Interactive companion to Liang, De Los Rios & Busiello · Phys. Rev. X (2026, accepted)
Thermodynamic space of chemical reaction networks
Living chemistry runs on fuel. The rate constants of a cell's reaction network are rarely known; the energy budget that drives it often is. This page walks through a paper that shows what the budget alone decides: how wide the window is in which every affinity and every reachable concentration ratio must sit. The rates only choose a place inside it.
Thermodynamic Space of Chemical Reaction Networks
Shiling Liang, Paolo De Los Rios, Daniel Maria Busiello · Physical Review X (2026, accepted) · preprint arXiv:2407.11498. Equation and figure numbers on this page follow the accepted version, arXiv:2407.11498v4.
How to read this page.
- Part I builds the paper's vocabulary (stoichiometric matrix, affinity, elementary flux modes, conservation laws) on one four-reaction example, the dimer network of the paper's Fig. 2. That network is the default state of Labs 1, 2 and 4.
- Part II derives the first result, a window for every affinity, and the paper's central tool, the chemical probe. Part III turns the windows into the thermodynamic space of concentrations and into a kinetics-free test of a network's topology. Part IV plays the same game on four models: bistability, chiral symmetry breaking, self-assembly, and reaction–diffusion patterns.
- Boxes marked LAB are interactive: move the sliders, press the buttons, drag on the figures. Each opens in the state of the paper's own figure (Lab 2 has no figure counterpart; Lab 8 opens with the rates of Fig. 8b–d on the geometry of Fig. 8a) and already shows its point.
- Every definition box carries a tag such as PRX · Eq. 20, so you can open the paper at that equation and find the same statement. Dotted terms show a definition on hover.
Conventions used everywhere
Energies are in units of \(RT\), so \(RT = 1\) in every lab and "\(\Delta\mu = 2\)" means \(2RT\); concentrations are dimensionless. Every reaction is reversible. Which direction is called forward is a convention: flipping it flips the sign of that reaction's affinity, of its entry in every flux mode, and of its window, and nothing else. Random rate constants in the labs are always thermodynamically consistent: forward constants are drawn log-uniformly, backward constants follow from local detailed balance (Eq. 11) with the standard potentials stated in each lab, chemostat concentrations set \(\Delta\mu\), and totals are drawn per the conservation law; these distributions are this page's choice, the paper says only "random rates and totals". One colour has one meaning on the whole page:
Part I · Anatomy of a driven reaction network (paper Sec. III)
The vocabulary, on one example
The paper's results hold for any network, but every definition below is introduced on the same four-reaction network, the one drawn in the paper's Fig. 2. Keep it in mind: it returns as the default state of Labs 1, 2 and 4.
1 · Species, reactions, and the stoichiometric matrix
Two monomers and a dimer. A is an inactive monomer, B an active one, C a dimer; these three are the internal species. A fuel F and a waste W are chemostatted: their concentrations are held fixed from outside, and the fuel-to-waste conversion is what drives the network. Keeping \(F\) abundant and \(W\) scarce is work that the reservoir pays for; the chemical-potential difference \(\Delta\mu = \mu_F - \mu_W\) is its price per turnover, the energy budget.
A reaction \(\rho\) converts reactants into products with forward and backward rate constants \(k_\rho^\pm\); all reactions are reversible. The stoichiometric matrix \(\bS\) has one column per reaction and one row per species; the entry \(\bS_{\sigma\rho}\) is the net number of molecules of species \(\sigma\) produced by one forward event of \(\rho\) (products count positive, reactants negative). Splitting the rows into internal and chemostatted species gives the two blocks \(\bS = [\bS^X;\ \bS^Y]\).
For the dimer network, with rows \(A, B, C\) above the line and \(F, W\) below, and columns for reactions 1 to 4:
| 1 | 2 | 3 | 4 | |
|---|---|---|---|---|
| A | −1 | 0 | −1 | −2 |
| B | +1 | −2 | −1 | 0 |
| C | 0 | +1 | +1 | +1 |
| F | −1 | 0 | 0 | 0 |
| W | +1 | 0 | 0 | 0 |
Read the columns: reaction 4 consumes two \(A\) and makes one \(C\), so its column is \((-2, 0, +1)\) on the internal rows and zero on the chemostatted rows.
Why "hypergraph"
A reaction with two reactants, or two copies of one reactant, cannot be an ordinary edge between two nodes. The paper draws each reaction as a box with a line to every species it touches, a hyperedge, with the number of line ends at a species equal to its stoichiometric coefficient (a forked end in Lab 1 means a coefficient 2). A network of such boxes is a hypergraph. A cycle in it is not a loop of edges but a set of reactions whose net effect on \(A, B, C\) is zero; Lab 1 shows four of them. Earlier bounds of this kind (Part V) only covered ordinary graphs, where every column of \(\bS^X\) has one \(+1\) and one \(-1\).
