible extension of the model are discussed.     

Ion fluxes,transmembrane potential,and osmotic stabilization: a new dynamic electrophysiological model for eukaryotic cells

Survival of mammalian cells is achieved by tight control of cell volume, while transmembrane potential has been known to control many cellular functions since the seminal work of Hodgkin and Huxley. Regulation of cell volume and transmembrane potential have a wide range of implications in physiology, from neurological and cardiac disorders to cancer and muscle fatigue. Therefore, understanding the relationship between transmembrane potential, ion fluxes, and cell volume regulation has become of great interest. In this paper we derive a system of differential equations that links transmembrane potential, ionic concentrations, and cell volume. In particular, we describe the dynamics of the cell within a few seconds after an osmotic stress, which cannot be done by the previous models in which either cell volume was constant or osmotic regulation instantaneous. This new model demonstrates that both membrane potential and cell volume stabilization occur within tens of seconds of changes in extracellular osmotic pressure. When the extracellular osmotic pressure is constant, the cell volume varies as a function of transmembrane potential and ion fluxes, thus providing an implicit link between transmembrane potential and cell volume. Experimental data provide results that corroborate the numerical simulations of the model in terms of time-related changes in cell volume and dynamics of the phenomena. This paper can be seen as a generalization of previous electrophysiological results, since under restrictive conditions they can be derived from our model.     

Implicit membrane treatment of buried charged groups: Application to peptide translocation across lipid bilayers

The energetic cost of burying charged groups in the hydrophobic core of lipid bilayers has been controversial, with simulations giving higher estimates than certain experiments. Implicit membrane approaches are usually deemed too simplistic for this problem. Here we challenge this view. The free energy of transfer of amino acid side chains from water to the membrane center predicted by IMM1 is reasonably close to all-atom free energy calculations. The shape of the free energy profile, however, for the charged side chains needs to be modified to reflect the all-atom simulation findings (IMM1-LF). Membrane thinning is treated by combining simulations at different membrane widths with an estimate of membrane deformation free energy from elasticity theory. This approach is first tested on the voltage sensor and the isolated S4 helix of potassium channels. The voltage sensor is stably inserted in a transmembrane orientation for both the original and the modified model. The transmembrane orientation of the isolated S4 helix is unstable in the original model, but a stable local minimum in IMM1-LF, slightly higher in energy than the interfacial orientation. Peptide translocation is addressed by mapping the effective energy of the peptide as a function of vertical position and tilt angle, which allows identification of minimum energy pathways and transition states. The barriers computed for the S4 helix and other experimentally studied peptides are low enough for an observable rate. Thus, computational results and experimental studies on the membrane burial of peptide charged groups appear to be consistent. This article is part of a Special Issue entitled: Interfacially Active Peptides and Proteins. Guest Editors: William C. Wimley and Kalina Hristova.     

Mathematical modeling of transmembrane calcium transport kinetics in smooth muscle cells

A mathematical model, which describes kinetics of transmembrane calcium transport in a smooth muscular cell, has been elaborated and investigated taking into account that the change of calcium cations concentration within a cell is determined by two mutually opposite processes: an increase of a carrying capacity of calcium channels of plasma membrane under signal substance action and calcium removal from the intracellular space by Mg2+, ATP-dependent calcium pump localized on the plasma membrane. The fundamental difference of the proposed model against the models analyzed in literature before is that the cellular system returns to the initial stationary state after enzyme-catalysed transformation of the signal substance. The results of calculations showed that this model really described the experimental kinetics of the transmembrane calcium transport. In this paper the influence of different parameters (Michaelis constant and ultimate rate of calcium pump, initial concentrations of signal substance and enzyme decomposing it, rate constants) on kinetics of calcium transport through the plasma membrane has been investigated in detail.     

