Optimal enzyme profiles in unbranched metabolic pathways

Footnotes

• Aside from many small changes to impove the manuscript, we added a simple but very accurate approximation formula to the thermodynamic rate law and explain its use in simple cell models.

• https://gitlab.com/elad.noor/optimal_enzyme_investment

• ↵² In fact, there are a number of similar arguments why the enzyme-control rules fails in this case: first, the state with infinite concentrations does not exist mathematically, so formally we cannot even refer to it as a metabolic state; second, even if this state existed, all elasticities would vanish, and the Jacobian matrix would not be invertible, so the control coefficient would not be defined; third, even if we argued that, logically, the first enzyme must have full flux control, the same argument would also apply to all other enzymes: each of the enzymes would have full flux control, thus violating the summation theorem.

• ↵³ Instead of a linear flux objective, a nonlinear function z(v) could be used. In this case, the rule (42) would remain the same, but with the gradient ∇_vz(v) replacing the weight vector z.. Likewise, in a model with separate density constraints a ·ε ≤ρ_ε and b ·s ≤ρ_c, we obtain a similar formulae, but with separate Lagrange multipliers for the two constraints.

• ↵⁴ For a simple constraint on the total mass, the weights are simply molecular weights. This could be a proxy for excluded volume (which would still ignore, for example, hydration shells). However, a “density constraint” need not represent space demands; it may also be related to osmotic effects or opportunity costs (e.g. energy demand for production of compounds in growing cells). Therefore the meaning of the weights in the density constraint may differ from model to model.

• ↵⁵ Incidentally, from the condition for rows we obtain another (alternative) set of sufficient conditions These conditions are satisfied, for example, if all elasticities are non-zero, if the product elasticity of the first reaction is equal or larger than the product elasticity of the second reaction, if the substrate elasticity of the last reaction is equal or larger that the substrate elasticity of the third reaction, and if . For a reversible mass-action kinetics v = e[k₊s − k₋p] the latter condition would mean: or where the “standard velocity” denotes the rate that a reaction l would show at unit metabolite and enzyme levels.