How molecular regulation became a problem of composition

Where the binding-catalysis model and its regime analysis came from: productive states, supplied totals, dominant terms and the lecturer's papers from 2020 to 2026.

CCBS · Lecture 8 · lineage · lecturer-written · revised 24 September 2026

The core of Lecture 8 builds a model class and a method for analyzing it. This page traces where their ingredients came from. Enzyme kinetics, allostery and promoter models supplied the productive state. Tight inhibition, titration and loading supplied the finite total. Power laws and design spaces supplied the method of dominant terms. The lecturer's papers from 2020 to 2026 joined these into the binding-catalysis model and its regime maps.

The sequence is conceptual, and each source carries its own date. It does not claim one uninterrupted line of influence. Historical measurements, theoretical constructions and the course's new calculations are identified separately.

The core uses the result as three tools. The exposition derives them and works their conditions in detail. Here each source is read for what it actually established, and for the step of the core's analysis that it prepares.

1Three historical questions lead to the core's four parts

Three questions organize the history behind the core. Which molecular state carries activity? Which physical amount does the experiment control? Which terms determine behavior in a given regime? The questions arose in different experimental and mathematical settings. The core combines their answers without claiming that the sources formed one research programme.

This page writes every comparison in the core's notation. A binding pair has free species E and S, a complex $C_{ES}$, totals $q_E=E+C_{ES}$ and $q_S=S+C_{ES}$, and the binding relation $K_dC_{ES}=ES$. A regime is named by its winner tuple, the species that dominates each total, listed in the order $(q_E,q_S)$. So $(E,S)$ is weak binding, $(E,C_{ES})$ and $(C_{ES},S)$ are the two regimes in which one total limits the complex, and $(C_{ES},C_{ES})$ is the singular ray where the totals match. Historical sources use other letters, and each is translated where it appears.

Each route through the history prepares specific steps of the core, and the map below pairs them. Routes I and II supply the ingredients of the model, which the core assembles in Parts 1 and 2 and reads out in Part 4. Route III supplies the method of Part 3. Route IV collects mechanisms that the core's §1 says effective laws miss, and several return in Part 4. Route V shows how the lecturer's papers joined the pieces.

The history is most useful where it shows why a distinction became necessary. A model's success under substrate excess does not establish its validity when substrate is depleted. A cooperative binding curve does not specify which occupied states produce. A good equilibrium allocation does not supply the time required to reach it. The sources below make these distinctions concrete.

Before comparing two historical models, fix the state that produces, the total the experiment controls and the terms each model keeps. Those three answers place a source on the core's pipeline.

2An enzyme intermediate connected occupancy to flux

Enzyme kinetics supplied the first distinction the core needs: total enzyme versus productive enzyme state. The early papers also show that one response formula can rest on different kinetic premises.

Michaelis and Menten settled the substrate total by excess

Michaelis and Menten's 1913 invertase paper connected reaction velocity to an enzyme-substrate intermediate. Following Henri, they assumed a labile sucrose-enzyme complex, so the rate of inversion must be proportional to that complex. They computed the complex from the law of mass action with a dissociation constant. Their summary also treats complexes with the products as a source of inhibition, so the modern initial-rate hyperbola is not the whole paper.¹

Their key simplification is a dominance statement. They wrote the mass-action law with free sucrose and then replaced it by total sucrose, because only a vanishingly small fraction of the sucrose is bound. They kept the enzyme total in full, as free plus bound enzyme. In the core's terms, excess settles the substrate total once, with free S as its winner. Only the enzyme total can change winner, from E to $C_{ES}$, as sucrose rises past the dissociation constant.

That is step 3 of the core's pipeline, done by hand in 1913 (core §9). The path runs from $(E,S)$ into $(C_{ES},S)$, and the complex saturates. The core's weak cut reproduces this shape at leading order (core §10). A rate proportional to the complex is the core's activity line, $v=K^{\mathrm{cat}}x$, in its smallest form (core §4).

Briggs and Haldane changed the premise behind the same curve

Briggs and Haldane's 1925 note put formation and removal of the complex into approximate balance. Removal includes both dissociation and productive conversion. In the core's rate constants the resulting scale is $K_M=(k_{-S}+k_{\mathrm{cat}})/k_{+S}$, rather than the binding ratio $K_{d,S}=k_{-S}/k_{+S}$.²

The two constants agree when catalysis is slow against unbinding, $k_{\mathrm{cat}}\ll k_{-S}$. That is the core's clock separation (core §3), and the core states the agreement in §6. A fitted hyperbola cannot tell the two premises apart. The core assumes the separation, so its low-occupancy coefficient is $k_{\mathrm{cat}}/K_d$, the second-order constant of the whole enzyme in the equilibrium reduction (core §10).

The present model inherits both lessons. A productive intermediate connects occupancy to flux. The constant in its rate law must still be read through the kinetic balance that produced it.

Read a saturation law with three things fixed: its productive intermediate, the total held in excess and the clock separation behind its constant. It is then one path through the core's regime map.

3Allostery made regulation a redistribution among states

Monod, Wyman and Changeux proposed a model in which an oligomer occupies alternative concerted conformations with different ligand affinities. Within each conformation, equivalent sites were intrinsically independent in the model. Coupling arose through the shared conformational transition, so the whole population could show cooperative behavior.³

The change in viewpoint is central to Lecture 8. A regulator need not alter every molecule's catalytic rate. It can redistribute a population among states with different activities. The output is then each state's activity weighted by its abundance. Bound fraction, conformation fraction and catalytic activity remain different observables.