A mathematical model of the P-glycoprotein pump as a mediator of multidrug resistance

Cells displaying the classic multidrug resistant (MDR) phenotype possess a transmembrane protein (p170 or P-glycoprotein) which can actively extrude cytotoxic agents from the cytoplasm. A mathematical model of this drug efflux pump has been developed. Outward transport is modeled as a facilitated diffusion process. Since energy-dependent efflux of cytotoxic agents requires that ATP also bind to p170, the model includes a dynamic calculation for efflux rate which considers Michaelis-Menten kinetics for both the substrate agent and ATP. The final system consists of one partial differential equation (PDE) for the facilitated diffusion of substrate agents out of the cell a 2×2 ordinary differential equation (ODE) system for the dynamic calculation of the ATP-ADP pool, and a dynamic algebraic calculation of the efflux rate given substrate levels at the interior cell membrane interface and ATP levels in the cell. A stability analysis of the ATP-ADP pool distribution and a simplistic closed form solution of the linearized PDE are included. Numerical simulations are also provided.     

Theoretical model for cellular shapes driven by protrusive and adhesive forces

The forces that arise from the actin cytoskeleton play a crucial role in determining the cell shape. These include protrusive forces due to actin polymerization and adhesion to the external matrix. We present here a theoretical model for the cellular shapes resulting from the feedback between the membrane shape and the forces acting on the membrane, mediated by curvature-sensitive membrane complexes of a convex shape. In previous theoretical studies we have investigated the regimes of linear instability where spontaneous formation of cellular protrusions is initiated. Here we calculate the evolution of a two dimensional cell contour beyond the linear regime and determine the final steady-state shapes arising within the model. We find that shapes driven by adhesion or by actin polymerization (lamellipodia) have very different morphologies, as observed in cells. Furthermore, we find that as the strength of the protrusive forces diminish, the system approaches a stabilization of a periodic pattern of protrusions. This result can provide an explanation for a number of puzzling experimental observations regarding cellular shape dependence on the properties of the extra-cellular matrix.     

Design and performance study of a novel immobilized hollow fiber membrane bioreactor

A dual-layer coaxial hollow fiber (DLHF) bioreactor for cell immobilization developed to overcome nutrients transport limitation is presented. Cells were contained in the annular space between two coaxial hollow fibers, and nutrients were supplied by a forced convective transport from the shell side through the annular space to the lumen side. With judicious selection of the membrane materials, a low operating transmembrane pressure of 50 kPa, and using E. coli as the model organism, a high cell density of 10(11) cells/mL annular space volume and a high cell viability of (up to 80%) were obtained.     

Effect of electric field induced transmembrane potential on spheroidal cells: theory and experiment

The transmembrane potential on a cell exposed to an electric field is a critical parameter for successful cell permeabilization. In this study, the effect of cell shape and orientation on the induced transmembrane potential was analyzed. The transmembrane potential was calculated on prolate and oblate spheroidal cells for various orientations with respect to the electric field direction, both numerically and analytically. Changing the orientation of the cells decreases the induced transmembrane potential from its maximum value when the longest axis of the cell is parallel to the electric field, to its minimum value when the longest axis of the cell is perpendicular to the electric field. The dependency on orientation is more pronounced for elongated cells while it is negligible for spherical cells. The part of the cell membrane where a threshold transmembrane potential is exceeded represents the area of electropermeabilization, i.e. the membrane area through which the transport of molecules is established. Therefore the surface exposed to the transmembrane potential above the threshold value was calculated. The biological relevance of these theoretical results was confirmed with experimental results of the electropermeabilization of plated Chinese hamster ovary cells, which are elongated. Theoretical and experimental results show that permeabilization is not only a function of electric field intensity and cell size but also of cell shape and orientation.     

An experimental and theoretical analysis of ultrasound-induced permeabilization of cell membranes

Application of ultrasound transiently permeabilizes cell membranes and offers a nonchemical, nonviral, and noninvasive method for cellular drug delivery. Although the ability of ultrasound to increase transmembrane transport has been well demonstrated, a systematic dependence of transport on ultrasound parameters is not known. This study examined cell viability and cellular uptake of calcein using 3T3 mouse cell suspension as a model system. Cells were exposed to varying acoustic energy doses at four different frequencies in the low frequency regime (20-100 kHz). At all frequencies, cell viability decreased with increasing acoustic energy dose, while the fraction of cells exhibiting uptake of calcein showed a maximum at an intermediate energy dose. Acoustic spectra under various ultrasound conditions were also collected and assessed for the magnitude of broadband noise and subharmonic peaks. While the cell viability and transport data did not show any correlation with subharmonic (f/2) emission, they correlated with the broadband noise, suggesting a dominant contribution of transient cavitation. A theoretical model was developed to relate reversible and irreversible membrane permeabilization to the number of transient cavitation events. The model showed that nearly every stage of transient cavitation, including bubble expansion, collapse, and subsequent shock waves may contribute to membrane permeabilization. For each mechanism, the volume around the bubble within which bubbles induce reversible and irreversible membrane permeabilization was determined. Predictions of the model are consistent with experimental data.     

