Prebiotic Molecule Accumulation Model

Version History

This repository contains two versions of a dynamical systems model tracking the accumulation of prebiotic organic molecules on the early Earth.

Version 1 (prebiotic_molecule_accumulation_model.ipynb) was developed as a introductory computational exploration. It implements a single-compartment linear ODE with constant production, linear decay, and a sinusoidal environmental input term. That model established the basic result: periodic environmental input and loss rate are the key determinants of whether molecules accumulate at all.

Version 2 (prebiotic_model_v2.ipynb) extends that foundation into a more physically and chemically realistic framework. It introduces nonlinear autocatalytic production, Arrhenius-based temperature-dependent degradation, UV photodegradation, wet-dry cycling, and a two-environment comparison between a surface hydrothermal pond and a submarine hydrothermal vent system.

Prebiotic Molecule Accumulation Model: Version 2

Scientific Background

The Accumulation Problem

Experimental work has established that the building blocks of life—amino acids, nucleotide precursors, sugars, and lipids—can form abiotically under early Earth conditions. Stanley Miller demonstrated this in 1953, producing amino acids within days by sparking electricity through a simulated early atmosphere (Miller, 1953). John Sutherland and colleagues later showed that RNA nucleotide precursors could form under plausible prebiotic conditions through an alternative synthetic pathway (Powner et al., 2009).

But formation is not accumulation. Prebiotic organic molecules face simultaneous competing pressures: they are produced by abiotic chemistry, but they are also destroyed by thermal degradation, photodegradation by ultraviolet radiation, and physical dilution into the surrounding environment. Whether a molecule persists long enough to participate in further chemistry depends entirely on the balance between these processes, and that balance is strongly environment-dependent.

This is the accumulation problem: Whether the abundant prebiotic molecule they can reach and sustain concentrations high enough to matter chemically. It is a necessary precondition for any origin-of-life scenario. A molecule that forms and immediately degrades contributes nothing to prebiotic chemistry regardless of how efficiently it was synthesised.

Two environments have dominated origin-of-life research as candidate settings for early prebiotic chemistry:

Surface hydrothermal ponds: shallow, geothermally heated pools subject to wet-dry cycling driven by evaporation and rainfall. First proposed by Darwin (1871) as a "warm little pond" and as a plausible setting for life's origin. The hypothesis was formalised computationally by Mulkidjanian et al. (2012), who identified geothermal fields as geochemically favourable environments for prebiotic synthesis, and developed experimentally by Damer and Deamer (2015), who proposed that wet-dry cycling drives combinatorial selection and concentration of organic polymers. The pond's key advantage is physical containment: molecules are retained in a semi-enclosed environment, concentrated by evaporation during dry phases, and periodically refreshed by rainfall during wet phases.
Submarine hydrothermal vents: continuous, mineral-rich fluid systems on the deep ocean floor, shielded from UV radiation and driven by serpentinization reactions. Proposed as a prebiotic setting by Russell and Hall (1997) and developed theoretically by Lane and Martin (2012), who argued that the proton gradients across alkaline vent pore walls provide a direct analogue to the chemiosmotic energy systems of modern cells. The vent's key advantage is sustained energy: continuous fluid flow delivers chemical energy and mineral catalysts indefinitely, shielded from UV radiation by the overlying ocean water column.

These two environments represent different hypotheses about what the accumulation problem requires. The pond hypothesis prioritises concentration through physical containment. The vent hypothesis prioritises sustained energy through geochemical flow. This model tests which environment, under physically motivated parameter choices, provides more favourable conditions for organic molecule accumulation.

Model Overview

Version 2 models the concentration of a generalised prebiotic organic molecule [X] in two contrasting early Earth environments: a surface hydrothermal pond and a submarine hydrothermal vent, using a system of ordinary differential equations (ODEs). The model tracks how [X] changes over time as a function of production, autocatalytic amplification, and three competing loss mechanisms.

The central scientific question is: under which environmental regime do organic molecules accumulate to concentrations sufficient to sustain chemistry, and why?