The core makes this the first step of its pipeline (core §9). For a machine with a free state E at activity $k_0$ and a bound state $C_{ES}$ at activity $k_1$, the output is $v=k_0q_E+(k_1-k_0)C_{ES}$. At fixed $q_E$, binding activates when $k_1>k_0$ and represses when $k_1<k_0$. The affinity alone therefore does not decide whether a ligand activates or represses. This is also the core's picture of cellular control in §2: regulation moves a catalyst between states of different activity.

Assign each state its activity before reading a binding curve as regulation. The activities set the sign of the response, and the affinity sets where it happens.

4Zero-order ultrasensitivity balances two saturated enzymes

A modification cycle asks how two opposed activities on one target can amplify a small change in capacity. Goldbeter and Koshland derived that amplification in 1981. They also asked how enzyme-bound target changes the measured response.⁴

Saturation of both converter enzymes produced the amplification

A target protein W is modified by one converter enzyme, $E_1$, and demodified by another, $E_2$. Goldbeter and Koshland called the key condition zero order, meaning saturation with respect to the protein substrate. In the core's terms the complex wins each converter-enzyme total: $C_{WE_1}$ wins $q_{E_1}$, and $C_{W^*E_2}$ wins $q_{E_2}$. Each flux then sits at its enzyme's capacity and depends only weakly on the target forms. A small difference between the two capacities can move most of the target into one form.

The switch itself is not a binding equilibrium. At steady state the two catalytic fluxes balance, $k_1C_{WE_1}=k_2C_{W^*E_2}$ in the core's symbols. The authors stress that the system reaches a steady state rather than an equilibrium. The amplification therefore belongs to the slow layer. The core's regime map supplies the two saturated fluxes, and the closed dynamics of the coda balance them.

The one-cycle model still has a unique steady state, because more modified target lowers one flux and raises the other. Its steep response is a regulatory building block, not yet a bistable switch. Lecture 9 asks when such balances have more than one solution.

The original modification paper already asked what the output counts

The 1981 paper also examines appreciable enzyme-substrate complexes and nonproductive binding. Its Figure 3 changes enzyme amounts relative to target and plots free target forms. The discussion asks whether enzyme-bound target might itself remain active, in which case a different response measure is appropriate.⁴

This is an early version of the readout distinction at the center of Lecture 8. If an assay counts modified molecules whether free or bound, its output includes the complexes. If function requires the free modified form, the same state calculation must be read differently. An apparent disagreement between response curves can come from counting different molecular populations.

Read zero-order ultrasensitivity as two saturated fluxes in balance. The regime map supplies the saturation, and the steady state belongs to the slow dynamics.

5Promoter state models made joint states the unit of regulation

A gene can integrate several regulators at one promoter. The mechanistic question is how their joint binding configurations set transcription. The wiring diagram alone supplies neither a sum nor a product of the isolated response curves.

Buchler, Gerland and Hwa addressed this question in a 2003 theory of combinatorial transcription control. They represented transcription through equilibrium RNA-polymerase occupancy and let regulators recruit or exclude one another through their site arrangement. They built response functions by choosing site strengths and interactions, including AND, OR and more complex logic.⁵

Their logic functions are defined at extremes. ON and OFF are read at low and high regulator concentrations, for example about 1 nM and 1 µM. Such extremes lie deep inside regimes, where one joint state wins the promoter total and the output is that state's activity. The core's §12 turns this into a rule. The candidates for the promoter total are the joint states, and at leading order the regime map is a truth table.

The claim was constructive and conditional. It showed how specified molecular arrangements could implement responses in the model. It did not establish that every natural promoter uses the same bridge from occupancy to expression. The possibility of unintended interactions also limited how freely those constructions could be composed across a genome.

Thermodynamic promoter models enumerate molecular configurations and assign them equilibrium weights. Bintu and colleagues systematized this approach, connecting regulator abundance and binding energies to promoter occupancy. The central simplification is to solve a molecular state distribution before calculating an expression response.⁶ Their full author preprint names its kinetic premise:

“Perhaps the most glaring assumption is that of equilibrium itself.”Bintu et al., full author preprint, 2004

The authors discuss the clocks of binding and open-complex formation, and separately the assumption that polymerase occupancy is proportional to expression. These are two model choices. A state sum can describe occupancy accurately while a further kinetic bottleneck changes the output generated from it.

Both treatments take regulator concentrations as given, which suits a single promoter. The core's pipeline states why. A species in trace cannot win another species' total, so step 3 settles each regulator's total without the promoter (core §9). Only the promoter total is left to change winner.

The core's joint-state comparison is a deliberately small instance of this tradition. Independent sites, mutual exclusion and an inactive double state can share one isolated dose curve and still differ when both regulators rise. Lecture 7 established what equilibrium state counting can explain. Lecture 8 adds finite totals and explicit state activities.

Treat combinatorial regulation as a question about which joint state wins the promoter total. Then check separately that occupancy sets expression.

6Tight inhibitors made the supplied amount part of the mechanism

Morrison's 1969 paper addressed reversible inhibitors effective at amounts comparable to the enzyme concentration. The introduction connects the problem to folate-related enzymes and substrate analogues. In that regime the enzyme can bind a substantial fraction of the inhibitor, so total inhibitor differs from free inhibitor.⁷

The paper develops generalized quadratic initial-rate equations, including inhibitor binding to several enzyme forms. It also distinguishes reversible tight inhibition from irreversible covalent inhibition. Tight binding describes an occupancy and depletion regime, not a guarantee that binding is chemically irreversible.

