Electromechanics and Volume Dynamics in Nonexcitable Tissue Cells

Cell volume regulation is fundamentally important in phenomena such as cell growth, proliferation, tissue homeostasis, and embryogenesis. How the cell size is set, maintained, and changed over a cell’s lifetime is not well understood. In this work we focus on how the volume of nonexcitable tissue cells is coupled to the cell membrane electrical potential and the concentrations of membrane-permeable ions in the cell environment. Specifically, we demonstrate that a sudden cell depolarization using the whole-cell patch clamp results in a 50% increase in cell volume, whereas hyperpolarization results in a slight volume decrease. We find that cell volume can be partially controlled by changing the chloride or the sodium/potassium concentrations in the extracellular environment while maintaining a constant external osmotic pressure. Depletion of external chloride leads to a volume decrease in suspended HN31 cells. Introducing cells to a high-potassium solution causes volume increase up to 50%. Cell volume is also influenced by cortical tension: actin depolymerization leads to cell volume increase. We present an electrophysiology model of water dynamics driven by changes in membrane potential and the concentrations of permeable ions in the cells surrounding. The model quantitatively predicts that the cell volume is directly proportional to the intracellular protein content.     

Cell Growth and Division: III. Conditions for Balanced Exponential Growth in a Mathematical Model

In a previous paper, we proposed a model in which the volume growth rate and probability of division of a cell were assumed to be determined by the cell's age and volume. Some further mathematical implications of the model are here explored. In particular we seek properties of the growth and division functions which are required for the balanced exponential growth of a cell population. Integral equations are derived which relate the distribution of birth volumes in successive generations and in which the existence of balanced exponential growth can be treated as an eigenvalue problem. The special case in which all cells divide at the same age is treated in some detail and conditions are derived for the existence of a balanced exponential solution and for its stability or instability. The special case of growth rate proportional to cell volume is seen to have neutral stability. More generally when the division probability depends on age only and growth rate is proportional to cell volume, there is no possibility of balanced exponential growth. Some comparisons are made with experimental results. It is noted that the model permits the appearance of differentiated cells. A generalization of the model is formulated in which cells may be described by many state variables instead of just age and volume.     

Endocytosis and exocytosis protect cells against severe membrane tension variations

The ability of cells to regulate their shape and volume is critical for many cell functions. How endocytosis and exocytosis, as important ways of membrane trafficking, affect cellular volume regulation is still unclear. Here, we develop a theoretical framework to study the dynamics of cell volume, endocytosis, and exocytosis in response to osmotic shocks and mechanical loadings. This model can not only explain observed dynamics of endocytosis and exocytosis during osmotic shocks but also predict the dynamics of endocytosis and exocytosis during cell compressions. We find that a hypotonic shock stimulates exocytosis, while a hypertonic shock stimulates endocytosis; and exocytosis in turn allows cells to have a dramatic change in cell volume but a small change in membrane tension during hyposmotic swelling, protecting cells from rupture under high tension. In addition, we find that cell compressions with various loading speeds induce three distinct dynamic modes of endocytosis and exocytosis. Finally, we show that increasing endocytosis and exocytosis rates reduce the changes in cell volume and membrane tension under fast cell compression, whereas they enhance the changes in cell volume and membrane tension under slow cell compression. Together, our findings reveal critical roles of endocytosis and exocytosis in regulating cell volume and membrane tension.     

Cellular Pressure and Volume Regulation and Implications for Cell Mechanics

In eukaryotic cells, small changes in cell volume can serve as important signals for cell proliferation, death, and migration. Volume and shape regulation also directly impacts the mechanics of cells and tissues. Here, we develop a mathematical model of cellular volume and pressure regulation, incorporating essential elements such as water permeation, mechanosensitive channels, active ion pumps, and active stresses in the cortex. The model can fully explain recent experimental data, and it predicts cellular volume and pressure for several models of cell cortical mechanics. Moreover, we show that when cells are subjected to an externally applied load, such as in an atomic force microscopy indentation experiment, active regulation of volume and pressure leads to a complex cellular response. Instead of the passive mechanics of the cortex, the observed cell stiffness depends on several factors working together. This provides a mathematical explanation of rate-dependent response of cells under force.     

