Prof. Dorilson Cambui
PTEN
Metabolism · Growth · Fibonacci

Metabolic Scaling, Fibonacci Dynamics and Ontogenetic Growth

Original mathematical models relating metabolic rate, body mass, development and Fibonacci dynamics.

This research line investigates the relationships between metabolic rate, body mass, and the metabolic scaling exponent, including models of my own development based on Fibonacci dynamics and ontogenetic growth. The Fibonacci sequence exhibits recursive growth, in which each stage is constructed from the preceding ones, allowing development to be represented progressively through stages. From this structure, the model can relate developmental stage to body mass and thereby calculate a stage-dependent metabolic exponent. These models were developed to explore new ways of representing the relationship between body mass, development, and metabolism, using a developmental variable associated with the golden ratio.

Interactive tool

Metabolic Calculator

Compare the classical Kleiber model with the simplified and refined Fibonacci formulations and contrast the results with a Brazilian empirical reference.

Open calculator ↓
Conceptual note

A classical relationship describes metabolic rate \(B\) as a function of body mass \(M\):

\[B=B_0M^b\]

where \(B_0\) is a normalization constant and \(b\) is the metabolic scaling exponent.

Central question: must the metabolic exponent remain constant throughout an organism's growth?

The golden ratio is

\[\varphi=\frac{1+\sqrt{5}}{2}\]

Body mass can be represented approximately as

\[M(n)\sim M_0\varphi^n\]

and therefore

\[n=\log_\varphi\left(\frac{M}{M_0}\right)\]

In this interpretation, \(n\) acts as a variable associated with developmental stage. The metabolic exponent is defined as a function of \(n\)

\[b=b(n)\]

A refined form for the stage-dependent metabolic exponent is

\[b(n)=\frac{(n-1)\ln\varphi-\ln\sqrt{5}}{n\ln\varphi-\ln\sqrt{5}}\]

For sufficiently large values of \(n\), this expression can be approximated by

\[b(n)\approx\frac{n-1}{n}\]
Interactive tool

Metabolic Calculator — Fibonacci Model

Compare the classical Kleiber model with the simplified and refined forms of the stage-dependent metabolic exponent.

Theoretical calculation
\[ n=\log_\varphi\left(\frac{M}{M_0}\right),\qquad B_{\rm K}=B_{0,\rm K}M^{0.75},\qquad B_{\rm Fib}=B_{0,\rm Fib}M^{b(n)} \]
Ontogenetic stage: —

Kleiber

\(b=0.75\)

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Simplified Fibonacci

\(b(n)=\frac{n-1}{n}\)

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Refined Fibonacci

\(b(n)=\frac{(n-1)\ln\varphi-\ln\sqrt5}{n\ln\varphi-\ln\sqrt5}\)

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Comparison with a Brazilian empirical reference

Compare the three estimates with mean BMR measured by indirect calorimetry in adults from Niterói, Brazil, grouped by sex and age range.

Empirical reference: —
ModelEstimated BDifference from reference
Kleiber——
Simplified Fibonacci——
Refined Fibonacci——

Normalization implied by the empirical reference

The values below show which \(B_{0,\mathrm{Fib}}\) would be required, at the entered body mass, for each form of the model to match the selected empirical mean exactly. This is a calibration diagnostic, not an independent validation.

Simplified
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Refined
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Source: probability sample of adults from Niterói, Rio de Janeiro, Brazil, with BMR measured by indirect calorimetry. This comparison uses group means by sex and age range rather than an individual clinical value.

Note: this is a theoretical tool for exploring the model equations. The results are not intended as a clinical, nutritional, or individual prescription. The Kleiber coefficient is kept separate from the adjustable \(B_{0,\mathrm{Fib}}\) coefficient and should not automatically be interpreted as equal to 70.

Article graphs

The original article graph is shown together with a complementary visualization. Taken together, they highlight both the theoretical trend of the model and its comparison with empirical estimates across growth stages.

Article graph comparing the empirical range, the WBE model, and the simplified and refined expressions
Main figure. Comparison between the empirical range of the metabolic scaling exponent, the simplified approximation, the refined expression, and the classical WBE value.
Complementary graph with points for the simplified and refined versions of b(n) across growth stages
Complementary figure. Points for the simplified and refined versions of b(n) distributed across growth stages, overlaid on the empirical range and on the baseline line \(b=0.75\).

The two figures together reinforce the main idea of the work: the metabolic exponent may be interpreted as development-stage dependent rather than strictly fixed.

Open DOIArticle link
Preprint · arXiv

Ontogenetic Growth: A Fibonacci-Based Ontogenetic Discretization of Body-Mass Growth Trajectories

In this work, the Fibonacci-based idea is extended from metabolic scaling to body-mass growth trajectories. Biological growth remains a continuous process; the discretization is applied to the ontogenetic coordinate used to represent development, rather than directly to body mass.