The model is not a reaction-specific chemical simulation. [X] is a generalised proxy for prebiotic organic polymers: proto-RNA strands, peptides, or nucleotide precursors, rather than a specific molecular species. This abstraction allows the model to focus on accumulation dynamics rather than species-specific reaction pathways, consistent with published dynamical models of prebiotic chemistry (Kauffman, 1993; Hordijk & Steel, 2004).

Key Upgrades from Version 1

Feature	Version 1	Version 2
Environments	Single surface environment	Surface pond + hydrothermal vent
Production	Constant linear term	Constant + nonlinear autocatalytic term
Degradation	Fixed constant	Arrhenius temperature-dependent rate
Environmental input	Smooth sinusoidal	Square-wave wet-dry cycling (pond) / constant (vent)
UV degradation	Not included	Explicit term. Active in pond, zero in vent
Sensitivity analysis	Single parameter variation	Two-parameter heatmap (T vs k_out)
Analytical steady state	Not computed	Derived and plotted alongside simulation

Mathematical Formulation

The accumulation of [X] in each environment is governed by a single ODE:

$$\frac{d[X]}{dt} = P_{\text{env}}(t) + R_{\text{auto}}([X]) - [X]\cdot\left(k_{\text{deg}}(T) + k_{\text{UV}} + k_{\text{out}}\right)$$

The right-hand side contains two production terms and three loss terms. Each term encodes a physically distinct process.

Production Terms

Environmental input: $P_{\text{env}}(t)$

Represents abiotic production of [X] from environmental chemistry.

Hydrothermal vent: constant continuous influx

$$P_{\text{env}}(t) = C_0$$

Surface pond: pulsed input following a square wave that alternates between wet phase (baseline delivery) and dry phase (delivery stops, evaporation concentrates existing molecules)

$$\text{wet-phase} = \frac{\text{square}(\omega t) + 1}{2}$$

$$P_{\text{env}}(t) = C_0 \cdot \text{wet-phase}$$

The square wave is mapped to oscillate between 1 (wet) and 0 (dry) rather than between +1 and -1, preventing physically impossible negative production rates.

Autocatalytic production: $R_{\text{auto}}([X])$

Represents the ability of existing molecules to catalyse their own synthesis, a central concept in prebiotic chemistry (Kauffman, 1993). A saturating nonlinear term is used to prevent unbounded growth:

$$R_{\text{auto}} = \frac{k_a [X]^2}{1 + K_m [X]^2}$$

The [X]² term reflects the requirement for two molecules to interact during autocatalytic replication; one acting as template, one as substrate. This produces sigmoidal (S-shaped) growth: slow at low concentrations, accelerating through a threshold, then saturating as K_m limits the ceiling. During dry phases in the pond, the concentration factor $\alpha$ amplifies autocatalysis to reflect evaporative crowding of existing molecules:

$$R_{\text{auto, dry}} = \frac{k_a [X]^2 \cdot (1 + \alpha\cdot \text{dry-phase})}{1 + K_m [X]^2}$$

Loss Terms

Thermal degradation: $k_{\text{deg}}(T)$

Molecular destruction rate as a function of temperature, modelled using the Arrhenius equation:

$$k_{\text{deg}}(T) = A \cdot e^{-E_a / RT}$$

A: pre-exponential frequency factor (converted to day⁻¹)
Ea: activation energy for degradation (J/mol)
R: universal gas constant (8.314 J/mol·K)
T: environmental temperature (Kelvin)

Higher temperatures exponentially increase degradation rate. The activation energy Ea = 105,000 J/mol reflects the covalent backbone bonds of prebiotic polymers such as proto-RNA and peptides, consistent with hydrolysis rate data for RNA and related molecules (Levy & Miller, 1998).

UV photodegradation: $k_{\text{UV}}$

A fixed degradation constant representing destruction by ultraviolet radiation. The early Earth lacked an ozone layer, exposing surface environments to significantly higher UV flux than today (Cnossen et al., 2007).

