The Single-Cell ODE — 13 Variables

The molecular cascade of Module 1 produces a complex network of state variables that change on timescales ranging from seconds (calcium spikes) to hours (carbon pools to equilibrium). To make this tractable, we model the cell as a system of 13 coupled ordinary differential equations(ODEs) and integrate them numerically. The variables are grouped into three subsystems — carbon, nutrients, and signalling — coupled through a single gating variable, the symbiotic exchange coupling Φ_sym.

State variables

The carbon subsystem

Carbon flows from light energy via plant photosynthesis into the plant pool, then partially across to the fungus when the symbiosis is open:

$$\frac{dC_{\text{plant}}}{dt} = \mu_P\, I_{\text{light}} \;-\; k_{\text{exch}}\,\Phi_{\text{sym}}\,(C_{\text{plant}} - C_{\text{fungus}}) \;-\; \lambda_C\, C_{\text{plant}}$$

The three terms model: (1) photosynthetic gain proportional to light intensity I_light; (2) symbiotic exchange driven by the concentration gradient between plant and fungus, gated by Φ_sym; (3) respiratory loss at constant rate λ_C. The fungal carbon pool obeys a similar equation with an inverted exchange term and its own loss.

The gating variable Φ_sym ∈ [0,1] is a sigmoid of the activated CCaMK fraction:

$$\Phi_{\text{sym}} \;=\; \sigma\!\left(\frac{\text{CCaMK} - \theta_{\text{sym}}}{k_\sigma}\right), \qquad \sigma(x)=\frac{1}{1+e^{-x}}$$

When CCaMK is below threshold θ_sym, Φ_sym ≈ 0 and the symbiosis is shut: no carbon or phosphorus crosses the interface. When CCaMK climbs past threshold, Φ_sym → 1 and the exchange opens up.

The nutrient subsystem

Phosphate moves from soil into the fungal hyphae against a steep concentration gradient (Module 6 explains the proton motive force that powers this), and is then transferred across the periarbuscular interface into the plant:

$$\frac{dP_{\text{fungus}}}{dt} = V_{\text{up}}\,\frac{P_{\text{soil}}}{K_M + P_{\text{soil}}} - k_{\text{P,exch}}\,\Phi_{\text{sym}}\,(P_{\text{fungus}} - P_{\text{plant}}) - \lambda_P P_{\text{fungus}}$$

The Michaelis–Menten form V_up·P_soil/(K_M + P_soil) captures the saturation of hyphal high-affinity phosphate transporters (K_M ≈ 3–30 µM in the real system). The plant phosphorus equation mirrors this. Nitrogen is treated analogously with its own rate constants.

The signal cascade

Strigolactones are produced by the plant at a rate that increases when soil phosphate is low (the plant signals it is hungry):

$$\frac{dSL}{dt} = k_{SL,0}\,\frac{1}{1 + P_{\text{plant}}/K_{SL}} - \lambda_{SL}\,SL$$

SL stimulates fungal LCO production:

$$\frac{dLCO}{dt} = k_{LCO}\,SL - \lambda_{LCO}\,LCO$$

LCO triggers nuclear calcium oscillations via a Hopf-like bifurcation in the Ca²⁺ release machinery. The simplest empirical model uses a Goldbeter-style oscillator forced by LCO; the precise functional form is in Module 3. The resulting Ca²⁺ trace is then convolved with the CCaMK memory kernel to drive the switch.

The defence channel

A coupled pair of jasmonate equations tracks stress-signal flow. Plant JA rises when light is unusually low (proxy for damage) and decays. JA is shuttled to the fungus, which propagates it to other plants in the network (in Module 4). Although a single cell sees only background JA, the equations are kept so the same ODE can be reused at the network level.

Parameters and units

The model is normalised so that pools are unitless fractions, time is in hours, and rate constants are of order 0.01 – 1 h⁻¹. Light I_light is given in µmol photons m⁻² s⁻¹ (typical canopy values are 500–1500). Soil phosphate is expressed in µM. The default parameter set reproduces the qualitative behaviour of laboratory measurements on Medicago-Rhizophagus irregularis symbiosis.

Why an ODE and not a stochastic model?

Individual molecular events — one LCO binding, one Ca²⁺ release event — are stochastic, and a full treatment would use a chemical master equation or Gillespie simulation. However, the relevant pools are large (millions of LCO receptors, ~10⁶ Ca²⁺ ions per nuclear release event) and the variance averaged over the cell is small. The deterministic ODE captures the mean trajectory faithfully and is many orders of magnitude faster to integrate. Stochasticity becomes important if you go to single- receptor experiments or single-arbuscule live-imaging studies.

Run it yourself

The panel below lets you change four key parameters with sliders — integration time, initial LCO concentration, soil phosphate, and light intensity — and watch the 13 variables evolve. CCaMK crosses threshold for some parameter combinations and not others; finding the boundary is itself an exercise in bifurcation analysis.