Supplementary information:

Description of the mathematical model and model parameters:

For our computational analyses we used the electrophysiological model of Berndt et al. in [1], where a more detailed description can be found. In short the model comprises as variables the membrane potential, intra- and extracellular ion concentrations of Na+, K+ and Cl-, and with , the net charge of non-diffusible organic cellular molecules (biomass) and the cell volume . Transport of ions across the membrane includes passive (= leak) currents (Na+, K+, Cl-), an inward Na+ current, , due to secondary-active transport of metabolites and neuro-transmitters, active transport driven by the Na+/K+-ATPase (Na+, K+), , and ion currents through voltage-gated channels (Na+, K+) described through the gated permeabilities . Excitation is achieved by opening ligand-gated Na-channels (= increase of the excitatory Na permeability). Osmotic pressure between the extra- and intracellular space,and , is equilibrated by the exchange of water. The fraction of the current of Na+ and K+ due to active secondary transport compared to total Na+ current at rest (in the absence of AP firing) is given by the constant . The downscaling of the pure leak currents is given by the constant denoting the remaining fraction with respect to the normal reference values.

The model describes the time dependent variation of the three ion species Na+, K+ and Cl-. The cellular ion concentrations can change either due to ion fluxes or changes in the cell volume:

(1)

Here is the charge of ion species and the Faraday constant. The external ion concentrations may also change due to ion fluxes and changes in the external volume :

(2)

The ion currents are the sum of three parts: A passive current, an active current, and a metabolic current:

(3)

The passive current describes electro-diffusion of ions across the cell membrane and is modeled via the Goldman-Hodgkin-Katz equation:

(4)

Here is the surface of the cell membrane, which is assumed to be constant and set to a value that is sufficient to allow the cell to expand to twice its resting volume.

(5)

with being the membrane potential, the universal gas constant, the temperature and 1000 the scaling factor since we measure the membrane potential in mV. The total permeability for ion species , , is the sum of the basal leak permeability times a scaling constant , the permeability due to voltage gated channels and the permeability due to ligand-gated channels :

(6)

The metabolic current is only carried by Na+ and given by

(7)

Here denotes the fraction of the metabolic Na+ current relative to the total Na+ current at rest (without AP firing), a constant and another scaling factor.

The voltage gated channels for Na+ and K+ are modeled by the standard model of Hodgkin and Huxley

(8)

(9)

where the gating variables obey the ordinary differential equation

(10)

with being a constant and and being voltage dependent functions.

The active transport of ions by the Na+/K+-ATPase is described by

(11)

and

(12)

The membrane is modeled as a capacitor with total capacitance C, related to specific capacitance and total charge via

(13)

Differentiation of (13) with respect to time gives:

(14)

With the additional assumption of constant membrane capacity, , this gives:

(15)

Volume changes are elicited by differences in the internal and external osmolarities, and

(16)

Here is the water permeability and the partial molar volume of water.

The osmolarities are given by

(17)

(18)

with denoting the osmolarity of the intracellular biomass.

All the constants and parameters are summarized in the following tables.

Parameters and constants

Table 1: constants

name / value / unit / equation / description
T
R
F / 293
8.3
96490 / K

/ (5)
(5)
(1),(2),(5) / temperature
universal gas constant
Faraday’s constant

Table 2: Geometry

name / value / unit / equation / description
/ / / - / cell radius
/ / / (4), (13),(16) / cell surface
/ / / (1) / cell volume
/ / / (2) / external volume

Table 3: initial ion concentration

name / value / unit / equation / reference / description
/ 8 / / (1),(4),(17) / [2] / bulk Cl- concentration
/ 140 / / (1),(4),(17) / [2] / bulk K+ concentration
/ 10 / / (1),(5) ,(11),(12),(17) / [2] / bulk Na+ concentration
/ 150 / / (2),(4),(18) / [2] / ext. Cl- concentrtaion
/ 4 / / (2),(4),(18) / [2] / ext. K+ concentration
/ 145 / / (2),(4),(18) / [2] / ext. Na+ concentration
/ 141 / / (17) / [2] / impermeable biomass

Table 4: capacitance and voltage

name / value / unit / equation / reference / description
/ / / (13) / [3] / spec. mem. capacitance
/ -70 / / (5),(13) / [4] / membrane potential
/ / / (13),(14),(15) / - / total capacitance
/ - / / (13),(14) / - / excess charge

Table 5: permeabilities and metabolic current

name / value / unit / equation / reference / description
/ / / (6) / [2, 5] / basal Cl- permeability
/ / / (6) / [2, 4] / basal K+ permeability
/ / / (6) / [2, 4, 5] / basal Na+ permeability
/ / / (6),(8) / [6], fit / gated K+ permeability
/ / / (6),(9) / [6], fit / gated Na+ permeability
/ / / (7) / see main text / metabolic Na+ current
/ variable
(see main text) / - / (7) / - / scaling for metabolic Na+ current

Table 6: HH-transition rates from eq. (10)

Table 7: pump kinetics

name / value / unit / equation / reference / description
/ 4 / - / (11),(12) / [7] / binding constant
/ / - / (11),(12) / fit / rate constant
/ 10 / / (11),(12) / [7] / binding constant

Table 8: parameters figure 4

panel / / / / / / / / in
A1/B1 / 0.5 / 1 / 1 / 200 / 1000 / 500 / 2 /
A2/B2 / 0.5 / 0.1 / 0.3 / 300 / 1000 / 500 / 1 /
A3/B3 / 0.5 / 0.01 / 0.2 / 300 / 1000 / 500 / 1 /

Literature

1. Berndt N, Hoffmann S, Benda J & Holzhutter HG (2011) The influence of the chloride currents on action potential firing and volume regulation of excitable cells studied by a kinetic model. J Theor Biol 276, 42-49.

2. Armstrong CM (2003) The Na/K pump, Cl ion, and osmotic stabilization of cells. Proceedings of the National Academy of Sciences of the United States of America 100, 6257-6262.

3. Hodgkin AL & Huxley AF (1952) A Quantitative Description of Membrane Current and Its Application to Conduction and Excitation in Nerve. Journal of Physiology-London 117, 500-544.

4. Kager H, Wadman WJ & Somjen GG (2007) Seizure-like afterdischarges simulated in a model neuron. Journal of Computational Neuroscience 22, 105-128.

5. Hodgkin AL & Horowicz P (1959) The Influence of Potassium and Chloride Ions on the Membrane Potential of Single Muscle Fibres. Journal of Physiology-London 148, 127-160.

6. Cohen LB (1973) Changes in Neuron Structure during Action Potential Propagation and Synaptic Transmission. Physiological Reviews 53, 373-418.

7. Zahler R, Zhang ZT, Manor M & Boron WF (1997) Sodium kinetics of Na,K-ATPase alpha isoforms in intact transfected HeLa cells. Journal of General Physiology 110, 201-213.