Surface pond: k_UV = 0.04 day⁻¹ - continuous UV exposure
Hydrothermal vent: k_UV = 0.0 - completely shielded by the overlying ocean water column

Advection and outflow: $k_{\text{out}}$

Represents physical loss through dilution and flushing:

Surface pond: k_out = 0.01 day⁻¹ - semi-enclosed, evaporative environment retains molecules
Hydrothermal vent: k_out = 0.08 day⁻¹ - continuous fluid flow flushes molecules into the surrounding ocean (Martin & Russell, 2007)

Analytical Steady State

Setting d[X]/dt = 0 and ignoring autocatalysis and cycling, the linear steady-state concentration is:

$$[X]^* = \frac{C_0}{k_{\text{deg}}(T) + k_{\text{UV}} + k_{\text{out}}}$$

This expression isolates the contribution of linear chemistry alone — production balanced against loss. It is plotted alongside the full simulation as a reference baseline, allowing direct visual assessment of how much autocatalysis and wet-dry cycling contribute to accumulation beyond what linear chemistry predicts.

Parameter Justification

All parameters are expressed in normalised concentration units and days. The pre-exponential factor A is converted from s⁻¹ to day⁻¹ by multiplying by 86,400 (seconds per day) to ensure unit consistency across the simulation.

Parameter	Symbol	Value	Justification
Universal gas constant	$R$	$8.314 \text{ J/mol}\cdot\text{K}$	Physical constant
Baseline production	$C_0$	$0.05 \text{ day}^{-1}$	Normalised abiotic production rate
Autocatalytic rate	$k_a$	$0.08 \text{ day}^{-1}$	Mathematical framework for autocatalytic sets (Kauffman, 1993)
Saturation constant	$K_m$	$5.0 \text{ a.u.}$	Michaelis-Menten type half-saturation concentration
Activation energy	$E_a$	$105,000 \text{ J/mol}$	Hydrolytic breakdown of prebiotic nucleobases (Levy & Miller, 1998)
Pre-exponential factor	$A$	$8.64 \times 10^{14} \text{ day}^{-1}$	Standard unimolecular degradation estimate, converted to day⁻¹
Pond temperature	$T_{\text{pond}}$	$303 \text{ K } (30^\circ\text{C})$	Continental geothermal field baseline (Mulkidjanian et al., 2012)
Pond UV rate	$k_{\text{UV pond}}$	$0.04 \text{ day}^{-1}$	Archean surface solar UV flux without ozone (Cnossen et al., 2007)
Pond outflow	$k_{\text{out pond}}$	$0.01 \text{ day}^{-1}$	Semi-enclosed evaporative basin (Damer & Deamer, 2015)
Dry-phase factor	$\alpha$	$2.5$	Evaporative concentration factor (dimensionless)
Cycling frequency	$\omega$	$2\pi/10 \text{ day}^{-1}$	Angular frequency for a 10-day wet-dry cycle (Damer & Deamer, 2015)
Vent temperature	$T_{\text{vent}}$	$363 \text{ K } (90^\circ\text{C})$	Core fluids of alkaline systems like Lost City (Kelley et al., 2005)
Vent UV rate	$k_{\text{UV vent}}$	$0.0 \text{ day}^{-1}$	Complete UV shielding at depth
Vent outflow	$k_{\text{out vent}}$	$0.08 \text{ day}^{-1}$	Open flow-through system (Martin & Russell, 2007)
Simulation duration	$t_{\text{end}}$	$365 \text{ days}$	One year. Sufficient to reach or approach steady state
Initial condition	$[X]_0$	$0.0 \text{ a.u.}$	No prior accumulation assumed. Conservative baseline

Results and Interpretation

Figure 1. Comparison of $[X]$ accumulation kinetics between a wet-dry surface pond and a hydrothermal vent. The surface pond simulation demonstrates rapid early accumulation, leveraging fluctuating wet-dry cycles and autocatalysis to stabilize at a dynamic plateau ($\approx 1.24$ a.u.) well above its linear steady state. The hydrothermal vent simulation rapidly reaches and maintains a low analytical steady state of $0.067$ a.u., suppressed by high-temperature thermal hydrolysis ($E_a = 105,000 \text{ J/mol}$) and open advective outflow ($k_{\text{out vent}} = 0.08 \text{ day}^{-1}$) which together isolate the system from nonlinear autocatalytic amplification.