Each reaction has a forward and a backward flux, \(J_\rho^+ = k_\rho^+ \prod_\sigma z_\sigma^{\nu^{+}_{\sigma\rho}}\) and likewise \(J_\rho^-\), where \(\nu^\pm\) are the stoichiometric coefficients of reactants and products and \(z_\sigma\) the concentrations. The net flux is \(J_\rho = J_\rho^+ - J_\rho^-\). Internal concentrations \(\bx\) evolve as \[ \frac{\dd\bx}{\dd t} = \bS^X \bJ , \] while the chemostats cancel whatever the reactions do to the external species. A steady state is a fixed point, \(\bS^X \bJ^{\ss} = \boldsymbol{0}\).
Scope of the whole paper
The results assume the network reaches a steady state (one or several). Multistability is fine; sustained oscillations are not covered.
2 · Local detailed balance and affinity
Thermodynamics enters through one relation between the two fluxes of a reaction and the free energy it releases. Write the chemical potential of a species as \(\mu_\sigma = \mu_\sigma^\circ + RT \ln z_\sigma\): a standard part and a concentration part.
\[ \frac{J_\rho^+}{J_\rho^-} = e^{-\Delta_\rho G/RT} = e^{A_\rho/RT}, \qquad \Delta_\rho G = \sum_\sigma \mu_\sigma \bS_{\sigma\rho}, \qquad A_\rho = -\Delta_\rho G . \] \(\Delta_\rho G\) is the free-energy change of one forward event; its negative, the affinity \(A_\rho\), is the thermodynamic force on the reaction. Because the fluxes are positive, the affinity has the sign of the net flux: a reaction runs forward exactly when its affinity is positive. This sign rule is the hinge of the whole proof in Part II.
Collecting all reactions, \(\bA = -\bS^{T}\bmu\). Splitting the chemical potentials into internal and external ones splits the affinity: \[ \bA = \bA^X + \bA^Y = -(\bS^X)^T \bmu^X - (\bS^Y)^T \bmu^Y . \] \(\bA^Y\) is fixed by the chemostats and is the only place the driving enters; \(\bA^X\) depends on the internal concentrations. For mass action, Eq. 8 becomes a condition on rate constants, \(k_\rho^+/k_\rho^- = e^{-\Delta_\rho G^\circ/RT}\) (Eq. 11): ratios of rate constants are thermodynamic, their magnitudes are not. The paper's results use only Eq. 8, not the mass-action form.
On the example: reaction 1 has \(A_1 = (\mu_F - \mu_W) + (\mu_A - \mu_B) = \Delta\mu + A_1^X\); the other three reactions never touch the chemostats, so \(A_\rho^Y = 0\) and \(A_\rho = A_\rho^X\) for \(\rho = 2, 3, 4\). This is why "the budget" of this network is the single number \(\Delta\mu\).
3 · Cycles, elementary flux modes, conservation laws
A steady state has \(\bS^X \bJ^{\ss} = 0\): the flux vector leaves every internal concentration unchanged. Any such vector is a closed route through the network.
A flux mode is any vector \(\bc\) with \(\bS^X \bc = \boldsymbol{0}\), i.e. an element of the kernel \(\Ker(\bS^X)\). The steady-state flux \(\bJ^{\ss}\) is one. An elementary flux mode \(\be\) is a flux mode whose support is minimal: no other flux mode uses a strict subset of its reactions. Its entry \(e_\rho\) counts how many times reaction \(\rho\) runs in one turn of the cycle, with a sign for the direction. \(\be\) and \(-\be\) are the same EFM run backwards. The set of all EFMs is written \(\Ecal\).
Along one turn of any EFM the internal species return to where they were, so only the chemostats can gain or lose free energy:
\[ A_{\be} = \be^{T}\bA = \underbrace{\be^{T}\bA^X}_{=0} + \be^{T}\bA^Y = A_{\be}^Y = -\Delta_{\be} G_Y . \] The cycle affinity is the free energy the reservoirs release in one turn of \(\be\). It is a structural number: it depends on the chemostats and the stoichiometry, not on the internal concentrations, and it holds in and out of steady state. A network relaxes to equilibrium if and only if \(A^Y_{\be} = 0\) for every EFM.
The dimer network has four EFMs. The paper draws the two that pass through reaction 4 and burn fuel (Fig. 2d): \(\be_1 = [2, 1, 0, -1]\) activates two monomers, dimerizes them through reaction 2 and returns the dimer to two \(A\) through reaction 4 run backwards, so it burns fuel twice and \(A^Y_{\be_1} = 2\Delta\mu\); \(\be_2 = [1, 0, 1, -1]\) activates one monomer and uses reaction 3, so \(A^Y_{\be_2} = \Delta\mu\). The remaining two, \(\be_3 = [0, 1, -2, 1]\) (no fuel, affinity 0) and \(\be_4 = [1, 1, -1, 0]\) (affinity \(\Delta\mu\)), were recomputed for this page and are shown in Lab 1.
Why all EFMs, and not a basis of the kernel
\(\bS^X\) has rank 2, so its kernel is two-dimensional, yet there are four EFMs: \(\be_3 = \be_1 - 2\be_2\) and \(\be_4 = \be_1 - \be_2\). They are not a basis and are not meant to be one. Part II needs to write the steady-state flux as a sum of cycles that never run any reaction against the flux, a conformal decomposition. Writing \(\bJ^{\ss}\) in a basis would require subtractions, and subtractions are exactly what would break the sign argument of the proof. That is why every EFM counts and why the bound of Part II takes extrema over all of them.