Theoretical model of thalassemic erythrocyte shape transformation

Our earlier model of reticulocyte shape transformation [Pawlowski, P.H., Burzynska, B., Zielenkiewicz, P., 2006. Theoretical model of reticulocyte to erythrocyte shape transformation. J. Theor. Biol. 243, 24-38] was applied to explain the morphological properties of thalassemic erythrocytes. Modification of the standard set of parameters of the model, describing minimal cell volume, membrane bending rigidity, and membrane tension, allowed for simulation of development of α- and β-thalassemic cells from splenectomized and nonsplenectomized individuals. This resulted in observation of thin rim discocytes, tailed erythrocytes and oval forms, as well as in differentiation of time of the cell shape metamorphosis. A comparative analysis of the susceptibility of thalassemic and normal erythrocytes to undergo deformation as well of their stability was performed.     

Shapes of bilayer vesicles with membrane embedded molecules

The interdependence of the lateral distribution of molecules which are embedded in a membrane (such as integral membrane proteins) and the shape of a cell with no internal structure (such as phospholipid vesicles or mammalian erythrocytes) has been studied. The coupling of the lateral distribution of the molecules and the cell shape is introduced by considering that the energy of the membrane embedded molecule at a given site of the membrane depends on the curvature of the membrane at that site. Direct interactions between embedded molecules are not considered. A simple expression for the interaction of the membrane embedded molecule with the local membrane curvature is proposed. Starting from this interaction, the consistently related expressions for the free energy and for the distribution function of the embedded molecules are derived. The equilibrium cell shape and the corresponding lateral distribution of the membrane embedded molecules are determined by minimization of the membrane free energy which includes the free energy of the membrane embedded molecules and the membrane elastic energy. The resulting inhomogeneous distribution of the membrane embedded molecules affects the cell shape in a nontrivial manner. In particular, it is shown that the shape corresponding to the absolute energy minimum at given cell volume and membrane area may be elliptically non-axisymmetric, in contrast to the case of a laterally homogeneous membrane where it is axisymmetric.     

Emerging new paradigms for ABCG transporters

Every cell is separated from its external environment by a lipid membrane. Survival depends on the regulated and selective transport of nutrients, waste products and regulatory molecules across these membranes, a process that is often mediated by integral membrane proteins. The largest and most diverse of these membrane transport systems is the ATP binding cassette (ABC) family of membrane transport proteins. The ABC family is a large evolutionary conserved family of transmembrane proteins (> 250 members) present in all phyla, from bacteria to Homo sapiens, which require energy in the form of ATP hydrolysis to transport substrates against concentration gradients. In prokaryotes the majority of ABC transporters are involved in the transport of nutrients and other macromolecules into the cell. In eukaryotes, with the exception of the cystic fibrosis transmembrane conductance regulator (CFTR/ABCC7), ABC transporters mobilize substrates from the cytoplasm out of the cell or into specific intracellular organelles. This review focuses on the members of the ABCG subfamily of transporters, which are conserved through evolution in multiple taxa. As discussed below, these proteins participate in multiple cellular homeostatic processes, and functional mutations in some of them have clinical relevance in humans.     

Glutaraldehyde mediated echinocyte/discocyte transformation is Ca2+ dependent.

Echinocytes, which were produced from freshly banked blood by repeated washes in phosphate buffered saline, undergo a transformation to the discoid shape within less than 30 seconds of incubation in isotonic 0.05% glutaraldehyde pH 7.4. This echinocyte/discocyte transformation is not associated with a change of cell volume or critical hemolysis volume although a slight decrease of cellular deformability and a 4-8 fold increase of K+ efflux within 1 hour after glutaraldehyde incubation provide evidence of the fixative's attack on the cell membrane. Trypsination prior to the incubation in isotonic glutaraldehyde could not inhibit the shape change. Hypertonic glutaraldehyde solutions partially prevent the E/D transformation with regard to both the osmolarity of the medium and the permeability of the cell membrane. The glutaraldehyde stimulated transformation is entirely inhibited in the presence of a chelating agent the efficiency of which is overcome by addition of a more-than-equivalent amount of Ca2+. The mutual action of either agent is discussed, however, the mechanism of the phenomenon remains unclear.     