Tight binding is the core's limiting pool. When binding is tight, the complex takes nearly all of the smaller total, and at leading order $C_{ES}\approx\min(q_E,\,q_S,\,q_Eq_S/K_d)$ (core §8). The core's optional finite-ligand case restates Morrison's problem as a change of winner (core §11). A tight inhibitor is mostly bound while $q_I<q_E$, so the enzyme titrates it. Once $q_I$ exceeds $q_E$, the inhibitor complex wins the enzyme total.

Validity analysis made the total the slow variable

The quasi-steady-state literature reached the same totals from the side of timescales. Segel in 1988 asked when the Briggs-Haldane reduction is valid. He proposed $\varepsilon=q_E/(K_M+q_S)\ll1$ as the criterion, with both totals taken at the start of the reaction.⁸

At leading order the criterion is a union of two regimes. The sum $K_M+q_S$ lies within a factor of two of its larger term, so $\varepsilon\ll1$ holds when $q_E\ll q_S$ or when $q_E\ll K_M$. The lecturer's 2026 validity paper derives the same union from regimes and reports it as consistent with earlier analyses.⁹ A leading-order reading thus recovers a classical validity condition.

Borghans, de Boer and Segel in 1996 then changed the variable itself. They introduced total substrate, free plus bound, and noted that complex formation cannot deplete it. Total substrate changes much more slowly than free substrate, and the change of variables enlarges the parameter domain in which the reduction holds.¹⁰ The core makes this choice for every species. Binding moves constituents without changing totals, catalysis changes totals, and the slow dynamics are written in totals (core §2, §4).

Treat a partner's depletion as part of the mechanism whenever its total is comparable to the binding pool. Write the slow dynamics in totals, which binding cannot deplete.

7Balance theory located equilibrium within a conserved class

The mathematical question behind finite-pool reconstruction is whether one set of totals selects one equilibrium. Balance theory answers it under structural hypotheses. It leaves the slow dynamics and the steepness of a response as separate problems.

Horn and Jackson found one equilibrium in each conserved class

Horn and Jackson's 1972 theory analyzed general mass-action systems with structural balance properties. Detailed balance is one member of a broader complex-balanced class. Their treatment uses a convex pseudo-Helmholtz function and relates its constrained minimum to an equilibrium in the reaction's conserved class.¹¹ Their Lemma 4B states the relevant conclusion under its hypotheses:

“each reaction simplex contains precisely one equilibrium concentration”Horn and Jackson, 1972, Lemma 4B

The conserved class matters as much as the word equilibrium. Initial totals restrict the accessible concentrations. One equilibrium per class still permits different equilibria at different totals. Totals label the classes rather than select one concentration vector for the whole network.

The core's §6 uses the version that the lecturer's thesis proves for binding networks. The map $\log x\mapsto(\log q,\log k)$ is a global change of coordinates, so each positive set of totals and constants fixes one positive state (Prop. 3.5.2, p. 96).¹² The exposition proves the same uniqueness with a convex potential.

A unique equilibrium can still switch

A unique binding equilibrium can be a steep function of its totals. Near equivalence, a small rise in one total can release much of a free species, with no second equilibrium at that total. In the core this is the singular ray $(C_{ES},C_{ES})$, where both totals change winner together and the free species jumps at leading order (core §8, §10). Steepness and multiplicity are different properties.

Once catalysis, production or removal changes the totals, the system moves among binding equilibria. Its slow vector field can then have more than one steady state. Uniqueness of the binding layer does not settle that question, and the core's coda hands it to Lecture 9.

State the conserved class behind a uniqueness claim. Then keep it apart from the steepness of the response and from the number of slow steady states.

8Titration placed a switch at equivalence, and yeast tested it

Finite sequestration connects one binding equilibrium to a sharp response without many cooperative sites. The theory and its synthetic test address different parts of that claim.

Molecular titration made equivalence the threshold

Buchler and Louis analyzed an active protein sequestered by a finite partner, with total active protein as input and free active protein as output.¹³ Write their active protein as E and the titrant as S, as in the core's second run of §10. The input is then $q_E$, and the output is free E.

Their three statements are the three regimes of that path. When $q_S>K_d$ and $q_E<q_S$, the titrant buffers the active protein, and $E\approx K_dq_E/q_S$. That is the regime $(C_{ES},S)$. At $q_E\approx q_S$, the equivalence point, a threshold appears, and that is the singular ray $(C_{ES},C_{ES})$. Beyond it $E\approx q_E-q_S$, and the path has entered $(E,C_{ES})$.

They also found that ultrasensitivity grows with $q_S/K_d$, and the regime map explains the number. At leading order free E jumps across the ray by the factor $q_S/K_d$, the held total over the dissociation constant. The core's tight cut shows the same factor as a hundredfold drop at $q_E=100K_d$ (core §10).

The one-to-one complex never grows faster than proportionally with either total. The free species can, because it is the small residual of an almost filled pool. That is why sequestration gives strong ultrasensitivity without a high-order binding event. The lecturer's 2026 sampling paper shows how narrow the condition is in totals. Switch-like points fill only a thin strip near matched totals well above $K_d$, about 1% of random draws.¹⁴

Buchler and Louis also examined production and degradation of monomer and complex. Their relative removal rates decide whether the sharp behavior survives at a driven steady state or appears mainly during a transient. An equilibrium titration curve and a steady cellular dose response are connected by the surrounding dynamics.

