Deletion of aquaporin-4 increases extracellular K(+) concentration during synaptic stimulation in mouse hippocampus

The coupling between the water channel aquaporin-4 (AQP4) and K⁺ transport has attracted much interest. In this study, we assessed the effect of Aqp4 deletion on activity-induced [K⁺]_o changes in acute slices from hippocampus and corpus callosum of adult mice. We show that Aqp4 deletion has a layer-specific effect on [K⁺]_o that precisely mirrors the known effect on extracellular volume dynamics. In CA1, the peak [K⁺]_o in stratum radiatum during 20 Hz stimulation of Schaffer collateral/commissural fibers was significantly higher in Aqp4^−/− mice than in wild types, whereas no differences were observed throughout the [K⁺]_o recovery phase. In stratum pyramidale and corpus callosum, neither peak [K⁺]_o nor post-stimulus [K⁺]_o recovery was affected by Aqp4 deletion. Our data suggest that AQP4 modulates [K⁺]_o during synaptic stimulation through its effect on extracellular space volume.

Electrodiffusive model for astrocytic and neuronal ion concentration dynamics

The cable equation is a proper framework for modeling electrical neural signalling that takes place at a timescale at which the ionic concentrations vary little. However, in neural tissue there are also key dynamic processes that occur at longer timescales. For example, endured periods of intense neural signaling may cause the local extracellular K⁺-concentration to increase by several millimolars. The clearance of this excess K⁺ depends partly on diffusion in the extracellular space, partly on local uptake by astrocytes, and partly on intracellular transport (spatial buffering) within astrocytes. These processes, that take place at the time scale of seconds, demand a mathematical description able to account for the spatiotemporal variations in ion concentrations as well as the subsequent effects of these variations on the membrane potential. Here, we present a general electrodiffusive formalism for modeling of ion concentration dynamics in a one-dimensional geometry, including both the intra- and extracellular domains. Based on the Nernst-Planck equations, this formalism ensures that the membrane potential and ion concentrations are in consistency, it ensures global particle/charge conservation and it accounts for diffusion and concentration dependent variations in resistivity. We apply the formalism to a model of astrocytes exchanging ions with the extracellular space. The simulations show that K⁺-removal from high-concentration regions is driven by a local depolarization of the astrocyte membrane, which concertedly (i) increases the local astrocytic uptake of K⁺, (ii) suppresses extracellular transport of K⁺, (iii) increases axial transport of K⁺ within astrocytes, and (iv) facilitates astrocytic relase of K⁺ in regions where the extracellular concentration is low. Together, these mechanisms seem to provide a robust regulatory scheme for shielding the extracellular space from excess K⁺.

When neurons generate electrical signals they release potassium ions (K⁺) into the extracellular space. During periods of intense neural activity, the local extracellular K⁺ may increase drastically. If it becomes too high, it can lead to neural dysfunction. Astrocytes (a kind of glial cells) are involved in preventing this from happening. Astrocytes can take up excess K⁺, transport it intracellularly, and release it in regions where the concentration is lower. This process is called spatial buffering, and a full mechanistic understanding of it is currently lacking. The aim of this work is twofold: First, we develop a formalism for modeling ion concentration dynamics in the intra- and extracellular space. The formalism is general, and could be used to simulate many cellular processes. It accounts for ion transports due to diffusion (along concentration gradients) as well as electrical migration (along voltage gradients). It extends previous, related formalisms, which have focused only on intracellular dynamics. Secondly, we apply the formalism to model how astrocytes exchange ions with the extracellular space. We conclude that the membrane mechanisms possessed by astrocytes seem optimal for shielding the extracellular space from excess K⁺, and provide a full mechanistic description of the spatial (K⁺) buffering process.