A vector \(\bl\) with \(\bl^{T}\bS = \boldsymbol{0}\) is a conservation law. In an open network only those that avoid the chemostatted species survive, \(\bl^X \in \Ker[(\bS^X)^T]\), and each gives a conserved quantity \((\bl^X)^T \bx\). The dimer network has one: \(\bl^X = [1, 1, 2]\), i.e. \(a + b + 2c\) is constant (the plane drawn in Fig. 2c). Among all conservation laws, those with minimal support are the elementary conservation laws. Part II will ask which reactions could exist in a network; the answer is "every column compatible with its conservation laws".
Part II · First result: every affinity has a window (paper Sec. IV A)
From one cycle to a window for every reaction
The steady-state affinities of a network depend on all the rate constants and are usually out of reach. The paper shows that each of them nevertheless lies in a window that only the cycle affinities set. The idea fits in two sentences: along any closed cycle the affinities add up to the cycle affinity, a number fixed by the chemostats; if every term in that sum has the same sign, no single term can exceed the total.
4 · Four steps to a window
Fix a reaction \(\rho'\) that is out of equilibrium at the steady state, so that \(A^{\ss}_{\rho'}\) and \(J^{\ss}_{\rho'}\) are non-zero with the same sign (Eq. 8).
- Pick one conformal EFM through \(\rho'\). Since \(\bJ^{\ss}\) has a conformal decomposition (Sec. 3), some EFM \(\be^*\) in it has \(e^*_{\rho'} \neq 0\). Rewrite its cycle affinity (Eq. 14) to isolate \(\rho'\): PRX · Eq. 15 \[ e^*_{\rho'} A^{\ss}_{\rho'} = A^Y_{\be^*} - \sum_{\rho \neq \rho'} e^*_\rho A^{\ss}_\rho . \]
- Drop the positive terms. Conformal means that every reaction in \(\be^*\) runs in the direction of its own steady-state flux, \(e^*_\rho J^{\ss}_\rho \gt 0\); by Eq. 8 the affinity has the flux's sign, so every term \(e^*_\rho A^{\ss}_\rho\) in the sum is positive. Dropping positive terms gives PRX · Eq. 16 \(\;0 \lt e^*_{\rho'} A^{\ss}_{\rho'} \lt A^Y_{\be^*}\). This already bounds the affinity, but it needs \(\be^*\), which needs the steady state, which needs the kinetics.
- Take the worst case over EFMs. Replace \(A^Y_{\be^*}/|e^*_{\rho'}|\) by its maximum over all EFMs that run \(\rho'\) in the same direction as \(\be^*\) (Eq. 17). Because the ratio \(A^Y_{\be}/e_{\rho'}\) is the same for \(\be\) and \(-\be\), either orientation of an EFM may be used when the extrema are taken.
- Allow both signs of \(e^*_{\rho'}\). Its sign is the sign of the unknown flux \(J^{\ss}_{\rho'}\). The case \(e^*_{\rho'} \gt 0\) gives an upper edge and a lower edge \(0\) (Eq. 18); the case \(e^*_{\rho'} \lt 0\) gives a lower edge and an upper edge \(0\) (Eq. 19). Combined, with the equality cases added by hand:
\[ \min\Big(0,\ \min_{\Ecal^-_\rho} \frac{A^Y_{\be}}{e_\rho}\Big) \;\le\; A^{\ss}_\rho \;\le\; \max\Big(0,\ \max_{\Ecal^+_\rho} \frac{A^Y_{\be}}{e_\rho}\Big), \] where \(\Ecal^\pm_\rho\) are the EFMs whose \(\rho\)-entry is positive (negative). Since the ratio is orientation-invariant, both extrema are in effect taken over every EFM through \(\rho\): each cycle affinity \(A^Y_{\be}\) divided by the signed entry \(e_\rho\), so an EFM that burns fuel while running \(\rho\) backwards contributes a negative number (reaction 4 in \(\be_1\): \(2\Delta\mu/(-1) = -2\Delta\mu\)). The \(0\) stands for the side of the window that the argument cannot see, the side opposite to the unknown flux. The window needs the stoichiometry and the chemostats, nothing else.
On the dimer network the windows read: reaction 1, \([0, \Delta\mu]\); reaction 2, \([0, 2\Delta\mu]\) (through \(\be_1\), which uses it once while burning fuel twice); reaction 3, \([-\Delta\mu, \Delta\mu]\); reaction 4, \([-2\Delta\mu, 0]\). Reaction 4 can never run forward at a steady state: every cycle through it that burns fuel runs it backwards. Lab 2 tests all four windows against random rate constants.
When is Eq. 20 tight?
(i) At equilibrium every \(A^Y_{\be} = 0\) and every window collapses to the point \(0\). (ii) A reaction that belongs to no cycle has empty \(\Ecal^\pm_\rho\), a window \(\{0\}\), and obeys detailed balance. (iii) A reaction whose net flux happens to vanish under driving has \(A^{\ss}_\rho = 0\) by Eq. 8; it touches the \(0\) edge of a window that stays open. Case (iii) shows that the \(0\) edge can be touched. The \(0\) itself is there because the argument sees only the side toward which the unknown flux points (step 4); the probe of Sec. 5 has no flux at all, and its bound will drop the \(0\).
