arXiv subjects

Riccardo Marchesi

Publications and source records attributed to Riccardo Marchesi.

4 recordsLinked to original sources

arXiv2026

Beyond Murray's Law: Non-Universal Branching Exponents from Vessel-Wall Metabolic Costs

Murray's cubic branching law ($α=3$) predicts a universal diameter scaling exponent for all hierarchical transport networks, yet arterial trees yield $α\sim 2.7-2.9$. We show that this discrepancy has a structural origin: Murray's universality is an artifact of cost homogeneity, not a biological property. Incorporating the empirical vessel-wall thickness law $h(r)=c_0 r^p$ ($p \approx 0.77$) introduces a third metabolic cost term $\propto r^{1+p}$ that renders the cost function inhomogeneous with incommensurate scaling exponents. By Cauchy's functional equation, homogeneity is necessary and sufficient for a universal branching exponent to exist; its absence implies non-universality, and Murray's law is identified as a singular degeneracy of the cost-function family rather than a general principle. We prove that the resulting scale-dependent exponent satisfies the strict bounds $(5+p)/2 < α^*(Q) < 3$ independently of flow asymmetry (Theorem 4, Corollary 5). The static wall-tissue mechanism bounds the symmetric bifurcation exponent to $α_t \in [2.90, 2.94]$ from measured parameters, marking a first-order symmetry breaking from Murray's law that narrows the empirical gap by one-third. The remaining discrepancy with the cardiovascular mean ($α_{exp} \approx 2.70$) is not a model failure but a mathematical necessity that signals the independent contribution of pulsatile wave dynamics. Additionally, the wall cost breaks Murray's topological degeneracy, bounding the optimal branching number to small finite integers; binary bifurcation emerges as the physiologically selected minimum under steric constraints.

physics.bio-ph↗

arXiv2026

The Incommensurability Principle in Biological Transport

Why does the mammalian vascular tree maintain a conserved branching exponent $α^* \approx 2.72$ across a $10^7$-fold range in body mass, despite a fundamental shift from viscous to wave-dominated transport? We prove this universality cannot emerge from local optimization: any junction-level coupling of incommensurable costs requires scale-dependent fine-tuning varying by $O(10^2$--$10^3)$ across the hierarchy. Real networks resolve this through structural heterogeneity, and vascular geometry emerges as a scale-free attractor of a network-level minimax principle. Grounding the fitness penalty in ATP stoichiometry, we prove a Topological Rigidity theorem: the optimal branching exponent depends only on dimensionless structural parameters $(G, N, p, α_w)$, independent of all metabolic quantities. A self-consistency condition on the viscous--inertial energy partition yields a dual-threshold framework with $\mathrm{Wo}_c^{\mathrm{fluid}} = \sqrt{3}$ and $\mathrm{Wo}_c^{\mathrm{wave}} = 3/\sqrt{2}$. The symmetric model yields $α^*_{\mathrm{model}} \approx 2.626$, in agreement with mammals near the allometric transition; morphometric heterogeneities shift large-mammal values toward $2.72$. The framework explains developmental stability of cardiovascular networks as a consequence of architecture being decoupled from biochemistry.

physics.bio-ph↗

arXiv2026

The Dynamic Origin of Kleiber's Law

The ubiquitous $3/4$ metabolic scaling exponent, known as Kleiber's law, has long been attributed to the minimization of viscous dissipation within fractal transport networks. In this paper, we invert this standard narrative, demonstrating that Kleiber's law is fundamentally a signature of pulsatile wave physics rather than steady-state geometry. By coupling local branching optimization to global allometry, we derive the exact generalized metabolic exponent $β= dα/(2d+α)$, which strictly maps local transport microphysics to global organismal scaling. We show that dynamic wave-impedance matching in the proximal vasculature uniquely enforces $β= 3/4$ in three dimensions. This bound is dynamically protected: no static optimization of a viscous network can reproduce it. Consequently, we analytically predict the critical body mass for the wave-to-viscous transition, successfully explaining the empirical shift to steeper allometric scaling ($β\approx 0.9$) in small mammals and invertebrates with no free parameters. Furthermore, we demonstrate that the classical West--Brown--Enquist (WBE) derivation is structurally divergent under its own geometric assumptions, failing at the required proximal-dominance limit. Our framework is validated across nine biological systems spanning five phyla -- including vertebrate vasculature, insect tracheae, plant xylem, and sponge canals -- accurately predicting empirical branching exponents from independent biophysical measurements. Ultimately, we establish a general allometric equation of state that organizes diverse biological networks into discrete universality classes, generating falsifiable predictions across clades from shrews to flatworms.

physics.bio-ph↗

arXiv2026

A Unified Variational Principle for Branching Transport Networks: Wave Impedance, Viscous Flow, and Tissue Metabolism

The branching geometry of biological transport networks is characterized by a diameter scaling exponent $α$. Two structural attractors compete: impedance matching ($α\sim 2$) for pulsatile flow and viscous-metabolic minimization ($α= 3$) for steady flow. Neither predicts the empirically observed $α_{\mathrm{exp}} = 2.70 \pm 0.20$ in mammalian arterial trees. Incorporating sub-linear vessel-wall scaling $h(r) \propto r^p$ ($p = 0.77$) into a three-term metabolic cost rigorously breaks Murray's cubic law -- via Cauchy's functional equation -- bounding the static optimum to $α_t \in [2.90, 2.94]$. We formulate a unified network-level Lagrangian balancing wave-reflection penalties against transport-metabolic costs. Because the operational duty cycle $η$ is uncertain over developmental timescales, we cast the optimization as a zero-sum game between network architecture and environment. Von Neumann's minimax theorem -- proved via strict monotonicity of the cost curves -- yields a unique saddle point $(α^, η^)$ satisfying an exact equal-cost condition. We further prove $N = 2$ uniquely maximizes the network stiffness ratio $κ_{\mathrm{eff}}(N)$, deriving binary branching as a structural consequence of the framework. For the porcine coronary tree ($G = 11$ generations), $α^* = 2.72$, within $0.1σ$ of morphometric data. Sensitivity analysis confirms $|Δα^*| < 0.01$ across physiological metabolic ranges; the prediction depends critically only on the histological exponent $p$ -- a zero-parameter derivation from fundamental scaling principles that simultaneously recovers a cumulative wave dissipation of 6.3%, consistent with independent clinical estimates.

physics.bio-ph↗