The order of a reaction

One log-log slope, five names, and the assumption underneath twenty-five years of argument.

Fangzhou Xiao · Westlake University · a working paper, August 2026. Companion to Structure is sparsity, Three schools, one question and The biomachine perspective.

Take any rate law that is positive and differentiable in the logarithm, and any variable it depends on. The derivative

α = ∂log v / ∂log x

is a pure number. It has no units, whatever the units of v and x. It is local, so it belongs to an operating point and not to a molecule. And it is the exponent you would fit if you decided to model v as proportional to a power of x.

This quantity has been discovered, named and argued over at least five separate times. It has been called the exponent on an active mass, the number of molecules taking part in a reaction, the order of a reaction, the Hill coefficient, the reflection coefficient, the kinetic order, and the elasticity. Two schools of theoretical biochemistry were built on it, twenty years apart, and then spent twenty-five years disputing whether one was a special case of the other. The dispute was conducted in the pages of Trends in Biochemical Sciences, Mathematical Biosciences and the Journal of Theoretical Biology, it was intermittently bitter, and it has never been cleanly settled in the secondary literature.

This essay does three things. It traces the object back past the systems age into chemistry, where it was a measurement before it was a theory. It reconstructs the dispute from both sides’ own papers rather than from either side’s summary of the other. And it argues that underneath metabolic control analysis sits exactly one substantive assumption, that this assumption is itself a reaction order, and that the assumption is a claim about binding which is false across a large part of the regimes that synthetic and regulatory biology occupy.

1The quantity that keeps coming back

Before any school claims it, the object is worth looking at on its own. It has three properties, and all three matter for what follows.

Let v be a rate and x a concentration it depends on. The log-log slope is

α ≡ ∂log v / ∂log x = (x / v) · ∂v / ∂x

It is dimensionless. The units of v cancel against v and the units of x against x, so the number survives any change of units and can be compared between a rate and a concentration, or between two systems that share no scale.

It is local. It is a derivative, so it is attached to an operating point, not to a molecule or a mechanism. A rate law has a different α at every concentration, and it is only constant if the rate law is exactly a power of x.

It is additive in logarithms, which is why it composes. If z depends on y and y on x, the chain rule in log coordinates reads α_zx = α_zy α_yx, and for a product of factors the slopes add. That last property is the reason every field that has found this quantity has found it useful: it turns multiplication into addition, and a rate law is usually a product.

The consequence people notice first is that α is the exponent of the best local power law. Write v ≈ α₀ x^α near a point, take logs, and the fit is a straight line whose slope is α. Reading an exponent off a straight line on log paper is an old and cheap experiment, and it is the reason this quantity was measured long before it was theorised.

2Order was fitted, not derived

The law of mass action was written as a product of powers from the beginning, and the exponents were numbers you obtained from data.

Guldberg and Waage’s 1864 paper introduces what they call the active mass of a substance and asserts that the force driving a reaction is a product of powers of the active masses¹. Written in modern notation, for a reaction between two species,

v = k [A]^a [B]^b

and the exponents a and b are quantities to be determined. This is a stronger statement than it looks. Nothing in the formulation says a and b are the stoichiometric coefficients of a balanced equation. They are whatever makes the expression fit, and the 1864 paper obtains them from titration data by trial.

The distinction between a fitted exponent and a counted molecule is therefore present from the first statement of the law, and it is not a modern gloss. It is worth being precise about what came later, because the standard potted history compresses two different acts into one.

31884: the slope, and what it was called

van ’t Hoff wrote the log-log slope as a formula and used it as a measurement method. He called the number it returns the number of molecules taking part in the reaction. Three years later Ostwald renamed it, and the rename is the first instance of this essay’s theme.

In Études de dynamique chimique, van ’t Hoff sets out a method for determining the exponent n in −dC/dt = k Cⁿ. Take two experiments at different concentrations, measure the rate in each, and form a ratio. Since the rate constant cancels, the exponent falls out of a ratio of logarithms. He writes it as a displayed equation, using l. for the logarithm.

Two things about this page are worth dwelling on. The first is that van ’t Hoff prefers this method to integrating a guessed rate equation, and says why: the integral method loses all its value if a disturbing influence acts on the course of the transformation, whereas the differential method is insensitive to that. This is a robustness argument for a local slope over a global fit, made in 1884, and it is the same argument both twentieth-century schools would make for working with logarithmic derivatives.

The second is the heading. van ’t Hoff calls the number he has just extracted the number of molecules taking part. His terms are monomoléculaire, bimoléculaire, trimoléculaire, and they classify reactions by the exponent, not by the count. Laidler’s history of the period is explicit that van ’t Hoff was aware of the difference, and glosses each of van ’t Hoff’s uses of “molecularity” with “[order]” to keep the modern reader straight³.

The subsequent history of chemical kinetics is largely a history of methods for getting n out of data. Ostwald’s method of isolation floods all reactants but one so that the order with respect to the remaining one can be read alone. His half-life method uses t_½ ∝ c¹⁻ⁿ to give

n = 1 + log[(t_½)₁/(t_½)₂]  /  log(c₂/c₁)

which is again a ratio of logarithms³. Every one of these is an estimator of the same derivative.

4The order in substrate runs from one to zero

The central textbook fact of enzyme kinetics is a statement about a slope that moves. Saying it that way makes the connection to everything downstream immediate.