Bacteria use such a threshold to time DNA replication. Citing Hansen and colleagues, Fu, Xiao and Jun describe initiator protein as first titrated by high-affinity DnaA boxes, with replication starting only after the chromosomal boxes are nearly saturated. Their estimate is about 300 boxes at a dissociation constant near 1 nM.¹⁵ The core reads this threshold as the rising titration curve of its second run.

A synthetic yeast circuit tested the sequestration mechanism

Buchler and Cross built a synthetic budding-yeast system with a CEBPα-based transactivator, a high-affinity dominant-negative inhibitor and a reporter promoter with one activator binding site. Varying inhibitor expression shifted the activation threshold and increased response steepness. The experiment connected the finite-pool theory to a controlled biological implementation.¹⁶

The single promoter site was a diagnostic design choice. It removed the option of explaining the sharpness through several cooperative DNA sites. Fluorescence and curve fitting still involved measurement and modeling choices, including background and high-expression effects. The experiment supports the sequestration mechanism under those conditions.

The reported apparent Hill coefficients reached 12 in the tested inhibitor-expression conditions. They came from fitting response curves with corrections for background and high-activator squelching. A fitted value of 12 does not mean twelve cooperative promoter sites. It is also not automatically the local logarithmic slope at the midpoint of an idealized model.

The exposition's reporter makes that distinction calculable. For a trace one-site reporter, the half-response free activator is fixed by the reporter's own binding scale. The total activator needed to reach it depends on how much inhibitor has been filled. The local slope follows by differentiating that total.

Locate a titration threshold on the singular ray, where the totals match. Test it with a design that removes other sources of steepness, and do not read a fitted Hill coefficient as a site count.

9Retroactivity wrote a connection in total coordinates

Del Vecchio, Ninfa and Sontag asked how a downstream system changes the signal of an upstream module. Downstream binding sites can capture an upstream protein. If that protein carries a changing signal, the captured pool alters its dynamics. A connection carries a physical load as well as an intended influence.¹⁷

Their derivation changes coordinates from free protein to total protein, free plus bound, and identifies the total as the slow variable. Fast binding then decides how much of a change in the total appears in the free signal. A buffering factor greater than one slows the free response under the stated production and removal law.

This is why rapid exchange does not guarantee modularity. A fast binding reaction can have an excellent equilibrium approximation and still sequester and buffer the signal. The core's accumulation line keeps this load by construction. Binding fluxes cancel from the totals, so $\dot q$ carries only catalysis and exchange (core §4). The core's optional A2 is the static version of the same load, in which a sponge moves the input that a promoter sees.

Assess a connection by the total it shares. Fast binding keeps the load, so a composed model must keep the shared total.

10Dominant terms became a method: select, solve and check

The core's second tool belongs to a longer effort to make nonlinear biochemistry analyzable. Two methods in that effort must be kept apart. One describes the local response at an operating point. The other simplifies the equations over a region where some terms dominate.

Local power laws quantify response at an operating point

Savageau's 1988 account explains the power-law formalism through a Taylor expansion in logarithmic coordinates, referring back to its earlier development. For a positive response f and a positive reference state $x^*$, the first-order form is

The exponents are local sensitivities. They are real numbers, not necessarily integers, and they do not count reactant molecules. The approximation preserves the first-order fractional response at the reference point. Savageau argued that S-system representations stay accurate over a wide range. The expansion itself, however, fixes the powers at one operating point and does not say where that range ends.¹⁸

Dominant terms produce candidate regimes and validity conditions

Savageau and colleagues' 2009 design-space analysis keeps the largest term on each side of a balance, solves the retained balance and tests the solution against the inequalities its dominance requires. Not every candidate passes its test. In logarithmic coordinates the boundaries between regimes are straight lines.¹⁹

Largest-term regions do not certify uniform accuracy. The 2009 article notes that discrepancies are greatest near the boundaries, and it emphasizes the analytic organization of the regions. The core turns this caution into two statements. Inside its region a regime's law holds within a bounded factor, since a sum of n positive terms lies between its largest term and n times it (core §7). For one pair, the exact equal-contribution curves, such as $q_S=K_d+q_E/2$, lie within a factor of two of the rays that the regime map draws (core §8).

The binding-catalysis model makes the terms structural

The core's §7 applies select, solve and check to conservation laws. Select a winner for each total, solve the binding relations as monomials, and check that each discarded term is smaller than its total's winner. The terms are the species inside a total, so each regime's exponents are fixed by the complex list and the equilibrium relations rather than fitted. The lecturer's thesis credits Savageau with the idea that reaction orders regulate catalytic fluxes. It notes that such work often took the orders as empirically useful quantities, and it asks for their mechanistic basis (p. 13).¹²

One candidate needs more than a check. When the same complex wins both totals, the leading equations $q_E\approx C_{ES}$ and $q_S\approx C_{ES}$ use one unknown twice and lose a constraint. The core calls such a regime singular and keeps the exact difference $q_E-q_S=E-S$ as the extra balance (core §7, §8). The titration switch of Buchler and Louis lives on that ray.

