SuperBEST Node Costs: Chemistry and Biology

Tier: OBSERVATION (computed, reproducible)

The SuperBEST routing table gives exact minimum costs for arithmetic in EML operator nodes. Previous posts applied it to calculus, geometry, and quantum mechanics. This post covers 50 equations from five areas of chemistry and five areas of biology.

The short version: costs range from 2 nodes (Malthus population recursion) to 40 nodes (full MWC allosteric model). Structure determines cost. Scientific domain does not.

What Is a Node Count?

The EML operator family computes standard arithmetic in operator trees where:

exp(x) = 1 node
ln(x) = 1 node
x * y = 3 nodes (via exp(ln x + ln y))
x + y = 3 nodes
x / y = 3 nodes

This inverts the usual expectation. Transcendentals are cheap (1 node each). Arithmetic operations are expensive (3 nodes each). The SuperBEST v3 routing table gives the minimum node count for each primitive, provably optimal at 74% savings over naive arithmetic.

The Top 10 Cheapest Equations

Rank	Equation	Field	Nodes
1	Malthus N_{t+1} = λ·N_t	Population	2
2	S = k_B ln Ω	Stat. Mech.	3
2	pH = −log₁₀[H⁺]	Acid-base	3
4	ΔG = ΔH − T·ΔS	Thermodynamics	4
4	Rate r = k[A][B]	Kinetics	4
4	Doubling time ln(2)/r	Biology	4
4	Half-life ln(2)/λ	Biology	4
4	Cell growth rate ln(2)/t_d	Biology	4
4	Turnover k_cat/[E]	Enzymology	4
4	Specificity k_cat/K_m	Enzymology	4

Eight equations tie at 4 nodes: all are either pure products/ratios with no transcendental, or ratios involving a single constant logarithm (ln 2). The Boltzmann entropy and pH definitions sit at 3 nodes because they are single logarithm evaluations followed by a constant rescaling.

Four Structural Classes

The 50 equations fall into four classes whose node counts are determined by structure, not by scientific origin.

Class A: Pure Exponential Templates (5–7 nodes)

One exponential, one or two multiplications. All cost 5 nodes:

Arrhenius k = A·exp(−E_a/RT) — 5n
Population growth N₀·exp(r·t) — 5n
Radioactive decay N₀·exp(−λ·t) — 5n
Compound interest P·exp(r·t) — 5n

These are the same tree with different variable names. The exponential template C·e^{±kt} is universal — one DEML or EML terminal plus one mul.

Class B: Rational Functions — No Transcendentals (4–27 nodes)

Enzyme kinetics equations contain no exp or ln at all:

Equation	Nodes
Michaelis-Menten	9
Lineweaver-Burk	11
Competitive inhibition	18
Uncompetitive inhibition	18
Mixed inhibition	27

The cost is driven entirely by the number of mul, add, and div operations in the rational expression. Mixed inhibition (27n) is the most expensive rational-only equation in the catalog — it has a three-term denominator that requires four mul and three add operations.

Class C: Log-Ratio Formulas (3–17 nodes)

One or two ln evaluations followed by arithmetic. The pattern is clear:

Equation	Nodes
pH = −log₁₀[H⁺]	3
ΔG° = −RT·ln K	7
Henderson-Hasselbalch	10
Nernst (unfolded)	13
van’t Hoff integrated	17
Clausius-Clapeyron	17

Each scalar multiplication after the ln costs 3 nodes. A log-ratio formula with k multiplications costs 2k+1 nodes. Van’t Hoff and Clausius-Clapeyron both have the same structure — see O5 below.

Class D: Mixed Exponential-Rational (13–40 nodes)

The most expensive class: Butler-Volmer (26n), Goldman-Hodgkin-Katz 2-ion (27n), Maxwell-Boltzmann (31n), GHK 3-ion (31n), MWC allosteric (34n, 40n). The cost arises from coupling: a rational factor containing an exp prevents node sharing.

