How equilibrium became a model of regulation

A history of compatible reactions, reversible probabilities and molecular states that predict biological activity.

CCBS · Fall 2026 · Lecture 7 · Lecturer-written lineage · Revised 22 September 2026

The attraction of equilibrium is a reduction in what must be measured. A regulatory protein may have many conformations and binding partners. A kinetic model asks how rapidly the system moves between those states. An equilibrium model can sometimes predict their relative populations without those individual clocks. The missing kinetic information is replaced by physical constraints, not by permission to choose arbitrary probabilities.

This method has several histories. Chemical kinetics asked whether different reaction routes could agree at equilibrium. Statistical mechanics asked how to assign probabilities to physical states. Probability theory asked when a stochastic history had the same law as its reverse. Molecular biology supplied a further question: which of those states changes the activity of an enzyme or a promoter?

The argument below follows the joining of these problems. The papers are not a single chain of discoveries or a list of first inventors. Where the historical sources explicitly connect to earlier work, that connection is identified. Elsewhere, the comparison is a modern reconstruction of shared mathematical structure. The core teaches the resulting framework. The exposition supplies its derivations.

1Predict without measuring every transition

A molecular-state model needs both probabilities and a readout. Let $s$ label a distinguishable configuration, $p_s$ its probability, and $r_s$ its activity. The average activity is $\bar r=\sum_s r_sp_s$. For example, a repressor-bound promoter may have little transcriptional activity. Knowing that activity rule does not yet tell us how often the bound state occurs.

Equilibrium offers a particular answer to the probability question. Assign each state a dimensionless statistical weight $w_s$, then normalize. The challenge is to justify those weights from a physical system. They must count the permitted configurations, include the relevant reservoirs and agree with reversible transitions. The papers in this essay gradually make those requirements usable.

Separate the probability model from the activity rule in a proposed equilibrium model of regulation.

2Wegscheider asks whether reaction routes can agree

A set of rate equations can settle without balancing each opposing reaction. Wegscheider's 1901 discussion of three interconverting isomers calculated stationary concentrations and compared the opposing reaction rates. His calculation exposes the distinction that now separates node balance from detailed balance. The selected historical passage is available in Niklas Manz's translation of section IV.¹

The consistency condition can be seen without solving the dynamics. In modern notation, take three elementary reversible isomerizations, each with its own microscopic constants:

At detailed balance, the three equilibrium concentration ratios are $B/A=k_{+1}/k_{-1}$, $C/B=k_{+2}/k_{-2}$ and $A/C=k_{+3}/k_{-3}$. Multiplying eliminates the concentrations:

The restriction belongs to physical equilibrium, not to linear algebra alone. Six positive constants that violate (2) still define a valid linear dynamical system with a stationary composition. They do not describe a closed equilibrium cycle with those steps interpreted as the complete physical mechanism. A reservoir-driven interpretation would require additional chemical bookkeeping.

Test a closed three-reaction cycle for equilibrium compatibility before solving its stationary concentrations.

3Gibbs changes the object from a trajectory to an ensemble

Gibbs begins his 1902 book by changing the question asked of mechanics. One can predict a particular system's configuration and velocity, or study a collection of systems distributed across possible mechanical states. The second object is an ensemble. Its distribution can remain stationary while individual members continue moving. This does not require the underlying state space to be discrete.²

The canonical distribution makes energy a probability coordinate. In chapter IV, Gibbs studies an exponential distribution in the energy and emphasizes its behaviour when systems with separate energies are combined. In present notation, discrete microstates $\alpha$ with energies $E_\alpha$ have probabilities

The denominator carries physical information. It counts every permitted state with its thermal weight. Changing the state list changes the prediction even if the energy of one selected state stays fixed. Gibbs's treatment requires a normalizable distribution. The formula is not a licence to assign a Boltzmann interpretation to every stationary histogram.

The relevant ensemble depends on what can be exchanged. An isolated system has a fixed energy constraint. A heat bath leads to canonical weights. Particle exchange adds chemical potentials, as in Gibbs's chapter XV. The familiar statements about entropy maximization and free-energy minimization therefore need their constraints attached. They are not interchangeable descriptions of every stationary system.