The lecturer's reaction-order programme asks a further question: which local exponents can the whole architecture reach? A power law at one point, a dominance regime and an attainable set of exponents are related objects, and each answers a different question (§19).

Run select, solve and check on totals, and read each regime's law as leading order. It holds within a bounded factor inside its region, and a singular ray needs one extra balance.

11Graph reduction marks where the equilibrium assumption ends

Gunawardena's 2012 framework describes molecular states as a labeled directed graph whose conditional dynamics are linear. With slow quantities held fixed in the edge labels, a strongly connected graph has one stationary distribution up to its conserved total. Tree-sum expressions make that distribution a positive rational function of the labels.²⁰

The framework includes equilibrium ensembles and also driven stationary systems. A stationary state satisfies node balance. Detailed balance additionally cancels every reversible edge current. The distinction is essential when reducing a promoter cycle, a modification network or a transporter that consumes a maintained free-energy difference.

The core's §3 assumes that binding clocks are fast against catalysis, so binding equilibrates at fixed totals. The linear framework shows what replaces that step when a fast subsystem is driven. The replacement is a stationary distribution that carries cycle currents, and it is still a rational function of the labels.

The core's equilibrium map has an additional task. It reconstructs free concentrations from finite totals that couple the state occupancies. Conditional graph linearity can simplify an internal state calculation while the outer finite-pool problem stays nonlinear. The two methods are complementary rather than competing claims about every biochemical model.

Use the equilibrium map where fast binding is undriven. Where a fast cycle carries a current, use a stationary graph reduction that keeps it.

12Rule-based models generate the complex list from local rules

A signaling protein with several sites can occupy many complexes and modification states. Rule-based modeling describes families of reactions through their local molecular requirements. Chylek and colleagues explain how a rule records the reactant features, the transformation and the rate law, with contextual requirements added when needed.²¹

This compression is earned when the omitted contexts share the same event rule. If another site changes an affinity or an activity, the context must reappear in the rule or the state representation. A short rule list is not evidence that a protein has no allosteric coupling.

The connection to Lecture 8 is the complex list. The core's §5 starts from an explicit list of complexes, $L=[I_d\mid L^{(c)}]$, and rule-based tools are how such lists are generated for proteins with many sites. The binding-catalysis construction then asks which subset equilibrates, which totals it preserves and which states act. Compact specification and dynamical reduction solve different parts of the modeling problem.

Judge a rule set by its molecular-context assumptions. Then read the complex list it generates as the core's binding layer, with its own totals and activities.

13Negative cooperativity exposed the role of inventory and readout

Ha and Ferrell's 2016 study asked how a familiar binding property changes when ligand can be depleted. Their theory separated free-ligand from total-ligand response curves, and independent site activity from activity that requires double occupation. Strong negative cooperativity and finite high-affinity ligand could then produce an inactive buffering region followed by a sharp rise in double occupancy.²²

Their account of the first site describes a titration inside one receptor. The first site acts as a stoichiometric buffer, soaking up the first increments of a depletable ligand without producing a response. In the core's terms the singly bound receptor wins the ligand total until the high-affinity sites fill. The response turns on near matched totals of ligand and receptor, which is the singular ray of the first site (core §10).

The experimental realization used DNA strands as a receptor and two distinct ligands. Their binding-site arrangement changed the cooperative interaction. This tested a specific physical mechanism behind the total-input threshold. It did not establish that every negatively cooperative receptor is ultrasensitive.

The conceptual addition combines three specifications: binding interaction, available ligand total and active readout. Positive cooperativity is one route to a steep response, but the measured input coordinate and the silent states can change the conclusion. The core's §1 lists negative cooperativity among the mechanisms that effective rate laws miss.

Read a cooperativity claim together with the experiment's input coordinate, finite total and active state.

14BMP mixtures tested how shared receptors combine signals

Antebi and colleagues' 2017 BMP study measured how cells with different receptor compositions interpret ligand combinations. Promiscuous binding had often suggested redundancy. The work instead showed that competition among complexes could support distinct forms of signal integration, including ratiometric and imbalance-sensitive responses.²³

The model gave complexes separate affinities and activities. Affinities determined the allocation among receptor combinations. Activities determined the output produced by that allocation. Changing receptor expression could alter response profiles without changing the underlying biochemical parameters.

The lesson is broader than depletion. The main model considered a large extracellular ligand reservoir and also examined finite-volume behavior. Even a well-maintained ligand bath does not decide how two ligands combine at shared receptors. The core's §12 makes the same point at a promoter: joint states are the candidates for a shared total, and their activities decide the output.

This result motivates a specific test of compositional models. Characterize isolated ligands, predict mixtures with the same receptor-state architecture, and then alter receptor expression. Success requires the shared-state mechanism to transfer across contexts, not merely to fit each curve.

Use mixed ligands and altered receptor composition to test the allocation and activity assumptions behind a signaling model.

15MultiFate carried binding-derived regulation into cell-state dynamics

Zhu and colleagues' 2022 MultiFate circuit made promiscuous dimerization part of a synthetic cell-fate architecture. Active homodimers supported self-activation. Heterodimers competitively sequestered the transcription factors. The resulting production-removal network supported multiple stable expression states in mammalian cells.²⁴

The supplementary model kept finite monomer-dimer totals and then used a Hill response in the reconstructed active homodimer. This is a useful precedent for the lecture's choice of resolution. A mechanistic description of one layer and an effective description of another can coexist when their roles and assumptions are visible. The core's optional A1 shows when such a Hill law holds, since the winner of the repressor total sets its degree.

