Source
Liang, De Los Rios and Busiello (2024; PRX 16, 041026, 2026), LIT-1886, §IV, eqs. 15–29, and §V, pp. 7–17 of arXiv 2407.11498v4; read in NOTE-1463.
What was actually shown
Affinities. A steady flux decomposes without cancellation into elementary flux modes (EFMs), minimal sets of reactions that leave the internal species unchanged. Along an EFM e conformal to the flux, every reaction pushes forward, and the affinities sum to the cycle affinity A^Y_e the chemostats supply (eq. 14). So no reaction’s affinity exceeds A^Y_e per use, and maximising over EFMs that contain the reaction gives a bound with no kinetic input (eq. 20). A virtual, infinitely slow “probe” reaction extends it to any interconversion the conservation laws allow (eq. 24).
Concentrations. Writing affinities through chemical potentials turns the bound into K^min_π ≤ Π_i x_i^{S_iρ} ≤ K^max_π, where K_π is the equilibrium constant along a pathway π including the free energy its chemostats supply (eq. 29). At equilibrium every pathway has the same K, and the space is a point.
Checks (mine). The Schlögl model with the paper’s k₁± = 8, k₂± = 1, b = 0.02 and a = 0.8 has fixed points 0.0234, 0.1787 and 0.5979, all in the bound [0.02, 0.8] of eq. 36. At a = 0.5 and 0.3 there is one fixed point, inside; at a = b the bound and the fixed point are both 0.02. In a reversible rewriting of CASE-189’s first construction, every fixed point lies in [(d′/d)[W], (k/k′)[S]] for each [W] tried.
What could have failed. A stationary state could have lain outside the bounds for some kinetics, as a state above the slope-1 ceiling of the network without 2B ⇌ C would (Fig. 4). The paper’s random-kinetics simulations and my two checks find none.
What this does not say
- Which state is occupied. The bounds are attained only when one pathway dominates kinetically, a time-scale-separation limit; otherwise the state is interior and set by kinetics. This is THEORY-692’s “thermodynamics supplies … not a principle selecting which structure appears”, made quantitative. It neither supports nor refutes maximum entropy production (THEORY-671).
- How much driving multistability needs. The space has positive width for any positive driving; a particular network may need much more. The rewriting of CASE-189 is monostable below 3.52 RT (THEORY-1266).
- Anything about oscillations or transients. The authors exclude persistent oscillations.
- That an organization’s lifetime is bounded by its driving. The chemostats are fixed. A13 of the exchange “Analyze nucleation concepts” cited this paper for that point; it belongs to LIT-1887 (THEORY-1250).