The Michaelis–Menten rate law is

v = V_max S / (K_m + S)

and its kinetic order in substrate is obtained by differentiating the logarithm:

∂log v / ∂log S = K_m / (K_m + S) = 1 / (1 + S/K_m)

This is the whole of the first-order-to-zero-order transition, written as one expression. The order is 1 when substrate is far below K_m, it is 0 when substrate is far above, and it equals exactly ½ when S = K_m. The Michaelis constant is, in these terms, the substrate concentration at which the reaction is half order.

The transition is slow. Going from order 0.9 to order 0.1 takes S/K_m from 1/9 to 9, which is 1.91 decades. An enzyme therefore spends most of its accessible concentration range at a fractional order, and the two round numbers that the textbooks name are the two ends of a curve rather than two regimes with a boundary.

5Hill’s coefficient is the same slope, at a different variable

The Hill coefficient is a log-log slope of a binding curve. It varies with saturation, and the number that gets reported is one sample from a curve.

Hill’s 1910 paper fits oxygen binding to haemoglobin with y = x^h/(K + x^h)⁷. Rearranged, y/(1−y) = x^h/K, so

n_H ≡ ∂log[y/(1−y)] / ∂log x

is h exactly, if the Hill equation is exactly true. It is not exactly true for any real cooperative protein, and the consequence is that n_H defined by that derivative is a function of x. The Adair equation, which assumes only that a tetramer binds four ligands with four association constants and no relation among them, gives a Hill plot that is sigmoid in log coordinates with slope tending to 1 at both extremes⁸.

The unit asymptotes are not an artefact. At very low saturation only the first binding step is populated, and one ligand binding to one site is first order. At very high saturation only the last vacancy remains, and the same argument applies. The cooperativity lives in the middle, and the reported n_H is conventionally the slope at half saturation, which is the maximum.

Ferrell and Ha’s three-part review is the modern statement of the same point, and it is the one to read if the question is which mechanisms generate which slopes^9,10,11. Their organising quantity is the local logarithmic slope of a response, and they use it to separate Michaelian responses from zero-order ultrasensitivity, multisite phosphorylation and cascades. That review is a survey of mechanisms indexed by the value of one derivative.

6The object, stated once

Everything from here is naming, aggregation, and one assumption. It is worth fixing the object first so the rest can be read as operations on it.

Three consequences will be used repeatedly.

The local power law is exact to first order. Given α at a point and the value there, the power law v̂ = α₀ x^α matches v in value and in first derivative at that point. It cannot match the second derivative unless the true law happens to be a power. This is the whole content of the tangency dispute in §13, and it is elementary.

A full set of slopes determines the law up to constants. If you know α as a function of x over a range, you can integrate d log f = α(x) d log x and recover f up to one multiplicative constant. Higgins made this point in 1959, and it matters for §24: the derivative does not destroy information about mechanism. Truncating to one value at one point does.

Homogeneity is a statement about slopes. A function is homogeneous of degree h in a set of variables if scaling all of them by t scales f by t^h. By Euler’s theorem that is exactly the statement that the slopes with respect to those variables sum to h. This equivalence is the hinge of §17, and it is the reason a summation theorem and an assumption about enzymes turn out to be the same sentence.

7Higgins: the ratio of relative changes

The systems-age object arrives in 1963, ten years before metabolic control analysis and six before biochemical systems theory. It arrives with a calculus, an identifiability theorem, and a lineage in wartime control engineering.

Higgins defines what he calls the stoichiometric reflection coefficient between any two variables of a reaction network as the ratio of their relative changes¹³. In the infinitesimal limit,

_aR_b = (a/b)(db/da) = ∂log b / ∂log a

which is the object of §6 with a different name. His own retrospective account, given at the 1989 NATO workshop where both sides of the later dispute were in the room, sets it out with the percentage-change reading attached.

Three features of the 1963 paper matter for everything downstream, and all three are usually left out of the potted history.

It is indexed by which variable you perturb. Higgins notes that reflection coefficients can be defined between any two variables, but that their values depend on which variable is taken as the primary variable, meaning the one experimentally changed. He carries a forescript to record it. This single piece of notation contains the distinction that metabolic control analysis would later draw between an elasticity, where the perturbed variable is a metabolite acting on an isolated enzyme, and a control coefficient, where it is an enzyme acting on the whole system. In Higgins the two are one object at two settings of an index.

The relations among coefficients depend on stoichiometry, not on rate constants. Higgins states this explicitly, and draws the consequence that the relations can be tested directly against measured concentrations without knowing any constant. This is the idea that Reder would formalise in 1988 (§16) and that the reaction-order formula of §23 turns into a computation. It is the oldest statement of the position that this essay ends up defending.

He proves a non-identifiability theorem. If two mechanisms yield reflection-coefficient relations of the same functional form over some subset of variables, then stationary-state experiments on that subset cannot tell them apart. That is a statement about what data can fix, made in 1963, and it belongs beside the modern versions.

Higgins also computes the reflection coefficient for a Michaelis–Menten step and gets _SR_v = K_m/(s+K_m), which is the expression of §4. He observes that the reflection coefficient gives a simpler result than the ordinary derivative, with fewer parameters and lower degree, and he uses that simplicity to try to discriminate mechanisms: his type I and type II sequences yield distinguishable relations. Hold on to this. In 1963 the log-log slope is a tool for finding out what the mechanism is.