The dimerization equilibrium did not itself establish the number of cell fates. The authors analyzed and tested the closed dynamics. Lecture 8 builds the regulatory input to that analysis, and its coda hands the stability and transition questions to Lecture 9.

Separate the binding mechanism that supplies regulation from the dynamical analysis and measurements that establish stable cell states.

16Control and growth required additional system-level balances

Finite pools recur in control and growth, but a shared binding mechanism does not make two system-level conclusions equivalent. Integration needs a specific balance law. A growth bottleneck needs a self-reproduction model.

Antithetic feedback connected sequestration to error integration

Briat, Gupta and Khammash's 2016 antithetic controller used equal inactivation of two controller species to cancel a shared flux from their difference equation. Their stochastic theory supplied conditions for robust regulation of a stationary mean. The essential object was an error-integrating balance embedded in a feedback loop, rather than a static binding curve.²⁵

Aoki and colleagues' 2019 implementation used sigma and anti-sigma factors to realize the controller in living cells. Their analysis included finite production capacity and dilution, which can turn an ideal integral balance into a leaky one.²⁶

Sequestration thus connects a molecular total to a dynamical control function. The connection holds only after the activity of the complex, the fate of bound molecules and all production and removal terms are stated. In the core those are the activity line and the accumulation line, closed in the coda.

Growth models gave competing resource limitations equal status

Roy, Goberman and Pugatch's 2021 growth framework organized bacterial autocatalytic cycles by the resources that limit them. Ribosome production, RNA polymerase production and tRNA charging were coupled parts of the construction. Minimum functions exposed different limitation regimes and their growth relations.²⁷

A minimum function resembles the limiting pool of a tight binding pair, $C_{ES}\approx\min(q_E,\,q_S,\,q_Eq_S/K_d)$, once resource units are matched (core §8). This is a correspondence between approximations. The growth framework does not derive its translation steps from the binding equilibrium model.

The paper also states an inference limit: a valid growth law does not fully reveal the physiological state. The same caution applies to any effective relation and the mechanism that generated it.

Carry finite-pool accounting into control or growth together with the balance and dynamical assumptions that make the claimed function possible.

17The lecturer's programme turned regimes into a model class

Seven papers from 2020 to 2026 carry the core's construction. The table lists them in order, with the problem each addressed, what it added and the core sections that use it. Every locator was checked in the full text. The programme's main paper has not yet appeared, and its supplement is cited only for three definitions that no published paper carries.

Regimes came before a model class

Marken, Xiao and Murray's 2020 preprint started from Hill-type rate laws and divided a circuit's behavior into saturation regimes. Each regime has a monomial law, so its behavior is a straight line on log axes. A regime counts only when its own consistency conditions hold.²⁸ This is select, solve and check, applied to effective laws.

The preprint states its limits plainly. It lacks rigorous proofs, and the approximation breaks down close to the borders between regimes (p. 15). Both limits return in the core. The thesis proves the structural results on which the regimes rest. The bounded-factor estimate of core §7 addresses the borders, and it is not yet published (§18).

Xiao, Khammash and Doyle's 2021 paper moved the regimes from effective laws to mechanism. Binding is fast and catalysis slow, because binding involves weak bonds such as hydrogen bonds while catalysis makes and breaks covalent ones. For one binding pair, the orders of the complex in the two totals fill a triangle whose vertices are regimes. The paper names a regime, $C_{ES}\approx q_S$, that the usual Michaelis-Menten reading misses, and calls that approximation one edge of the triangle.²⁹ Its constant K stands for $K_d$ or variants such as $K_M$. The core fixes $K_d$ and relates the two constants in §6.

The thesis stated the model class

The lecturer's 2022 thesis made binding's regulation of catalysis the core structure of biomolecular reactions (pp. 2, 24). It separated the two by timescale and named time-scale separation as its only assumption (p. 16). It wrote catalysis as a change of totals, because a consumed substrate may come from its free or its bound form (p. 39). It wrote the binding layer with a conservation matrix L and an equilibrium matrix N, called them dual configurations (pp. 88–89), and proved that each positive set of totals and constants fixes one positive state (p. 96).¹²

The thesis also found the regimes in this structure. For one pair, the complex has three vertex regimes (Eq. 3.43, p. 109). For free enzyme, a further extreme case appears when both totals are mostly bound and nearly equal. Along it the sensitivity of free enzyme to total substrate grows without bound, so the regime is a ray rather than a vertex (Eq. 3.45, p. 113). The general statement follows later in the thesis: in the singular case, the regime is a ray (Lemma 3.8.1, p. 130). The core's singular ray $(C_{ES},C_{ES})$ is this object.

Two conventions in the thesis differ from the core. The thesis orients each binding relation in the dissociation direction, $N\log x=\log k$ with dissociation constants (Eq. 3.11, p. 91). The core uses association constants, and its basis change $N=N_2N^{\mathrm{aa}}$ covers both conventions (core §6). The thesis also puts rates on squiggly arrows (Eq. 1.2, p. 25), which the course reserves for composite processes and never labels with a rate.