An ensemble minimum and a kinetic relaxation law answer different questions. The modern excess-free-energy identity $\mathcal F[p]-\mathcal F[p^{\mathrm{eq}}]=k_BT\sum_i p_i\ln(p_i/p_i^{\mathrm{eq}})$ identifies the equilibrium distribution as a minimum. Showing that a chosen dynamics approaches that minimum requires another argument. The exposition derives that Lyapunov argument for a detailed-balanced generator. This is a modern connection between the course's energy and probability tools, not an attribution of the later Markov-chain proof to Gibbs.

The distinction becomes essential for living systems. A maintained enzyme cycle can approach a stationary distribution while continuing to convert substrate and release free energy. Its probability distribution stops changing without its operation stopping. The later reversibility and cycle theories address the kinetic information absent from an equilibrium ensemble calculation.

Distinguish prediction of an individual trajectory from prediction of a population of mechanical states.

4A binding state includes entropy, not only a bond energy

A named molecular configuration usually groups many microstates. A promoter described as “free” leaves its repressors many places to be. A bound configuration constrains one of those molecules. Summing the microscopic Boltzmann weights within each class defines an effective state free energy. If all $g_s$ microstates in a class have the same energy $E_s$, the result is especially transparent:

Ligand concentration enters through the reservoir. For dilute free ligand at concentration $c$, exchanging one molecule contributes a chemical-potential term containing $k_BT\ln(c/c^\circ)$. Let $K_d$ denote the dissociation constant, in concentration units. A standard-state binding free energy $\Delta G^\circ_{\mathrm{bind}}$ gives the relative bound weight

The concentration factor cannot be dropped. The bound state's probability does not follow from a bond energy alone. A declared standard concentration makes (5) dimensionless. Changing that convention shifts the numerical standard free energy but cannot change the physical probability. Bintu and colleagues make the counting argument concrete by enumerating specific and nonspecific DNA-binding configurations.⁸

Construct a molecular state weight from energetic preference, multiplicity and the specified reservoir.

5Langmuir gives a saturation curve a physical setting

Langmuir's 1918 problem was adsorption on material surfaces. His simple limiting model has one kind of site, one adsorbed molecule per site, and adsorption balanced by evaporation. In modern concentration notation, occupied fraction $\theta$ obeys $k_+c(1-\theta)=k_-\theta$, giving $\theta=(c/K_d)/(1+c/K_d)$. The same algebra describes single-site receptor occupancy.³

Langmuir did not offer this as a universal adsorption law. Before deriving the simple case, he stresses that different physical conditions need different limiting equations. He also treats relaxation separately from the equilibrium amount. Those distinctions matter when the formula appears in biology: an equilibrium binding isotherm and a Michaelis–Menten turnover law can share a rational form while describing different observables and assumptions.

Distinguish a binding isotherm from a turnover law even when both have the same rational form.

6Kolmogorov compares histories with their reverses

Kolmogorov's 1936 paper asks when reversed conditional probabilities reproduce the original law. It first defines a finite-state process whose future conditional probabilities do not depend on earlier history once the present state is specified. It then constructs inverse probabilities and identifies a reversibility condition. The inspected source is the English translation in his collected works, pages 182–187.⁴

The course's generator is a continuous-time version of this question. Let $q_{ij}$ be the transition hazard from state $j$ to state $i$. The diagonal entry is $q_{jj}=-\sum_{i\ne j}q_{ij}$. For a finite irreducible chain, let $p^*$ be its normalized stationary probability column. Its entries are positive. For the time-dependent probability column $p$,

Detailed balance equates the two stationary dynamics. The condition $q^{\mathrm{rev}}_{ij}=q_{ij}$ is exactly $q_{ij}p_j^*=q_{ji}p_i^*$. It compares probabilities of histories, not just the histogram of states. Constant hazards additionally assume a time-homogeneous model. They do not follow from the Markov property alone.

Test reversibility by comparing forward and reversed transition hazards, rather than a stationary histogram alone.

7Cycle compatibility makes a state potential possible

Multiplying detailed-balance ratios around a closed path removes the state probabilities. For the finite, connected, time-homogeneous chain with positive reverse transitions used here, the resulting cycle criterion is also sufficient. Compatible ratios define a positive state weight independently of the chosen path. This is the rate-language counterpart of assigning one free energy to each state.^{4, 11}

The potential must agree with every rate ratio. Writing $G_i=-k_BT\ln p_i^*$ for an arbitrary positive stationary distribution does not establish (7). A driven ring can have a uniform stationary histogram and still prefer clockwise histories. The constraint is kinetic compatibility, not the ability to take a logarithm.