8Savageau: a Taylor series in log coordinates

Expand the logarithm of a rate in the logarithms of its arguments, keep two terms, and every rate law becomes a power law. The exponents are what chemistry already called kinetic orders, and Savageau says so.

The construction is one line. For a rate v depending on x₁ … x_n, expand log v as a Taylor series in the variables log x_j about an operating point, and truncate after the linear term:

log v ≈ log v⁰ + ∑_j g_j (log x_j − log x_j⁰) ⇒ v ≈ α ∏_j x_j^g_j

with g_j = ∂log v/∂log x_j at the operating point. Savageau, Voit and Irvine state the identification plainly: in chemical and biochemical kinetics the g_ij exponents and the α_i coefficient have traditionally been called kinetic orders and rate constant²³. The genealogy of §2 to §3 is not reconstructed here. It is acknowledged by the person doing the naming.

9What aggregation buys

The choice that separates the two schools is not the derivative. It is what you sum before you take it. Here is what one side gets for summing first.

A metabolite pool is fed by several reactions and drained by several others. Biochemical systems theory, in its S-system form, aggregates all the production into one net synthesis term and all the consumption into one net degradation term, and applies the power-law representation to each aggregate:

dX_i/dt = α_i ∏_j X_j^g_ij − β_i ∏_j X_j^h_ij

Set the derivative to zero and take logarithms. Because each side is a single product of powers, the logarithm of each side is a sum, and the steady-state condition becomes linear in y_j = log X_j:

A y = b, a_ij = g_ij − h_ij, b_i = log(β_i/α_i)

That is the payoff, and it is a large one. A nonlinear steady-state problem has become a linear system whose matrix is built from differences of kinetic orders. Three things follow immediately and none of them has a counterpart on the other side^16,17.

• Existence. A steady state exists and is unique when A restricted to the dependent variables has full rank, which is a determinant condition on kinetic-order differences.
• Stability. Local stability reduces to eigenvalue conditions on a matrix assembled from the same exponents.
• An explicit symbolic solution. y_d = L y_i + M b, so every steady-state concentration and flux is an explicit function of the parameters, obtained by one matrix inversion.

Now the alternative. Savageau’s own taxonomy of variants turns on exactly this choice, and places metabolic control theory as one cell of it: power law explicit or implicit, aggregation by pools or by reactions, minimal equations or redundant ones³⁶. Generalized mass action, which is also a biochemical-systems-theory variant, applies the power law to each reaction separately and keeps the sum over reactions:

dX_i/dt = ∑_r ± v_r, v_r = α_r ∏_j X_j^g_rj

Setting this to zero gives the logarithm of a sum, which does not simplify. There is no closed-form steady state, hence no existence theorem and no stability condition stated in the parameters. Metabolic control analysis makes this same choice, which is why it inherits the same limitation and why Savageau kept pressing on it.

10Kacser and Burns, Heinrich and Rapoport

Two independent papers in 1973 and 1974 define the same two coefficients. Both credit Higgins, in their own text, for the object.

Kacser and Burns define what they first called a sensitivity coefficient, now the flux control coefficient¹⁸,

C^J_i = ∂log J / ∂log E_i

and the elasticity of a single reaction to a metabolite it sees,

εⁱ_x = ∂log v_i / ∂log x

The two differ in what is perturbed and in what is held fixed, which is Higgins’ primary-variable index. The control coefficient perturbs an enzyme and lets the whole system re-equilibrate; the elasticity perturbs a metabolite and looks at one reaction in isolation. Both are log-log slopes.

The acknowledgement is in their own text. Cornish-Bowden quotes it in 1989 to settle the priority question: Kacser and Burns wrote that Higgins had presented a kinetic treatment of sequential reactions analysing system responses in terms of reflection coefficients, and that their own treatment was based on essentially the same approach²⁷. Heinrich and Rapoport likewise compared their control strength with Higgins’¹⁹.

11The two theorems

Stated bare first, then the reading that made them famous, which is where the trouble starts.

For an unbranched pathway of n enzymes carrying flux J, with internal metabolites x:

The reading that made them famous is the conservation reading. If the flux control coefficients are positive and sum to one, then control is a conserved unit shared out among the enzymes of a pathway. A coefficient of 1 for one enzyme forces 0 for all others. The average coefficient in a pathway of n enzymes is 1/n, so in a long pathway no single enzyme can matter much. The idea of a rate-limiting step, on this reading, is not merely unsupported but arithmetically excluded.

Fell’s survey is the standard account of how far that reading was taken and what was done with it³³. That reading did a great deal of useful work. It is also the part of metabolic control analysis that does not survive branching, and both sides came to say so. §15 separates the three distinct questions that get run together here, because the answers differ.

121987: where is the theory?

The dispute opens in a discussion forum in Trends in Biochemical Sciences. Savageau’s objection is not about priority. It is about an assumption, and he names the right one on the first page.

Kacser and Porteous open the forum with a programmatic piece arguing that qualitative questions about which enzyme controls a pathway should be replaced by quantitative ones, answerable by measuring control coefficients²¹. In setting up the framework they write that we expect, from our knowledge of enzymology, a change in local reaction rate proportional to any change in enzyme concentration.

