# Can a 0.1% chiral excess become dominant?

Yes, in this explicitly specified open-system Frank-type mechanism simulation. The result depends on reaction competition and noise; it is not a general prediction for an arbitrary chemical mixture.

## Definition and initial state

Enantiomeric excess is e=(L-D)/(L+D). An initial e=0.001 means 0.1% ee, or 50.05% L and 49.95% D within the active chiral pool. Initial dimensionless concentrations are x=0.5005, y=0.4995, with total x+y=1. “Dominant” means |e|>=0.90: at least 95% of the active L+D pool has one handedness. It does not refer to the fraction of all material including precursor and inactive product.

## Full-text verification and noise distinction

Verified against the user-supplied HTML of Michael Stich, Josep M. Ribó and David Hochberg, *Chiral autocatalysis: reaction noise, micro-reversibility and chiral inhibition in mirror symmetry breaking*, arXiv:1601.04152v1 (16 January 2016): https://arxiv.org/html/1601.04152. This is a different paper from the originally cited arXiv:0811.3124; it supplies direct support for the Frank scheme and criterion below. The supplied HTML SHA-256 is 90f3db5228949044c0b38e6947327d6504bba7b8d4a757b965786779b4b88576, matching the archive's source.json. This verifies file consistency, not independent authentication of the download.

### Reaction and assumption audit

| Item in the simulation | Check against the supplied paper |
|---|---|
| A+L -> 2L and A+D -> 2D | Section I, Eq. (1), explicitly gives these reactions. |
| L+D -> P | Section I, Eq. (2), gives irreversible mutual inhibition with constant k_2. Our k_i is this k_2. |
| Reverse steps 2L -> A+L and 2D -> A+D | Section I displays reversible enantioselective autocatalysis and explicitly discusses its combination with Eq. (2) as a reversible-autocatalysis Frank model. Our model includes these reverse steps. |
| g=k_-a/k_i; amplification for 0<g<1 | Section I explicitly states g=k_-a/k_2 and the condition 0<=g<1 for that Frank model. Our positive-g choices lie in its bounded steady-state regime. The g=0 limit was not simulated. |
| Constant precursor A, irreversible P sink | Our specified open-system idealization is compatible with the paper's description of open-system Frank models, but is not a verbatim reactor protocol reproduced from this paper. There is no explicit L/D washout or flow-rate model. With no return reaction from P, its explicit removal does not change the L/D equations; it explains the maintained sink physically. |
| No spontaneous A <-> L,D conversion; no limited-enantioselectivity reactions | These are deliberate omissions. The paper's main stochastic model contains direct production/decay and has different constraints and outcomes. |
| g=1 control | This is a **deterministically neutral control**, dx=x(1-x-y), dy=y(1-x-y). The reverse and inhibition reactions still exist in its chemical interpretation; their effects on ee cancel. It is not a reaction-deletion control, and its chemical noise cannot be inferred from its net drift alone. |
| Open-system thermodynamics | Sections II.2, IV and V stress that rate constants for alternative pathways cannot simply be chosen independently of microreversibility. Reservoirs and sinks must be explicitly justified. Our irreversible sink is a kinetic/open-system approximation, not a complete microscopically reversible chemical model. |

The deterministic drift was independently checked in the Notebook by summing the five reaction stoichiometries times their mass-action rates. It matches the implemented equations exactly. The ee growth criterion also follows directly from those equations. No numerical trajectory changes are required by this source audit.

### Why the paper's racemic conclusion is not a contradiction

The abstract and principal noise calculations concern enantioselective autocatalysis coupled to direct production/linear decay, subject to detailed balance, without Frank's nonlinear heterochiral inhibition. Sections II.2 and V emphasize equal forward/reverse equilibrium-constant ratios for pathways with the same net chemical conversion; their main model remains racemic even with reaction noise and in the driven case they analyze.

That result is not a statement that every Frank model must racemize. Section I explicitly distinguishes the reversible-autocatalysis Frank model with L+D inhibition and quotes the g<1 symmetry-breaking condition. Conversely, our noisy simulation must not be presented as contradicting or reproducing the paper's main stochastic calculations.