5 · The chemical probe: a voltmeter for chemistry
Eq. 20 speaks only about reactions the network contains. The paper's central trick extends it to any conversion the conservation laws allow, whether or not a reaction performs it.
The reaction space \(\Rcal\) is the set of every reaction column orthogonal to all conservation laws of the full matrix \(\bS\), present in the network or not. Take a reaction \(\hat\rho \in \Rcal\) that the network does not contain and add it as a column, \(\tilde{\bS} = [\bS,\ \hat{\bS}_{\hat\rho}]\), infinitely slowly: its net flux is negligible, \(J_{\hat\rho} \simeq 0\), so the original steady state and all its affinities are undisturbed, but its own affinity \(A_{\hat\rho} = RT\ln(J^+_{\hat\rho}/J^-_{\hat\rho})\) is finite and reads the chemical-potential difference between its two sides. This is a voltmeter: infinite resistance, no current, a reading.
Why the 0 of Eq. 20 disappears. For a reaction in the network the proof pins only the side its own unknown flux points to; the \(0\) covers the other side. For a probe the paper builds two cycles. The first, \(\be^{(F)}\), passes through the probe and runs every reaction it uses along its affinity; it exists because \(\hat\rho \in \Rcal\) closes at least one cycle with a chain of reactions of the original network, a pathway in the sense made precise in Sec. 6 (Fig. 3a). Step 2 of Sec. 4 applied to it gives an upper edge (Eq. 22). The second, \(\be^{(B)}\), passes through the probe in the direction of the probe's affinity but runs every other reaction against its steady-state affinity (Fig. 3b); it exists because the probe's vanishing flux leaves the original affinities untouched, so a flux mode of the original network can be sent backwards through the same pathway and closed by another one. In Eq. 15 all its dropped terms are now negative, which gives a lower edge (Eq. 23). Both edges are cycle affinities per pass, so no \(0\) is needed:
\[ \min_{\Ecal^-_{\hat\rho}} \frac{A^Y_{\be}}{e_{\hat\rho}} \;\le\; A^{\ss}_{\hat\rho} \;\le\; \max_{\Ecal^+_{\hat\rho}} \frac{A^Y_{\be}}{e_{\hat\rho}} , \] with the extrema over the EFMs of the extended network that pass through the probe. It is tighter than Eq. 20 because the probe is known to carry no flux. In Part III the same window will be read as a window on concentrations, with pathways of the original network: the probe never existed physically.
Part III · Second result: the thermodynamic space (paper Sec. IV B–C)
From affinities to concentrations
An affinity is a chemical-potential difference, and a chemical potential is a logarithm of a concentration. So a window on an affinity is a window on a ratio of concentrations. The paper names the region cut out by all such windows the thermodynamic space.
6 · The thermodynamic space
Two objects from Part II are needed once more. Removing the external part \(A^Y_\rho\) from Eq. 20 bounds the internal part of an affinity by pathways (Eq. 21), and the same for a probe (Eq. 25):
A pathway in \(\Pi^X_{\rho^-}\) converts the internal species from the right side of \(\rho\) to its left: it is either \(\rho\) itself run backwards, or an EFM through \(\rho\) with \(\rho\) deleted and divided by \(e_\rho\) (one conversion); its affinity is \(-A^Y_\rho\), or \(A^Y_{\be}/e_\rho - A^Y_\rho\). Then \(A^{X,\ss}_\rho \le \max_{\Pi^X_{\rho^-}} A^Y_{\bpi}\). For a probe the pathways are built from the original network only. \(\Pi^X_{\rho^+}\) is the mirror set, converting left to right.
\[ A^{X,\ss}_\rho = -\Delta_\rho G^\circ_X - RT \sum_i \bS^X_{i\rho} \ln x_i^{\ss}, \qquad A^Y_{\bpi} = -\Delta_{\bpi} G_Y , \] where \(\Delta_\rho G^\circ_X\) is the standard free-energy change of the internal species along \(\rho\) and \(\Delta_{\bpi} G_Y\) the free-energy change of the reservoirs along \(\bpi\) (Eq. 26). A pathway run backwards has the opposite affinity, so \(\max_{\Pi^X_{\rho^-}} A^Y_{\bpi} = -\min_{\Pi^X_{\rho^+}} A^Y_{\bpi}\) (Eq. 27). This duality is the hinge: an upper bound on the affinity becomes a lower bound on the concentration quotient.