The physico-chemical mechanism of mediated transport. I. Facilitated diffusion

On the basis of the currently accepted model for the cell membrane structure, a physico-chemical model for mediated transport is developed and solved for the case of polar non-electrolyte migration through the cell membrane. The model considers the interstitial space defined by the transport protein subunits to be the migration pathway for polar solutes. A Langmuir-type adsorption equilibrium is assumed at the interfaces and a multicomponent diffusion mechanism of solute and water is postulated within the migration pathway, where the polar residues of the transport protein represent another component of the system. Membrane selectivity is governed by the adsorption constants, which are shown to affect strongly the kinetics of transport. Isosmotic transport and the volume change of the cell are important features incorporated in the model, which is shown to fulfill the peculiar properties of facilitated diffusion systems. It is concluded that the same type of pathway can be used for the transport of other polar solutes through existing or induced hydrophilic channels, for which a similar approach is suggested.     

The relationship between valinomycin-induced alterations in membrane phospholipid fatty acid turnover, membrane potential, and cell volume in the human erythrocyte

The relationship between alterations in transmembrane potential, cell volume, and phospholipid fatty acid turnover has been examined in human erythrocytes by treating the cells with the monovalent cation ionophore valinomycin. Valinomycin increases the cellular uptake of tetra[3H]phenylphosphonium ion by erythrocytes, indicating membrane hyperpolarization, and causes net loss of potassium chloride and water from the cells leading to a decrease in cell volume. Treatment of erythrocytes with valinomycin also enhances incorporation of [9, 10-(3)H]oleic acid into phospholipids, primarily diacylphosphatidylethanolamine. After replacing intracellular chloride with sulfate and treating cells with the anion transport inhibitor 4,4'-diisothiocyanostilbene-2,2'-disulfonate, exposure to valinomycin results in uptake of tetra[3H]phenylphosphonium ion and stimulation of [9, 10-(3)H]oleic acid incorporation, but, because anion efflux is prevented, no decrease in cell volume occurs. When tetra[3H]phenylphosphonium ion uptake is also prevented by suspending these cells in 125 mM KCl to dissipate the transmembrane potassium gradient, valinomycin still enhances [9, 10-(3)H] oleic acid incorporation into phospholipid. These results suggest that the presence of valinomycin in the membrane directly alters phospholipid fatty acid turnover and that some of the effects of this ionophore on cellular function previously attributed to alterations in transmembrane potential or cellular potassium content may instead be due to altered phospholipid turnover. Since it is possible that valinomycin may directly perturb phospholipid fatty acid turnover in other cells, the possibility that valinomycin-induced alterations in cellular function are due to altered phospholipid turnover rather than membrane hyperpolarization or altered potassium content should be considered in the interpretation of studies employing this ionophore.     

Exact solutions of a two parameter flux model and cryobiological applications

Solute-solvent transmembrane flux models are used throughout biological sciences with applications in plant biology, cryobiology (transplantation and transfusion medicine), as well as circulatory and kidney physiology. Using a standard two parameter differential equation model of solute and solvent transmembrane flux described by Jacobs [The simultaneous measurement of cell permeability to water and to dissolved substances, J. Cell. Comp. Physiol. 2 (1932) 427-444], we determine the functions that describe the intracellular water volume and moles of intracellular solute for every time t and every set of initial conditions. Here, we provide several novel biophysical applications of this theory to important biological problems. These include using this result to calculate the value of cell volume excursion maxima and minima along with the time at which they occur, a novel result that is of significant relevance to the addition and removal of permeating solutes during cryopreservation. We also present a methodology that produces extremely accurate sum of squares estimates when fitting data for cellular permeability parameter values. Finally, we show that this theory allows a significant increase in both accuracy and speed of finite element methods for multicellular volume simulations, which has critical clinical biophysical applications in cryosurgical approaches to cancer treatment.     

