The growth and development of living organisms from the thermodynamic point of view

This paper applies Prigogine's thermodynamics of open systems to formulate a law for individual organism growth and development, demonstrating through experimental comparison that specific entropy decreases across the evolutionary sequence from yeast to birds.

The Gist

Imagine a living organism, like a human, a bird, or even a tiny yeast cell, not just as a bag of bones and cells, but as a busy, high-tech factory that never stops running. This paper by Alexei Zotin and Vladimir Pokrovskii tries to understand how this factory grows and changes by looking at the rules of energy and heat (thermodynamics).

Here is the story of the paper, broken down into simple concepts and everyday analogies.

1. The Factory vs. The Chemical Soup

Most chemical reactions (like rust forming on a car) just happen until they run out of fuel. But a living organism is different. It's an open system, meaning it constantly eats food (energy in) and releases heat and waste (energy out).

The authors compare a living thing to a famous theoretical model called the "Brusselator" (a chemical reaction that creates patterns). But there's a catch: The Brusselator is like a chaotic kitchen where ingredients mix randomly. A living organism is more like a factory with a strict manager.

• The Manager: This is your DNA. It's the "internal program" that tells the factory exactly how to build a heart, a wing, or a leaf.
• Homeostasis: The factory has a thermostat. If it gets too hot outside, the factory turns on the sprinklers (sweating) to keep the inside temperature perfect. It fights to stay organized, unlike a pile of leaves that just rots.

2. The Three Types of "Workers" (Internal Variables)

To explain how this factory works, the authors divide the "workers" inside the organism into three groups based on how long they stay busy:

• Group 1: The Architects (The DNA)
  These are the blueprints. They are packed into the nucleus of every cell. They don't change much; they just hold the instructions on how to build the organism. They are the "boss" that never leaves the building.
• Group 2: The Builders (The Structure)
  These are the actual organs, tissues, and shapes being built. Think of them as the walls, windows, and machinery being assembled. They take a long time to build and, once built, they stay there for the organism's whole life. They represent the complexity of the organism.
• Group 3: The Sweat (The Heat)
  These are the tiny, fast-moving chemical reactions happening right now. They are the friction of the gears, the heat from the engines, and the waste products. They appear and disappear very quickly. This group is responsible for entropy (disorder/heat).

3. The Golden Rule of Growth

The paper proposes a simple equation for growth. Imagine the organism is a bank account:

• Income: Energy from food.
• Expenses: Heat released into the air.

Usually, in a steady state (like an adult human sitting still), Income = Expenses. You eat a sandwich, and your body burns it all off as heat.

But what about growth?
If the organism is growing (like a baby or a seedling), the math says:

Growth = (Energy In) - (Heat Out)

If you eat 100 calories but only release 90 as heat, that missing 10 calories didn't disappear. It was used to build new structure (Group 2). It turned into a new muscle cell or a new leaf.

The authors call this missing energy the $\psi_u$-function. It's the "construction fund."

4. The Mystery of "Specific Entropy"

Here is the most fascinating part. In physics, Entropy usually means "disorder." A messy room has high entropy; a clean room has low entropy.

The authors calculated the "Specific Entropy" (disorder per gram of weight) for different animals:

1. Yeast (simple single cell)
2. Insects
3. Reptiles
4. Birds (complex, warm-blooded)

The Result: As animals evolved from simple to complex, their Specific Entropy went DOWN.

The Analogy:
Think of a pile of bricks (high entropy/disorder) vs. a beautiful cathedral (low entropy/order).

• A yeast cell is like a small pile of bricks.
• A bird is like a magnificent cathedral.

The paper suggests that as life evolves, it gets better at organizing itself. It becomes more efficient at turning energy into structure rather than just wasting it as heat. A bird is a "lower entropy" creature than a yeast because it is more highly ordered and complex.

5. The Big Conclusion

The paper argues that living things follow a universal law of growth.

1. They take in energy.
2. They release some as heat (Group 3).
3. They use the rest to build complex structures (Group 2) according to a genetic plan (Group 1).

The most important finding is that growth is essentially a process of reducing disorder. As an organism grows and evolves, it becomes a more efficient machine at organizing matter. The "specific entropy" (disorder per pound) stays surprisingly constant during an individual's life, but it gets lower and lower as we look at more advanced species in the history of evolution.

In a nutshell: Life is a battle against chaos. We eat energy to build order. The more complex we are (like a bird vs. a bug), the better we are at keeping that order, and the less "messy" (low entropy) we are per pound of our body.

Technical Summary

Here is a detailed technical summary of the paper "The growth and development of living organisms from the thermodynamic point of view" by Alexei A. Zotin and Vladimir N. Pokrovskii.

1. Problem Statement

The paper addresses the challenge of formulating a rigorous phenomenological (thermodynamic) theory for the growth, development, and aging of living organisms (ontogenesis). While previous theories analyzed energy exchange between organisms and their environment, they struggled to reconcile the observation that incoming energy fluxes often exceed heat production during growth. The authors aim to:

• Define the thermodynamic relationship between the "surplus" energy (the difference between incoming energy and heat output) and the rate of mass growth.
• Integrate the concept of an internal "control program" (genetics/homeostasis) into non-equilibrium thermodynamics.
• Derive a thermodynamic equation of growth that can be tested against experimental data to estimate the specific entropy of living systems across different evolutionary stages.