Aquaporin-4 regulates extracellular space volume dynamics during high-frequency synaptic stimulation: a gene deletion study in mouse hippocampus

Little is known about the physiological roles of aquaporin-4 (AQP4) in the central nervous system. AQP4 water channels are concentrated in endfeet membranes of astrocytes but also localize to the fine astrocytic processes that abut central synapses. Based on its pattern of expression, we predicted that AQP4 could be involved in controlling water fluxes and changes in extracellular space (ECS) volume that are associated with activation of excitatory pathways. Here, we show that deletion of Aqp4 accentuated the shrinkage of the ECS that occurred in the mouse hippocampal CA1 region during activation of Schaffer collateral/commissural fibers. This effect was found in the stratum radiatum (where perisynaptic astrocytic processes abound) but not in the pyramidal cell layer (where astrocytic processes constitute but a minor volume fraction). For both genotypes the ECS shrinkage was most pronounced in the pyramidal cell layer. Our data attribute a physiological role to AQP4 and indicate that this water channel regulates extracellular volume dynamics in the mammalian brain.

Dependence of spontaneous neuronal firing and depolarisation block on astroglial membrane transport mechanisms

Exposed to a sufficiently high extracellular potassium concentration ([K( + )]₀), the neuron can fire spontaneous discharges or even become inactivated due to membrane depolarisation ('depolarisation block'). Since these phenomena likely are related to the maintenance and propagation of seizure discharges, it is of considerable importance to understand the conditions under which excess [K( + )]₀ causes them. To address the putative effect of glial buffering on neuronal activity under elevated [K( + )](o) conditions, we combined a recently developed dynamical model of glial membrane ion and water transport with a Hodgkin-Huxley type neuron model. In this interconnected glia-neuron model we investigated the effects of natural heterogeneity or pathological changes in glial membrane transporter density by considering a large set of models with different, yet empirically plausible, sets of model parameters. We observed both the high [K( + )]₀-induced duration of spontaneous neuronal firing and the prevalence of depolarisation block to increase when reducing the magnitudes of the glial transport mechanisms. Further, in some parameter regions an oscillatory bursting spiking pattern due to the dynamical coupling of neurons and glia was observed. Bifurcation analyses of the neuron model and of a simplified version of the neuron-glia model revealed further insights about the underlying mechanism behind these phenomena. The above insights emphasise the importance of combining neuron models with detailed astroglial models when addressing phenomena suspected to be influenced by the astroglia-neuron interaction. To facilitate the use of our neuron-glia model, a CellML version of it is made publicly available.

Astrocytic mechanisms explaining neural-activity-induced shrinkage of extraneuronal space

Neuronal stimulation causes ∼30% shrinkage of the extracellular space (ECS) between neurons and surrounding astrocytes in grey and white matter under experimental conditions. Despite its possible implications for a proper understanding of basic aspects of potassium clearance and astrocyte function, the phenomenon remains unexplained. Here we present a dynamic model that accounts for current experimental data related to the shrinkage phenomenon in wild-type as well as in gene knockout individuals. We find that neuronal release of potassium and uptake of sodium during stimulation, astrocyte uptake of potassium, sodium, and chloride in passive channels, action of the Na/K/ATPase pump, and osmotically driven transport of water through the astrocyte membrane together seem sufficient for generating ECS shrinkage as such. However, when taking into account ECS and astrocyte ion concentrations observed in connection with neuronal stimulation, the actions of the Na⁺/K⁺/Cl⁻ (NKCC1) and the Na⁺/HCO₃⁻ (NBC) cotransporters appear to be critical determinants for achieving observed quantitative levels of ECS shrinkage. Considering the current state of knowledge, the model framework appears sufficiently detailed and constrained to guide future key experiments and pave the way for more comprehensive astroglia–neuron interaction models for normal as well as pathophysiological situations.