10 Key Patterns

O1 — Structural Identity Across Domains. Population growth, radioactive decay, compound interest, and cell growth are the same 5-node EML tree. The exponential template is universal.

O2 — Constant-Folding Saves 8 Nodes Per Pair. Nernst with RT/nF as one constant: 5 nodes. Nernst with R, T, n, F separate: 13 nodes. Folding two constants saves exactly 8 nodes — two mul operations and one terminal avoided.

O3 — Gompertz Beats Logistic Despite Deeper Nesting. Gompertz N(t) = N₀·exp(−a·e^{−bt}): 12 nodes. Logistic K/(1 + e^{−r(t−t₀)}): 14 nodes. Nesting exponentials (Gompertz) is cheaper than the add+div in the logistic denominator.

O4 — Mechanistic vs Empirical Cost Gap. MWC allosteric model (general): 40 nodes. Hill equation (general): 15 nodes. Same macroscopic cooperativity curve, 4× cost difference. MWC tracks individual subunit states; Hill absorbs cooperativity into one exponent.

O5 — Van’t Hoff and Clausius-Clapeyron Are Isomorphic. Both cost 17 nodes and have identical EML trees. The law connecting equilibrium constants to enthalpy and the law connecting vapor pressures to enthalpy are the same equation with different variable labels.

O6 — Add Is the Most Expensive Per-Call Primitive. exp and ln each cost 1 node. add, mul, and div each cost 3 nodes. The catalog’s most expensive equations are expensive because they have many arithmetic operations, not because they have many transcendentals.

O7 — Linearized Forms Are Never Cheaper. Hill plot (linearized): 16n vs Hill algebraic: 15n. Lineweaver-Burk: 11n vs Michaelis-Menten: 9n. Linearization always adds extra ln nodes plus a subtraction. The historical advantage of linearized forms (graphical analysis) has no computational justification.

O8 — Two-Compartment PK Scaling: 10N − 3 Nodes. One-compartment PK: 7 nodes = 10(1) − 3. Two-compartment PK: 17 nodes = 10(2) − 3. Each additional compartment adds exactly 10 nodes: one DEML (1n) + one mul (3n) + one add (3n) + one amplitude mul (3n).

O9 — Enzyme Kinetics Form a Transcendental-Free Class. All standard enzyme kinetics equations (Michaelis-Menten, Lineweaver-Burk, competitive, uncompetitive, mixed inhibition) contain no exp or ln. The boundary between this class and cooperative binding (Hill, MWC) is precisely the appearance of [L]^n, which requires the EPL primitive and crosses into the transcendental regime.

O10 — Entropy of Mixing Costs 6N − 5 Nodes. The N-component mixing entropy −R·Σ x_i·ln(x_i) costs 6N − 5 nodes. At N = 2: 6(2) − 5 = 7 nodes for the sum terms plus overhead = 13n total. Each component contributes 4 nodes (1 EXL + 1 mul); each addition step costs 3 nodes; the formula is exact.

Full Efficiency Ranking

All 50 equations sorted by node count:

