| """Mass-balanced equilibrium for a peptide connecting two protein partners. |
| |
| All concentrations and dissociation constants use nM. Alpha is dimensionless. |
| The six species are A, B, L, AL, BL, and ABL. This model assumes negligible |
| unliganded A:B association, monovalent binding, and solution equilibrium. |
| """ |
| from __future__ import annotations |
| import numpy as np |
| from scipy.optimize import least_squares, minimize_scalar |
|
|
|
|
| def equilibrium(a_total: float, b_total: float, l_total: float, |
| kd_a: float, kd_b: float, alpha: float) -> dict[str, float]: |
| values = np.array([a_total,b_total,l_total,kd_a,kd_b,alpha],float) |
| if not np.all(np.isfinite(values)) or np.any(values[:3] < 0) or np.any(values[3:] <= 0): |
| raise ValueError("invalid concentrations or equilibrium constants") |
| totals = values[:3] |
| if np.any(totals == 0): |
| |
| a,b,l = totals |
| al=bl=0. |
| if a>0 and l>0: |
| s=a+l+kd_a;al=2*a*l/(s+np.sqrt(s*s-4*a*l));a-=al;l-=al |
| elif b>0 and l>0: |
| s=b+l+kd_b;bl=2*b*l/(s+np.sqrt(s*s-4*b*l));b-=bl;l-=bl |
| return dict(A=float(a),B=float(b),L=float(l),AL=float(al),BL=float(bl),ABL=0.) |
| def species(logs): |
| a,b,l=np.exp(logs) |
| return np.array([a,b,l,a*l/kd_a,b*l/kd_b,alpha*a*b*l/(kd_a*kd_b)]) |
| def residual(logs): |
| a,b,l,al,bl,abl=species(logs) |
| return (np.array([a+al+abl,b+bl+abl,l+al+bl+abl])-totals)/totals |
| fit=least_squares(residual,np.log(totals)-1.,bounds=(np.log(totals)-80,np.log(totals)), |
| xtol=1e-12,gtol=1e-12,ftol=1e-12,max_nfev=1000) |
| if np.max(np.abs(residual(fit.x)))>1e-7: |
| raise RuntimeError("equilibrium solver failed mass conservation") |
| return dict(zip(["A","B","L","AL","BL","ABL"],map(float,species(fit.x)))) |
|
|
|
|
| def fit_cooperativity(totals, abl_nm, kd_a, kd_b, sd_nm=None): |
| """Fit alpha with binary Kd values fixed from separate binding measurements. |
| |
| totals has shape (N,3). abl_nm and sd_nm have shape (N,). This fitter expects |
| calibrated ternary concentrations. Raw BRET needs its own observation model. |
| """ |
| t=np.asarray(totals,float);y=np.asarray(abl_nm,float) |
| s=np.ones_like(y) if sd_nm is None else np.asarray(sd_nm,float) |
| if t.ndim!=2 or t.shape[1]!=3 or y.shape!=(len(t),) or s.shape!=y.shape or np.any(s<=0): |
| raise ValueError("invalid concentration-grid dimensions") |
| if len(y)==0 or not np.all(np.isfinite(t)) or not np.all(np.isfinite(y)) or not np.all(np.isfinite(s)) or np.any(y<0): |
| raise ValueError("invalid ternary data") |
| def objective(log_alpha): |
| pred=np.array([equilibrium(*row,kd_a,kd_b,np.exp(log_alpha))["ABL"] for row in t]) |
| return float(np.sum(((pred-y)/s)**2)) |
| result=minimize_scalar(objective,bounds=(-12.,12.),method="bounded",options={"xatol":1e-8}) |
| return {"alpha":float(np.exp(result.x)),"weighted_sse":float(result.fun), |
| "boundary":bool(abs(result.x)>11.9),"n":len(y)} |
|
|
