Biological τ — Metabolism and Breath as Temporal

Abstract

We apply the τ framework (τ ≡ E/c³ ≡ m/c) to human metabolism. Oxidation of macronutrients converts bodily mass into CO₂ and H₂O while releasing energy as work and heat. Expressed in τ units, breathing is a τ-exchange with the environment. We provide balanced reactions, mass/energy accounting, and practical, quantitative tests (indirect calorimetry, CO₂ mass capture, isotopic methods) to validate the τ interpretation in vivo.

1. Introduction

“Weight loss” is often discussed energetically, yet mass conservation reveals that most lost mass exits as exhaled CO₂ and H₂O. In parallel, energy is dissipated as heat and exported via biochemical work. In a τ-first view, where m = cτ, E = c³τ, metabolic processes move τ between body and environment through gases and photons, yielding testable relations between gas exchange, heat production, and body-mass change.

2. Metabolic Stoichiometry & Mass Balance

Representative fat oxidation (average triacylglycerol):

C₅₅H₁₀₄O₆ + 78 O₂ → 55 CO₂ + 52 H₂O + energy

Mass balance (per mole of C₅₅H₁₀₄O₆ ≈ 861.4 g): O₂ consumed ≈ 78×32 = 2496 g; products: CO₂ ≈ 55×44.01 = 2420.6 g, H₂O ≈ 52×18.015 = 936.8 g. Total reactants ≈ 861.4 + 2496 ≈ 3357.4 g = total products ≈ 3357.4 g.

Thus, most mass leaves as CO₂ (gas) and H₂O (vapor/liquid), while the energy of oxidation is exported as heat/work.

3. τ-Formulation for Metabolism

Define τ as the “temporal charge” carried by matter/energy:

τ = m/c = E/c³, Δτ = Δm/c = ΔE/c³

A breath exchanges Δm_air with the environment, hence Δτ_air = Δm_air/c. Summed over time, net mass loss Δm_body = −(m_CO₂ + m_H₂O + other excreta) corresponds to Δτ_body = Δm_body/c.

Gas-exchange observables (VO₂, VCO₂) thus provide direct τ-flux estimates.

4. Energy & Entropy Accounting

With respiratory quotient RQ = VCO₂ / VO₂, substrate mix is inferred (fat ≈ 0.7, carbs ≈ 1.0). Energy expenditure (EE) by indirect calorimetry can be estimated (Weir-type):

EE ≈ 3.941·VO₂ + 1.106·VCO₂ (kcal·day⁻¹; VO₂,VCO₂ in L·day⁻¹)

Because E = c³τ, measured EE maps to a τ-rate: \dot τ_E = EE / c³, while gas mass flow maps to \dot τ_m = \dot m / c. A consistent account requires \dot τ_E ≈ \dot τ_m after storage and mechanical work are included.

5. Quantitative Benchmarks & Examples

5.1 Per-breath CO₂ mass (at rest)

If VCO₂ ≈ 200 mL·min⁻¹ and respiratory rate ≈ 12 min⁻¹, then per breath ≈ 16–20 mL CO₂. At 1 atm, 37 °C, that is roughly 30–40 mg CO₂ per breath (order-of-magnitude), i.e., a per-breath τ export of ≈ (3–4)×10⁻⁵ kg / c.

5.2 “10 kg fat” oxidation example

Using the stoichiometry above, metabolizing 10 000 g of C₅₅H₁₀₄O₆ (~11.6 mol) consumes ~29 kg O₂ and produces ~28 kg CO₂ and ~11 kg H₂O. In τ units:

Δτ_CO₂ ≈ 28 000 g / c, Δτ_H₂O ≈ 11 000 g / c, Δτ_body ≈ −10 000 g / c.

6. Implications

Mass loss is directly measurable via gas exchange; τ renders this as a conserved flow between body and environment.
Proper accounting must include oxygen mass in, not only CO₂/H₂O mass out.
Calorimetry (energy) and spirometry (mass) provide two projections of the same τ-exchange.

7. Conclusion

Breathing is a τ-transport process: oxidation converts stored τ (mass/energy) into gaseous τ (CO₂/H₂O) and thermal τ (heat). This yields concrete, testable relations linking VO₂/VCO₂, heat, and body-mass change—placing everyday physiology within the same τ-first substrate as fundamental physics.

References

Standard biochemistry texts on oxidative metabolism and respiratory quotient (RQ).
Indirect calorimetry (Weir-type) formulations for estimating energy expenditure from VO₂ and VCO₂.
Stoichiometric balances for fatty-acid oxidation (e.g., palmitate analogies) and average TAG models.

Appendix A — τ-First Biological Dictionary

A.1 Core definitions

τ ≡ E/c³ ≡ m/c

Δτ_body = −(Δm_CO₂ + Δm_H₂O + …)/c, Δτ_env = −Δτ_body

A.2 Gas exchange & fluxes

RQ = VCO₂ / VO₂

ṁ_CO₂ = ρ_CO₂ · \dot V_CO₂, ṁ_O₂ = ρ_O₂ · \dot V_O₂ (adjusted to BTPS/STPD as applicable)

A.3 Energy link

EE ≈ 3.941·VO₂ + 1.106·VCO₂ (kcal·day⁻¹)

\dot τ_E = EE / c³, \dot τ_m = (ṁ_out − ṁ_in)/c

A.4 Example stoichiometry

C₅₅H₁₀₄O₆ + 78 O₂ → 55 CO₂ + 52 H₂O + energy

A.5 Operational identities

Δm_body ≈ −(m_CO₂ + m_H₂O + m_other), ΔE_body ≈ −(Q + W)

Consistency: Δτ_body ≈ (ΔE_body)/c³ ≈ (Δm_body)/c

Appendix B — Test Protocols (Checklist)

B.1 Lab Protocols (Human)

Test	Observable	Procedure	Outcome
Indirect calorimetry	VO₂, VCO₂ (L·min⁻¹)	Metabolic cart; steady-state measures (rest/exercise)	Compute EE; infer τ-rate `\dot τ_E`; compare to mass flux `\dot τ_m`
CO₂ capture & weighing	ṁ_CO₂ (g·h⁻¹)	Soda-lime or molecular sieve; gravimetric difference	Quantify mass-out via CO₂; cross-check with VCO₂
Humidity & water loss	ṁ_H₂O (g·h⁻¹)	Hygrometry of inspired/expired air; body weight change	Account for H₂O fraction of mass-out
Doubly labeled water	CO₂ production (mol·day⁻¹)	^2H/^18O isotopic washout (free-living)	Integrate to long-term τ-flux; compare with weight change
Direct calorimetry (optional)	Heat Q (kJ)	Calorimeter chamber or high-precision thermal sensors	Energy-out check vs EE; τ consistency

B.2 Analysis & Reporting

Report mass-in (O₂) and mass-out (CO₂, H₂O) explicitly.
Convert to τ via τ = m/c and τ = E/c³; verify τ-consistency across energy and mass channels.
State conditions (temp, pressure, BTPS/STPD corrections) and uncertainty.
Include a one-line balance: Δτ_body + Δτ_env ≈ 0 (within measurement error).

B.3 Worked Example Template

Given: VO₂, VCO₂, duration T → EE(T), m_CO₂(T), m_H₂O(T).
Predict: Δm_body(T) ≈ −(m_CO₂ + m_H₂O + ...), ΔE_body(T) ≈ −EE(T).
Check: Δm_body/c ≈ EE(T)/c³.