Light, Gravity and Auxin Bending

What is modelled

A stem senses two things at once: where the light is and which way is down. Blue light moves growth hormone, auxin, to the shaded side, where cells elongate faster and the stem bends toward the light; the sensed tilt from vertical does the opposite and pulls it upright. The stem settles where the two turning rates cancel. Roots only sense gravity and turn until they point straight down. Spinning the pot on a clinostat averages gravity away and leaves only the light.

Equations

• Turning law of the stem tip dθ/dt = −g sin θ + λ I sin(α − θ) (sine laws of gravitropism and phototropism); roots dφ/dt = −gr sin φ.
• Balance angle tan θ* = λ I sin α / (g + λ I cos α); the return rate to it is g cos θ* + λ I cos(α − θ*).
• Bending happens only in the growing tip, so older parts keep their shape: the tip moves (v sin θ, v cos θ) with v = 6 mm/h for the stem and 4 mm/h for the root.
• With constant turning rate ω the stem traces an arc of radius v/ω; with no gravity the exact light-only solution is tan((α − θ)/2) = tan((α − θ0)/2) e−λ I t.

Checks run in the page

• Integrated stem angle against the balance formula.
• Return rate after a small nudge against the linearised rate.
• Arc radius and path length of a stem turning at constant rate.
• Light-only turning against the exact solution on a clinostat.

Honest simplifications

The stem and root are reduced to a growing tip with a turning rate; proprioception, the real distribution of auxin transporters and the lag between stimulus and response are left out. The auxin ratio shown is a simple readout of the turning rate with an assumed proportionality, not a measurement, and the gains are illustrative values chosen so that bending takes hours.

• Press Tilted pot, dark: the stem rights itself and the root turns down.
• Press Light from the side and read the balance angle in the stem plot.
• Press Clinostat: no net gravity: the stem bends fully to the light.