Erratum to: Mathematical properties of pump-leak models of cell volume control and electrolyte balance

Homeostatic control of cell volume and intracellular electrolyte content is a fundamental problem in physiology and is central to the functioning of epithelial systems. These physiological processes are modeled using pump-leak models, a system of differential algebraic equations that describes the balance of ions and water flowing across the cell membrane. Despite their widespread use, very little is known about their mathematical properties. Here, we establish analytical results on the existence and stability of steady states for a general class of pump-leak models. We treat two cases. When the ion channel currents have a linear current-voltage relationship, we show that there is at most one steady state, and that the steady state is globally asymptotically stable. If there are no steady states, the cell volume tends to infinity with time. When minimal assumptions are placed on the properties of ion channel currents, we show that there is an asymptotically stable steady state so long as the pump current is not too large. The key analytical tool is a free energy relation satisfied by a general class of pump-leak models, which can be used as a Lyapunov function to study stability.     

Mathematical models of the balance between apoptosis and proliferation

As our understanding of cellular behaviour grows, and we identify more and more genes involved in the control of such basic processes as cell division and programmed cell death, it becomes increasingly difficult to integrate such detailed knowledge into a meaningful whole. This is an area where mathematical modelling can complement experimental approaches, and even simple mathematical models can yield useful biological insights. This review presents examples of this in the context of understanding the combined effects of different levels of cell death and cell division in a number of biological systems including tumour growth, the homeostasis of immune memory and pre-implantation embryo development. The models we describe, although simplistic, yield insight into several phenomena that are difficult to understand using a purely experimental approach. This includes the different roles played by the apoptosis of stem cells and differentiated cells in determining whether or not a tumour can grow; the way in which a density dependent rate of apoptosis (for instance mediated by cell-cell contact or cytokine signalling) can lead to homeostasis; and the effect of stochastic fluctuations when the number of cells involved is small. We also highlight how models can maximize the amount of information that can be extracted from limited experimental data. The review concludes by summarizing the various mathematical frameworks that can be used to develop new models and the type of biological information that is required to do this.     

Mathematical models of cell cycle regulation

The cell division cycle is a fundamental process of cell biology and a detailed understanding of its function, regulation and other underlying mechanisms is critical to many applications in biotechnology and medicine. Since a comprehensive analysis of the molecular mechanisms involved is too complex to be performed intuitively, mathematical and computational modelling techniques are essential. This paper is a review and analysis of recent approaches attempting to model cell cycle regulation by means of protein-protein interaction networks.     

Mathematical models in mammalian cell biology

A report on the Conference on Systems Biology of Mammalian Cells, Dresden, Germany, 22-24 May 2008.     

Perturbation of the pump-leak balance for Na(+) and K(+) in malaria-infected erythrocytes

In humanerythrocytes infected with the mature form of the malaria parasitePlasmodium falciparum, the cytosolic concentration ofNa⁺ is increased and that of K⁺ is decreased.In this study, the membrane transport changes underlying thisperturbation were investigated using a combination of⁸⁶Rb⁺, ⁴³K⁺, and²²Na⁺ flux measurements and a semiquantitativehemolysis technique. From >15 h postinvasion, there appeared in theinfected erythrocyte membrane new permeation pathways (NPP) that causeda significant increase in the basal ion permeability of theerythrocyte membrane and that were inhibited by furosemide (0.1 mM). The NPP showed the selectivity sequenceCs⁺ > Rb⁺ > K⁺ > Na⁺, with the K⁺-to-Na⁺permeability ratio estimated as 2.3. From 18 to 36 h postinvasion, the activity of the erythrocyte Na⁺/K⁺ pumpincreased in response to increased cytosolic Na⁺ (aconsequence of the increased leakage of Na⁺ via the NPP)but underwent a progressive decrease in the latter 12 h of theparasite's occupancy of the erythrocyte (36-48 h postinvasion). Incorporation of the measured ion transport rates into a mathematical model of the human erythrocyte indicates that the induction of the NPP,together with the impairment of the Na⁺/K⁺pump, accounts for the altered Na⁺ and K⁺levels in the host cell cytosol, as well as predicting an initial decrease, followed by a lytic increase in the volume of the host erythrocyte.