2. Methodology

The authors utilize Prigogine's thermodynamics of open systems, treating the living organism as an open system in a non-equilibrium state. The methodology involves:

• Entropy Balance Equation: Starting from Prigogine's formulation, they define the change in entropy ($S$) of an open system at temperature $T$ as:
  $$T dS = \Delta Q - \sum \Xi_j \Delta \xi_j + \sum \eta_\alpha \Delta N_\alpha$$
  Where $\Delta Q$ is heat flux, $\Delta N_\alpha$ is particle flux, $\eta_\alpha$ is chemical potential, and $\Xi_j \Delta \xi_j$ represents changes in stored energy due to internal variables ($\xi_j$).

• Classification of Internal Variables: To account for the biological control mechanisms (homeostasis and development), the authors categorize internal variables into three groups based on their relaxation times ($\tau$):

  1. Group 1 (Hereditary/Control): Variables representing DNA and genetic instructions. These are constant (or slowly changing) and control the system. They do not contribute to immediate entropy production.
  2. Group 2 (Metastable Structure): Variables representing the organism's structure (organs, tissues, protein conformations). Their relaxation times are comparable to the organism's lifespan. They define the "complexity" of the organism and do not contribute significantly to current entropy dissipation.
  3. Group 3 (Fast Relaxation): Variables representing gradients (temperature, concentration) and chemical reaction degrees of completion. These relax rapidly and are responsible for the dissipation of energy as heat (entropy production).

• Derivation of the Growth Equation: By applying the conservation of mass and the entropy balance to the specific entropy ($S/M$), the authors derive a linear relationship between the specific growth rate and the $\psi_u$-function. The $\psi_u$-function represents the net energy available for construction:
  $$\psi_u = \frac{1}{M} \left( \frac{\Delta Q}{\Delta t} - \sum \Xi_i \frac{d\xi_i}{dt} + \sum \eta_\alpha \frac{\Delta N_\alpha}{\Delta t} \right)$$
  This leads to the Thermodynamic Equation of Growth:
  $$\frac{1}{M} \frac{dM}{dt} = \sigma + \kappa \psi_u$$
  Where $\sigma$ is the rate of change of specific entropy and $\kappa$ is a proportionality constant.

3. Key Contributions

• Integration of Control Mechanisms: The paper successfully incorporates the biological concept of an "internal program" (homeostasis and development) into thermodynamics by distinguishing between fast-relaxing variables (dissipative) and slow-relaxing variables (structural/complexity).
• Definition of Complexity via Entropy: The authors propose that complexity is inversely related to specific entropy ($S/M$). As an organism grows and develops under a genetic program, its structural complexity increases, implying a decrease or stabilization of specific entropy.
• The $\psi_u$-Function as a Growth Driver: The paper formally identifies the difference between incoming energy flux and heat production (the $\psi_u$-function) as the thermodynamic driver of mass growth, quantifying the energy required for biosynthesis and structural organization.
• Universal Growth Law: The derived equation connects the thermodynamic approach with established empirical growth models (like the Bertalanffy equation), suggesting a universal law governing growth across diverse species.

4. Results

The authors tested their theoretical equation against experimental data from direct and indirect calorimetry for various species:

• Species Studied: Yeast (Saccharomyces cerevisiae), cricket larvae (Acheta domestica), lizard embryos (Lacerta agilis), and chicken embryos (Gallus domesticus).
• Correlation: A statistically significant positive correlation ($p < 0.01$) was found between the specific growth rate and the $\psi_u$-function, validating the linear relationship proposed in Equation (7).
• Specific Entropy Estimates: By calculating the coefficient $\kappa$, the authors estimated the specific entropy ($S/M$) for each species. The results showed a distinct trend in the evolutionary row:
  • Yeast: $365 \times 10^6$ J/g·K
  • Insects: $4,400$ J/g·K
  • Reptiles: $3,350$ J/g·K
  • Birds: $300$ J/g·K
• Constant Specific Entropy: For individual organisms during significant growth intervals, the specific entropy ($S/M$) was found to remain approximately constant (the parameter $\sigma$ was not significantly different from zero).

5. Significance

• Evolutionary Thermodynamics: The results demonstrate a reduction in specific entropy (and a corresponding increase in structural complexity) as one moves up the evolutionary ladder from simple unicellular organisms to complex vertebrates. This provides a thermodynamic metric for evolutionary progress.
• Validation of Non-Equilibrium Thermodynamics: The study confirms that Prigogine's framework, when adapted to include biological control variables, accurately describes the energetics of living systems.
• New Diagnostic Tool: The ability to estimate specific entropy from growth and energy exchange data offers a new thermodynamic characteristic for comparing the complexity of different biological systems.
• Resolution of the Energy Paradox: The paper resolves the apparent paradox where energy input exceeds heat output during growth by mathematically attributing the difference to the construction of low-entropy, high-complexity structures (Group 2 variables) rather than dissipation.

In conclusion, the paper establishes a robust thermodynamic foundation for ontogenesis, linking the physical laws of energy dissipation with the biological imperatives of growth and evolutionary complexity.