### Two distinct noise models

**Implemented extension: phenomenological environmental noise.** The added Ito terms are sigma*x*dW_L and sigma*y*dW_D, with independent Wiener processes and an adjustable dimensionless sigma. These represent assumed species-specific fluctuations in per-capita growth. Their covariance per unit time is

    B_environment = sigma^2 * [[x^2, 0], [0, y^2]].

Neither sigma nor these independent noise channels was derived from molecular reaction propensities. No system volume or molecule-number scale was specified. The reported reversal frequencies and the 16.17% dominant endpoints in the strongest-noise control belong only to this extension. They are not predictions of intrinsic chemical noise. Being multiplicative does not make a noise model chemical.

**Chemical reaction noise: not simulated here.** For the implemented five-reaction scheme, independent *reaction channels* would have stoichiometric jumps (+1,0), (0,+1), (-1,0), (0,-1), and (-1,-1), with leading-order dimensionless density rates x, y, g*x^2, g*y^2, and x*y. Consequently, at large system-size scale Omega, the chemical Langevin covariance would be

    B_chemical = (1/Omega) * [[x+g*x^2+x*y, x*y],
                            [x*y, y+g*y^2+x*y]].

This is a stoichiometric derivation for our simplified scheme, not a quoted equation or reproduced noise model from the paper. Both species share the same inhibition event, so that event produces a positive off-diagonal covariance. Reaction amplitudes scale with the square roots of propensities; finite molecule counts require the corresponding discrete pair-count factors. Omega would relate counts to the dimensionless concentrations. A Gillespie simulation or a chemical Langevin approximation would require specifying it and the elementary reactions consistently.

The paper's Section III and Appendix A derive intrinsic noise from chemical kinetics for their specified network; that procedure must not be replaced by choosing an arbitrary sigma. Chemical and environmental noise could coexist, but we simulated only the latter. Perfectly common multiplicative environmental noise would also behave differently from the independent perturbations chosen here.

The report's environmental-noise results remain an illustrative sensitivity analysis. No real reaction, biological handedness, or origin-of-life inference is established.

## Documented mechanism and explicit reduction

The model is a simplified Frank-type scheme. Its reactions and amplification criterion have now been checked directly against the supplied Stich, Ribó and Hochberg (2016) full text (see verification below). Related stability analysis is discussed in the open preprint by Ribo and Hochberg (2008), *Stability of racemic and chiral steady states in open and closed chemical systems*: https://arxiv.org/abs/0811.3124 (journal DOI: https://doi.org/10.1016/j.physleta.2008.10.079).

The reaction scheme implemented here is:
- A+L -> 2L and A+D -> 2D, rate constant k_a;
- 2L -> A+L and 2D -> A+D, reverse constant k_-a;
- L+D -> P, rate constant k_i, with inactive P removed.

A is chemostatted: its concentration is maintained by an external supply. Rate constants use the mass-action convention in which the reverse concentration loss term is k_-a L^2. No explicit precursor depletion, spatial gradients, spontaneous chiral production, or reverse production from P is included.

Let a=k_a[A], dimensionless time tau=a*t_physical, x=k_i[L]/a, y=k_i[D]/a, and g=k_-a/k_i. The implemented equations, derived directly from the above scheme, are

    dx/dtau = x - g*x^2 - x*y
    dy/dtau = y - g*y^2 - x*y.

These equations and their derivation are fully specified here for independent reproduction. The original 2008 source was retrieved only as metadata and abstract. The subsequently supplied 2016 full text has now been inspected and supports the reaction scheme and g<1 criterion; see the full-text audit below. The chemostat realization and environmental-noise extension remain our explicit modeling choices, not a reproduction of that paper's stochastic simulations.

With s=x+y and e=(x-y)/s, direct substitution gives

    de/dtau = (1-g)*s*e*(1-e^2)/2.