     

Mathematical models of specificity in cell signaling

Cellular signaling pathways transduce extracellular signals into appropriate responses. These pathways are typically interconnected to form networks, often with different pathways sharing similar or identical components. A consequence of this connectedness is the potential for cross talk, some of which may be undesirable. Indeed, experimental evidence indicates that cells have evolved insulating mechanisms to partially suppress "leaking" between pathways. Here we characterize mathematical models of simple signaling networks and obtain exact analytical expressions for two measures of cross talk called specificity and fidelity. The performance of several insulating mechanisms--combinatorial signaling, compartmentalization, the inhibition of one pathway by another, and the selective activation of scaffold proteins--is evaluated with respect to the trade-off between the specificity they provide and the constraints they place on the network. The effects of noise are also examined. The insights gained from this analysis are applied to understanding specificity in the yeast mating and invasive growth MAP kinase signaling network.     

Mathematical models of cell colonization of uniformly growing domains

During the development of vertebrate embryos, cell migrations occur on an underlying tissue domain in response to some factor, such as nutrient. Over the time scale of days in which this cell migration occurs, the underlying tissue is itself growing. Consequently cell migration and colonization is strongly affected by the tissue domain growth. Numerical solutions for a mathematical model of chemotactic migration with no domain growth can lead to travelling waves of cells with constant velocity; the addition of domain growth can lead to travelling waves with nonconstant velocity. These observations suggest a mathematical approximation to the full system equations, allowing the method of characteristics to be applied to a simplified chemotactic migration model. The evolution of the leading front of the migrating cell wave is analysed. Linear, exponential and logistic uniform domain growths are considered. Successful colonization of a growing domain depends on the competition between cell migration velocity and the velocity and form of the domain growth, as well as the initial penetration distance of the cells. In some instances the cells will never successfully colonize the growing domain. These models provide an insight into cell migration during embryonic growth, and its dependence upon the form and timing of the domain growth.     

Mathematical models and lymphatic filariasis control: monitoring and evaluating interventions

Monitoring and evaluation are crucially important to the scientific management of any mass parasite control programme. Monitoring enables the effectiveness of implemented actions to be assessed and necessary adaptations to be identified; it also determines when management objectives are achieved. Parasite transmission models can provide a scientific template for informing the optimal design of such monitoring programmes. Here, we illustrate the usefulness of using a model-based approach for monitoring and evaluating anti-parasite interventions and discuss issues that need addressing. We focus on the use of such an approach for the control and/or elimination of the vector-borne parasitic disease, lymphatic filariasis.     

Mathematical models and lymphatic filariasis control: endpoints and optimal interventions

The current global initiative to eliminate lymphatic filariasis is a major renewed commitment to reduce or eliminate the burden of one of the major helminth infections from resource-poor communities of the world. Mathematical models of filariasis transmission can serve as an effective tool for guiding the scientific development and management of successful community-level intervention programmes by acting as analytical frameworks for integrating knowledge regarding parasite transmission dynamics with programmatic factors. However, the power of these tools for supporting control interventions will be realized fully only if researchers address the current uncertainties and gaps in data and knowledge of filarial population dynamics and the effectiveness of currently proposed filariasis intervention options.     

Mathematical modeling of living cell metabolism using the method of steady-state stoichiometric flux balance

This approach uses a set of algebraic linear equations for reaction rates (the method of steady-state stoichiometric flux balance) to model the purposeful metabolism of the living self-reproducing biochemical system (i.e. cell), which persists in steady-state growth. Linear programming (SIMPLEX method) is used to derive the solution for the model equations set (determining reaction rates which provide flux balance at given conditions). Here, we demonstrate the approach through the mathematical modeling of steady-state metabolism in Saccharomyces cerevisiae mitochondria.