Savageau’s reply, titled “Control of metabolism: where is the theory?”, fastens on that sentence²². His objections, in the order he makes them:

1. Rates are not generally linear in enzyme. Enzyme-enzyme interaction, channelling and structural organisation are documented and widespread, and where they occur the Michaelis–Menten formalism does not apply to the enzyme as it functions in the intact cell.
2. Enzyme levels are not always parameters. In enzyme-proenzyme cascades, in hormonal systems and in the regulation of gene expression, enzyme levels are dependent variables the experimentalist cannot set directly, so the perturbation that defines a control coefficient cannot be performed.
3. No existence or stability results. Metabolic control analysis assumes its models have stable steady states rather than establishing it, whereas that is exactly what §9’s closed-form solution delivers.
4. The derivation is circular. Summation is shown to follow from expressions for the individual control coefficients in terms of elasticities, but deriving those expressions required the summation property in the first place.
5. The formalisms are the same. He asserts that the formalism underlying his theory and the one Kacser and Porteous describe is the same, and that they have failed to realise it.

Objection 1 is the one to watch. It is the assumption that Giersch proves is equivalent to the summation theorems a year later (§17), and it is the reaction order this essay is about. Savageau got to it first, in the first exchange, and then spent five years arguing about other things.

13Tangency, and who was right

The central factual dispute is whether elasticities and kinetic orders are the same quantity. The answer is that they are the same derivative used under two different quantifiers, and the record shows one side misreading the other on exactly this point, with a referee objecting in advance.

Savageau, Voit and Irvine assert the identity: the elasticities proved to be identical to the kinetic orders in biochemical systems theory²³. Cornish-Bowden replies that this is misleading, and states the alternative precisely.

Sorribas and Savageau devoted a paper of their own to the comparison, reaching the opposite verdict from the same algebra³⁵. Tangency is exactly right, and it is worth making quantitative, since the whole disagreement turns on how wide the agreement is.

So the agreement is exact at a point, good to a few percent over a factor of two, and gone by an order of magnitude. Both sides can quote this correctly and mean different things by it. That is a real disagreement about how to use a number, not a misunderstanding of what the number is.

Except that in this instance it partly was a misunderstanding, and the record is unusually candid about it. Cornish-Bowden’s argument requires that biochemical systems theory treats kinetic orders as constants. A referee told him that Savageau and colleagues nowhere say so. He published anyway, with a footnote.

14The theorems are orthogonality relations

Savageau’s deflationary reading of summation and connectivity is correct, and Cornish-Bowden concedes it. They disagree about what follows.

In biochemical systems theory the steady state is A y = b with explicit solution involving M = A_d⁻¹. Then M A_d = I is a matrix identity, and writing it componentwise gives a family of relations between the systemic sensitivities (the entries of M) and the molecular parameters (the entries of A). Savageau, Voit and Irvine show that summation and connectivity are members of that family²⁴. In the same spirit Sauro and colleagues wrote the metabolic-control-analysis theorems as C ε = I.

The two statements are the same observation. Both say the theorems are the orthogonality relations of a matrix and its inverse. And Cornish-Bowden agrees without reservation: given that the analysis is within linear algebra and that one knows the established results of linear algebra, the truth of C ε = I becomes so natural as to be hardly in need of proof²⁷.

He then reframes the question, and the reframing is better than it first appears. The puzzle is not why biochemical systems theory failed to prove the relations. It is why it did not think them worth pointing out. His answer is the objectives argument: if your goal is to model and predict, relations that add no information to a model you already have are decoration. If your goal is to dislodge the idea of a key enzyme or a metabolic bottleneck, a relation forcing the coefficients to sum to one is precisely the tool, because it converts a vague claim into an arithmetic contradiction.

15The branch: three questions, three answers

The claim that metabolic control analysis works only for unbranched chains is the most repeated thing said about this dispute. It contains three separate questions, and they do not have the same answer.

The three questions are: do the theorems remain true when the pathway branches; do they remain sufficient to determine the coefficients; and does the conservation reading of §11 survive. Take them in that order on one concrete network.

(a) Do the theorems remain true? Yes, exactly. Every row of the control matrix sums to 1 to within 2×10−10, and the connectivity relation ∑i CJki εiX1 = 0 holds to 1×10−9, for all three fluxes. Giersch demonstrates the same thing analytically, and the reason is instructive: J2 is not homogeneous of degree 1 in the enzymes of its own branch, but it is homogeneous of degree 1 in all the enzymes of the network. The summation must run over every enzyme, not over the branch. Cornish-Bowden makes the same point about C ε = I remaining true for multiple fluxes in a branched pathway27.

(b) Do they remain sufficient? No, and Savageau said so first and correctly. The relationships are not sufficient to obtain the explicit relationships between systemic and molecular parameters, except in the special case of sequential chains of simple enzymatic reactions, and one has to introduce the auxiliary relationships of Heinrich and Rapoport in the case of branching systems^24,20. This is checkable as a rank count, and the count is exact: nine unknown coefficients, and the 1973 statement of summation and connectivity supplies a system of rank 6. The deficiency is three. Solving the deficient system by least norm gives coefficients wrong by up to 1.92 in absolute value, so this is not a technicality.