Insert Eq. 26 into Eq. 21, use Eq. 27, and exponentiate. The sum of logarithms becomes the mass-action quotient of the internal species across \(\rho\), and each pathway becomes a number:
\[ \prod_i \big(x_i^{\ss}\big)^{\bS^X_{i\rho}} \;\ge\; \min_{\Pi^X_{\rho^+}} \underbrace{\exp\!\Big[-\frac{\Delta_\rho G^\circ_X + \Delta_{\bpi^X_{\rho^+}} G_Y}{RT}\Big]}_{K_{\bpi^X_{\rho^+}}} \;\equiv\; K^{\min}_{\bpi^X_{\rho^+}} . \] \(K_{\bpi}\) is what the quotient would settle to if \(\bpi\) were the only pathway: the standard equilibrium constant of \(\rho\), shifted by the reservoir free energy that \(\bpi\) releases while converting the left side of \(\rho\) into its right side. The mirror argument, starting from the lower bound on the affinity, gives the upper edge.
\[ K^{\min}_{\bpi^X_{\rho^+}} \;\le\; \prod_i \big(x_i^{\ss}\big)^{\bS^X_{i\rho}} \;\le\; K^{\max}_{\bpi^X_{\rho^+}} , \] the extrema over all pathways that convert the left side of \(\rho\) into its right side. The statement holds for every reaction of the network and, through the probe (Eq. 25), for every conversion \(\hat\rho\) in the reaction space, with pathways built from the original network. Writing it for every such conversion carves out the accessible region of concentrations: the thermodynamic space. Quotients that no conservation-law-compatible conversion sets (\(x_C/x_A\) in the dimer network, since \(A \rightleftharpoons C\) would change \(a + b + 2c\)) have no window.
On logarithmic axes each window is a strip of fixed width between two parallel lines (the paper's Fig. 1b): the red edge is the pathway with the largest effective \(K\), the blue edge the one with the smallest, and the width, \(RT\ln(K^{\max}/K^{\min})\), is the reservoir free energy separating the two extremal pathways. Three consequences, all from the same equation:
- Equilibrium collapses the space. With \(A^Y_{\be} = 0\) for every cycle, every pathway has the same \(K = e^{-\Delta_\rho G^\circ_X/RT}\); the strip becomes a line and the concentrations are the unique equilibrium fixed by free energies and conservation laws. A non-zero width is a signature of driving.
- Each edge can be saturated, in a pseudo-equilibrium state with vanishing flux, when the extremal pathway dominates the kinetics; Labs 5 and 6 show this happening through a timescale separation.
- The space is the room for complexity. Bistability, symmetry breaking and patterns need distinct stationary concentrations; all of them must fit inside the space, so its width is a prerequisite for them (Part IV).
7 · Turning the bound around: a kinetics-free test of a topology
Eq. 29 needs the network's stoichiometry as input. Read backwards, it interrogates that stoichiometry: the thermodynamic space contains every steady state that any kinetics compatible with a proposed set of reactions can reach, so a measured quotient outside it falsifies the whole proposal at once, with no rate constant fitted.
On the dimer network, ask how much the dimer can be amplified over its equilibrium value, \(\ln[x_C/x_A^2] - \ln K^\circ\) with \(K^\circ = e^{-(\mu^\circ_C - 2\mu^\circ_A)/RT}\). The pathways converting \(2A\) into \(C\) are reaction 4 itself (no fuel, \(K = K^\circ\)) and the EFMs through reaction 4 with reaction 4 removed: from \(\be_1\), activate both monomers and dimerize them, which burns fuel twice; from \(\be_2\), activate one monomer, which burns fuel once; from \(\be_3\), no fuel at all.
Full network \(\mathcal{N}_1\) (reactions 1–4): \(\;0 \le \ln\dfrac{x_C^{\ss}}{(x_A^{\ss})^2} - \ln K^\circ \le \dfrac{2\Delta\mu}{RT}\) (Eq. 30), the ceiling set by \(\be_1\).
Reduced network \(\mathcal{N}_2\) (reaction 2 removed): \(\;0 \le \ln\dfrac{x_C^{\ss}}{(x_A^{\ss})^2} - \ln K^\circ \le \dfrac{\Delta\mu}{RT}\) (Eq. 31), because removing \(2B \rightleftharpoons C\) destroys \(\be_1\) and the best remaining mode, \(\be_2\), meets the driven reaction once.
A single added hyperedge doubles the exponent. A measured amplification in the inference window \([\Delta\mu/RT,\ 2\Delta\mu/RT]\) cannot come from \(\mathcal{N}_2\) and reveals the active-monomer dimerization. The test is one-sided: exceeding a ceiling falsifies a topology, while a missing reaction that does not re-use the driving leaves every bound unchanged and stays invisible. Re-using a driven reaction is impossible in an ordinary graph, so this is a signature of hypergraphs.
Part IV · Four applications (paper Sec. V)
The same game on four models
Each application follows the same three moves: write the stoichiometric matrix, find the elementary flux modes and their cycle affinities, and read off the windows. The lab then shows the model's actual fixed points sitting inside them.