This equivalence needs the stated state space. Mechanical time reversal can also reverse velocity or momentum. The simple transition symmetry above concerns the configurational jump states of this lecture. Its identification with thermodynamic equilibrium further requires physically resolved reverse mechanisms and compatible reservoirs.

Check a proposed state potential against every transition ratio, not merely against the observed probabilities.

8Schnakenberg connects graph circulation to reservoirs

A biological transport cycle needs more than a probability graph. Schnakenberg's 1976 review begins with systems coupled to reservoirs that need not be mutually equilibrated. Its examples include an ion moving through a membrane pore and a chemical reaction system. The graph organizes both the stationary solve and the physical forces that maintain circulation.⁵

The stationary current and its driving force are different quantities. On an oriented edge, define $J_{i\to j}=q_{ji}p_i^*-q_{ij}p_j^*$. On a single ring, node balance makes this current the same on every edge. Call that common current $J_{\mathrm{cycle}}$. Its dimensionless affinity is the logarithm of the product of forward hazards divided by the reverse product:

The thermodynamic reading depends on the mechanism. When local rate ratios correctly encode reservoir exchange, $k_BT\mathcal A$ measures the chemical work available per oriented cycle, and $\dot S_{\mathrm{tot}}$ is the total entropy-production rate. A maintained substrate-to-product chemical-potential difference can drive an enzyme wheel. Constant enzyme-state probabilities then coexist with throughput. Stationarity says nothing has accumulated, not that nothing has been consumed.

The review already distinguishes complete balance from cancelling channels. In its chemical example, separate transition mechanisms can carry opposing currents between the same observed counts. Balancing their sum is weaker than balancing each mechanism. The modern warning about hidden dissipation has a clear precursor in this explicit channel-resolved construction.

Metabolic pathway models turn this distinction into a resource question. Flamholz and colleagues' 2013 comparison of glycolytic strategies asks why organisms use pathways with different ATP yields. Their analysis connects the free-energy drop of a reaction to opposing forward and reverse conversion and the enzyme needed for a fixed net flux. In their constrained optimization, several steps in classical glycolysis operate with a driving force around $2.9\,\mathrm{kJ/mol}$. This is a model-derived bottleneck scale under stated concentration and solution constraints, not a constant of glycolysis.¹⁷

The thermodynamic cost and the protein cost are not the same bill. For the same net conversion, a smaller free-energy drop dissipates less energy per product but can require more opposing traffic and enzyme. Kinetic capacities and substrate occupancy still matter. The core's DHAP/GAP playground uses the published energy scale with deliberately uncalibrated kinetics. The exposition derives its exact current and power. Neither calculation measures the enzyme investment or power consumption of a real cell.

The flux–force relation isolates the thermodynamic part of enzyme demand. In the overall reversible-conversion description used by Flamholz and colleagues, let $j_+$ and $j_-$ denote one-way conversion fluxes and $\mathcal A=-\Delta_rG/(RT)>0$ the molar driving. Then $j_-/j_+=e^{-\mathcal A}$. Subtracting the reverse conversions gives

The forward capacity per enzyme, $r_{\mathrm{conv}}^+$, has inverse-time units. At a matched forward capacity and desired net flux, $(1-e^{-\mathcal A})^{-1}$ is the enzyme penalty due to opposing conversion. The original paper also includes substrate-saturation and catalytic-capacity factors. The directional conversion fluxes here are not the raw jumps on one binding edge of the three-state enzyme.

The pathway comparison exposes a tradeoff rather than a universal preference for dissipation. Classical glycolysis conserves more energy in ATP. The alternative Entner–Doudoroff route can avoid some near-equilibrium bottlenecks and require less enzyme for a given throughput in the authors' model. ATP yield, reaction driving, saturation and protein investment must be accounted for together. This is why a statement that equilibrium costs no sustained dissipation does not imply that a cell should run every productive reaction arbitrarily close to equilibrium.¹⁷

Connect a cycle current to reservoir work, then distinguish its dissipated power from the kinetic and protein costs of maintaining that current.

9MWC explains how binding affects a distant site

The allosteric problem was how one binding event could affect another distant site. Monod, Wyman and Changeux opened their 1965 paper with haemoglobin and extended the question to regulatory enzymes. Spatially separated sites made a purely local account inadequate. Their proposal connected ligand affinity to a collective conformational change of an oligomeric protein.⁶

The model makes a strong structural choice. Equivalent subunits share one of two conformations. Binding sites are independent within a fixed conformation, but their affinities differ between conformations. A ligand preferentially stabilizes the conformation to which it binds more strongly. Apparent cooperation can therefore arise from redistribution between conformational populations.

