Unlocking Life’s Secrets: Four Core Mathematical Models of Biological Systems

Understanding the complexity of living systems requires more than observation; it demands deep theoretical guidance. By abstracting biological phenomena into mathematical models, we can uncover hidden regularities. This article, referencing Bertalanffy’s General System Theory, introduces four basic models in biology: open system steady‑state, automatic regulation dynamic balance, allometric growth, and the Bertalanffy growth model.

These models interrelate, providing a relatively complete framework for understanding metabolism, growth, and regulation.

1. Open System and Steady‑State Model

In nature, organisms continuously exchange matter and energy with their environment, essential for life. The core idea is that when input and output of matter and energy balance, key internal variables (temperature, pH, ion concentration) remain relatively constant—a steady state.

This can be described by a differential equation where the concentration of a substance changes according to input and output rates; equilibrium is reached when rates are equal, explaining cellular homeostasis and metabolic balance.

While effective for explaining dynamic balance, the model assumes system stability and linear behavior, which may not hold in nonlinear or extreme conditions.

2. Automatic Regulation Dynamic Balance Model

Organisms can restore balance after disturbances through feedback mechanisms. A classic example is body‑temperature regulation: shivering generates heat in cold, sweating dissipates heat in heat. This is modeled by a feedback equation where the current state, target state, and feedback gain determine adjustments.

The model assumes feedback always works, but in pathological states or extreme environments feedback may fail, limiting explanatory power.

3. Allometric Growth Model

Biological size and function are not linearly related. Allometric growth describes how certain traits (e.g., metabolic rate) change non‑linearly with body mass, revealing large energy‑consumption differences among organisms of different sizes.

The model is expressed as metabolic rate = k·mass^b, where k is a constant and b the allometric exponent; many mammals show metabolic rate proportional to body surface area.

The model ignores inter‑species complexity and environmental influences, requiring adjustments for cross‑species or extreme conditions.

4. Bertalanffy Growth Model

The Bertalanffy model describes organism growth over time, based on metabolic processes rather than simple curve fitting, providing a theoretical framework for growth phenomena.

Its basic form relates body weight to anabolic and catabolic constants and an exponent, allowing prediction of growth patterns under specific conditions.

The model excels in describing long‑term growth trends, especially in vertebrates like fish, but it overlooks environmental changes, nutrition, and genetics.

These four models offer important theoretical tools for understanding metabolism, growth, and regulation, yet they are simplifications—not complete descriptions—of reality. They serve as exploratory instruments, guiding prediction and further research while requiring continual refinement.