Cellular Pressure and Volume Regulation and Implications for Cell Mechanics

In eukaryotic cells, small changes in cell volume can serve as important signals for cell proliferation, death, and migration. Volume and shape regulation also directly impacts the mechanics of cells and tissues. Here, we develop a mathematical model of cellular volume and pressure regulation, incorporating essential elements such as water permeation, mechanosensitive channels, active ion pumps, and active stresses in the cortex. The model can fully explain recent experimental data, and it predicts cellular volume and pressure for several models of cell cortical mechanics. Moreover, we show that when cells are subjected to an externally applied load, such as in an atomic force microscopy indentation experiment, active regulation of volume and pressure leads to a complex cellular response. Instead of the passive mechanics of the cortex, the observed cell stiffness depends on several factors working together. This provides a mathematical explanation of rate-dependent response of cells under force.     

Condition-dependent cell volume and concentration of Escherichia coli to facilitate data conversion for systems biology modeling

Systems biology modeling typically requires quantitative experimental data such as intracellular concentrations or copy numbers per cell. In order to convert population-averaging omics measurement data to intracellular concentrations or cellular copy numbers, the total cell volume and number of cells in a sample need to be known. Unfortunately, even for the often studied model bacterium Escherichia coli this information is hardly available and furthermore, certain measures (e.g. cell volume) are also dependent on the growth condition. In this work, we have determined these basic data for E. coli cells when grown in 22 different conditions so that respective data conversions can be done correctly. First, we determine growth-rate dependent cell volumes. Second, we show that in a 1 ml E. coli sample at an optical density (600 nm) of 1 the total cell volume is around 3.6 μl for all conditions tested. Third, we demonstrate that the cell number in a sample can be determined on the basis of the sample's optical density and the cells' growth rate. The data presented will allow for conversion of E. coli measurement data normalized to optical density into volumetric cellular concentrations and copy numbers per cell--two important parameters for systems biology model development.     

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.     

Effects of stiffness and volume on the transit time of an erythrocyte through a slit

By using a fully coupled fluid–cell interaction model, we numerically simulate the dynamic process of a red blood cell passing through a slit driven by an incoming flow. The model is achieved by combining a multiscale model of the composite cell membrane with a boundary element fluid dynamics model based on the Stokes flow assumption. Our concentration is on the correlation between the transit time (the time it takes to finish the whole translocation process) and different conditions (flow speed, cell orientation, cell stiffness, cell volume, etc.) that are involved. According to the numerical prediction (with some exceptions), the transit time rises as the cell is stiffened. It is also highly sensitive to volume increase inside the cell. In general, even slightly swollen cells (i.e., the internal volume is increased while the surface area of the cell kept unchanged) travel dramatically slower through the slit. For these cells, there is also an increased chance of blockage.     

A General Model for the Dynamics of the Cell Volume

The conservation of the cell volume within values compatible with the overall cell functions represents an ubiquitous property, shared by cells comprising the whole biological world. Water transport across membranes constitutes the main process associated to the dynamics of the cell volume, its chronic and acute regulations therefore represent crucial aspects of cell homeostasis. In spite of the biological diversity, the dynamics of the cell volume exhibits common basic features in the diverse types of cells. The purpose of this study is to show that there is a general model capable to describe the basic aspects of the dynamics of the cell volume. It is demonstrated here that the steady states of this model represent asymptotically stable configurations. As illustrations, several cases of non-polarized (i.e., symmetrical) and polarized (e.g., epithelial) cells performing water transport are shown here to represent particular cases of the general model. From a biological perspective, the existence of a general model for the dynamics of the cell volume reveals that, in spite of physiological and morphological peculiarities, there is a basic common design of the membrane transport processes. In view of its stability properties, this basic design may represent an ancestral property that has proven to be successful regarding the overall homeostatic properties of cells.     