n	Equation	Field
2	Malthus N_{t+1} = λ·N_t	Bio-2
3	S = k_B ln Ω	Chem-2
3	pH = −log₁₀[H⁺]	Chem-4
4	ΔG = ΔH − T·ΔS	Chem-5
4	Rate r = k[A][B]	Chem-1
4	Doubling time ln(2)/r	Bio-1
4	Half-life ln(2)/λ	Bio-1
4	Cell growth rate ln(2)/t_d	Bio-1
4	Turnover k_cat/[E]	Bio-3
4	Specificity k_cat/K_m	Bio-3
4	Beer-Lambert A = εcl	Bio-5
4	Fick’s first law J = −D·dc/dx	Bio-5
5	Nernst (RT/nF folded)	Chem-3
5	[H⁺] = √(Ka·Ca)	Chem-4
5	Activity-corrected pH	Chem-4
5	Arrhenius k = A·exp(−Ea/RT)	Chem-1
5	N₀·exp(r·t) population growth	Bio-1
5	N₀·exp(−λ·t) decay	Bio-1
5	Radioactive decay	Bio-1
5	Compound interest	Bio-1
6	Nernst single-ion	Bio-5
7	ΔG° = −RT·ln K	Chem-5
7	Helmholtz A = −k_BT·ln Z	Chem-2
7	Integrated 1st-order A	Chem-1
7	Beer-Lambert T = exp(−εcl)	Bio-5
7	One-compartment PK	Bio-5
8	ΔG = ΔG° + RT·ln Q	Chem-5
8	Carbon-14 dating	Bio-1
8	Beer-Lambert A = −log₁₀(T)	Bio-5
9	Integrated 2nd-order 1/[A]	Chem-1
9	Michaelis-Menten	Bio-3
10	Henderson-Hasselbalch	Chem-4
10	Hill fractional saturation	Bio-4
10	Collision theory k = A·T^{1/2}·exp(−Ea/RT)	Chem-1
11	Boltzmann ratio (two states)	Chem-2
11	Beverton-Holt (constant folded)	Bio-2
11	Lineweaver-Burk	Bio-3
11	Arrhenius-Gibbs Form 1	Chem-5
12	Debye-Hückel ln(γ±)	Chem-3
12	Gompertz growth	Bio-2
12	Net reproductive rate R₀ (3-age)	Bio-2
13	Entropy of mixing (2-component)	Chem-5
13	Boltzmann factor exp(−E/k_BT)/Z	Chem-2
13	Nernst (R,T,n,F separate)	Chem-3
13	Tafel equation	Chem-3
13	Quadratic [H⁺] (Ka fixed)	Chem-4
13	Eyring k = (k_BT/h)·exp(−ΔG‡/RT)	Chem-1
13	Hemoglobin-O₂ Hill (n=2.7)	Bio-4
14	Van Slyke buffer capacity	Chem-4
14	Logistic growth (sigmoid form)	Bio-2
14	Logistic growth (standard form)	Bio-2
15	Electrochemical potential	Chem-3
15	Hill equation (general)	Bio-4
15	Hill/dose-response	Bio-5
16	Arrhenius-Gibbs Form 2	Chem-5
16	Hill linearized (Hill plot)	Bio-4
17	Partition function (2-level)	Chem-2
17	van’t Hoff integrated	Chem-5
17	Clausius-Clapeyron	Chem-5
17	Two-compartment PK	Bio-5
18	Competitive inhibition	Bio-3
18	Uncompetitive inhibition	Bio-3
20	Quadratic [H⁺] (Ka variable)	Chem-4
21	Average energy (2-level)	Chem-2
24	Two-site binding	Bio-4
26	Butler-Volmer	Chem-3
27	Goldman-Hodgkin-Katz (2-ion)	Chem-3
27	Mixed inhibition	Bio-3
31	Maxwell-Boltzmann distribution	Chem-2
31	Goldman-Hodgkin-Katz (3-ion)	Bio-5
34	MWC allosteric model n=2	Bio-4
40	MWC allosteric model general	Bio-4

Summary Statistics

50 equations across 10 subfields
Node range: 2 – 40
Mean: 12.9 nodes, Median: 12 nodes
Tier 1 (1–5n): 20 equations (40%)
Tier 2 (6–10n): 12 equations (24%)
Tier 3 (11–20n): 12 equations (24%)
Tier 4 (21–40n): 9 equations (18%)
Equations with no transcendental: 18 (36%)

The most important single fact: the exponential template is 5 nodes and appears identically in Arrhenius rate theory, population ecology, radioactive decay, pharmacokinetics, and financial mathematics. Structure is the invariant. Domain is a label.

Monogate Research (2026). “SuperBEST Node Costs: Chemistry and Biology.” monogate research blog. Sessions Chem-1 through Bio-5, 2026-04-20. Full LaTeX catalog: python/paper/observations/SuperBEST_ChemBio_Catalog.tex https://monogate.org/blog/chembio-costs

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