Thus g<1 amplifies a nonzero excess; g>1 restores balance; g=1 preserves ee. The symmetric steady state has s=2/(1+g), and its small-ee growth rate is (1-g)/(1+g). For 0<g<1 the stable pure-handed boundary states have (x,y)=(1/g,0) or (0,1/g). Exact e=0 remains exactly zero without a perturbation. A deterministic nonzero e cannot cross zero under these symmetric equations.

Opposite-handed inhibition consumes one molecule of each type, a larger proportional loss for the minority. Reverse autocatalysis penalizes the more abundant type through its squared concentration. Their relative strengths determine whether imbalance grows or shrinks.

## No-amplification control

At g=1 both species have the identical per-capita growth rate:

    dx/dtau=x*(1-s), dy/dtau=y*(1-s).

This is also interpretable as common density-regulated growth with no composition-selective feedback. It is a control for amplification of ee, not a claim that every underlying reaction is absent. Starting at s=1, both deterministic concentrations are constant. The same noise prescription is applied to the control.

## Phenomenological environmental-noise extension (not reaction noise)

The stochastic extension is illustrative, specified here rather than fitted to chemical data:

    dx=f_x*d_tau + sigma*x*dW_L
    dy=f_y*d_tau + sigma*y*dW_D.

These are Ito equations with independent standard Wiener processes for L and D. Noise is unbiased in handedness and acts throughout the run. It represents fluctuating species-specific per-capita growth; it is not a Gillespie simulation or a model of intrinsic molecular counting noise. Perfectly shared noise would have different effects on ee.

Log concentrations are advanced with Euler–Maruyama:
d(log x)=(1-g*x-y-sigma^2/2)*d_tau+sigma*dW_L, and analogously for y. This preserves positive concentrations without clipping. For the control, log(x/y)=log(x0/y0)+sigma*(W_L-W_D) exactly, so stochastic wandering can change ee even without deterministic amplification.

## Study design

Deterministic runs: initial ee -0.1%, 0%, 0.01%, 0.1%, and 1%; g=0.2, 0.8, 1, 1.2, and 2. Adaptive log-coordinate integration uses relative tolerance 1e-10 and absolute tolerance 1e-12.

Stochastic runs: initial ee 0%, 0.01%, 0.1%, and 1%; g=0.2, 1, and 2; sigma=0, 0.0002, 0.002, 0.02, and 0.2. There are 600 realizations per condition, ending at tau=60 with timestep 0.01 and seed 20261009. Replicates within a condition are independent. The same Brownian paths are reused across conditions for paired comparisons; conditions are consequently not independent samples. Zero-noise rows contain identical deterministic realizations, not 600 independent pieces of evidence.

## Observed results

Starting from 0.1% ee:

| Reaction parameter | Final ee at tau=60 | Interpretation |
|---|---:|---|
| g=0.2 | Approximately 100% | Strong amplification |
| g=0.8 | 62.2146% | Slower amplification; not yet dominant |
| g=1 | 0.1% | No-amplification control |
| g=1.2 | 0.000424% | Lead decays |
| g=2 | Approximately 2.10e-10% | Effectively balanced |

For g=0.2 without noise, 0.01%, 0.1%, and 1% initial ee first cross 90% ee at approximately tau=15.00, 11.55, and 8.10, respectively. These times are resolved on a 0.05-unit output grid. A -0.1% start gives the mirror-image result. A zero start stays zero. “100%” is rounded numerical saturation, not evidence of exact molecular extinction.

With initial 0.1% ee and g=0.2:

| Noise sigma | Initially trailing D dominates at tau=60 | 95% Wilson interval |
|---|---:|---:|
| 0.0002 | 0/600 (0%) | 0–0.64% |
| 0.002 | 145/600 (24.17%) | 20.91–27.75% |
| 0.02 | 282/600 (47.0%) | 43.04–51.00% |
| 0.2 | 298/600 (49.67%) | 45.68–53.66% |

In these amplified ensembles every run finishes dominant in one direction. Noise can overturn the tiny early lead, after which feedback amplifies the new leader. Stronger starting excess resists reversal: at sigma=0.002 the negative-dominance frequency is 47.0%, 24.17%, and 0% for initial 0.01%, 0.1%, and 1% ee.