Two applications tested the split

Xiao, Li and Doyle's 2023 paper carried the split into metabolism. Because binding regulates the exponents of catalytic fluxes, it proposed the rule that cells regulate those exponents rather than the fluxes themselves, and called the approach flux exponent control (p. 1189). It separated fast binding from slower catalysis in a metabolic network and wrote external exchange fluxes as a separate stoichiometric term (p. 1191).³⁰ The core's accumulation line keeps the same separate term, $S^{\mathrm{ext}}v^{\mathrm{ext}}$ (core §4). The paper's state variables are metabolite concentrations rather than binding totals.

Fu, Xiao and Jun's 2023 paper treated replication initiation in E. coli as protein counting. Initiator protein is first titrated by high-affinity DnaA boxes, and replication starts only after the boxes are nearly saturated (p. 013011-2). The model assumes tight binding outright, with initiation when the initiator count reaches the binding-site count (p. 013011-4).¹⁵ It has no regime vocabulary. The core reads the same threshold on the singular ray of a titration path (core §10).

Validity and sampling made regimes computable

Liu, Ren and Xiao's 2026 paper asks when a reduced law is valid. Dominance choices in the conservation laws define the regimes, written as tuples such as $(q_E,q_S)\approx(E,S)$ in the course's letters. The one-pair table has four regimes, and the fourth has zero area in parameter space. The procedure enumerates the choices, solves each and substitutes the solution back into its inequalities. It then measures the share of parameter space in which a function holds, its realizability index (p. 2). Validity depends on which variables the experiment controls (pp. 2–3).⁹

Two facts about that paper's status matter for citation. The version read here is the third bioRxiv version, posted 20 May 2026. The paper is accepted at the 23rd IFAC World Congress, whose proceedings have not yet appeared. Its validity results use a Michaelis constant where the core writes $K_d$.

Liu and colleagues' flux-species sampling paper, posted 22 July 2026, uses the model behind the scenes. It states the binding-catalysis model in four lines, without an external term (Eq. 1, p. 5). It treats many reduced steady-state descriptions as leading-order descriptions of dominance regimes (p. 5). Sampling species rather than parameters finds switches throughout the complex-dominant region in about 23% of draws, where parameter sampling finds them in a thin strip in about 1% (p. 3).¹⁴ The paper states uniqueness of the binding state without proof, which is the thesis's result. It cites the main paper as in preparation.

The main paper's supplement carries three definitions

The programme's main paper, on the universal polyhedral geometry of biomolecular reaction systems, has not yet appeared. Its supplement is archived as an unpublished snapshot of 21 September 2026. It carries three definitions that no published paper has. Faithful-atomic composition requires distinct composition columns (Def. 3.7, p. 26). The bipartite composition graph has L as its biadjacency matrix (p. 61). Orienting that graph by the winners of the totals gives an acyclic graph exactly when a regime is realizable (Def. 7.3, p. 71).³¹ The core cites the supplement for these three items only, and so does this page.

The 2025 course asked what binding can build

The 2025 teaching material posed a constructive question: how can one binding reaction generate the diversity of bioregulation? Its Lecture 3 developed saturation, bottleneck and ultrasensitive free-species responses. The associated assignment combined reactions in competition, in a sponge and in series, and asked which parameters tune the result.

Lecture 10 of that course used gene regulation and titration to motivate molecular computation and cell-state selection. Binding supplied nonlinear response functions, while production and removal supplied their dynamics. Homodimer activation and heterodimer sequestration motivated additional stable cell states.

These materials set the course's direction rather than a general theorem. A finite-affinity titration curve approximates a threshold function over a specified range. Several binding configurations do not by themselves establish several stable states or a computation. The later lectures inherit those constructions.

Read the programme in order: regimes on effective laws in 2020, on mechanism in 2021, the model class in 2022, two applications in 2023, and validity and sampling in 2026. Cite a published paper for each core step, and the supplement only for its three definitions.

18The present formulation assembles published pieces into four lines

The core's model has four lines. Each line has a published source, but no single paper prints all four in the core's form.

Conservation and equilibrium come from the thesis. It pairs L and N as dual configurations and proves the equilibrium map unique.¹² The core restates the pair in a canonical association basis, $N^{\mathrm{aa}}=[-(L^{(c)})^{\top}\mid I_r]$, and calls it a Gale pair (core §6).

Activity is the oldest line. Michaelis and Menten made the rate proportional to the complex, and the thesis writes each catalytic flux as a rate constant times its active species (p. 39). The matrix $K^{\mathrm{cat}}$ collects one such constant per active state.

Accumulation in totals has the most scattered record. The thesis writes catalysis as a change of totals (p. 39) and says that the species at the catalysis timescale are totals (p. 218). The 2023 metabolism paper writes exchange fluxes as a separate term.³⁰ The 2026 sampling paper prints $\dot q=Sv$, without the exchange term (Eq. 1, p. 5).¹⁴ The factorization $S=L\Gamma_{\mathrm{cat}}$, with a separate exchange term, is the core's own statement (core §4).

Composition is where the published record ends

The binding layer is specified by its complex list, and here the record is thinnest. The thesis defines isomer-atomic networks, which drop a no-isomer condition, so distinct states may share a composition column (Def. 3.3.7, p. 86).¹² The unpublished supplement adds faithful-atomic composition, with distinct columns, and the bipartite composition graph.³¹ The core uses both in §5 and orients the graph by winners in §7.

The composition graph records what every state contains before any assembly pathway is chosen. At equilibrium, that composition and compatible weights determine the allocation. The matrix L alone does not determine the weights, the activities or the relaxation times. Those remain separately specified parts of the model.