This is more informative than inserting a Hill exponent. The model specifies which states exist, which symmetries are preserved and which affinities change. It also excludes mixed-conformation oligomers in its concerted construction. These are hypotheses about a protein, not consequences of equilibrium itself. The original title's description of a plausible model is an appropriate reminder of that scope.

Explain how a binding event can regulate a distant site through a shift in conformational populations.

10The state sum distinguishes activity from occupancy

The original MWC calculation separates a conformational fraction from a ligand-bound fraction. In course notation, call the conformations active and inactive, with $n$ equivalent sites, dissociation constants $K_A,K_I$, and unliganded inactive-to-active weight ratio $L$. At free ligand concentration $c$, the binomial sums give

Counting occupied sites gives a different numerator. If $\theta$ is the fraction of ligand sites occupied, then

For two sites there are eight molecular configurations, grouped into six conformation-and-occupancy classes. The multiplicity of the singly occupied class is two. Activity can be identified with $p_A$ only when the chosen active-state readout is justified. The ligand-binding experiment measures $\theta$, not automatically that activity.

Calculate active fraction and ligand occupancy as different averages over the same molecular ensemble.

11A successful curve does not identify every microscopic feature

Population measurements can leave the underlying transitions undetermined. Any detailed-balanced graph can have an opposing rate pair multiplied by a common positive factor without changing its equilibrium weights. Such a change modifies relaxation and waiting times. Even a perfect occupancy fit therefore leaves kinetic questions open.

A sharp dose-response also does not uniquely count binding sites. A Hill coefficient summarizes a response slope under a selected normalization. Conformational bias, affinity ratios and the activity assigned to each state affect that slope. Equations (9)–(10) show why a response fitted by a compact empirical function need not reveal its state list.

The historical achievement was a testable mechanism, not a unique explanation of every sigmoidal curve. Structural observations, ligand-binding measurements and activity assays constrain different parts of an allosteric model. Their joint agreement is stronger than fitting only one curve. Razo-Mejia and colleagues later encountered a related practical issue: multiple allosteric parameter combinations could explain one induction curve, so they used additional calibration information.¹⁰

Identify which kinetic and structural questions an occupancy curve leaves unanswered, then choose complementary measurements.

Historical laboratory: what the original experiments were asking

The three familiar laws did not begin as interchangeable fitting functions. Invertase kinetics asked how catalytic speed depends on substrate and product. Haemoglobin experiments asked how oxygen pressure and chemical conditions change binding. MWC joined binding to conformational regulation. This interlude returns to the original observations to distinguish those questions. It is a comparison across historical programmes, not a claim that each paper directly caused the next.

Michaelis and Menten made the optical measurement match the kinetic question

The 1913 invertase study began with a measurement problem. Hydrolysis converts optically rotating sucrose into glucose and fructose. The products' optical rotation also changes as their forms equilibrate. A continuously observed change in rotation can therefore mix enzyme conversion with product mutarotation. Michaelis and Menten explicitly distinguished these processes. They stopped sampled reactions with alkali and allowed the optical signal to settle before taking the reading. Temperature and hydrogen-ion concentration were controlled.^MM

The experiment tested a complex-formation hypothesis already associated with Henri. The proposed catalytic rate was proportional to enzyme bound to sucrose. Rapid binding equilibrium then predicted a hyperbolic substrate dependence. Michaelis and Menten used initial rates to reduce the influence of accumulated products, but their paper did not stop there. They separately tested the inhibitory effects of glucose and fructose and incorporated those effects into the progress calculation.

Their parameter extraction was graphical and explicitly tied to the response scale. They plotted rate against logarithmic sucrose concentration and used the shape around half-saturation to establish an ordinate normalization and concentration scale. The translation's pp. 12–13 describe the procedure and report about 0.0167 M for the principal series. Our nonlinear least-squares reanalysis of the seven printed initial slopes gives 0.01789 M. That difference compares procedures on a rounded historical table. It is not a correction of the original experiment or a new estimate with known sampling uncertainty.

