 Solution

In this two-pool calcium model, $c$ is the cytosolic calcium ion concentration, and $c_s$ measures calcium in an intracellular store, such as the endoplasmic reticulum. The variables in these balances are expressed using compatible effective volumes, so internal transfers enter their two equations with equal and opposite signs. If actual compartment concentrations use different reference volumes, the corresponding volume ratio must first be included in their balances.

The flux $J_{+}$ pumps calcium from the cytosol into the store, $J_{-}$ releases it back, and $k_s{c}_s$ is passive store leakage back to the cytosol. The external input $r$ supplies cytosolic calcium, while $kc$ exports it from the biological cell. Thus $k,k_s$ have units of inverse time and all five fluxes have units of concentration per time. Their directions can be sketched asAdding the two ordinary differential equations verifies the total calcium balanceInternal uptake, release and leakage redistribute calcium; only the exchanges across the cell membrane change this total. In particular any equilibrium must have $c=r/k$ when $k>0$.