p is raw material (precursors), r the protein machinery that makes proteins (ribosomes), both as shares of the cell. E is how good the food is. α, the one thing the cell controls, is the share of new protein it makes as ribosomes rather than as metabolic enzymes. Growth μ is ribosomes times how well fed they are. Time is in units of the ribosome's own rate.
A dot means the rate of change in time: ṗ = dp/dt, ṙ = dr/dt, given by the first equation above. V is the most extra biomass (in logs) still obtainable from where the cell stands. Because α enters the equation linearly, the best α is always a corner: all ribosomes where one more ribosome is worth something (∂V/∂r > 0), none where it is not. The amber curve is where ∂V/∂r = 0. Near the target the optimal cell switches back and forth across it: the chattering the control strip shows. The optimal policy is computed on a grid, so shares within about 0.2% of 100% are ties.
| optimal | solves Bellman's equation on the whole plane and follows its policy |
| on–off (ppGpp) | switches ribosome making fully on below the teal dashed curve g(p), fully off above it. E. coli's alarm molecule ppGpp works this way |
| precursor-only | sets α = g(p): the share that would be best at the current level of raw material |
| nutrient-only | jumps at once to the best steady share for the new food, and stays there |
| no reaction | keeps the share that was best for the old food |