The full-progress analysis tested consistency across a changing mixture. The enzyme was allocated among free, sucrose-bound and product-bound forms. As products accumulated, those nonproductive bound forms changed the available catalytic population. Michaelis and Menten asked whether one conversion constant could remain consistent across substrate amounts and reaction times. Johnson and Goody's 2011 translation and reanalysis brought this often-neglected part of the original paper back into view.

The experiment could not measure turnover per enzyme directly. The molar enzyme concentration was unknown. A fitted maximum rate therefore did not yield a turnover frequency per enzyme by itself.

A hyperbola also did not uniquely identify rapid equilibrium. Briggs and Haldane's two-page 1925 note retained conversion in complex balance and obtained the same rate-law form with $K_M=(k_-+k_{\mathrm{cat}})/k_+$. They pointed out that the rate curve cannot determine the ratio of unbinding to conversion. Their contribution altered the interpretation of the constant, not the visible shape alone.^BH

Hill compared hypotheses about haemoglobin, not only exponents

Hill's 1910 problem involved discrepant estimates of haemoglobin aggregation. Osmotic-pressure measurements suggested different molecular sizes under different solution conditions. Hill explored whether salt-dependent aggregation could explain oxygen-dissociation curves. He used observations by Barcroft and Camis, first writing responses for mixtures of aggregate sizes. The historical monomer/dimer hypothesis is not the modern tetrameric description of haemoglobin.^Hill

The simpler Hill expression was introduced partly to make comparisons manageable. After the aggregate-mixture calculation, Hill asked whether a two-parameter curve could summarize the observations. He explicitly limited the immediate physical interpretation of those parameters. For the Na₂HPO₄ series, his mixture and his $n=1.670$ Hill formula nearly agree. A useful description of a response need not settle the molecular hypothesis that motivated it.

The 1913 follow-up made intermediate states explicit. Hill considered a dimer with zero, one and two bound oxygen molecules. The singly occupied form enters the mass-action calculation. If it has little equilibrium population, the remaining terms can approximate an all-or-none curve. He also discussed why treating oxygenation as a simultaneous collision would be problematic. Reducing this history to “Hill assumed that several ligands bind in one step” loses the distinction between a populated intermediate and a reaction pathway.^{Hill 1913}

What survives in modern usage is a response measure, not the original aggregation account. The logarithmic slope of binding odds can quantify cooperativity under declared conditions. It need not be constant, integer-valued or equal to the number of sites. The original tables remain useful because students can see both the successful response description and the limits of its physical interpretation.

MWC asked how one molecular conformation could organize several binding sites

MWC's 1965 starting point was communication between distinct sites. Haemoglobin's haem–haem interaction and regulatory enzymes suggested that ligand binding could affect a distant function. The model represented this through alternative, symmetry-preserving conformations of an oligomer. Binding sites were independent within a conformation but coupled through their effect on the conformation population. The construction retained partially occupied states and imposed a concerted conformational constraint.^MWC

The original paper distinguished conformation from saturation before comparing with data. Its two functions measure the fraction of protein in one conformation and the fraction of sites bound by ligand. This distinction matters experimentally. Oxygen saturation can be measured directly, while an enzyme's catalytic rate reports occupancy only through additional assumptions about which bound states convert substrate and how rapidly they do so.

The authors did not treat a good fit as proof of their assumptions. Beside Figure 3 they explicitly qualify the evidential value of numerical agreement. Their subsequent argument also examines effectors, changes in the shape of saturation curves and structural observations. Their haemoglobin idealization acknowledges nonidentical alpha and beta chains. A model built around equivalent sites and concerted conformations was a constrained hypothesis, not a direct transcription of every structural detail.

Reconstructing the published fit reveals what was and was not specified. The printed $L,c$ pair determines a dimensionless saturation shape. It does not supply the unprinted absolute affinity scale. Our reconstruction therefore normalizes oxygen pressure by half-saturation. The resulting midpoint Hill slope is about 2.853 despite four binding sites. Equal affinities in the two conformations would instead produce a hyperbolic fractional occupancy, still with four sites. These are modern calculations of the original model, not additional historical measurements.

The inherited lesson is to test connected predictions. Invertase products challenged an enzyme allocation. Haemoglobin solution conditions challenged a binding description. Allosteric effectors challenged a shared-conformation model across response curves. None of those tests can be replaced by naming a fitted law. The next historical step carries this same state-and-readout discipline to gene regulation, where the molecular ledger becomes more elaborate but the inference problem remains.