8 · Schlögl model: a strip that closes at equilibrium
The minimal model of chemical bistability has one internal species \(X\) and two chemostatted ones, \(A\) and \(B\):
\(\bS^X = [\,1, -1\,]\), so the only EFM is \(\be = [1, 1]\): one \(A\) becomes one \(B\) through \(X\), with cycle affinity \(A_{\be} = \mu_A - \mu_B\). Both affinities satisfy \(0 \le A^{\ss}_\rho \le \mu_A - \mu_B\) (Eq. 35). Reaction 1 changes the internal species by a net \(+X\) (its \(\bS^X\) column is \([1]\)), so Eq. 29 written for it bounds \(x^{\ss}\) itself. The pathways from its left side \(2X\) to its right side \(3X\) are reaction 1 forward (\([1, 0]\)) and reaction 2 backward (\([0, -1]\)); their effective equilibrium constants bound the stationary concentration (Eq. 36): \[ e^{-(\mu^\circ_X - \mu_B)/RT} \;\le\; x^{\ss} \;\le\; e^{-(\mu^\circ_X - \mu_A)/RT} . \]
With the rate constants of the paper's Fig. 5, \(k_1^\pm = 8\), \(k_2^\pm = 1\), \(b = 0.02\), all standard potentials are equal, so the strip is simply \(b \le x^{\ss} \le a\) and \(A_{\be}/RT = \ln(a/b)\). At equilibrium (\(a = b\)) the strip is a point and only one fixed point can exist: bistability needs driving. The upper edge is approached when reaction 1 is much faster than reaction 2 (reaction 1 at quasi-equilibrium), the lower edge in the opposite limit. At the driving of Fig. 5 both limits leave a single fixed point (with reaction 2 fast the fold only moves to a much larger driving, beyond this lab's range): bistability needs driving, and at a given budget it also needs the two time scales not too far apart.
9 · Chiral symmetry breaking: how much imbalance can a budget buy
Two mirror-image molecules \(R\) and \(S\) with the same standard chemical potential are made autocatalytically from an achiral precursor \(A\) and destroyed by pairing into a dimer \(C\); \(A\) and \(C\) are chemostatted. This is a reversible version of Frank's model of homochirality.
The single EFM \(\be = [1, 1, 1]\) converts \(2A\) into \(C\), with cycle affinity \(A_{\be} = \Delta\mu = 2\mu_A - \mu_C\). No reaction converts \(S\) into \(R\), so the paper adds the probe \(S \rightleftharpoons R\) (the probe drawn in Fig. 6a). The extended network has three EFMs through the probe, \(\hat{\be}_1 = [2, 0, 1, 1]\), \(\hat{\be}_2 = [1, -1, 0, 1]\), \(\hat{\be}_3 = [0, -2, -1, 1]\) (Eq. 38), with cycle affinities \(\Delta\mu\), \(0\), \(-\Delta\mu\). The probe reads \(A^{\ss}_{\hat\rho} = \mu_S - \mu_R = RT\ln(s/r)\) since \(\mu^\circ_R = \mu^\circ_S\), and Eq. 24 gives \[ -\frac{\Delta\mu}{RT} \;\le\; \ln\frac{r^{\ss}}{s^{\ss}} \;\le\; \frac{\Delta\mu}{RT} . \]
The model bifurcates: below a critical driving the only fixed point is achiral (\(r = s\)); above it two mirror chiral states appear and the achiral one loses stability. Both chiral states stay inside the wedge \(|\ln(r/s)| \le \Delta\mu/RT\), so perfect homochirality (\(s \to 0\)) would need infinite driving.
PRX · Appendix A, Eqs. A2–A8 further
The model is exactly solvable. Subtracting the two stationarity conditions factorizes as \((r - s)[k_0^+ a - k_0^-(r + s)] = 0\), so a chiral state has \(r + s = P = k_0^+ a/k_0^-\), and summing them gives \(rs = Q = k_1^- c/(k_1^+ - k_0^-)\), which requires \(\kappa = k_1^+/k_0^- \gt 1\) (the slider of Lab 6 starts at 1.5; below \(\kappa = 1\) only the achiral state exists). With \(\beta = 4Q/P^2 = \dfrac{4\kappa}{\kappa - 1} e^{-\Delta\mu/RT}\) and \(\kappa = k_1^+/k_0^-\), the order parameter is \(\ln(r^{\ss}/s^{\ss}) = 2\,\mathrm{artanh}\sqrt{1 - \beta}\) and the pitchfork sits at \(\Delta\mu_{\rm crit}/RT = \ln[4\kappa/(\kappa - 1)]\). On the chiral branch the dimerization step has the stationary affinity \(A^{\ss}_3 = RT\ln[\kappa/(\kappa - 1)]\) exactly (Eq. A8), and the distance to the fence is the sum of the other affinities around the cycle \(\hat{\be}_3\), \(\Delta\mu - RT\ln(r/s) = 2A^{\ss}_2 + A^{\ss}_3\); as the minority enantiomer vanishes, \(A_2^{\ss} \to 0\) and \[ \frac{\Delta\mu}{RT} - \ln\frac{r^{\ss}}{s^{\ss}} \;\to\; \ln\frac{\kappa}{\kappa - 1} \] (Eqs. 40, A7). The asymptotic gap is a purely kinetic knob: making the dimerization fast at fixed \(k_1^+/k_1^-\) (\(\kappa \to \infty\)) leaves \(\Delta\mu\) untouched, brings \(R + S \rightleftharpoons C\) to pseudo-equilibrium, and lets the chiral branches come as close to the fence as one likes: the gap \(\ln[\kappa/(\kappa - 1)]\) vanishes only in the limit.
