Cell metabolism

Economic Principles in Cell Biology · Ch. 3 · 10.5281/zenodo.8156823

How fast: enzyme kinetics

An enzyme's rate depends on how much substrate is around. At low substrate the enzyme is starved and the rate climbs almost linearly; at high substrate every enzyme is busy and the rate saturates at a maximum. The Michaelis-Menten law captures both regimes with two constants: V_max, the saturated rate, and K_m, the substrate level at which the enzyme runs half-speed.

v = V_max · S / (K_m + S)

; Michaelis-Menten: rate as a function of substrate
(define Vmax 10.0)   ; saturated rate
(define Km    2.0)   ; half-saturation substrate level

(define (rate S) (/ (* Vmax S) (+ Km S)))

(for-each
  (lambda (S)
    (display "S = ") (display S)
    (display "\tv = ") (display (rate S)) (newline))
  (list 0.5 2 8 50))

; at S = Km the enzyme runs at exactly half Vmax
(display "half-max at S=Km? ") (display (rate Km))

At S = K_m the rate is exactly V_max/2, confirming what the constant means. Doubling substrate from there buys less and less. It's the first hint of diminishing returns, a theme the economics chapters make precise.

Which way: reaction thermodynamics

A reaction's direction isn't fixed by its chemistry alone. It depends on how far the current mixture sits from equilibrium. The driving force is the Gibbs free energy change, ΔG = ΔG° + RT ln Q, where Q is the reaction quotient, the ratio of product to substrate concentrations for this reaction. Net flux runs in whichever direction makes ΔG negative. Pile up product and you can stall a reaction, or push it backward, no matter how good the enzyme.

; Reaction direction from thermodynamics: dG = dG0 + RT ln(Q)
(define R 8.314)      ; J / (mol K)
(define T 298.15)     ; K
(define dG0 -7500.0)  ; standard free energy, J/mol (mildly favorable)

(define (dG Q) (+ dG0 (* R T (log Q))))   ; Q = [product]/[substrate]

(for-each
  (lambda (Q)
    (let ((g (dG Q)))
      (display "Q = ") (display Q)
      (display "\tdG = ") (display (/ g 1000)) (display " kJ/mol -> ")
      (display (if (< g 0) "forward" "reverse")) (newline)))
  (list 0.01 1 100))

; A favorable reaction still reverses once product accumulates enough.

The enzyme sets the speed; thermodynamics sets the sign. A cell that lets product pile up is like a firm whose warehouse is full. The line backs up regardless of how fast the workers are. Both facts return in Chapter 6, where the cost of a flux turns out to depend on how close to equilibrium the reaction is forced to run.

Adaptation notes

The source frames metabolism as a dynamical system and develops rate laws and reaction thermodynamics in depth, including regulation and the dynamics of metabolite pools. We take saturating kinetics and free-energy-driven direction as the two load-bearing facts, the minimum needed to make later flux and cost models honest. Reversible rate laws, allosteric regulation, and the full dynamics are in the source.