(c) Does the conservation reading survive? No. The control matrix at this operating point is

The last row sums to one and tells you nothing you could describe as sharing out a conserved unit. One enzyme has 164% of the control of the flux through a reaction it does not catalyse, another has minus 157%. The mechanism is transparent: J₃ is a small difference between two nearly cancelling terms, so a small absolute change in it is a large relative change. Savageau lists exactly this: coefficients need not be positive fractions when cascades or branches are present, and when they can be negative and exceed unity, the ability to test a model by summing them is lost²⁴.

Cornish-Bowden concedes the point and then bounds it, which is the most useful thing anyone said in the whole exchange. He grants that this aspect of metabolic control analysis has been somewhat over-sold, and grants separately that summing model-computed coefficients is no test of a model since they sum to one whether the model is right or wrong. His repair is quantitative: the intuition survives in weaker form, with a typical enzyme’s control over the flux in its own branch approximately the reciprocal of the number of enzymes in that branch, rather than in the system²⁷.

16Reder, and the correct count

The sufficiency complaint had a one-year life. Reder computes the control matrices from the stoichiometry matrix and the elasticities for any topology, and the reason the 1973 count fell short turns out to be a simple miscount of how many summation relations there are.

Reder’s formulation uses the stoichiometry matrix N, a matrix K whose columns are a basis of ker(N), and the link matrix L relating dependent to independent metabolites under conservation²⁶. Her Theorem 5 characterises both control matrices by two equalities each.

Now count. Let r be the number of reactions and m₀ = rank(N). The summation relations number dim ker(N) = r − m₀, and the connectivity relations number m₀. Together

(r − m₀) + m₀ = r

which is exactly the number of columns of the matrix being determined. The system is exactly determined for any topology, with no auxiliary relations.

Two footnotes to this. First, metabolic control analysis thereby arrives at one matrix inversion over a structurally defined subspace, which is what biochemical systems theory had been doing since 1969. The disagreement about method narrows to almost nothing at this point, and what remains is §13’s question about whether you then leave the operating point. Second, Reder’s formulation was not immediately absorbed. Smallbone showed in 2013 that a standard implementation of the algorithm still mishandled a rank condition twenty-five years on, and had to be corrected³⁴.

17Giersch names the one assumption

The summation theorems are not merely implied by the assumption that rates are first order in their own enzyme. They are equivalent to it.

The proof is Euler’s theorem, in the form of §6: a function is homogeneous of degree h in a set of variables exactly when its log-log slopes with respect to them sum to h. Since C^J_i = ∂log J/∂log E_i, the statement ∑_i C^J_i = 1 is the statement that J is homogeneous of degree 1 in the enzymes. Nothing has been proved so much as translated, and the translation is the point: it converts a theorem about control into an assumption about kinetics, and lets you ask whether the assumption is true.

Giersch asks, and answers no. The section is headed “Are biochemical systems homogeneous?” and the argument is about binding. Adding δE_i raises the enzyme-substrate complex by δES_i and, by the same amount, lowers the free substrate. It will also perturb free magnesium, pH, and any other species that binds the enzyme. To make the relation homogeneous you would have to add, along with the enzyme, matched amounts of every metabolite, cofactor and ion that interacts with it. His conclusion is that for real biochemical systems v_i is certainly not, strictly speaking, a homogeneous function of E_i, and the reason he gives is that enzyme and metabolite concentrations are not independent variables, because equilibrium relations tie the free substrate, the complexes and the free enzyme together.

He gives a named counterexample: rubisco, the most abundant enzyme on Earth, whose active-site concentration in the chloroplast stroma is comparable to that of its substrate. Its rate law is non-homogeneous in enzyme, and by the equivalence the summation theorem correspondingly fails there.

18Dominance, which is what was really at stake

The exchange turned bitter over genetics, not kinetics. Both sides were arguing about whether a century-old evolutionary explanation had been overturned by an arithmetic identity.

Kacser and Burns argued in 1981 that the dominance of wild-type alleles needs no selective explanation³¹. The argument runs through summation: because flux control coefficients are small and sum to one, the flux-versus-enzyme curve is concave and nearly flat over most of its range, so halving an enzyme’s activity in a heterozygote moves the flux hardly at all. Dominance is then a systemic property of pathways, not an adaptation, and Fisher’s modifier theory is unnecessary.

Savageau and Sorribas attacked the derivation, and Savageau’s 1992 reply to Kacser is titled “Dominance according to metabolic control analysis: major achievement or house of cards?”^28,30. The substance of the attack is the substance of §15 and §17: if coefficients can be negative or exceed one, the concavity argument loses its guarantee, and if rates are not first order in enzyme, the summation relation the argument rests on does not hold.

The modern view is that both the systemic and the selective accounts contribute, and that the question is quantitative rather than either-or. Bagheri and colleagues make the case that the metabolic-control argument establishes a tendency rather than a mechanism, and that enzyme kinetics with realistic parameters can produce a range of dominance relations³².

19Scoring it

On the specific claims, with each verdict traced to the text or computation that carries it.

Two modern surveys reach compatible conclusions from opposite starting points: Voit’s review of biochemical systems theory³⁷, and Sauro’s recent side-by-side comparison, which comes from the metabolic-control-analysis side and concludes that the frameworks are complementary rather than nested³⁸.