Volumetric response of vertebrate hepatocytes challenged by osmotic gradients: a theoretical approach

In this study we use a theoretical approach to study the volumetric response of goldfish hepatocytes challenged by osmotic gradients and compared it with that of hepatocytes from another teleost (the trout) and a mammal (the rat). Particular focus was given to the multiple non-linear interactions of transport systems enabling hypotonically challenged cells to trigger a compensatory response known as volume regulatory decrease or RVD. For this purpose we employed a mathematical model which describes the rates of change of the intracellular concentrations of main diffusible ions, of the cell volume, and of the membrane potential. The model was fitted to experimental data on the kinetics of volume change of hepatocytes challenged by anisotonic media. In trout and rat hepatocytes, experimental results had shown that hypotonic cell swelling was followed by RVD, whereas goldfish cells swelled with no concomitant RVD (M.V. Espelt et al., 2003, J. Exp. Biol. 206, 513-522). A comparison between data predicted by the model and that obtained experimentally suggests that in trout and rat hepatocytes hypotonicity activates a sensor element and this, in turn, activates an otherwise silent efflux of KCl - whose kinetics could be successfully predicted - thereby leading to volume down-regulation. In contrast, with regard to the absence of RVD in goldfish hepatocytes the model proposed suggests that either a sensor element triggering RVD is absent or that the effector mechanism (the loss of KCl) remains inactive under the conditions employed. In line with this, we recently found that extracellular nucleotides may be required to induce RVD in these cells, indicating that our model could indeed lead to useful predictions.     

Regulatory volume decrease and P receptor signaling in fish cells: mechanisms, physiology, and modeling approaches

For animal cell plasma membranes, the permeability of water is much higher than that of ions and other solutes, and exposure to hyposmotic conditions almost invariably causes rapid water influx and cell swelling. In this situation, cells deploy regulatory mechanisms to preserve membrane integrity and avoid lysis. The phenomenon of regulatory volume decrease, the partial or full restoration of cell volume following cell swelling, is well-studied in mammals, with uncountable investigations yielding details on the signaling network and the effector mechanisms involved in the process. In comparison, cells from other vertebrates and from invertebrates received little attention, despite of the fact that e.g. fish cells could present rewarding model systems given the diversity in ecology and lifestyle of this animal group that may be reflected by an equal diversity of physiological adaptive mechanisms, including those related to cell volume regulation. In this review, we therefore present an overview on the most relevant aspects known on hypotonic volume regulation presently known in fish, summarizing transporters and signaling pathways described so far, and then focus on an aspect we have particularly studied over the past years using fish cell models, i.e. the role of extracellular nucleotides in mediating cell volume recovery of swollen cells. We, furthermore, present diverse modeling approaches developed on the basis of data derived from studies with fish and other models and discuss their potential use for gaining insight into the theoretical framework of volume regulation.     

Convergent extension by intercalation without mediolaterally fixed cell motion

We construct and implement a stochastic model of convergent extension, using a minimal set of assumptions on cell behavior. In addition to the basic assumptions of volume conservation, random cell motion, and cell-cell and cell-ECM adhesion, and a non-standard assumption that cytoskeletal polymerization generates an internal pressure tending to keep cells convex, we find that we need only two conditions for convergent extension. (1) Each cell type has a particular aspect ratio towards which it regulates its geometry. We do not require that cells align in a specific orientation, e.g. to be oriented mediolaterally. (2) The elongating tissue is composed of cells that prefer to be elongated, and these cells must be accompanied by cells which prefer to be round. The latter effectively provide a boundary to capture. In simulations, our model tissue extends and converges to a stacked arrangement of elongated cells one cell wide, an arrangement which is seen in ascidian notochords, but which has not been observed in other models. This arrangement is achieved without any direct mediolateral bias other than that which is provided by the physical edge of the adjacent tissue.     