12Ackers, Johnson and Shea build a lambda operator ledger

The lambda switch posed a concrete cooperative-binding problem. Its right operator has three sites, $O_{R1},O_{R2},O_{R3}$. Repressor dimers bind with different affinities and can interact across neighbouring sites. Ackers, Johnson and Shea's 1982 paper combined those rules with statistical thermodynamics to connect operator binding to the repression of the divergent promoters $P_R$ and $P_{RM}$.⁷

The model's detail resides in its state list. Three occupied-or-empty sites give eight configurations. The cooperative energy for sites 2 and 3 is included only when site 1 is empty. Triple occupancy includes the 1–2 cooperative term, not both pair terms. The readout rule also matters: occupancy of site 1 or 2 represses $P_R$, whereas occupancy of site 3 represses $P_{RM}$.

The calculation keeps free dimers separate from total repressor. Write $R_1$ for free monomer concentration and $C_{R_1R_1}$ for free dimer concentration. The total operator concentration is $q_G$. The total repressor concentration $q_R$ counts monomer equivalents, including operator-bound dimers. For configuration $s$ containing $j_s$ dimers, the dimensionless weight and exact bookkeeping constraint are

Here $\mathcal R$ is the molar gas constant, and the standard concentration is $c^\circ=1\,\mathrm M$. The authors also use $C_{R_1R_1}=K_aR_1^2$, with $K_a=5\times10^7\,\mathrm M^{-1}$. The totals equation is a conservation statement. The equilibrium weights and dimerization relation are the constitutive assumptions that close it.

The paper distinguishes sufficiency from proof of mechanism. Binding energies were inferred from in-vitro site-specific protection measurements and then used to predict regulatory properties. The discussion explicitly acknowledges that another mechanism might explain the available physiological observations. It also identifies the free-to-total conversion as a place where unmodelled nonspecific binding could matter.

Build an operator-state ledger and connect its binding weights to promoter repression without confusing free and total repressor.

13Bintu and colleagues make the recipe transferable

The 2005 “by the numbers” articles organize promoter models around reusable physical operations. Enumerate configurations. Count their multiplicities. Assign binding and interaction energies. Normalize the weights. Then calculate the probability of the configurations associated with transcription. The models article explicitly situates this approach in earlier lac and lambda work.⁸

The simple-repression calculation also exposes the hidden bridge to expression. Suppose a promoter can be empty, polymerase-bound or repressor-bound, with relative weights $1,w_P,w_R$. Polymerase and repressor exclude one another. If expression is proportional to polymerase occupancy, the fold-change relative to no repressor is

The proportionality assumption is not supplied by the partition function. Polymerase binding, open-complex formation, transcript initiation and the eventual reporter signal need not share one clock. The article explicitly separates the equilibrium approximation from the assumption relating occupancy to expression. A failure at the reporter level can therefore implicate the readout mechanism, not only the binding energies.

Derive an expression fold-change only after specifying the occupancy-to-expression approximation.

14Garcia and Phillips separate calibration from prediction

The 2011 simple-repression study made parameter transfer an experimental test. Garcia and Phillips changed operator sequence and repressor abundance in engineered E. coli strains. In one analysis route, a reference strain's measured repressor count and fold-changes determined operator binding energies. Those energies were then held fixed while expression measurements inferred repressor counts in other strains. Quantitative immunoblots provided a separate check.⁹

“Parameter-free prediction” begins after calibration, not before it. The binding energies were inferred from data. The model's achievement was that the same energies could be reused across changed repressor backgrounds, rather than re-fitting an unrelated response curve for every strain. The paper also explores a global-fit route. These are different uses of the data and should not be described as the same validation.

The quantities being counted must also stay consistent. A protein tetramer, its DNA-binding heads and the number of active binding units are not interchangeable counts. Concentration-based and copy-number-based weights can agree, but only after the reservoir convention and molecular unit have been specified. This is the same modelling discipline required when translating the rate equation to the master equation.

Distinguish parameter calibration from a new prediction by naming which measurements and parameters were held fixed.