Read down that column and a pattern shows. Savageau was right about nearly every technical particular and wrong about the conclusion he drew from them, which was that the other programme was therefore derivative and unnecessary. Metabolic control analysis was wrong about several particulars, corrected most of them within two years, and was right about the thing that made it useful: that a coefficient attached to a single enzyme is a quantity an experimenter can go and measure.

The one item on the list that was never repaired is the last one, because it is not a defect to be repaired. It is a modelling assumption, it is a reaction order, and whether it holds is an empirical question about binding.

20The order with respect to the enzyme

Two of this essay’s threads meet here, and the meeting is the argument.

Part I established that ∂log v/∂log x is a local exponent that moves with the operating point, and that a century of chemistry and enzymology is about measuring it rather than assuming it. Part III established, via Giersch, that metabolic control analysis rests on exactly one assumption of its own, which is that each rate is homogeneous of degree 1 in its own enzyme.

Put them together. Homogeneous of degree 1 in its own enzyme means

∂log v_i / ∂log E_i = 1

which is a reaction order, with respect to the enzyme, fixed at 1. So the foundation of metabolic control analysis is a particular value of the very quantity the subject measures. Every other order in the theory is a variable to be determined by experiment. This one is a constant, set in advance.

It is worth being clear that this is not a reading imposed from outside. Kacser and Porteous state the assumption in their own words in the paper that opened the dispute: a change in local reaction rate proportional to any change in enzyme concentration²¹. Savageau objects to it in the first reply²². Giersch proves it equivalent to the theorems a year later²⁵. All three parties agree about what the assumption is. They disagree about how often it fails.

21When it is not one

Giersch states the physical condition in words. It is worth having as a number, because the number decides whether any of this matters in practice.

Giersch’s condition is that total enzyme be small compared to total substrate, and his mechanism is sequestration: adding enzyme creates complex, and the complex it creates is removed from free substrate²⁵. Make that quantitative with the smallest model that contains it. One enzyme, one substrate, conserved totals, and the rate proportional to the complex:

E + S ⇆ C, q_E = E + C, q_S = S + C, v = k_cat C

Solving the resulting quadratic for C and differentiating gives the enzyme order exactly. The textbook Michaelis–Menten law is the limit of this as q_E/q_S → 0, and in that limit the order is 1. Away from it, it is not.

So Giersch’s two-to-three orders of magnitude is the right rule of thumb, and it is a demanding one. Two decades of separation buys you an assumption good to a quarter of a percent. One decade buys 3%. Parity is useless.

This is why the assumption is not a technicality. Metabolic control analysis was built for central metabolism, where substrates are millimolar and enzymes micromolar, and there its assumption is excellent. The regimes that regulatory and synthetic biology care about are chosen, deliberately, to sit where it fails.

That leaves a question the essay has to answer rather than gesture at. If the enzyme order is not 1, what can it be? A curve through one family of models does not settle that. The next three sections do, and the answer is a shape.

22Separate binding from catalysis, and the assumption becomes exactly true

The regime problem of §21 comes from applying one exponent to two different physical processes at once. Split them and the confusion disappears, because one of the two really does have order 1.

A binding-catalysis mechanism has two layers, and they are not the same kind of chemistry³⁹.

Read the second definition again with §20 in hand. A catalytic rate is, by construction,

v = ∑_j c_j x_j, c_j ≥ 0

so it is exactly first order in each active species. Metabolic control analysis’s assumption is not merely a good approximation at the catalysis layer. It is true there, identically, by the definition of what catalysis is.

This is the sense in which the framework resolves the dispute rather than joining it. Both schools compute log-log slopes of rates with respect to totals, and a slope of that kind is a composite of two things with different characters: a catalytic coefficient that is structurally 1, and a binding response that is not. Neither school separates them, so neither can say which part of a measured elasticity is chemistry and which part is bookkeeping about who is bound to whom.

23The reaction-order formula

Once binding is a layer of its own, its reaction orders have a closed form that needs no rate constants and no solved equilibrium. It is a matrix inverse, and it has the same shape as the identity metabolic control analysis discovered.

Set up the binding layer. Let x > 0 list the species, let the rows of L ≥ 0 count the conserved constituents so that q = L x are the physical totals, and let N be a Gale dual of L, a matrix whose rows span the orthogonal complement, so that binding equilibrium reads N log x = log k with k the affinities. Two objects do the work.

Three things about this deserve to be said in order.

It is the same shape as Cε = I. Metabolic control analysis discovered that its control matrix is the inverse of its elasticity matrix (§14), and both camps eventually agreed the theorems are that identity written componentwise. The formula above says the binding layer has the identical structure. The difference is what sits in the matrix being inverted. In metabolic control analysis it is elasticities, which must be measured. Here the top block is Φ, which is composed of concentration shares, and the bottom block is N, which is stoichiometry. Neither requires a rate constant.

It is Higgins’ observation, made computable. §7 recorded that Higgins in 1963 noticed the relations among reflection coefficients depend on the stoichiometry of the mechanism and not on the values of the rate constants¹³. The formula is that remark with the matrix written down.

It turns a question about kinetics into a question about a polytope. N is fixed by the network. So as the state ranges over all positive concentrations, the reaction-order matrix ranges over the image of a set of Φ matrices under inversion, and the question of what orders are achievable becomes the question of what shape the Φ live in. That shape is the subject of §24.

