Separable rate laws and enzyme cost functions

According to Eq (3), reversible rate laws can be factorized into four terms: the enzyme level E, its forward catalytic constants kcat+, and two efficiency factors [22]. In Fig 6 we add a non-competitive allosteric inhibitor x. While the enzyme level and kcat+ are not directly affected by the concentration of metabolites (although kcat+ can vary with conditions such as pH, ionic strength, or molecular crowding in cells), the efficiency factors are concentration-dependent, unitless, and can vary between 0 and 1. The reversibility factor η^rev depends on the driving force (and thus, indirectly, on metabolite levels), and the equilibrium constant is required for its calculation. The saturation factor η^sat depends directly on metabolite levels and contains the K_M values as parameters. Allosteric regulation yields additive or multiplicative terms in the rate law denominator, which in our example can be captured by a separate factor η^reg. In general, η^sat and η^reg can be combined into one kinetic factor η^kin, as depicted in Eq 6.

For a reaction S ⇌ P with reversible Michaelis-Menten kinetics, a driving force θ = −Δ_rG′/RT, and a prefactor for non-competitive allosteric inhibition, the rate law can be written as with inhibitor concentration x. In the example, with non-competitive allosteric inhibition, the kinetic factor η^kin could even be split into a product η^sat ⋅ η^reg. The first two terms in our example, E·kcat+, represent the maximal velocity (the rate at full substrate-saturation, no backward flux, full allosteric activation), while the following factors decrease this velocity for different reasons: the factor η^rev describes a decrease due to backward fluxes (see Figure A in S1 Text) and the factor η^kin describes a further decrease due to incomplete substrate saturation and allosteric regulation (see Fig 1b). The inverse of all these terms appear in the equation for enzyme demand, q, which is given by the enzyme level multiplied by the burden of that enzyme, h_E.

The second equation in Fig 6 describes the enzyme cost for a flux v, and contains the terms from the rate law in inverse form multiplied by the enzyme burden h_E. The left-hand part of the equation, hEv/kcat+, defines a minimum enzyme cost, which is then increased by the following efficiency factors. Again, 1/η^kin can be split into 1/η^sat ⋅ 1/η^reg. By omitting some of these factors, one can construct simplified enzyme cost functions with higher specific rates, or lower enzyme demands (compare Fig 1b). Since both rate and enzyme demand are a product of several terms, it is convenient to depict them as a sum on a logarithmic scale (Fig 7), where the simplified functions are seen as upper/lower bounds on the more complex rate/demand functions.

(a) Starting from the logarithmic enzyme level (dashed line on top), we add the terms logkcat+, log η^rev, and log η^kin, and obtain better and better approximation of the rate. In the example shown, kcat+ has a numerical value smaller than 1. The more precise approximations (with more terms) yield smaller rates. The EMC4 arrows refer to other possible rate laws with additional terms in the denominator. (b) Enzyme demand is shaped by the same factors (see Eq (5)). Starting from a desired flux (bottom line), the predicted demand increases as more terms are considered.