GOE Model

This project explores oxygen dynamics during the Great Oxidation Event (GOE) using a simplified dynamical model. It documents the progression from an initial rate-based model (v1) through two redox-informed revisions (v2, v3), motivated by feedback from Professor David Catling (University of Washington).

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Repository Structure

• GOE_model.ipynb → Initial model (v1)
• GOE_model_v2_redox.ipynb → Redox-revised model (v2)
• GOE_model_v3_redox.ipynb → Fully redox-balanced model (v3, current)

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Version 3: Fully Redox-Balanced Model (Current)

Key Improvement

Every process that adds oxidising power to the atmosphere now has an equal and opposite term removing reducing power from the geological reservoir:

• Organic carbon burial (β·α·B): Source in dO/dt, equal sink in dR/dt
• H₂ escape (ε): Appears symmetrically in both equations
• Geochemical oxidation (γOR): Depletes both O and R simultaneously

This enforces Catling's central argument: oxygen cannot accumulate unless the system experiences a net loss of reducing power through physical processes — burial and H₂ escape — not biological production alone.

Variables and Parameters

Symbol	Value	Description
B(t)	—	Cyanobacteria biomass
O(t)	—	Atmospheric oxygen
R(t)	—	Reduced geological reservoir (volcanic reductants + reactive reduced species)
r	0.05	Cyanobacteria intrinsic growth rate
K	1.0	Carrying capacity
α	0.05	Gross photosynthesis rate (O₂ + CH₂O per biomass)
β	0.30	Burial efficiency (fraction of CH₂O permanently buried)
γ	0.5	Geochemical oxidation rate (pre-GOE reductant sink)
ε	0.010	H₂ escape flux (baseline — constant net oxidation driver)
V	0.031	Volcanic reductant flux
δ	0.017	Oxidative weathering rate (post-GOE O₂ sink)

Parameter logic: V is tuned so that burial alone (β·α·K = 0.015) cannot deplete R, since V/2 = 0.0155 > 0.015. Only when H₂ escape (ε) is added does the net reductant loss exceed replenishment, triggering the GOE.

Governing Equations

$$\frac{dB}{dt} = rB\left(1 - \frac{B}{K}\right)$$

$$\frac{dO}{dt} = \underbrace{\beta\alpha B}_{\text{organic burial}} + \underbrace{\varepsilon}_{\text{H}_2\text{ escape}} - \underbrace{\gamma OR}_{\text{geochemical sink}} - \underbrace{\delta O}_{\text{weathering}}$$

$$\frac{dR}{dt} = \underbrace{V}_{\text{volcanic input}} - \underbrace{\gamma OR}_{\text{oxidation}} - \underbrace{\beta\alpha B}_{\text{burial removes reductant}} - \underbrace{\varepsilon}_{\text{H}_2\text{ escape}}$$

Redox balance check: Every term in dO/dt has a mirror term in dR/dt with opposite sign. This enforces conservation of redox state globally.

Redox Balance Verification

Process	Effect on dO/dt	Effect on dR/dt
Organic burial (β·α·B)	+β·α·B	−β·α·B
H₂ escape (ε)	+ε	−ε
Geochemical oxidation (γOR)	−γOR	−γOR
Volcanic input (V)	—	+V
Oxidative weathering (δO)	−δO	—

Note: γOR appears with the same sign in both equations, i.e., the oxidation reaction consumes both O₂ and the reductant simultaneously.

Methodology

The ODE system was solved numerically using Python's scipy.integrate.solve_ivp. Three analyses are presented:

1. Time evolution: Tracks B(t), O(t), and R(t) through three phases: anoxic buffering, GOE transition, and stable oxic state
2. Phase-space analysis: Visualises O versus R to identify the tipping point independent of time
3. H₂ escape sensitivity: Demonstrates that ε = 0 produces permanent anoxia regardless of burial, reproducing the key result of Claire et al. (2006) Fig. 8

Results

Figure 1: System dynamics (v3)

The simulation produces three distinct phases:

Phase I - Anoxic buffering: Cyanobacteria grow logistically but oxygen remains suppressed. The geochemical sink (γOR) dominates because R is large and volcanic input (V) continuously replenishes it. Burial and H₂ escape are insufficient to overcome this sink because V exceeds burial alone.

Phase II - GOE transition: Cumulative H₂ escape slowly drains the reduced reservoir. As R falls, geochemical suppression weakens and O begins to rise non-linearly — the tipping point.

Phase III - Stable oxic state: R is depleted. O stabilises at a new equilibrium where burial + H₂ escape is balanced by oxidative weathering (δO).

Figure 3: H₂ escape sensitivity

When ε = 0, the system remains permanently anoxic regardless of cyanobacterial growth or burial — directly reproducing Claire et al. (2006) Fig. 8. H₂ escape is a necessary, not merely contributory, driver of the GOE.

Conclusion (v3)

The fully redox-balanced model confirms that the GOE cannot be explained by biological productivity alone. The transition to an oxic atmosphere requires a persistent net loss of reducing power from the Earth system, driven by organic carbon burial and, critically, the escape of hydrogen to space. Without H₂ escape, the model remains permanently anoxic, consistent with the theoretical framework of Catling (2014) and the biogeochemical modelling of Claire et al. (2006).

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Appendix - Model History

Version 1: Rate-Based Model

The initial model used a coupled system of ODEs to describe interactions between cyanobacterial biomass, atmospheric oxygen, and a reduced geological reservoir. The goal was to investigate whether biological oxygen production alone could explain the timing and structure of the GOE.

