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Local optimization of oxygen transport gives rise to Kleiber's law

This paper proposes that Kleiber's law and other metabolic scaling phenomena arise from a general physical theory based on the Metabolic Holon, a locally optimized oxygen-supply unit that predicts absolute metabolic rates and lifespan across diverse organisms without empirical calibration.

Original authors: Pelz, P. F., Meck, T. C.

Published 2026-09-09

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CC BY 4.0

Original authors: Pelz, P. F., Meck, T. C.

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

For nearly a century, biologists have watched a quiet pattern emerge across the animal kingdom: the larger an animal is, the slower its metabolism grows relative to its size. A mouse burns energy at a frantic pace compared to an elephant, but the relationship is not a simple straight line. In tiny creatures, metabolism rises directly with weight, but as animals grow larger, the rate of energy use begins to lag, following a specific curve that has puzzled scientists since the 1930s. This pattern, known as Kleiber's law, suggests that nature has a hidden rulebook for how living things are built, yet the physical reason behind it has remained a mystery. Is it a matter of how blood vessels branch? Is it about heat loss? Or is there a more fundamental physical limit at play?

A team of researchers at Technische Universität Darmstadt has now proposed a solution that bypasses the complex branching of the entire circulatory system to look at a much smaller, repeating unit. They suggest that the secret lies in the "Metabolic Holon," a tiny, self-contained module where blood delivers oxygen to a small patch of tissue. By treating the body not as a single, sprawling network, but as a collection of these identical, optimized building blocks, the researchers found that the famous three-quarter power law emerges naturally from the physics of a single unit. They did not need to fit their model to existing data; instead, they started with the basic laws of fluid flow and diffusion, optimized the shape of this tiny unit to be as efficient as possible, and let the math predict the metabolic rates of animals ranging from microscopic bacteria to massive whales.

The core of their discovery is a shift in perspective. Previous theories often tried to explain the whole animal by analyzing the entire tree of blood vessels from the heart down to the smallest capillaries. These researchers argued that the most important physics happens locally, within a single functional unit where a capillary meets the tissue it feeds. They imagined this unit as a cylinder of tissue surrounding a blood vessel. Inside this cylinder, oxygen must travel from the blood, diffuse through the tissue, and be consumed by cells. The researchers asked a simple question: if nature were to design this cylinder to be perfectly efficient, what would it look like?

To answer this, they applied three competing goals that any biological system must balance. First, the system must use as little blood vessel material as possible to build the pipe. Second, it must minimize the energy lost to friction as the blood flows through that pipe. Third, it must extract as much oxygen as possible from the blood before it leaves the unit. When these three objectives are balanced mathematically, the optimal shape of the cylinder and the speed of the blood flow fall into a specific relationship. This relationship dictates that the amount of oxygen a single unit can deliver scales with the size of that unit raised to the power of three-quarters.

This result is significant because it means the famous three-quarter exponent does not require a complex, organism-wide network to explain. It arises from the physics of a single, optimized piece of tissue. The researchers then showed that a whole animal is simply a repetition of these identical units. If a small animal has one unit and a large animal has a billion, the total metabolic rate is just the sum of all those units. Because the number of units in a specific group of animals—like mammals or reptiles—remains constant regardless of the animal's size, the scaling law holds true. The size of the animal changes the size of each unit, not the number of units.

The theory also explains why different types of animals have different metabolic levels. While the shape of the scaling curve is the same for all animals, the vertical position of that curve depends on how many of these metabolic units an animal has packed into its body. The researchers calculated that mammals, which are warm-blooded and highly active, have roughly 500 million of these units, while cold-blooded animals like reptiles have only about 100,000. This massive difference in the number of repeating units accounts for why a mammal burns energy at a much higher rate than a reptile of the same size. The theory predicts these numbers using independent measurements of lung surface area and capillary density, without ever looking at the metabolic rates themselves to adjust the answer.