A key experimental observation associated with the astroglia–neuron interaction is the shrinkage of the extracellular space (ECS) that occurs in response to enhanced neuronal activation. Although well documented to be present in mammalian brains, this phenomenon has resisted a proper explanation since it was first reported. We present here a mathematical conceptualization that may explain the main mechanisms behind ECS shrinkage and provide a framework for a theoretical-experimental research programme that may help to reach a consensus explanation. Effective clearance of K⁺ is essential for normal brain function because an inappropriate increase in extracellular K⁺ will enhance neuronal excitability and promote neuronal afterdischarges and increase the probability of epileptic episodes. The shrinkage of the ECS usually appears in conjunction with K⁺ clearance and must be taken into account in a model of how astrocytes clear excess K⁺ following trains of action potentials. The present model allows us to make several clear and testable predictions addressing the relationship among potassium clearance, water transport, and ECS shrinkage. Among these are predictions concerning water transport functions of aquaporins in astrocytes, involvement of cotransporters in potassium clearance, and effects of particular knockouts on ECS shrinkage and ion concentrations.

A stochastic model of cancer initiation including a bystander effect

A stochastic model of cancer initiation is considered. The model is used to evaluate whether a bystander effect may be important in the pre-malignant and malignant stages of carcinogenesis, and furthermore, on the basis of epidemiological data, to estimate the mutation rates of genes involved in the development of oral leukoplakias. The bystander effect is defined here as the capability of oncogenic mutations to increase the mutation probability of neighbouring (bystander) cells, thus leading potentially to a cascade of neighbouring mutated and neoplastic cells as a pre-stage in the development to leukoplakias and cancer. We find that incidence data for oral cancer are indeed in accordance with a significant bystander effect, operating either alone or in combination with genomic instability in the early stages of carcinogenesis, i.e. the development of neoplasia. Simulations performed gave a picture of how mutations and neoplasia may spread in a tissue, to form characteristic leukoplakias with a core of neoplastic cells. The model also showed that the probability of finding at least one neoplastic cell in the tissue after a given number of years is more sensitive to changes in genomic instability within the cell itself than to changes in a bystander effect. Based on epidemiological data we also calculate the maximum number of oncogenic genes that may be involved in the bystander effect and development of genomic instability. Even if capable of explaining the initial development of oncogenic mutations towards neoplastic cells, the bystander model could not reproduce the observed incidence rates of leukoplakia without assuming a carcinogen mutation probability per cell per year of neoplastic cells practically equal to one. This means that the bystander effect, to be of substantial importance in the final development of neoplastic cells towards leukoplakias, requires a very significant increase in mutation probabilities for bystanders to neoplastic cells. Alternatively, additional mechanisms such as abnormal cell differentiation and uncontrolled proliferation and apoptotis in the neoplastic stage may be of major importance during the development to cancerization.

Mathematical modeling of granulocyte reconstitution after high-dose chemotherapy with stem cell support: effect of post-transplant G-CSF treatment

Cancer patients treated with high-dose chemotherapy and autotransplanted with peripheral blood progenitor cells most often reconstitute neutrophils (> 0.5 x 10(9)c/l) 8-16 days after the initiation of treatment. By means of a mathematical model of human granulopoiesis, the present work assesses the effect of administering granulocyte colony stimulating factor (G-CSF) post-transplant to reduce engraftment time, and also assesses the effect of delaying initiation of G-CSF treatment relative to a general schedule. Hematopoietic progenitor cells from 21 breast cancer patients were mobilized by chemotherapy followed by G-CSF injections. Purified CD34+ cells taken from the mobilized blood were infused 3 days after termination of chemotherapy. Patients were given subcutaneous injections of G-CSF post-transplant (5 microg/kg every 12 h). Neutrophil counts calculated from a mathematical model were compared with data from individual patients. These results were also compared with data and modeling results from a group of 19 lymphoma patients given no post-transplant G-CSF therapy. The observed engraftment times were associated with the number of CFU-GM cells in the reinfused blood graft and the administration of post-transplant G-CSF. The latter finding was most predominant in patients given < 5.0 x 10(5) CFU-GM/kg bw. These tendencies were well captured by the model. Interestingly, the model showed no major differences in time to engraft neutrophils if the initiation of G-CSF was postponed for up to 5 days after transplantation. Our findings indicate that the present mathematical model of neutrophil recovery following high-dose therapy correlates with clinical observations and can potentially be used to predict time to neutrophil recovery.