24The dominance polytope, and an exact bound

The set of dominance profiles is a convex polytope whose vertices are combinatorial: each one records which species holds the majority of each conserved total. Nothing weaker than this shape is the right answer, because this shape is attained.

Fix the binding network, so L is fixed. As the concentrations range over all positive values, where does Φ(x) go?

The extreme cases are easy to name. Make the concentrations wildly separated in scale, and each row of Φ concentrates all its mass on the single largest species in that row’s support. The result is a 0-1 matrix D_σ that records, for each conserved total, which species carries it. Call that a dominance regime. Every ranking σ of the species produces one.

Read the two inclusions in order, because they say different things. The lower one says every dominance regime is attained, in the limit of wide concentration separation. The upper one says nothing else is: every achievable dominance profile is a convex combination of regimes, so the closure of the image is sandwiched between the vertex set and its hull. Together they identify the polytope exactly, and the uniqueness clause is what licenses calling it the bound rather than a bound.

Now do the smallest case explicitly, because it is the case §21 was about. For E + S ⇆ C with species order (S, E, C),

L = [[1,0,1],[0,1,1]], N = [1, 1, −1]

Write e = E/K, s = S/K and Z = 1 + e + s. Inverting the chart matrix gives every reaction order in the network at once:

∂log(S,E,C) / ∂log(q_S,q_E,K) = (1/Z) [[(1+e)(1+s), −e(1+s), e], [−s(1+e), (1+e)(1+s), s], [1+e, 1+s, −1]]

The two rows this essay needs decompose over the dominance regimes:

∂log C / ∂log(q_S,q_E) = (e/Z)(1,0) + (s/Z)(0,1) + (1/Z)(1,1)

∂log E / ∂log(q_S,q_E) = ((1+e)/Z)(0,1) + (s/Z)(−1,1) + (es/Z)(−1,1)

In the first line the three weights sum to 1, so the complex’s achievable set is the closed triangle with vertices (1,0), (0,1), (1,1). In the second the first two weights sum to 1 and the third is a nonnegative ray coefficient, so the free species’ set is an unbounded strip: the segment from (0,1) to (−1,1), extended in the direction (−1,1).

That strip is the region drawn as Mechanism B in the companion tutorial’s figure 8, where it appears as the signature of repression by sequestration⁴⁰. It is the same object arrived at from the other direction: there as a design pattern to be recognised in a circuit, here as the achievable set of an exponent that two theories of metabolism disagreed about.

Two readings of that figure are worth stating, because they are what the geometry buys over a curve.

Metabolic control analysis’s assumption is a face, not a point. The value ∂log C/∂log q_E = 1 is not an isolated special case. It is the top edge of the triangle, a one-dimensional face of a two-dimensional set, and it is exactly the set of states in which the free enzyme carries a negligible share of the enzyme total. Everything metabolic control analysis says is true on that face.

The failure is a single dominance weight. From the decomposition, the deficit is

1 − ∂log C / ∂log q_E = e/Z

and e/Z is precisely the barycentric weight on the vertex (C,E), the regime in which the complex holds the substrate pool while free enzyme holds the enzyme pool. Verified as an identity to 3×10⁻¹⁶ over six decades of both totals. So the error in metabolic control analysis’s foundational assumption is not a vague approximation error. It is the weight the operating point puts on one named regime.

25What that says about both schools

Not that either was wrong. That the object they share has a computable achievable set, and that each of them fixed a different part of it.

Biochemical systems theory treats every exponent as a variable that moves with the operating point, and says so explicitly (§8). Its difficulty is the opposite one: having made the exponents free, it must hold them fixed to solve, and the solution is then used where they have moved (§13). What it never supplies is a constraint on where the exponents can go. A power-law model with arbitrary real exponents is more general than chemistry allows, and the extra generality is not free, because it is what makes the parameters hard to identify.

Metabolic control analysis fixes one exponent at 1 and lets the rest float. That buys parameters attached to manipulable objects and a pair of theorems with real content, at the cost of a domain restriction it did not always state. The restriction is not about branches, which §15 and §16 settle. It is about which face of the polytope the cell is standing on.

Read that way, the twenty-five-year dispute of Part III looks less like a disagreement about theories and more like two projects working on different factors of one product without a notation that separates them. Both were computing ∂log v/∂log q. That derivative factors, and the two factors have entirely different characters: one is structurally 1, the other has a geometry. Neither school had the factorisation, so neither could locate the assumption they were arguing about.

26The slope you can get, and the mechanism you cannot

The opening claim needs correcting in one place before it can be stated properly. The correction is what the sources forced, and it makes the claim sharper.

The pattern in Part I looks like this. van ’t Hoff measures a log-log slope and calls it the number of molecules taking part. Ostwald renames it “order” to say it is not that number. Hill fits an exponent to a binding curve, and it is widely read as a count of sites, which it is not. In both cases a measurable exponent stood in for an unobservable count, and in both cases the field eventually said so by introducing a word.

The tempting generalisation is that the derivative is where mechanism goes to die: that taking a log-log slope is an admission that you cannot get the mechanism, and that each rename records another such admission. That generalisation is wrong, and Higgins is the counterexample.