Intercellular communication via intracellular calcium oscillations

In this letter, we present the results of a simple model for intercellular communication via calcium oscillations, motivated in part by a recent experimental study. The model describes two cells (a "donor" and "sensor") whose intracellular dynamics involve a calcium-induced, calcium release process. The cells are coupled by assuming that the input of the sensor cell is proportional to the output of the donor cell. As one varies the frequency of calcium oscillations of the donor cell, the sensor cell passes through a sequence of N : M phase-locked regimes and exhibits a "Devil's staircase" behavior. Such a phase-locked response has been seen experimentally in pulsatile stimulation of single cells. We also study a stochastic version of the coupled two-cell model. We find that phase locking holds for realistic choices for the cell volume.     

The interrelationship of cell growth and division in haploid and diploid cells of Saccharomyces cerevisiae

The coordination of cell growth and division has been examined in isogenic haploid and diploid strains of Saccharomyces cerevisiae. The average cell volume of the haploid and diploid cells was unaffected by a range of environmental conditions and generation times. For most environments and generation times the mean cell volume of diploid cells was between 1.52 and 1.83 of the haploid cell volume. Both haploid and diploid cell volumes were reduced drastically when the cells were grown in the chemostat with glucose as the limiting substrate. In this environment diploid cells have the same mean cell volume as haploid cells. Diploid cells are more elongated than haploid cells, and the characteristic shape (eccentricity) of the cells is unaffected by all environmental conditions and generation times tested. Mother cell volume increased during the cell cycle, although the pattern of this increase was affected by the environmental conditions. Under most growth conditions detectable mother cell volume increase occurred only during the budding phase, whereas under conditions of carbon limitation detectable increase only occurred during the unbudded phase. A consequence of this result is that the mean cell volume of haploids at bud initiation is relatively constant in all environments, including carbon limitation. This suggests that there is a critical size for bud initiation for haploids which is constant and independent of environmental conditions. The results for diploids are more complex. Coordination of growth and division in haploid cells can be explained by a simple model initially developed for prokaryotes by Donachie. A modification of this model is proposed to account for the results with diploids.     

Molecular crowding defines a common origin for the Warburg effect in proliferating cells and the lactate threshold in muscle physiology

Aerobic glycolysis is a seemingly wasteful mode of ATP production that is seen both in rapidly proliferating mammalian cells and highly active contracting muscles, but whether there is a common origin for its presence in these widely different systems is unknown. To study this issue, here we develop a model of human central metabolism that incorporates a solvent capacity constraint of metabolic enzymes and mitochondria, accounting for their occupied volume densities, while assuming glucose and/or fatty acid utilization. The model demonstrates that activation of aerobic glycolysis is favored above a threshold metabolic rate in both rapidly proliferating cells and heavily contracting muscles, because it provides higher ATP yield per volume density than mitochondrial oxidative phosphorylation. In the case of muscle physiology, the model also predicts that before the lactate switch, fatty acid oxidation increases, reaches a maximum, and then decreases to zero with concomitant increase in glucose utilization, in agreement with the empirical evidence. These results are further corroborated by a larger scale model, including biosynthesis of major cell biomass components. The larger scale model also predicts that in proliferating cells the lactate switch is accompanied by activation of glutaminolysis, another distinctive feature of the Warburg effect. In conclusion, intracellular molecular crowding is a fundamental constraint for cell metabolism in both rapidly proliferating- and non-proliferating cells with high metabolic demand. Addition of this constraint to metabolic flux balance models can explain several observations of mammalian cell metabolism under steady state conditions.     