15Razo-Mejia and colleagues combine allostery with repression

The 2018 induction study joins two previously separate state calculations. An inducer shifts LacI between DNA-binding-active and inactive conformations. The active fraction then changes the effective repressor pool seen by a simple-repression promoter. This modular construction links inducer concentration, repressor abundance and operator affinity to one expression response.¹⁰

The composed formula makes the transfer of information visible. Let $N_R$ count repressor dimers, $N_{\mathrm{NS}}$ count nonspecific DNA sites and $\Delta\epsilon_R$ denote the specific-minus-nonspecific operator binding free energy. With active fraction $p_A(c)$ from the MWC state sum, weak polymerase occupancy and negligible inactive-repressor binding give

Changing inducer concentration changes the conformational population. Changing repressor expression changes the dimer count. Changing operator sequence changes the DNA binding energy. Each experimental intervention therefore has a designated place in the formula. Promoter depletion and changes in the reporter's downstream kinetics remain outside this simplified composition.

The study tests transfer across a family of circuits. Previous measurements supplied repressor counts and operator energies. An earlier dataset constrained the conformational free-energy difference. Two inducer dissociation constants were inferred from one strain's induction curve. With those choices fixed, the authors predicted the remaining 17 strains' induction profiles without further fitting.

The calibration was more specific than fitting all three allosteric parameters to one curve. Earlier fold-change data constrained the conformational energy difference to $4.5\,k_BT$. The O2 strain with 260 dimers then supplied central affinity estimates of 139 μM and 0.53 μM for active and inactive conformations. The much smaller inactive-state affinity constant makes inducer favor inactivity and relieve repression. The paper counts dimers, while the earlier count calibration reported tetramers. The factor of two is a molecular bookkeeping requirement, not an adjustable correction.¹⁰

The discrepancies matter as much as the collapse. The paper reports systematic deviations for some weak-operator and high-repressor constructs. A later global fit improved some comparisons, but that is not the same evidence as the original held-out predictions. The historical advance is a compact, physically interpretable set of parameters that can be tested across conditions, not a demonstration that equilibrium models never fail.

Combine allosteric and promoter weights, then test their transfer across changed repressor counts and operator affinities.

16The linear framework keeps the graph when equilibrium fails

A non-equilibrium stationary state still has an organized calculation. Nam, Martinez-Corral and Gunawardena's 2022 account of the linear framework connects several biochemical formulas through graph Laplacians and directed spanning trees. It brings equilibrium binding and driven fast-subsystem models into one mathematical setting without treating them as physically identical.¹¹

Each state's stationary weight is a sum over trees pointing into it. A tree contains every state, has no cycle and gives each non-root state one outgoing edge. Multiply its edge hazards, then sum over trees rooted at $i$ to obtain $T_i$. For a finite strongly connected graph, $p_i^*=T_i/\sum_jT_j$.

The simplification is therefore structural. Without detailed balance, altering one transition can alter state probabilities throughout the graph. With detailed balance, compatible ratios provide path-independent weights. Neither method removes the need to solve free concentrations from finite molecular totals. That conservation layer becomes central when binding modules are assembled into a larger reaction network.

Construct stationary tree weights when detailed balance is unavailable and identify the further simplification that equilibrium permits.

17Phillips argues for equilibrium as a local approximation

A living cell need not be at equilibrium for an equilibrium binding model to be useful. Phillips's 2015 review confronts that objection directly. It examines circumstances in which occupancy models agree with kinetic descriptions and asks which relative time scales justify the approximation. The review's memorable title records an objection to applying equilibrium ideas to life, not a claim that life as a whole is equilibrated.¹²

The relevant claim concerns a subsystem under specified surroundings. A fast binding layer may equilibrate while a slower catalytic process changes molecular totals. Its state weights are then conditional on the current slow variables. Continuous fuel use outside that layer is compatible with the approximation. Fuel-coupled circulation inside it is a different matter.

Speed and balance are separate requirements. Fast relaxation can justify replacing dynamics by a stationary solve. It does not by itself justify equilibrium weights. The fast process must also satisfy the appropriate physical balance conditions, and its relaxation must keep up with changes in its inputs. This separation explains why Lecture 6's reduction needs Lecture 7's additional physics.

The same review warns that matching a response curve is not decisive. Phillips compares a kinetic repression calculation with the thermodynamic model and obtains the same dependence on repressor abundance through an effective parameter. A successful sigmoid or saturation fit can therefore leave the physical interpretation underdetermined. Perturbing the kinetic quantities hidden in that parameter can discriminate models. This is why the core's epilogue asks for a subsystem boundary and a timescale argument, not only a plausible-looking occupancy curve.

State the subsystem, surroundings, observable and time-scale ordering needed for a local equilibrium approximation.