Beyond just energy use, the model successfully predicts other biological traits that have long been linked to body size. It forecasts that heart rate should decrease as animals get larger, following a specific power law, and that lifespan should increase. The researchers tested their predictions against real-world data spanning eighteen orders of magnitude in size, from tiny single-celled organisms to the largest whales. The model matched the observed metabolic rates with remarkable accuracy, capturing the transition from small animals that rely on simple diffusion to large animals that require a pumping circulatory system. It also correctly predicted the heart rates and lifespans of mammals, using only one empirical fact about the total number of heartbeats in a lifetime to set the scale for how long an animal lives.

The work challenges the idea that the complexity of the entire vascular network is the primary driver of metabolic scaling. Instead, it suggests that the rules are set locally, at the interface where blood meets tissue. By optimizing this tiny interface, nature creates a building block that, when repeated, automatically produces the scaling laws seen across the animal kingdom. The researchers did not invent new physics; they simply applied the known laws of fluid dynamics and diffusion to a specific, repeating structure and showed that the result matches the reality of life on Earth. This approach offers a unified explanation for how size, energy, and time are linked in living things, suggesting that the grand patterns of biology are rooted in the simple, local efficiency of a single oxygen-supply unit.

Technical Summary: Local Optimization of Oxygen Transport Gives Rise to Kleiber's Law

Problem Statement
For nearly a century, the physical origin of metabolic scaling has remained unresolved. While metabolic rate is proportional to body mass ( $m^1$ ) in small, diffusion-dominated organisms, it follows Kleiber's three-quarter-power law ( $m^{3/4}$ ) in larger animals. Existing theories, such as the West-Brown-Enquist (WBE) model, primarily address allometric scaling through organism-level network architecture. However, these models often fail to derive the linear regime, the transition to the Kleiber asymptote, and group-specific metabolic levels from a single mesoscale transport-reaction mechanism. Furthermore, they typically treat terminal capillary characteristics as body-mass invariant without deriving this invariance from local physical constraints. The central challenge is to provide a general physical theory that predicts absolute metabolic rates across 18 orders of magnitude, explains the transition between regimes, and accounts for group-specific differences (e.g., between ectotherms and endotherms) without fitting parameters to metabolic-rate data.

Methodology
The authors propose the Metabolic Holon (MH) framework, a locally optimized capillary–tissue oxygen-supply unit that couples convection, diffusion, and cellular oxygen consumption. The methodology proceeds in three distinct steps:

Physical Similarity and Topology: Using dimensional analysis and Buckingham's $\Pi$ theorem, the authors argue that within a metabolic similarity group, the effective number $N$ of repeated MHs must be body-mass invariant ( $N \propto m^0$ ). This is derived from the requirement that physical laws remain invariant under transformations of the base-unit system. The group-specific magnitude of $N$ is inferred independently from geometric and physiological observations (e.g., respiratory surface area, capillary density) rather than fitted to metabolic data.
Local Optimization at the Mesoscale: For a prescribed MH volume $v = V/N$ , the geometry and operating state are determined by minimizing vascular material volume and viscous dissipation (Murray's law) while maximizing oxygen utilization. The optimization is closed by a physiological constraint: the wall shear rate is matched to the characteristic shear rate of the von Willebrand factor (vWF)–platelet response ( $\dot{\gamma}'_{vWF} \sim 10^3 \text{ s}^{-1}$ ), which fixes the timescale parameter $\tau$ .
Cross-Scale Mapping: The optimized solution for a single MH is mapped to the organismal macroscale through kinematic repetition ( $V = Nv$ , $Q_0 = Nq$ ). The theory incorporates independent inputs for tissue density ( $\rho$ ), energy density ( $e$ ), and oxygen capacity ( $\zeta$ ). No multiplicative coefficients are fitted to the target observables (metabolic rate, heart rate).

