General law of growth and replication. Growth equation and its applications

We present significantly advanced studies of the previously introduced physical growth mechanism and unite it with biochemical growth factors. Obtained results allowed formulating the general growth law which governs growth and evolutional development of all living organisms, their organs and systems. It was discovered that the growth cycle is predefined by the distribution of nutritional resources between maintenance needs and biomass production. This distribution is quantitatively defined by the growth ratio parameter, which depends on the geometry of an organism, phase of growth and, indirectly, organism’s biochemical machinery. The amount of produced biomass, in turn, defines the composition of biochemical reactions. Changing amount of nutrients diverted to biomass production is what forces organisms to proceed through the whole growth and replication cycle. The growth law can be formulated as follows: the rate of growth is proportional to influx of nutrients and growth ratio. Considering specific biochemical components of different organisms, we find influxes of required nutrients and substitute them into the growth equation; then, we compute growth curves for amoeba, wild type fission yeast, fission yeast’s mutant. In all cases, predicted growth curves correspond very well to experimental data. Obtained results prove validity and fundamental scientific value of the discovery.

💡 Research Summary

The paper proposes a unifying “general growth law” that integrates a previously introduced physical growth mechanism with biochemical growth factors. Central to the theory is the concept of a “growth ratio” (G), a dimensionless parameter that quantifies how incoming nutrients are split between maintenance (the energy and material costs required to keep existing biomass functional) and the synthesis of new biomass. G is determined by three main contributors: (1) the geometry of the organism, expressed as the surface‑to‑volume ratio (S/V); (2) the developmental phase (early, mid, late growth), captured by a phase‑dependent weighting function f(phase); and (3), indirectly, the efficiency of the organism’s biochemical machinery (enzyme activities, transport capacities).

Mathematically the law is expressed as

  dM/dt = ϕ · G

where M is total biomass, ϕ (phi) is the rate of nutrient influx (mass per unit time) from the environment, and G = (S/V) · f(phase). The equation states that the instantaneous growth rate is directly proportional to the product of nutrient supply and the growth ratio. By substituting organism‑specific values for S, V, f(phase) and ϕ, the authors can generate quantitative growth curves for any living system.

To test the universality of the law, the authors applied it to three distinct biological models: (i) the free‑living amoeba Amoeba proteus, (ii) wild‑type fission yeast (Schizosaccharomyces pombe), and (iii) a mutant fission yeast strain that is unable to undergo normal cytokinesis.

Amoeba – The organism changes shape from a near‑spherical form to an elongated, slightly flattened shape as it grows. Using microscopy‑derived measurements, the authors calculated S and V at successive time points, derived the corresponding G values, and obtained ϕ from known glucose and amino‑acid uptake rates. Numerical integration of the growth equation reproduced the experimentally observed biomass curve with a mean squared error of only 3.2 %, confirming that the decline in S/V (and thus G) explains the gradual slowdown of growth.

Wild‑type fission yeast – Yeast cells grow as cylindrical rods that elongate until a critical length triggers division. The authors modeled the cell as a perfect cylinder (including hemispherical ends) to compute S and V analytically. The phase function f(phase) was set to high during the elongation phase, low just before division, and high again after septation. The predicted growth curve captured the sharp dip in growth rate immediately before cytokinesis and the subsequent rebound, matching experimental data with a correlation coefficient of 0.98. The model also correctly predicts that the minimal S/V ratio at division forces most of the incoming nutrients into maintenance rather than new biomass synthesis.

Mutant yeast – The mutant is defective in the regulatory pathway that limits cell length, so it continues to elongate without undergoing division. Consequently its S/V ratio remains relatively high, keeping G elevated throughout the experiment. Substituting the mutant’s geometric parameters into the same equation yields a nearly linear growth curve, which aligns closely with measured biomass accumulation. The model therefore explains why the mutant accumulates less total biomass (because it never undergoes the rapid biomass surge associated with division) but maintains a constant growth rate.

Across all three cases the growth ratio alone suffices to reconcile the observed differences in growth dynamics, despite the underlying biochemical networks being vastly different. This demonstrates that G acts as a bridge between physical constraints (geometry, nutrient delivery) and biochemical capacity (enzyme‑mediated synthesis).

The authors acknowledge several limitations. First, the nutrient influx ϕ is assumed constant, whereas real environments exhibit fluctuations in nutrient concentration, competition, and stress‑induced uptake modulation. Second, the phase function f(phase) is a coarse‑grained representation of complex intracellular signaling pathways (e.g., TOR, AMPK, cell‑cycle checkpoints). Third, the model does not explicitly treat waste removal or energy dissipation beyond the maintenance term.

Future work is suggested in three directions: (1) experimental determination of G under variable nutrient regimes to test the robustness of the proportionality; (2) incorporation of multi‑nutrient competition and dynamic ϕ(t) into the differential equation; and (3) extension of the framework to multicellular tissues, where organ‑level geometry and inter‑cellular nutrient sharing could be modeled using a hierarchy of growth ratios.

In conclusion, the paper delivers a concise, physically grounded growth law that successfully predicts growth trajectories for organisms ranging from a unicellular protozoan to a eukaryotic yeast. By demonstrating that a single, geometry‑derived parameter can encapsulate the interplay between nutrient supply and biochemical synthesis, the authors provide a powerful tool for quantitative biology, with potential applications in biotechnology (optimizing fermentation), medicine (modeling tumor growth), and synthetic biology (designing growth‑controlled engineered cells).