Cholesterol depletion facilitates recovery from hypotonic cell swelling in CHO cells

The maintenance of cell volume homeostasis is critical for preventing pathological cell swelling that may lead to severe cellular dysfunction or cell death. Our earlier studies have shown that volume-regulated anion channels that play a major role in the regulation of cell volume are facilitated by a decrease in cellular cholesterol suggesting that cholesterol depletion should also facilitate regulatory volume decrease (RVD), the ability of cells to recover from hypotonic swelling. In this study, we test this hypothesis using a novel methodology developed to measure changes in cell volume using a microfluidics chamber. Our data show that cholesterol depletion of Chinese Hamster Ovary (CHO) significantly facilitates the recovery process, as is apparent from a faster onset of the RVD (162±10 s. vs. 114±5 s. in control and cholesterol depleted cells respectively) and a higher degree of volume recovery after 10 min of the hypotonic challenge (41%±6% vs. 65%±6% in control and cholesterol depleted cells respectively). In contrast, enriching cells with cholesterol had no effect on the RVD process. We also show here that similarly to our previous observations in endothelial cells, cholesterol depletion significantly increases the stiffness of CHO cells suggesting that facilitation of RVD may be associated with cell stiffening. Furthermore, we also show that increasing cell stiffness by stabilizing F-actin with jasplakinolide also facilitates RVD development. We propose that cell stiffening enhances cell mechano-sensitivity, which in turn facilitates the RVD process.     

Is competition between mesocarp cells of peach fruits affected by the percentage of wild species genome?

The number of cells and the mean cell volume in the mesocarps of fruits from peach genotypes with different percentages of the genome of Prunus davidiana, a wild, related, species, were evaluated. The mesocarp mass varied greatly between the four groups of genotypes. The mean cell volume and the number of cells were negatively correlated within each group. This correlation can be interpreted as a relationship of competition between cells. In order to describe the type of competition in the different groups, we tried to adjust a model of competition for resources proposed by Lescourret and Génard (Ecoscience 10:334–341, 2003). To estimate the values of the three parameters of the model for the different groups, we applied model selection. Within nested models, we identified a single best model with six parameter values. This model was roughly accurate, but it allowed us to describe the general relationship for each group. The parameter values revealed a strong and under-compensating density-dependence effect for all groups. The percentage of P. davidiana genome appeared to influence the maximal number of cells and the strength of the competition, but no effect was found on the maximal mean volume of cells.     

Cell cycle modelling

Models able to describe the events of cellular growth and division and the dynamics of cell populations are useful for the understanding of functional control mechanisms and for the theoretical support for automated analysis of flow cytometric data and of cell volume distributions. This paper reports on models that we have developed with this aim for different kinds of cells. The models are composed by two subsystems: one describes the growth dynamics of RNA and protein, and the second accounts for DNA replication and cell division, and describe in a rather unitary frame the cell cycle of eukaryotic cells, like mammalian cells and yeast, and of prokaryotic cells. The model is also used to study the effects of various sources of variability on the statistical properties of cell populations, and we find that in microbial cells the main source of variability appears to be an inaccuracy of the molecular mechanism that monitors cell size. In normal mammalian cells another source of variability, that depends upon the interaction with growth factors which give competence, is apparent. An extended version of the model, which comprises also this additional variability, is presented and used to describe the properties of mammalian cell growth.     

3-Dimensional microscopy as a method for volume measurement in cells undergoing apoptosis

Different patterns of cell volume perturbations are commonly used for modes of cell death: necrosis (cell swelling) and apoptosis (cell shrinkage). In this study we employed recently developed three dimensional microscopy for the measurement of the volume of attached vascular smooth muscle cells transfected with E1A-adenoviral protein. These cells undergo rapid apoptosis in the absence of growth factors or in the presence of staurosporine. In 30–60 min of serum deprivation the volume of these cells is increased by ～40% that corresponds to the time point of maximal activation of caspase 3 and chromatin cleavage. In 10–15 min swollen cells exhibit morphological collapse indicated by formation of apoptotic bodies. In contrast to serum-deprived cells, staurosporine leads to attenuation of cell volume by 30%. In this case, apoptotic bodies are detected in ～2.5 h after maximal shrinkage. Thus, our results show that cell shrinkage can not be considered as universal hallmark of apoptosis. The role of stimulus-specific cell volume perturbation in the triggering of the cell death machinery should be examined further.