Surface Pond

The surface pond simulation produces a rapid early accumulation that decelerates gradually into a stable plateau. Starting from zero, [X] rises steeply through the first 50 days, driven by the combined effect of environmental input and early autocatalytic amplification. Growth slows as the system approaches the autocatalytic saturation ceiling, stabilising at approximately 1.24, well above the analytical steady state of 0.987.

The plateau is sustained by the interplay between wet-dry cycling and autocatalysis. During dry phases, outflow drops to zero and evaporative concentration amplifies autocatalytic production. During wet phases, fresh precursor input refreshes the system. The result is a stable dynamic equilibrium that significantly exceeds what linear chemistry alone predicts.

Hydrothermal Vent

The hydrothermal vent simulation reaches its analytical steady state of 0.067 within approximately five days and remains there for the entire 365-day simulation without deviation. The concentration ceiling is approximately 18 times lower than the pond's peak accumulation.

This result reflects the thermal degradation trap inherent to high-temperature vent environments. At 90°C, the Arrhenius-computed $k_{\text{deg}}$ alone exceeds both the baseline production rate ($C_0$ = 0.05 day⁻¹) and the maximum autocatalytic rate constant ($k_a$ = 0.08 day⁻¹). Autocatalysis never meaningfully engages because [X] cannot reach the threshold concentration required for the nonlinear term to become significant. The vent reaches equilibrium between continuous production and continuous loss, and stays there.

The vent's continuous production and UV shielding provide real advantages not captured by this model. But under an accumulation-only framework, thermal degradation and rapid outflow together prevent the concentration buildup that autocatalytic chemistry requires.

Sensitivity Analysis

Figure 2. Sensitivity analysis of steady-state concentration $[X]^*$ as a function of environmental temperature and system outflow rate ($k_{\text{out}}$). The color gradient denotes the steady-state concentration ceiling, ranging from low accumulation (dark purple, $\approx 0.0$ a.u.) to high accumulation (bright yellow, $\geq 2.0$ a.u.). The blue marker indicates the parameterized regime of the subaerial surface pond ($30^\circ\text{C}$, $k_{\text{out}} = 0.01 \text{ day}^{-1}$), positioning it within a favorable chemical niche for autocatalytic amplification. Conversely, the red marker indicates the hydrothermal vent regime ($90^\circ\text{C}$, $k_{\text{out}} = 0.08 \text{ day}^{-1}$), illustrating how high temperatures and rapid advection pull the system deep into a kinetic dead zone where autocatalysis cannot engage.

The heatmap shows steady-state concentration [X]* across a broad temperature and outflow parameter space, computed #analytically without autocatalysis or cycling. Accumulation is maximised in the lower-left region of the plot—low temperatures and low outflow rates. Moving right (higher temperature) or upward (higher outflow) reduces accumulation rapidly. The transition from high to low accumulation is steep along the temperature axis and more gradual along the outflow axis, indicating that temperature is the dominant control on steady-state concentration under these parameter choices.

The two environment markers illustrate this directly. The pond marker sits within the mid-bright accumulation zone, consistent with the simulation result. The vent marker sits deep in the dark low-accumulation zone, far from any meaningful accumulation threshold.

The white region spanning 0°C to 20°C in the lower-left corner indicates where steady-state accumulation ([X]^*) exceeds the colorbar ceiling of 2.0 a.u. — extreme low-temperature, low-outflow conditions where thermal degradation is near zero and molecules accumulate without bound in the linear model. These conditions are mathematically predicted but physically unrealistic for early Earth. In actual geochemical environments below 20°C, the forward activation energy barriers for prebiotic synthesis ($C_0$) and autocatalytic pathways ($k_a$)) would severely reduce production rates, preventing such extreme concentrations. This is acknowledged as a model limitation.

Together, the three panels: pond simulation, vent simulation, and sensitivity heatmap, support a consistent conclusion that accumulation of prebiotic organic molecules is strongly favoured by cool temperatures, physical containment, and periodic concentration events. The surface pond satisfies all three. The hydrothermal vent satisfies none under these parameter regimes.