Research Question

How does the interaction between cyanobacterial growth, reduced geological materials, and volcanic fluxes control the accumulation of atmospheric oxygen on early Earth?

Conceptual Framework

Figure 1: Conceptual structure of the dynamical model. image by Author.

The model tracks three interacting state variables:

1. Cyanobacterial biomass (B): the population of oxygen-producing organisms
2. Atmospheric oxygen (O): oxygen concentration in the atmosphere
3. Reduced geological reservoir (R): available geochemical materials that consume oxygen

Processes included:

• Oxygen production via photosynthesis
• Oxidation reactions of reduced materials
• Logistic growth of cyanobacteria
• Replenishment of the reduced reservoir via volcanic outgassing

Variables and Parameters

Variable	Parameters
B(t): Cyanobacteria biomass	r: Cyanobacteria growth rate
O(t): Atmospheric oxygen concentration	K: Carrying capacity of cyanobacteria
R(t): Reduced geological reservoir	α: Oxygen production rate per biomass
	γ: Oxidation reaction rate constant
	V: Volcanic flux of reduced gases

Governing Equations

$$\frac{dB}{dt} = rB\left(1 - \frac{B}{K}\right)$$

$$\frac{dO}{dt} = \alpha B - \gamma OR$$

$$\frac{dR}{dt} = V - \gamma OR$$

Assumptions

• Cyanobacteria follow logistic growth
• Oxygen production is proportional to cyanobacteria biomass
• Oxidation of reduced materials depends on both oxygen concentration and the reduced reservoir
• Volcanic input replenishes the reduced reservoir at a constant rate
• Other geological processes (sedimentation, tectonics) are not explicitly modelled

Methodology

The ODE system was solved numerically using Python's scipy.integrate.solve_ivp. The analysis consists of three components:

1. Time evolution — tracking B(t), O(t), and R(t) through the anoxic-to-oxic transition
2. Phase-space analysis — visualising O versus R to examine non-linear coupling and identify the tipping point
3. Sensitivity analysis — assessing the effect of volcanic flux (V) on the timing and extent of oxygenation

Results

Figure 2: Temporal dynamics of the GOE model (v1).

The model produces a delayed oxygenation pattern driven by the gradual depletion of the reduced reservoir. Oxygen accumulation occurs once biological production exceeds the system's capacity to buffer oxygen through geochemical sinks.

Conclusion (v1)

The model suggests that the GOE can be interpreted as a non-linear transition driven by gradual biological growth coupled to finite geological sinks. Once the reduced reservoir is depleted, the system reaches a new, more oxidised equilibrium.

Limitation of v1

• The model did not explicitly enforce global redox balance
• Oxygen accumulation was treated as a local balance problem rather than a system-wide redox process
• Oxygenic photosynthesis was treated as a net O₂ source, which is incorrect on geological timescales, since CO₂ + H₂O ⇌ O₂ + CH₂O is reversible and redox-neutral

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Version 2: Redox-Revised Model

Motivation

Following expert review, v2 incorporates system-level redox accounting. The key insight, grounded in Catling (2014) and Claire et al. (2006), is:

Redox conservation is as inviolable as mass or energy conservation. Life on its own cannot change the net redox state of the surface of the Earth on a geological timescale. Every O₂ is balanced by organic matter (CH₂O). Net O₂ accumulation requires removal of reducing power from the system, either by burial of organic carbon or escape of hydrogen to space.

Revised Understanding

• Oxygen accumulation is not controlled by production alone
• Oxygenic photosynthesis produces both O₂ and CH₂O as a coupled redox pair — it does not change net atmospheric oxygen unless the reducing counterpart is removed
• Net O₂ accumulation requires:
  • Burial of organic carbon (CH₂O)
  • Escape of hydrogen (H₂) to space

Therefore, oxygen accumulation depends on the net removal of reducing power from the Earth system, not biological production alone.

Changes from v1

• Added organic carbon burial term (β·α·B)
• Added hydrogen escape term (ε)
• Added oxidative weathering sink (δO)
• Recast system as a redox-balanced evolution rather than a simple rate competition

Limitation of v2

Although v2 introduced the correct terms, H₂ escape (ε) was added as a free source in dO/dt without a corresponding reduction of R, violating the redox balance it set out to enforce. The burial term was also not consistently mirrored across both equations.

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References

Catling, D. C. (2014). The Great Oxidation Event Transition. In Treatise on Geochemistry (2nd Ed.), edited by H. D. Holland and K. K. Turekian, vol. 6, Elsevier, Oxford, 177–195. https://faculty.washington.edu/dcatling/Catling2014_GreatOxidationEvent.pdf

Claire, M. W., Catling, D. C., & Zahnle, K. J. (2006). Biogeochemical modelling of the rise in atmospheric oxygen. Geobiology, 4(4), 239–269. https://faculty.washington.edu/dcatling/Claire2006-Geobiology.pdf

Holland, H. D. (2006). The oxygenation of the atmosphere and oceans. Philosophical Transactions of the Royal Society B: Biological Sciences, 361(1470), 903–915. https://doi.org/10.1098/rstb.2006.1838

Lyons, T. W., Reinhard, C. T., & Planavsky, N. J. (2014). The rise of oxygen in Earth's early ocean and atmosphere. Nature, 506, 307–315. https://doi.org/10.1038/nature13068