Isomers make this concrete. Two states can share a composition column and still differ in affinity or activity (core §5). At a fixed equilibrium weight ratio they can sometimes be pooled into one effective state, whose activity is the ratio-weighted average of theirs. A bulk flux then constrains only that average, not the two activities.

Several of the core's framings are not yet published

The core also uses framings that no published paper states. They come from the lecturer's unpublished manuscripts, or they are the course's own. They are the open end of the record, and they become citable when the main paper appears.

• The words winner and winner tuple. The 2026 validity paper already writes regimes as tuples (p. 2).
• Regular and singular regimes as the core defines them: a regime is singular when one species wins two totals. The thesis has the singular case only as a ray (p. 130).
• The factorization $S=L\Gamma_{\mathrm{cat}}$ with a separate term $S^{\mathrm{ext}}v^{\mathrm{ext}}$.
• The bounded-factor estimate of core §7, and the factor-of-two statement for one pair in core §8.
• The five-step pipeline of core §9. The validity paper's four-step procedure is its closest published relative (p. 2).
• The calculus analogy of core §7 and the circuit reading of the four lines in core §4. The thesis speaks of rules of calculus for positive variables (p. 48) and cites mechanics and Kirchhoff's laws as models (p. 16).
• The regime readings of three familiar mechanisms: the joint-state truth table of core §12, the competitor threshold in max form in core §11 and the response window of core A3. The mechanisms themselves are published. Reading them through winners is the course's own.

Attribute each line of the model to its published source. Present the listed framings as the course's own until the main paper appears.

19Reaction-order geometry asks what all operating points can realize

A single dose-response curve shows one path through parameter and concentration space. The lecturer's reaction-order programme asks instead which local logarithmic responses a fixed binding architecture can reach across all positive operating points. The question changes from fitting one curve to describing a constrained set of regulatory shapes.¹²

For one binding pair the orders of the complex in the two totals fill an open triangle, and its vertices are the three regular regimes. List the orders as (in $q_E$, in $q_S$), as Figure 11 does. The vertex $(1,1)$ is weak binding, the regime $(E,S)$. The vertex $(1,0)$ is $C_{ES}\approx q_E$, the regime $(C_{ES},S)$. The vertex $(0,1)$ is $C_{ES}\approx q_S$, the regime $(E,C_{ES})$. Finite positive operating points lie inside.^{12, 29}

The singular regime has no vertex of its own. On the ray $q_E\approx q_S\gg K_d$ both totals are mostly bound, so the complex's orders lie near the tight-binding edge that joins $(1,0)$ and $(0,1)$. The thesis gives the tight-binding limit $C_{ES}=\min(q_E,q_S)$ for that edge (p. 47). The free species carries the singular behavior, and its orders run off along a ray (Eq. 3.45, p. 113).

The derivatives are taken with respect to controlled totals, so they carry the inventory constraints into stability and response calculations. Lecture 11 develops the general geometry. Lecture 8's core uses the regimes that the vertices name (core §8) and the paths that cross them (core §10).

Read total-order geometry as the set of local responses an architecture can reach. Its vertices are the core's regular regimes, and the singular ray sits on the tight-binding edge.

20The record establishes a model class and leaves three questions open

The historical strands converge on a conditional question: what can a specified molecular architecture do under specified physical conditions? The record now answers part of it, marks one boundary and leaves three questions open. Take them in that order.

The published record supports a model class and a method

Two claims now rest on published work. First, the binding-catalysis model is a model class, and its four lines contain enzyme saturation, titration switches and joint-state promoter logic as leading-order regimes. Second, leading-order dominance is its method of analysis. Select, solve and check came from design spaces, the thesis made the terms structural, and the 2026 validity and sampling papers made the regimes computable.

Equilibrium reduction must not erase a physical current

A chemical state model describes transport only after its transitions are connected to exchanges with named reservoirs. Loaded-carrier reorientation, empty return and a leak route have different physical consequences even when their rates share units. Counting state occupancy alone cannot identify useful cargo throughput.

The extension makes this explicit with a six-state ion-cargo symporter. Compatible binding chains equilibrate within the outward and inward orientations. Slow orientation changes remain dynamic and carry cargo, ion or both. A stationary graph calculation keeps their currents, while partial equilibrium reconstructs the state occupancies.

The same principle protects the driven mechanisms of Lecture 7. A fast driven cycle cannot be equilibrated wholesale without erasing the current whose function is being studied. Partial equilibrium is useful precisely when its boundary leaves the necessary slow exchanges and drive in place.

Three questions remain open

Three questions remain, about architecture, accuracy and dynamics. Take them in turn.

The first is which architecture a cell uses. A genome suggests proteins and transformations, but the relevant complexes, conformations and kinetic contexts need further evidence. Equilibrium affinity data also do not fix binding relaxation speeds. Solving a specified architecture is not the same as discovering which one biology uses.

The second is how accurate leading order is in large networks. The bounded factor grows with the number of terms in a total, and near boundaries the exact curves depart from the rays. The core shows the size of that departure for one pair. Its general published form awaits the main paper.

The third is what the closed dynamics can do. Production, removal and catalysis change the totals, and the binding layer then supplies a vector field. Lecture 9 asks which steady states exist and whether they are stable. Lectures 10 to 12 take up adaptation, reaction-order geometry and computation.

Use the model class to pose an accountable prediction, and keep solving a specified architecture apart from discovering which architecture biology uses.

References