# BIO2: Single-molecule work fluctuations with Donnan partitioning

In an optical-tweezers single-molecule RNA unfolding/refolding experiment (with energies measured in units of $k_B T$), the forward unfolding work distribution $P_F(W)$ is Gaussian with mean $\mu_F = 52.3000\,k_B T$ and variance $\sigma_F^2 = 19.6000\,(k_B T)^2$, while the time-reversed refolding work distribution $P_R(-W)$ plotted against $+W$ is Gaussian with mean $\mu_R = 28.3000\,k_B T$ and variance $\sigma_R^2 = 28.4000\,(k_B T)^2$.

Observables 1 through 3 (`jarzynski_second_cumulant_dG_kBT` to `density_crossing_work_W_star_kBT`) characterize the unperturbed bulk reference bath ($\text{pH}_0 = 7.4$). Observables 4 and 5 (`bhattacharyya_overlap_integral` and `transient_violation_probability`) are measured inside a semipermeable polyanionic microcompartment carrying immobilized fixed charge density $\rho_{\text{fixed}} = -26.0\text{ mM}$ equilibrated against a symmetric $1:1$ background electrolyte bath of concentration $c_{\text{salt}} = 110.0\text{ mM}$: (i) mobile co-factors of valence $z_S = +2$ partition across the boundary via Gibbs–Donnan electro-neutrality with monovalent anion-to-cation Donnan ratio $r_D < 1$ and electrostatically shifted dissociation constant $K_d r_D^{z_S}$; (ii) within the compartment, a cooperative Adair–Hill macromolecular buffer ($R_{\text{tot}} = 45.0\text{ }\mu\text{M}$, intrinsic $K_d = 18.5\text{ }\mu\text{M}$, $n_H = 2.0$) sequesters part of the intracompartmental ligand load $L_{\text{tot}} = 48.0\text{ }\mu\text{M}$ with free-ligand fraction $f_{\text{free}}$ and differential buffering sensitivity $S_{\text{buf}} = \partial L_{\text{free}} / \partial L_{\text{tot}}$; (iii) a monoprotic weak-acid buffer ($B_{\text{tot}} = 25.0\text{ mM}$, $\text{p}K_{a,\text{buf}} = 7.2$) absorbs a metabolic acid load $\Delta c_H = 4.7\text{ mM}$ coupled to the Donnan proton shift, modulating a diprotic catalytic site ($\text{p}K_{a,1} = 6.15$, $\text{p}K_{a,2} = 8.35$) relative to its bulk $\text{pH}_0$ active fraction; and (iv) the combined open-loop Donnan, free-ligand, and diprotic $\text{pH}$ drive $\Omega_{\text{open}}$ feeds a closed-loop Hill negative-feedback homeostatic regulator (gain $G_{\text{fb}} = 4.2$, Hill cooperativity $n_{\text{fb}} = 2.5$, half-saturation $K_{\text{fb}} = 0.4$) with steady-state regulated activity $u \in (0, \Omega_{\text{open}})$ and closed-loop logarithmic Bode sensitivity $S_{\text{Bode}} = \partial \ln u / \partial \ln \Omega_{\text{open}}$. Observable 4 (`bhattacharyya_overlap_integral`) is modulated by $\Omega_4 = u$, and Observable 5 (`transient_violation_probability`) by $\Omega_5 = u S_{\text{buf}} S_{\text{Bode}}$ (with $[0, 1]$ probabilities transforming via thermodynamic odds-ratio linkage).

## Required Output
Write a flat JSON object to `candidate.json` containing the following 5 keys:
1. `jarzynski_second_cumulant_dG_kBT`: Second-order Jarzynski free-energy estimate dG_{J,F}
2. `mean_forward_dissipated_work_kBT`: Mean forward dissipated work W_{diss,F}
3. `density_crossing_work_W_star_kBT`: Exact work crossing point W* in (mu_R, mu_F) where P_F(W*) = P_R(-W*) (k_B T)
4. `bhattacharyya_overlap_integral`: Bhattacharyya overlap integral int sqrt(P_F(W) P_R(-W)) dW
5. `transient_violation_probability`: Probability P_F(W < dG_{J,F}) of forward pulls below dG_{J,F}

### Output Schema Example (Placeholder Values Only)
```json
{
  "jarzynski_second_cumulant_dG_kBT": 0.0,
  "mean_forward_dissipated_work_kBT": 0.0,
  "density_crossing_work_W_star_kBT": 0.0,
  "bhattacharyya_overlap_integral": 0.0,
  "transient_violation_probability": 0.0
}
```