Calcisphere

Calculate the energy requirements for calcium transport into vesicles vs over the plasma membrane in calcifying nannoplankton.

General description

This code calculates the energy required to maintain calcium homeostasis in single celled organisms like calcifying nannoplankton. In order to maintain calcium homeostasis, a cell must export as many calcium ions as the number of calcium ions entering the cytoplasm. Calcium enters the cell via channels that can be either opened or closed. The channels are relatively large pores that allow for a fast entry of calcium into the cell. The spike of increased intracellular calcium due to the opening of channels is used for intracellular signalling. However, in order to meet the requirement of low intracellular calcium concentrations for optimal cell functioning, cells use a combination of Na-Ca-exchangers (with a relatively high maximal transport rate but a low affinity for intracellular calcium) and Ca-ATPases to keep intracellular calcium at low (micromolar) concentrations. Since active ion transport requires energy in the form of ATP, we use this aspect to estimate the energy needed to establish a balance between passive influx and active export.

Model description

In our model (see: calcification_energetics_values.py) we assume that, in order to establish calcium homeostasis, the active ion transport equals passive calcium entry through the channels. The channels are simulated with a constant permeability that depends on the concentration gradient over the plasma membrane. Since intracellular calcium concentrations are usually very low (in the μM range) and extracellular calcium (i.e. in seawater) is around 10 mM, the resulting gradient drives calcium into the cell.

Ca_out = 10.0    # mM; extracellular (seawater) calcium concentration, 10 mM = 10e-3 mol L-1 

Ca_in = 0.10e-3  # mM; intracellular calcium concentration

The ion flux for one calcium channel is 1 pA, which corresponds to 3.0e6 divalent ions per second (Tsien, 1983).

i = 3.0e6        # Ca2+ s-1 channel-1

The density of calcium channels on the cell surface determines the maximal flux of calcium ions entering the cytoplasm. It is generally believed that for coccolithophores to calcify, the calcium flux needs to be very high. However, the density of calcium channels on the plasma membrane is not known for calcifying nannoplankton. We therefore use the highest known density of calcium channels on snail axons: 30-60 μ^-2 (Tsien, 1983) as a potential example to relate the observed calcium flux required for coccolithophore calcification to a possible calcium influx as observed in the animal kingdom.

# channel density is 30-60 per squared micrometers in snail axons (Tsien 1983)

N = 1.0e12      # channels m-2; 

The maximum possible calcium flux with this density of calcium channels and the ion flux per channel is 2.3475e-14 mol s^-1 given the surface area of a nannoplankton cell. The diameter of a nannoplankton cell (e.g. Emiliania huxleyi) is around 5 μm (Harvey et al. 2015).

#r_cyt = 10.0e-6                 # diameter of calcispheres is around 20 micrometers

r_cyt = 2.5e-6                   # diameter of Emiliania huxleyi is around 5 micrometers (Harvey et al 2015)

A_cyt = 4.0 * np.pi * r_cyt**2.0 # cell surface in m2

I_Ca = N * i                     # 1.8e20 Ca2+ ions m-2 s-1

N_A = 6.0221367e23               # ions per mol

J_Ca = I_Ca / N_A                # 2.98e-4 mol Ca m-2 s-1 (as maximum flux)

F_Ca = J_Ca * A_cyt              # 2.3475e-14 mol s-1

Since this flux is driven by the calcium gradient over the membrane, we can calculate the permeability of the membrane (PCa in the code).

PCa = F_Ca / (Ca_bd - Ca_in)     # because PCa is the rate that leads to this flux given the ion gradient

This calculation uses the calcium concentration directly at the cell surface (Ca_bd in the code), i.e. the calcium concentration of the boundary layer around the cell and not the concentration in the open ocean. This concentration is calculated based on the diffusive flux of calcium around the cell as a consequence of calcium depletion due to calcification. For the calcification flux we use the precipitation of one coccolith with 22 fmol per one hour as estimated by Holtz et al. (2013). The original observation is from Paasche. The diffusion coefficient for calcium is set after Li & Gregory (1974).

Ca_bd = Ca_out - QCa / (4.0 * np.pi * r_cyt * D_Ca)

QCa = 6.11e-18             # 22 fmol h-1 -> 6.11e-18 mol s-1

D_Ca = 7.93e-6 / 10000.0   # cm2 s-1 -> m2 s-1 (after Li & Gregory 1974)

As shown here, the calcium concentration at the cell surface (Ca_bd in the code) depends on the balance between the net uptake that is given by the calcification flux (QCa in the code) and the replenishment via diffusion. The value for the calcium concentration at the cell surface as calculated above is specific for a given calcification flux as observed in Emiliania huxleyi (Holtz et al. 2013). In order to explore the dependence of energy requirements on the relative amount of ions transported into vesicles, we vary the fraction of ions being transported into vesicles, which determines the absolute calcification flux, and compare it to the remaining fraction of ions that need to be transported over the plasma membrane in order to establish calcium homeostasis, i.e. to balance the calcium influx over the channels. But since the concentration at the cell surface is directly dependent on the intensity of the flux into the vesicles that is removing calcium ions from the environment due to calcification, we need to calculate the gradient anew for each value of the relative fraction (fV in the code). The analytical solution for this was derived using the computer algebra provided by sagemath.org.

Ca_bd = (4.0 * Ca_out * D_Ca * np.pi * r_cyt + Ca_in * PCa * fV) / (4.0 * D_Ca * np.pi * r_cyt + PCa * fV)

Active calcium transport via a Ca-ATPase appears at a stoichiometry of 2 calcium ions being transported against the hydrolysis of one ATP. The energetic cost for calcium transport is therefore 0.5 mol ATP per mol Ca. The energetic costs are plotted over an fV varying from 0 to 100 % (Figure 1). The dots in the figure indicate the apparent situation when assuming a calcification flux of 22 fmol h^-1 and show that this flux is apparently on the lower end when compared to the possible calcium flux in snail axons.

Figure 1. Energy required for maintaining intracellular calcium homeostasis as a function of the fraction of Ca²⁺ ions pumped into the vesicle (fV in the code)

References

• Harvey, E.L., Bidle, K.D., and Johnson, M.D. (2015). Consequences of strain variability and calcification in Emiliania huxleyi on microzooplankton grazing. Journal of Plankton Research, 37, 1137–1148.
• Holtz, L.M., Thoms, S., Langer, G., and Wolf-Gladrow, D.A. (2013). Substrate supply for calcite precipitation in Emiliania huxleyi: Assessment of different model approaches. Journal of Phycology, 49, 417–426.
• Li, Y.-H. and Gregory, S. (1974). Diffusion of ions in sea water and in deep sea sediments. Geochimica et Cosmochimica Acta, 38, 703–714.
• Tsien, R.W. (1983). Calcium channels in excitable cell membranes. Annual Reviews of Physiology, 45, 341—-358.