Limitations

Well-mixed compartments. Each environment is modelled as a single spatially homogeneous box. Spatial gradients within environments, such as temperature and pH gradients across a vent pore wall, are not captured.
Single-species proxy. [X] represents a generalised prebiotic organic molecule rather than a specific chemical species. Degradation kinetics are represented by a single average activation energy. In reality, different molecular species degrade at different rates under different conditions.
Abundant precursors assumed. Raw chemical feedstocks are assumed to be non-limiting and constant. This isolates accumulation dynamics from upstream supply constraints but does not reflect environments where precursor availability is itself a limiting factor.
No exchange between environments. The pond and vent are modelled as independent systems. Material transport between surface and subsurface environments is not included.
Idealised wet-dry cycling. The pond's wet-dry cycling is represented as a square wave with a fixed 10-day period. Real Hadean cycling would have been irregular and climate-dependent.
Arrhenius degradation uses fixed activation energy. A single Ea is applied across all degradation pathways. In reality, different molecular species and bond types degrade at different rates under different temperatures.
Dry-phase concentration factor. During dry phases, the concentration factor α amplifies the autocatalytic production term to reflect evaporative crowding of existing molecules. This is a simplification—physical evaporation would scale local concentration across all ODE terms simultaneously. A more rigorous implementation would define a time-dependent local concentration variable γ(t) = α during dry phases, applied globally. This is noted as a direction for future refinement.
Sensitivity analysis is two-dimensional. Temperature and outflow are varied simultaneously, but UV degradation rate, autocatalytic parameters, and production rate are held fixed. A fuller sensitivity analysis would explore the full parameter space.
Accumulation-only framework. The model does not capture redox chemistry, mineral catalysis, or reaction network complexity, all of which are genuine advantages of hydrothermal vent environments not represented here.

References

Bada, J.L. & Lazcano, A. (2002). Some like it hot, but not the first biomolecules. Science, 296(5575), 1982–1983.

Cnossen, I. et al. (2007). Habitat of early life: Solar X-ray and UV radiation at Earth's surface 4–3.5 billion years ago. Journal of Geophysical Research, 112, E02008.

Damer, B. & Deamer, D. (2015). Coupled phases and combinatorial selection in fluctuating hydrothermal pools: A scenario to guide experimental approaches to the origin of cellular life. Life, 5(1), 872–887.

Hordijk, W. & Steel, M. (2004). Detecting autocatalytic, self-sustaining sets in chemical reaction systems. Journal of Theoretical Biology, 227(4), 451–461.

Kauffman, S.A. (1993). The Origins of Order: Self-Organization and Selection in Evolution. Oxford University Press.

Kelley, D.S. et al. (2005). A serpentinite-hosted ecosystem: The Lost City hydrothermal field. Science, 307(5714), 1428–1434.

Lane, N. & Martin, W.F. (2012). The origin of membrane bioenergetics. Cell, 151(7), 1406–1416.

Levy, M. & Miller, S.L. (1998). The stability of the RNA bases: Implications for the origin of life. PNAS, 95(14), 7933–7938.

Martin, W. & Russell, M.J. (2007). On the origin of biochemistry at an alkaline hydrothermal vent. Philosophical Transactions of the Royal Society B, 362(1486), 1887–1926.

Miller, S.L. (1953). A production of amino acids under possible primitive Earth conditions. Science, 117(3046), 591–592.

Mulkidjanian, A.Y. et al. (2012). Origin of first cells at terrestrial, anoxic geothermal fields. PNAS, 109(14), E821–E830.

Powner, M.W., Gerland, B. & Sutherland, J.D. (2009). Synthesis of activated pyrimidine ribonucleotides in prebiotically plausible conditions. Nature, 459, 239–242.

Russell, M.J. & Hall, A.J. (1997). The emergence of life from iron monosulphide bubbles at a submarine hydrothermal redox and pH front. Journal of the Geological Society, 154(3), 377–402.