Principles of cell growth

Economic Principles in Cell Biology · Ch. 7 (Golan, Hindley, de Jong, Köbis, Pascual Garcia, Weiße) · 10.5281/zenodo.8156979

Ribosomes make ribosomes

A cell grows by making protein, and protein is made by ribosomes, which are themselves protein. Devote a fraction φ_R of the proteome to ribosomes and the rest to the metabolic enzymes that feed them precursors. Too few ribosomes and synthesis is slow; too many and they starve for precursor because ribosomes crowd out the enzymes that supply it. Growth is the throughput of these two coupled sectors, and it peaks somewhere in between.

The environment sets where. Rich medium makes precursors cheap, so the metabolic sector can be small and the cell should pack in ribosomes; poor medium demands more enzymes and fewer ribosomes. Search the split and watch the optimum move.

; Balanced growth: proteome fraction phi-R to ribosomes, the rest to metabolism.
; Ribosomes demand precursor; metabolism supplies it. Growth couples the two.
(define gamma 3.0)   ; ribosome efficiency (precursor -> protein)

(define (growth phi-R nutrient)
  (let ((supply (* nutrient (- 1 phi-R)))   ; metabolic sector makes precursor
        (demand (* gamma phi-R)))           ; ribosomal sector consumes it
    (/ (* supply demand) (+ supply demand))))

(define (best-phi nutrient)
  (define (scan i bi bm)
    (if (> i 99)
        (list (/ bi 100.0) bm)
        (let ((g (growth (/ i 100.0) nutrient)))
          (if (> g bm) (scan (+ i 1) i g) (scan (+ i 1) bi bm)))))
  (scan 1 1 0))

(for-each
  (lambda (n)
    (let ((r (best-phi n)))
      (display "nutrient=") (display n)
      (display "  optimal ribosome fraction=") (display (car r))
      (display "  growth=") (display (cadr r)) (newline)))
  (list 1.0 4.0))

Richer nutrient pushes the optimal ribosome fraction up, and the growth rate with it. This is the content of the bacterial growth laws: measured across conditions, ribosome content rises almost linearly with growth rate. The linear relation is what optimal allocation looks like when you plot it. The cell isn't consulting a rule that says "make more ribosomes when you grow fast."

import numpy as np
gamma = 3.0
def growth(phiR, nutrient):
    s, d = nutrient*(1-phiR), gamma*phiR
    return s*d/(s+d)

grid = np.linspace(0.01, 0.99, 99)
for n in (1.0, 4.0):
    phi = grid[np.argmax([growth(p, n) for p in grid])]
    print(f"nutrient={n}: phi_R*={phi:.2f}  growth={growth(phi, n):.3f}")

Self-replication as autocatalysis

The cell is an autocatalytic system: its product is itself. That's the deep reason growth is exponential, and why an optimum exists at all. Double the machinery and you double the rate of making machinery. The source develops this from first principles: it builds up the minimal self-fabricating system and the assumptions any growth model must make, then shows the growth laws fall out as consequences rather than assumptions built in. The allocation instinct from Chapter 1 turns out to be forced by the arithmetic of self-reproduction.

Adaptation notes

The source chapter states the foundational assumptions of microbial growth and derives the growth laws, then builds toward mechanistic coarse-grained models linking gene expression to growth. We use the smallest coupled-sector model that reproduces the qualitative law (ribosome fraction rising with nutrient quality) as a runnable follow-through on Chapter 1. The formal derivations and the link to the experimental growth laws are in the source.