The continuous description begins with the relation between body mass and the ontogenetic coordinate:

\[m(t)=m_0\varphi^{n(t)}\]

One possible saturating dynamics for this coordinate is

\[n(t)=n_\infty\left[1-\exp\left(-k(t-t_0)\right)\right]\]

The formulation starts from a continuous coordinate \(n(t)\) and a discretized version divided into small substages. When the resolution \(\Delta n\) is small, the discretized trajectory approaches the continuous curve. The visible steps should therefore be interpreted as a finite-resolution representation of developmental progression, not as real biological jumps in body mass.

Main result: the Fibonacci-based formulation reproduced the general shape of the growth trajectories for guinea pig, guppy, hen and cow. In the quantitative comparison, the Fibonacci formulation was favored in two of the four cases (Guppy and Hen), whereas the WBE model was favored for Guinea pig and Cow. Even in those cases, the Fibonacci curves closely followed the empirical growth patterns.
arXiv page Open PDF

Growth-trajectory visualization

The GIFs below illustrate, across organisms with very different body-mass and developmental time scales, how the Fibonacci-based discretized description can follow the continuous growth trajectory as the ontogenetic resolution is refined.

Guinea pig growth GIF comparing observed data, WBE and Fibonacci discretization
Guinea pig. Comparison between observed data, the continuous reference curve and the Fibonacci-based ontogenetic discretization.
Guppy growth GIF comparing observed data, WBE and Fibonacci discretization
Guppy. A small-body-mass example showing the discretized representation approaching the continuous trajectory.
Hen growth GIF comparing observed data, WBE and Fibonacci discretization
Hen. The discretization represents progression through ontogenetic substages without implying real jumps in body mass.
Cow growth GIF comparing observed data, WBE and Fibonacci discretization
Cow. The same formalism is applied to a trajectory with much larger body mass and growth time.

The aim is not to replace classical growth models universally, but to provide an alternative ontogenetic coordinate for organizing and analyzing growth trajectories across biological systems.

Discrete Ontogenetic States

In this theoretical model, growth is represented as a sequence of transitions between discrete ontogenetic states. The index j identifies the level occupied by the organism; it is an effective state in a coarse-grained description of development and should not be interpreted directly as a cell cycle or a specific anatomical stage.

\[j=0,1,2,\ldots,J_{\max}\]
\[m_j=m_{\mathrm{ref}}\varphi^{j\Delta n}\]
\[j\longrightarrow j+1\]

Each level has a stage-dependent metabolic rate. In the principal Markovian formulation, the transition rate to the next level is written as

\[B_j=B_0\left(\frac{m_j}{m_{\mathrm{ref}}}\right)^{b_j}, \qquad b_j=b(n_j)\]
\[\lambda_j(t)=\lambda_b\,F_B(B_j)\,F_E(E(t))\]
\[F_B(B_j)=\frac{B_j}{B_j+B_c}, \qquad F_E(E(t))=\frac{E(t)}{E(t)+E_c}\]

Thus, the transition propensity depends on the current metabolic state and on available energy, without an explicit dependence on the elapsed residence time in the Markovian formulation.

Master equation

If \(P_j(t)\) is the probability that the organism occupies level \(j\) at time \(t\), the gain-loss dynamics is described by

\[ \frac{dP_j}{dt}= \begin{cases} -\lambda_0(t)P_0(t), & j=0,\\[6pt] \lambda_{j-1}(t)P_{j-1}(t)-\lambda_j(t)P_j(t), & 1\le j\le J_{\max}-1,\\[6pt] \lambda_{J_{\max}-1}(t)P_{J_{\max}-1}(t), & j=J_{\max}. \end{cases} \]

The mean body mass predicted by the state distribution is

\[\langle m(t)\rangle= \sum_{j=0}^{J_{\max}} P_j(t)m_j\]

Energy-threshold interpretation

A transition can be interpreted as occurring when the energy accumulated for growth reaches the energetic cost required to produce the next mass increment. The energetic threshold is

\[\epsilon_j=\eta\,(m_{j+1}-m_j)\]

Under approximately stationary energetic conditions at level \(j\), this interpretation yields the effective transition rate

\[\lambda_j^{(\epsilon)}\simeq \frac{\alpha_j B_j}{\eta\,(m_{j+1}-m_j)}\]

For approximately time-independent effective rates, the mean time required to traverse the full chain is

\[\langle T\rangle= \sum_{j=0}^{J_{\max}-1}\frac{1}{\lambda_j}\]

Continuous limit

When the ontogenetic steps become small, an effective velocity \(v(n,t)=\lambda(n,t)\Delta n\) can be defined, and the discrete description leads to

\[v(n,t)=\lambda(n,t)\Delta n\]
\[\frac{dm}{dt}=m\ln(\varphi)\,v(n,t)\]

Illustrative simulation of the stochastic dynamics

Individual trajectories have a step-like structure because mass changes only when a transition between neighboring levels occurs. Because transition times differ among individuals, the population mean can become smooth.

Step-like individual stochastic trajectories and smooth population mean
Representative stochastic trajectories and population mean. The figure illustrates how a discrete individual-level dynamics can generate a smooth macroscopic mean curve.