The g=2 restoring system has about half its noisy runs end with negative ee, but none reach dominance in the tested conditions. Those sign changes mainly show loss of the original directional memory around balance, not wholesale replacement by a nearly pure opposite-handed pool.

At sigma=0.2 the no-amplification control reaches positive dominance in 49/600 and negative dominance in 48/600 runs: 97/600, or 16.17%, total. Its continuous noise can generate extreme endpoints without deterministic amplification. The amplifier has 600/600 dominant endpoints under the same conditions. A nearly balanced ensemble mean can also hide individually dominant mixtures of opposite signs; distributions, not only means, are reported.

## Numerical checks

An initial direct concentration solver allowed tiny negative concentrations as the minority approached zero; it was replaced with log-coordinate integration before producing results. No concentration clipping was used.

Tests passed for exact racemic invariance, mirrored initial states, deterministic control invariance, and finite log concentrations. Tightening deterministic tolerances changed the ee trajectory by at most 7.75e-11. The zero-noise Euler and adaptive solver endpoints differed by at most 7.25e-12.

Coupled refinement from timestep 0.01 to 0.005 used 600 fresh paths (seed 80317) for each of g=0.2, 1, 2 and sigma=0.002, 0.2, with initial ee=0.1%. There were no final sign or dominance classification disagreements. The largest mean absolute final-ee difference was 1.41e-4. This checks selected endpoints, not every noise path or every possible parameter. Sampling uncertainty remains.

## Figure captions

**chiral-deterministic.png.** Left: starting imbalance sweep at g=0.2, no noise; exact symmetry remains stationary while either signed nonzero lead grows. Right: reaction parameter sweep starting at 0.1% ee, no noise; positive ee is plotted on a logarithmic axis to expose decay. Time is dimensionless.

**chiral-noise.png.** Upper left: the first 12 prespecified realizations, not outcome-selected paths, at initial ee=0.1%, g=0.2, sigma=0.002. Upper right: negative endpoint frequencies for initial ee=0.1%, with 95% Wilson intervals; 600 runs per condition at tau=60. Lower left: negative-dominant endpoint percentages for g=0.2, varying starting imbalance and noise; for the zero-start row, “negative” is symmetry breaking rather than reversal of an existing lead. Lower right: empirical cumulative endpoint distributions at initial ee=0.1%, sigma=0.2, for g=0.2, 1, and 2; dotted lines mark +/-90% ee. All stochastic panels use dt=0.01; sign change and dominance are distinct metrics.

## Limits and reproducibility

This is a mechanism simulation. No rate constants, noise amplitudes, molecular identities, measured time scales, or yields were calibrated to an experiment. It does not reproduce the Soai reaction or any other real chemical experiment. It neither establishes a prebiotic pathway nor explains the origin of life or why biology selected particular handedness. The open-system supply/removal assumptions matter: they are not a closed equilibrium system.

For context, Blanco, Stich and Hochberg, *Temporary mirror symmetry breaking and chiral excursions in open and closed systems* (2011), https://arxiv.org/abs/1104.2229, DOI https://doi.org/10.1016/j.cplett.2011.02.032, explicitly discusses the importance of open versus closed conditions. Hochberg, *Mirror symmetry breaking and restoration: the role of noise and chiral bias* (2009), https://arxiv.org/abs/0906.3391, DOI https://doi.org/10.1103/PhysRevLett.102.248101, studies another noise treatment; its reported restoration behavior is not a result reproduced here. Only metadata and abstracts for these contextual sources were retrieved.

The downloadable Notebook contains the numerical source, captured computed tables and embedded figures. Its plotting setup uses standard Matplotlib defaults for portability; the live Notebook used the figure-style helper. Re-running reproduces the numerical procedure; exact rendering may vary. The results archive contains all 36,000 stochastic endpoints, summary probabilities, deterministic trajectories, the first 12 stochastic paths per condition, convergence results and configuration.

Executed environment: Python 3.12.14; NumPy 2.5.3; pandas 2.3.3; SciPy 1.18.0; Matplotlib 3.11.1.