10 · Dissipative self-assembly: a probe for a reaction that does not exist
A monomer \(X_1\), a dimer \(X_2\) and a trimer \(X_3\); fuel drives the dimerization, the rest is spontaneous:
One EFM, \(\be = [1, 1, 1]\), with \(A_{\be} = \mu_F - \mu_W = \Delta\mu\); every affinity lies in \([0, \Delta\mu]\) (Eq. 42). Eq. 29 on the three reactions gives (Eq. 43) \[ e^{-\frac{\mu^\circ_2 - 2\mu^\circ_1}{RT}} \le \frac{x_2}{x_1^2} \le e^{-\frac{\mu^\circ_2 - 2\mu^\circ_1 - \Delta\mu}{RT}}, \quad e^{-\frac{\mu^\circ_3 - \mu^\circ_2 - \mu^\circ_1 + \Delta\mu}{RT}} \le \frac{x_3}{x_1 x_2} \le e^{-\frac{\mu^\circ_3 - \mu^\circ_2 - \mu^\circ_1}{RT}}, \quad e^{-\frac{3\mu^\circ_1 - \mu^\circ_3 + \Delta\mu}{RT}} \le \frac{x_1^3}{x_3} \le e^{-\frac{3\mu^\circ_1 - \mu^\circ_3}{RT}} . \] The conservation law \(x_1 + 2x_2 + 3x_3 = \) const (elementary conservation law \([1, 2, 3]\)) also allows the conversion \(3X_2 \rightleftharpoons 2X_3\) (Eq. 44), which no reaction performs; added as a probe it closes three EFMs, \(\hat{\be}_1 = [1, -2, 0, 1]\), \(\hat{\be}_2 = [3, 0, 2, 1]\), \(\hat{\be}_3 = [0, -3, -1, 1]\), and gives (Eq. 45) \[ e^{-\frac{2\mu^\circ_3 - 3\mu^\circ_2 + 3\Delta\mu}{RT}} \;\le\; \frac{x_3^2}{x_2^3} \;\le\; e^{-\frac{2\mu^\circ_3 - 3\mu^\circ_2}{RT}}, \qquad 0 \le A^{\ss}_{\hat\rho} \le 3\Delta\mu \ \text{(Eq. 46)} . \]
Note which edge is the equilibrium one. Fuel pumps \(2X_1 \to X_2\), so it pushes \(x_2/x_1^2\) up (the driven pathway is the upper edge), but it pushes \(x_3/(x_1 x_2)\), \(x_1^3/x_3\) and \(x_3^2/x_2^3\) down: along those conversions the fuel-burning pathway runs reaction 1 backwards, so the equilibrium constant is the upper edge. The probe strip is the widest, \(3\Delta\mu\), because \(\hat{\be}_2\) runs the driven reaction three times.
11 · Reaction–diffusion patterns: the widest cut through the strip
The last application leaves the well-mixed setting. Two species \(U\) and \(V\) interconvert through a spontaneous pathway and a fuel-driven catalytic one, and diffuse:
\[ mU \underset{k_-}{\overset{k_+}{\rightleftharpoons}} V, \qquad W + (n + m)U \underset{w_-}{\overset{w_+}{\rightleftharpoons}} nU + V + F , \] with \(w_+' = w_+[W]\) and \(w_-' = w_-[F]\) absorbing the chemostats and cycle affinity \(A_e = \Delta\mu = \mu_F - \mu_W\). The well-mixed steady states are the reactive nullcline \(f(u, v) = 0\), and Eq. 29 applied to the conversion \(mU \rightleftharpoons V\) traps that nullcline in a strip (Eq. 52): \[ K^{\min} = \frac{w_+[W]}{w_-[F]} \;\le\; \frac{v}{u^m} \;\le\; K^{\max} = \frac{k_+}{k_-}, \qquad \frac{\Delta\mu}{RT} = \ln\frac{K^{\max}}{K^{\min}} . \]
PRX · Eqs. 48–50 further
The dynamics is \(\partial_t u = D_u \nabla^2 u + m f\), \(\partial_t v = D_v \nabla^2 v - f\) with \(f = -k_+ u^m + k_- v - w_+' u^{n+m} + w_-' u^n v\) (Eqs. 48–49). With no-flux boundaries the combination \(\eta = u + m(D_v/D_u)v\) is harmonic at stationarity, hence constant on a connected domain (Eq. 50).