Key Contributions

Derivation of Kleiber's Law from Local Physics: The paper demonstrates that the characteristic $3/4$ exponent emerges from the physics of a single optimized MH in the convective-diffusive regime ( $q \propto v^{3/4}$ ), prior to any organism-level repetition. This challenges the view that the exponent is solely a consequence of fractal network architecture.
Absolute Prediction: Unlike scaling relations that describe relative mass dependence, the MH theory predicts absolute metabolic rates in physical units across 18 orders of magnitude (from bacteria to whales) without empirical normalization.
Explanation of Group-Specific Levels: The theory attributes the discrete shifts in metabolic levels between groups (e.g., planarians, ectotherms, endotherms) to the group-specific, mass-invariant number of repeated MHs ( $N$ ). The ratio $N_{endo}/N_{ecto} \sim 10^4$ predicts a $\sim 10$ -fold offset in metabolic rates, consistent with observations.
Unified Framework for Scaling Laws: The framework simultaneously derives the linear regime ( $m^1$ ), the transition near the diffusion limit ( $m_{diff}$ ), the Kleiber asymptote ( $m^{3/4}$ ), and the associated scaling for heart rate ( $m^{-1/4}$ ) and lifespan ( $m^{1/4}$ ).

Results

Metabolic Rate: The general MH model reproduces the linear regime for small organisms, the transition to convection, and the $3/4$ -power asymptote for large organisms. When compared with data spanning 18 orders of magnitude, the model achieves a coefficient of determination $R^2 = 97.7\%$ without fitting any parameters to the metabolic-rate observations.
Heart Rate and Lifespan: The theory predicts basal heart rate scaling ( $f_0 \propto m^{-1/4}$ ) with $R^2 = 76.8\%$ . Lifespan scaling ( $T \propto m^{1/4}$ ) is derived using one empirical constraint (the lifetime-heartbeat number), yielding $R^2 = 37.4\%$ , with the lower value attributed to extrinsic mortality factors outside the metabolic model.
Geometry and Topology: The model predicts that the number of functional oxygen-supply domains ( $N$ ) is invariant with body mass within a group. Independent checks using respiratory surface area, capillary density, and whole-organism estimates support the inferred values of $N_{endo} \approx 5 \times 10^8$ and $N_{ecto} \approx 10^5$ .
Transition Mechanism: The transition from diffusion-limited to convection-limited transport occurs when the organism-scale Damköhler number approaches unity. The theory shows that in the Kleiber asymptote, diffusion and cellular reaction become asymptotically matched within the MH ( $\text{Da}_\delta \to 1$ ), operating like a "just-in-time" system.

Significance and Claims
The authors claim that Kleiber's law is not an independent empirical rule but one asymptotic consequence of a general physical theory of organismal aerobic metabolism. The significance of the MH framework lies in its ability to:

Unify Scaling Regimes: It connects the microscale (cellular consumption), mesoscale (optimized MH), and macroscale (organismal repetition) in a single, closed system.
Distinguish Optimization from Innovation: The framework separates the optimization of geometry within a fixed topology (growth) from evolutionary innovations that change the topology (e.g., the evolution of oxygen-binding pigments or different $N$ values).
Provide Falsifiable Predictions: The theory makes specific, independent predictions regarding the mass invariance of $N$ , the scaling of capillary diameters, and the absolute magnitude of metabolic rates, which can be tested without fitting scaling coefficients.
Resolve the "Absolute Level" Problem: By determining the absolute metabolic level through physical inputs (density, oxygen capacity, shear scale) and topology ( $N$ ), the theory avoids the need for empirical normalization constants that characterize previous scaling laws.

The paper concludes that the vascular network is essential for connecting repeated MHs but need not determine the leading-order metabolic law; rather, that law is established within a single optimized MH and propagated by repetition. This provides a physical signature of optimal oxygen-supply organization that explains the universality and diversity of metabolic scaling.

Technical Summary: Local Optimization of Oxygen Transport Gives Rise to Kleiber's Law

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