In 1963 Higgins uses reflection coefficients to discriminate mechanisms. His type I and type II sequences give relations of different functional form, so the coefficients tell them apart. He proves the converse as a theorem: when two mechanisms give relations of the same functional form over a subset of variables, stationary-state experiments on that subset cannot distinguish them¹³. And he observes that the relations among coefficients are differential equations whose solutions are the steady-state equations, so that integrating them recovers the rate law with the constants appearing as constants of integration.

Read that way, Ostwald’s 1887 coinage and Cornish-Bowden’s 1989 footnote are the same act a century apart. Ostwald introduced a word to stop an exponent being read as a molecule count. Cornish-Bowden wrote a footnote to stop a derivative being read as a constant. Both are attempts to prevent a local measurement being taken for a global property, and in both cases the attempt was only partly successful, since van ’t Hoff never adopted Ostwald’s convention and each school kept reading the other’s exponent under its own quantifier.

This connects to the companion essay’s argument, which is about the same boundary drawn in a different place⁴¹. There the question is what counts as a model’s structure and what counts as its parameters, and the answer is that structure is the support of the parameter vector, so the split is a threshold rather than a kind. Here the question is what counts as a mechanism and what counts as a measured exponent, and the answer has the same shape: the exponent is what the mechanism looks like at one point, and calling it a different kind of thing is what generates a century of renaming.

27What the record establishes

What is settled, what is still argument, and what in this document to distrust.

Settled by the primary texts. One derivative underlies chemical reaction order, the Hill coefficient, the reflection coefficient, the kinetic order and the elasticity, and the participants say so in their own papers rather than being convicted of it retrospectively. van ’t Hoff wrote the log-log slope as a formula in 1884 and used it as his preferred measurement method. Ostwald introduced “order” in 1887 to separate it from molecularity. Both twentieth-century schools inherited the log-log habit from control engineering, independently, through Chance and through Bode.

Settled by computation here. The summation and connectivity theorems hold exactly in a branched network, to ten decimal places. The classical statement of them is nonetheless insufficient there, by a rank deficiency of exactly three on a nine-coefficient problem, and Reder’s statement closes it exactly because a branched network has more summation relations than a chain, not fewer. Control coefficients in that network reach +1.64 and −1.57 while summing to 1. A power law fitted at an operating point is within 5% of a Michaelis–Menten law over a two-fold range and 74% off at ten-fold. The enzyme order departs from 1 by 1% at an enzyme-to-substrate ratio of 0.04 and by 28% at parity.

Settled by the geometry. Once binding and catalysis are separated, the assumption of §20 stops being a modelling choice with a fuzzy domain and becomes a face of a computable set. A catalytic rate is first order in its active species by definition. A binding reaction order is [Φ;N]⁻¹, needing no rate constant. And the achievable set is the image of the dominance polytope, whose vertices name which species carries which conserved total. For one binding reaction the complex’s orders fill a triangle and the free species’ fill a strip, and the deficit in metabolic control analysis’s assumption is exactly the barycentric weight on one named vertex, verified as an identity to 3×10⁻¹⁶.

Still argument. Whether the regimes where the enzyme order departs from 1 are common enough to matter for metabolism, as opposed to for regulation and synthetic biology, is not settled here and would need a census rather than examples. The polytope says what is achievable; it does not say where cells actually stand in it, and that is an empirical question this essay leaves open. Whether biochemical systems theory’s practice of holding exponents fixed while solving is a defect or a legitimate approximation depends on how far from the operating point the solution is used, which is a case-by-case question. And the dominance dispute of §18 is a genuine open question in evolutionary genetics rather than a resolved one.

The last thing worth saying is about the shape of the dispute rather than its content. Two groups spent twenty-five years arguing about whether one was a special case of the other, and the answer turned out to be that they were two uses of one derivative under different quantifiers, with one of them additionally fixing a coordinate. Almost every technical particular was resolvable, and most were resolved within two years of being raised, by third parties: Giersch and Reder, both in 1988, neither of them a partisan.

What was not resolvable by those means was the question underneath, which is which parameters a theory should be built out of. Metabolic control analysis chose the ones an experimenter can turn. Biochemical systems theory chose the ones a system can be solved in. Those are different goods, they conflict, and no theorem settles which to want.

They stop conflicting when the derivative is factored. ∂log v/∂log q is a product of a catalytic factor that is structurally 1 and a binding factor with a geometry, and the two schools were each holding one end of it. Metabolic control analysis’s parameters are the manipulable ones because they sit at the catalysis layer, where first order in the catalyst is a definition rather than an approximation. Biochemical systems theory’s exponents are free because they absorb the binding layer, which is genuinely variable. Neither had the notation to say which was which, so a disagreement about a factorisation was conducted as a disagreement about theories.

The thing that was missing in 1987 is a bound on the free factor. Not an interval, and not a fitted range, but the actual set: a polytope whose vertices are the regimes of a binding network, computed from its stoichiometry before any constant is measured. That set is what an exponent is allowed to be. Everything in this essay from 1864 onward is people measuring points inside it without knowing its shape.

References

Every entry is a full text in literature/, read rather than cited from an abstract. Grouped by what they carry. The seven page images in the text are crops from these files, produced by analysis/reaction_order_snippets.py, which records the page and the crop for each.