18Proofreading and sensing restore the kinetic questions

Proofreading asks about product errors, not merely bound fractions. Hopfield's 1974 model adds a driven intermediate and another opportunity for rejection before product formation. Ninio's 1975 paper approaches discrimination through completion probabilities and explicitly compares its mechanism with Hopfield's. The shared lesson is that reaction timing can amplify discrimination beyond a single selection stage. The squared-error limit belongs to a restricted ordering of commitment and rejection rates, not to every fuel-consuming mechanism.^{13, 14}

Sensing asks how much information a finite observation supplies. Berg and Purcell's 1977 analysis treats diffusive molecular arrivals and fluctuations in receptor occupancy. A stationary mean response does not determine the precision of a concentration estimate. Observation duration, diffusion and correlations between repeated observations enter the answer. Numerical prefactors also depend on the detector and estimator being modelled.¹⁵

Neither branch makes occupancy models obsolete. They change the question and therefore the information required. An equilibrium response may still provide the mean signal, while kinetics determines the error, delay or throughput. The proofreading and sensing calculations develop these branches with their limiting assumptions attached.

Recognize when product fidelity or sensing precision requires restoring kinetic information that occupancy weights omit.

19Hidden channels can conceal dissipation

An observed transition need not correspond to one physical mechanism. Suppose two channels connect the same observed states. Channel $\alpha$ favours one direction and channel $\beta$ the other. Adding their hazards can produce an apparently reversible two-state process while each channel carries a nonzero stationary current. Schnakenberg's distinction between complete and summed balance already makes room for this possibility.⁵

Eliminating fast states can create a subtler thermodynamic mismatch. Hartich and Godec's 2023 molecular-motor example shows that a finite time-scale separation can produce apparently Markovian reduced dynamics while substantially misrepresenting dissipation. Rare reverse transition paths matter. Their result does not deny the fixed-driving infinite-separation limit. It distinguishes a useful kinetic approximation from an equally accurate thermodynamic one.¹⁶

The magnitude is visible in their explicit example. At driving of 20 thermal energy units per fuel conversion, the fast and slow hazard sums differ by roughly two orders of magnitude. Yet the specified lumped-state estimate recovers only about half of the entropy production. The long-time waiting-time density looks nearly exponential. Short, rare backward events carry information that this apparent Markov behavior does not certify. The paper contrasts ordinary state lumping with a milestone construction that retains the relevant transition information.¹⁶

The course enzyme supplies a simpler, separate accuracy test. The exposition's exact calculation compares rapid-binding populations, current and rare-state activity. It is not a reproduction of the motor study or a proof about all coarse graining. It shows why the modeller must state which prediction the time-scale separation is supposed to preserve.

Measurement resolution is therefore part of the equilibrium claim. A stationary histogram is the weakest test. Reversible observed histories are stronger. A physical equilibrium interpretation additionally asks whether fuel exchanges, reverse channels and eliminated variables have been accounted for. These are different levels of evidence.

Audit hidden channels and eliminated states before interpreting observed reversibility as physical equilibrium.

20The inherited method is an experimental programme

The historical payoff is a sequence of testable commitments. Specify molecular states and reservoirs. Build compatible weights. State which configurations contribute to the observable. Calibrate parameters using named measurements. Then change a condition and ask whether those same parameters predict the outcome. This order connects the conceptual framework to the experimental achievements of promoter modelling.

Different failures call for different repairs. Depletion requires a total-pool constraint. A delayed response requires a kinetic clock. An incorrect reporter prediction may require a different activity rule. A sustained cycle current requires reservoir-driven kinetics. Hidden channels require a richer physical description. No universal recommendation to add driving follows from a failed fit.

The next modelling step is composition. A single binding module produces occupancies or complex concentrations conditional on free molecular concentrations. In a network, those free concentrations are shared and constrained by totals. Combining exact conservation, fast equilibrium reconstruction and slower catalytic fluxes is the binding–catalysis construction that follows. Equilibrium earns its place there by replacing selected kinetic details while preserving their physical meaning.

Choose an experimental test that can distinguish a failed equilibrium assumption from a failed conservation or activity model.

Sources and reading routes

The original papers support the historical claims. Translations and author-preprint versions are identified below. Figures 2–5 and 9–10 are pedagogical calculations. Figures 6–7 and 8A reconstruct published models using reported parameter estimates. No synthetic curve is presented as measured data.