Every stationary pattern therefore lies on the flux-balance line \(D_u u + m D_v v = \) const (Eq. 51), of slope \(-D_u/(mD_v)\) in the \((u, v)\) plane. Following the phase-space geometry of pattern formation that the paper cites, the pattern's extreme concentrations \(u_{\min}, u_{\max}\) are where this line meets the reactive nullcline. Since the nullcline lies inside the strip, these points lie between the line's crossings of the strip edges, \(u^*_{\min}\) and \(u^*_{\max}\). That cut is widest when the line is horizontal (\(D_u/D_v \to 0\)), the only case in which it is independent of the constant, where \(u^*_{\max}/u^*_{\min} = e^{\Delta\mu/mRT}\) (Eq. 53); hence the diffusion-independent bound on the contrast (Eq. 54) \[ \Omega_u = \frac{u_{\max} - u_{\min}}{u_{\max} + u_{\min}} \;\le\; \tanh\frac{\Delta\mu}{2mRT} . \]
Part V · Beyond the labs (paper Sec. IV D, Sec. VI and appendices)
What the page leaves out
12 · Older results, experimental data, software further
Older results as special cases (paper Sec. IV D). In a linear or catalytic network every column of \(\bS^X\) holds one \(+1\) and one \(-1\): the network is an ordinary graph, the conservation law is \(\sum_i x_i = \) const, and steady states can be written with spanning trees. Choosing the probe \(X_i \rightleftharpoons X_{i'}\) in Eq. 29 and normalizing by the conserved total, \(p_i = x_i/\sum_j x_j\), gives the bound on stationary probabilities known from stochastic thermodynamics (Eq. 32), \[ K^{\min}_{\bpi_{X_i \to X_{i'}}} \le \frac{p^{\ss}_{i'}}{p^{\ss}_{i}} \le K^{\max}_{\bpi_{X_i \to X_{i'}}} , \] first found by Maes and Netočný, rederived by Lin through an electric-circuit analogy, extended to catalytic networks by Liang, De Los Rios and Busiello, and proven along trajectories by Liang and Pigolotti, whose flux-asymmetry bound \(|\ln J^+_\rho/J^-_\rho| \le \max_{\be \ni \rho} A^Y_{\be}/RT\) (Eq. 33) is Eq. 20 read through Eq. 8. Lab 3 was an instance of Eq. 32. Breaking free of the spanning-tree machinery is what lets the present paper reach hypergraphs.
Two data sets (paper Sec. VI, Fig. 9). For the chaperone GroEL acting on the enzyme malate dehydrogenase, the ratios native dimer to native monomer squared and native dimer to intermediate squared sit on the equilibrium line even with ATP present, while the ratio of native dimer to misfolded protein squared is displaced from equilibrium and falls inside the thermodynamic space: within the budget, ATP hydrolysis acts as if it raised the free energy of the misfolded state. For a light-driven photoacid (a protonated merocyanine cycling through four forms), the measured concentrations are incompatible with the equilibrium line but lie inside the space computed for the light-driven step; read backwards, Eq. 29 infers the smallest light-driving compatible with the observed ratios and recovers up to 75–85 % of the measured value. The paper labels these consistency bounds within the coarse-grained models it inherits from the original studies.
Software and what is not needed. Finding the space does not require listing every elementary flux mode: only pathways through driven reactions matter, which the paper frames as a targeted search left for future work. The authors release TACOS (paper Appendix C), an open-source Python package that computes the thermodynamic space of small- to medium-size networks and supports chemical probes.
Not covered by the theory. Networks that never settle (limit cycles, chaos); the paper names bounds on such attractors as an open direction.
Dictionary
The same object in the paper's words, with its equation, and on this page.
Self-check
Each answer can be confirmed in the lab named with it.
Q1In the dimer network at \(\Delta\mu = 3RT\), what is the largest possible value of \(\ln[x_C/x_A^2] - \ln K^\circ\) with reaction 2 present, and without it?
\(6\) with reaction 2 (\(2\Delta\mu/RT\), Eq. 30) and \(3\) without (\(\Delta\mu/RT\), Eq. 31). Lab 4: drag the cursor to \(\Delta\mu = 3\).
Q2Can reaction 4, \(2A \rightleftharpoons C\), have a positive stationary affinity in the dimer network?
No. Its window is \([-2\Delta\mu, 0]\): every EFM through it that burns fuel (\(\be_1\), \(\be_2\)) runs it backwards, and \(\be_3\) carries no fuel. Lab 2: the marker of reaction 4 never crosses zero.
Q3Schlögl model with \(b = 0.02\) and \(\mu^\circ_X = \mu^\circ_A = \mu^\circ_B\) as in Fig. 5, at \(A_{\be} = 4RT\): what is the strip?
\(0.02 \le x^{\ss} \le 0.02\,e^{4} = 1.092\). Lab 5: set \(A_{\be} = 4\); with the paper's rate constants the three fixed points are near \(0.026\), \(0.10\) and \(0.97\).
Q4Frank model with \(\kappa = 3.4\): what gap to the fence remains at very large driving, and how large must \(\kappa\) be for that asymptotic gap to drop below \(0.1\)?
\(\ln(3.4/2.4) = 0.348\). An asymptotic gap \(\ln[\kappa/(\kappa - 1)] \lt 0.1\) needs \(\kappa \gt 1/(1 - e^{-0.1}) = 10.5\); at finite driving the actual gap is larger by \(2A^{\ss}_2\). Lab 6: raise \(\kappa\) past 10.5 and read the row \(\ln\kappa/(\kappa-1)\); the row above it is the full gap at the current driving, larger by \(2A_2^{\ss}\).
Q5Of the four strips of the self-assembly network, which is the widest, and why?
\(x_3^2/x_2^3\), width \(3\Delta\mu\): the pathway from \(\hat{\be}_2 = [3, 0, 2, 1]\) runs the fuel-driven reaction three times. Lab 7, default view.
Q6Reaction–diffusion with \(m = 2\) and \(\Delta\mu = \ln 30 = 3.40\,RT\): what is the contrast bound, and does a finite \(D_u/D_v\) reach it?
\(\tanh(3.40/4) = 0.691\). A finite ratio tilts the flux-balance line and gives a narrower cut: at \(D_u/D_v = 0.1\) and constant \(0.8\), \(\Omega^* = 0.672\). Lab 8.