Turgor-Driven Cell Growth

What is modelled

A plant cell grows by letting water in and letting the wall give way, not by pushing out new wall everywhere. Water enters because the cell sap is saltier than outside; the incoming water inflates the cell against the wall, which makes turgor. Only the part of the pressure above a yield threshold stretches the wall for good. As the cell grows the sap is diluted, so the cell has to import solutes to keep going.

Equations

• Wall growth (Lockhart) (1/V) dV/dt = φ max(0, P − Y).
• Water uptake (1/V) dVw/dt = Lp(π − πout − P), pressure dP/dt = ε(uptake − growth), ε = 10 MPa, and dilution dπ/dt = −gπ + s.
• With the osmotic drive D = π − π_out held fixed: g = φ Lp(D − Y)/(φ + Lp), P = (LpD + φY)/(φ + Lp); below Y growth stops and P = D.
• Hoop stress in the wall σ = P r / h (exact mean of the Lamé solution); growth direction share along the cell φax/(φax + 2φrad) with microfibrils resisting stretch along their length.

Checks run in the page

• Steady state against the closed formulas.
• Step halving of the coupled solver converges.
• Average Lamé hoop stress against P r / h.
• Drive below the yield threshold stops growth and P equals the drive.

Honest simplifications

The cell is one cylinder with uniform wall; real walls are heterogeneous and the yield threshold and extensibility change with expansins and pH. The anisotropy rule (0.1 + 0.9 cos²) is illustrative, the wall strength of 120 MPa is a round number, and the solute supply is a constant rather than a regulated import.

• Press Stiff wall, high yield: the cell stops when turgor sinks to the yield threshold.
• Press Drought: salty outside to see turgor fall and growth stall.
• Press Microfibrils along the cell to see growth turn into fattening.