# 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.

## Documented mechanism and explicit reduction

The model is a simplified Frank-type scheme 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 connector returned the cited source's bibliographic record and abstract. A direct arXiv HTML request returned HTTP 403, so its full text was not inspected in this session; this work does not claim a verbatim reproduction of a particular published simulation.

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.

## Noise extension

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.
