NHE transporters

Sodium-hydrogen exchangers (NHE) are membrane transport proteins that exploit the influx of Na+ to export H+ ions. The sodium concentration gradient is maintained by the ATP-dependent Na+/K+ pump^19,46 so that the activity of NHE indirectly depends on ATP availability. This implies that as long as ATP is available the flux of H+ due to NHE is essentially unidirectional. It has also been reported that NHE activity is inhibited by hypoxia^10,19 and that, in the long-term, hypoxia inhibits the expression of NHE proteins. Energy and oxygen tune NHE activity and as in the previous model of tumor cell metabolism and growth^6–8, here we take into account these regulatory circuits by means of the two variables SensATP and SensO2 that assume real values in the interval [0, 1].

Experimental observations indicate that NHE activity is described by a Hill equation^9,47,48 and hence the unidirectional flux of H+ from the cell to the environment due to NHE transport is modeled by the equation:

where νmaxNHE=VmaxNHE·SC and fPHeNHE is a phenomenological function that tunes the activity of NHE transport as a function of extracellular pH:

Indeed, it has been observed that extracellular acidity enhances H+ transport through NHE^9,19,49. In the Supplementary Material we discuss how we fix parameter values and define the function fPHe on the basis of experimental observations.