Stability of a model of human granulopoiesis using continuous maturation

A class of mathematical models involving a convection-reaction partial differential equation (PDE) is introduced with reference to recovering human granulopoiesis after high dose chemotherapy with stem cell support. The stability properties of the model are addressed by means of numerical investigations and analysis. A simplified model with proliferation rate and mobilization rate independent of maturity shows that the model is stable as the maturation rate grows without bounds, but may go through stable and non-stable regimens as the maturation rate varies. It is also shown that the system is stable when parameters are chosen to approximate a real physiological situation. System characteristics do not change profoundly by introduction of a maturity-dependent proliferation and mobilization rate, as is necessary to make the model operate more in accordance with hematological observations. However, by changing the system mitotic responsiveness with respect to changes in cytokine level, the system is still stable but may show persistent oscillations much resembling clinical observations of cyclic neutropenia. Furthermore, in these cases, changes in the model feedback signal caused by, for instance, an impaired effective cytokine elimination by cell receptors may enforce these oscillations markedly.

Mathematical modeling of human granulopoiesis: the possible importance of regulated apoptosis

Steady state human granulopoiesis was modeled by a convection-reaction differential equation of the Rubinow type for the bone marrow granulocyte precursors and an ordinary differential equation for the blood granulocytes. Measured values reported from several laboratories were used as sources for the model proliferation, maturation, and mobilization rates. Due to the large variability in the measured input data, four alternative models were constructed initially, each one with a specific combination of proliferation rate and maturation rate. They were all able to produce output values for the bone marrow neutrophil count and turnover rate close to accepted data, but neither of them could reproduce good values for the differential fractions of the neutrophil precursor stages. The model output was especially sensitive to changes in transit time in the mitotic relative to the postmitotic precursor compartments. When the net proliferation rate was modeled to optimize the bone marrow differential fractions according to published data, the total bone marrow neutrophil count would not fit with published data. However, a composite model optimizing differential fractions, bone marrow neutrophil count, and turnover rate yielded plausible output values and a reduced proliferation rate in the myelocyte stage. This result opens for a possibly substantial apoptosis rate at the myelocyte stage in accordance with results from earlier investigators. However, the result was based on a special choice of precursor transit times, taken from the literature. More precise data concerning granulocyte precursor cycle times, transit times, and differential fractions would radically improve the model's ability to clarify the role of apoptosis during granulocyte production and storage.

A mathematical model for reconstitution of granulopoiesis after high dose chemotherapy with autologous stem cell transplantation

High dose chemotherapy supported with hematopoietic progenitor cells gives a characteristic neutropenic period (blood neutrophils < 0.5 x 10(9) c/l) ranging from 10 to 16 days. The question of a correlation between the CFU-GM content of the transplanted CD34+ cells and time to neutrophil recovery by patients having been given high-dose chemotherapy (HD-CT) with stem cell support was addressed by means of a mathematical model of granulopoiesis. The model utilizes a convection-reaction partial differential equation (PDE) with feedback from a cytokine compartment on proliferation, maturation, and mobilization of granulocytes from bone marrow to blood. The observed number of CFU-GM cells in the transplanted CD34+ cell autograft was used as input to the model. Using this approach, the observed gross relationship between CFU-GM content in the reinfused blood product and engraftment time could be reproduced. At the same time, the effects of assumed physiological mechanisms, especially some of the effects of G-CSF on proliferation rate, maturation rate, mobilization, and cell death, could be investigated and discussed relative to observed engraftment. The model makes it possible to explain how cytokines interfere with progenitor cell mobilization from bone marrow to blood, and it points out the implications of a regulating mechanism for the granulocyte maturation rate.