Red blood cell shape and deformability in the context of the functional evolution of its membrane structure

It is proposed that it is possible to identify some of the problems that had to be solved in the course of evolution for the red blood cell (RBC) to achieve its present day effectiveness, by studying the behavior of systems featuring different, partial characteristics of its membrane. The appropriateness of the RBC volume to membrane area ratio for its circulation in the blood is interpreted on the basis of an analysis of the shape behavior of phospholipid vesicles. The role of the membrane skeleton is associated with preventing an RBC from transforming into a budded shape, which could form in its absence due to curvature-dependent transmembrane protein-membrane interaction. It is shown that, by causing the formation of echinocytes, the skeleton also acts protectively when, in vesicles with a bilayer membrane, the budded shapes would form due to increasing difference between the areas of their outer and inner layers.     

Molecular dynamics simulations to the bidirectional adhesion signaling pathway of integrin αVβ3

The bidirectional force transmission process of integrin through the cell membrane is still not well understood. Several possible mechanisms have been discussed in literature on the basis of experimental data, and in this study, we investigate these mechanisms by free and steered molecular dynamics simulations. For the first time, constant velocity pulling on the complete integrin molecule inside a dipalmitoyl-phosphatidylcholine membrane is conducted. From the results, the most likely mechanism for inside-out and outside-in signaling is the switchblade model with further separation of the transmembrane helices.     

Coarse-grained molecular dynamics simulations of the energetics of helix insertion into a lipid bilayer

Experimental and computational studies have indicated that hydrophobicity plays a key role in driving the insertion of transmembrane alpha-helices into lipid bilayers. Molecular dynamics simulations allow exploration of the nature of the interactions of transmembrane alpha-helices with their lipid bilayer environment. In particular, coarse-grained simulations have considerable potential for studying many aspects of membrane proteins, ranging from their self-assembly to the relation between their structure and function. However, there is a need to evaluate the accuracy of coarse-grained estimates of the energetics of transmembrane helix insertion. Here, three levels of complexity of model system have been explored to enable such an evaluation. First, calculated free energies of partitioning of amino acid side chains between water and alkane yielded an excellent correlation with experiment. Second, free energy profiles for transfer of amino acid side chains along the normal to a phosphatidylcholine bilayer were in good agreement with experimental and atomistic simulation studies. Third, estimation of the free energy profile for transfer of an arginine residue, embedded within a hydrophobic alpha-helix, to the center of a lipid bilayer gave a barrier of approximately 15 kT. Hence, there is a substantial barrier to membrane insertion for charged amino acids, but the coarse-grained model still underestimates the corresponding free energy estimate (approximately 29 kT) from atomistic simulations (Dorairaj, S., and Allen, T. W. (2007) Proc. Natl. Acad. Sci. U.S.A. 104, 4943-4948). Coarse-grained simulations were then used to predict the free energy profile for transfer of a simple model transmembrane alpha-helix (WALP23) across a lipid bilayer. The results indicated that a transmembrane orientation was favored by about -70 kT.     

Interaction and fusion dynamics between cellular blebs

Membrane blebbing, as a mechanism for cells to regulate their internal pressure and membrane tension, is believed to play important roles in processes such as cell migration, spreading and apoptosis. However, the fundamental question of how different blebs interact with each other during their life cycles remains largely unclear. Here, we report a combined theoretical and experimental investigation to examine how the growth and retraction of a cellular bleb are influenced by neighboring blebs as well as the fusion dynamics between them. Specifically, a boundary integral model was developed to describe the shape evolution of cell membrane during the blebbing/retracting process. We showed that a drop in the intracellular pressure will be induced by the formation of a bleb whose retraction then restores the pressure level. Consequently, the volume that a second bleb can reach was predicted to heavily depend on its initial weakened size and the time lag with respect to the first bleb, all in quantitative agreement with our experimental observations. In addition, it was found that as the strength of membrane-cortex adhesion increases, the possible coalescence of two neighboring blebs changes from smooth fusion to abrupt coalescence and eventually to no fusion at all. Phase diagrams summarizing the dependence of such transition on key physical factors, such as the intracellular pressure and bleb separation, were also obtained.