Assembly Equilibrium Fit Calculator

Calculator

Model

Choose the mathematical description you want to fit.

Fit mode

For n-mer: constant only, or constant plus n.

Ct unit in data

Keep units consistent across all rows.

Fixed n (n-mer)

Used when fit mode is single constant.

log10 parameter min

Lower search bound for log10(Ka/K/C50).

log10 parameter max

Upper search bound for log10(Ka/K/C50).

n range min (n-mer)

Used only in advanced fit mode.

n range max (n-mer)

Keep the range realistic for your system.

n step (n-mer)

Smaller steps search longer, but can improve fit.

Hill h min (Hill)

Used only for Hill model fitting.

Hill h max (Hill)

Increase if you expect very cooperative transitions.

Hill h step (Hill)

Grid spacing for h coefficient search.

Temperature (K)

Used for an approximate ΔG° estimate where applicable.

Data (Ct, fobs [, weight])

Paste rows separated by new lines. Use comma, tab, or spaces. Optional third column is a non-negative weight.

Or upload CSV

CSV columns: Ct, fobs, optional weight.

Load example data if textarea is empty

For n-mer and isodesmic models, constants assume a 1 M standard-state approximation. Keep units consistent, and interpret ΔG° as an estimate.

Example data table

Ct (µM)	Observed fraction assembled	Optional weight
1	0.05	1
2	0.09	1
5	0.22	1
10	0.38	1
20	0.58	1
50	0.78	1
100	0.88	1

You can paste these rows into the textarea as: 1,0.05 2,0.09 5,0.22 10,0.38 20,0.58 50,0.78 100,0.88

Formula used

Single-step n-mer assembly

Reaction: n M ⇌ M_n with association constant K_a. The model uses:

[M_n] = K_a[M]ⁿ
C_t = [M] + n[M_n]
f = n[M_n] / C_t

The calculator solves the mass balance for [M] by bisection, then computes f.

Isodesmic polymerization

With equal stepwise association constant K, the total concentration satisfies:

C_t = [M]/(1 - K[M])²
f = 1 - ([M]/C_t) = 1 - (1 - K[M])²

The calculator solves for x = K[M] in x = K C_t(1-x)².

Hill assembly curve

f = C_t^h / (C50^h + C_t^h)

This is a flexible curve-fit that summarizes cooperativity, not a full mass-balance assembly model.

Fitting minimizes the weighted sum of squares: SSE = Σ w_i(f_pred - f_obs)², then reports RMSE = √(SSE/n) and an AIC-style score.

How to use this calculator

Select a model that matches your assembly mechanism.
Choose your concentration unit and keep it consistent.
Paste or upload data as Ct and assembled fraction.
Set realistic log10 bounds for the fitted parameter.
Press Fit equilibrium to compute best-fit values.
Review residuals and re-fit with wider bounds if needed.
Download CSV or PDF to share or archive results.

If you want to emphasize certain points, add a third column weight. Larger weights make those rows influence the fit more.

Fit meaning in equilibrium terms

The fit translates concentration dependent yield into mechanistic parameters for self assembly. In the single step n mer model, the association constant Ka and stoichiometry n control equilibrium between monomer M and oligomer Mn under a 1 M standard state. Ka has units of M^(1−n), so changing n changes the apparent scale. The reported fraction assembled is n[Mn]/Ct.

Data requirements and formatting

Provide total concentration Ct and an observed assembled fraction f for each condition. Ct must be strictly positive and consistent in units across rows; the unit selector converts to molar for computation and back for reporting. Aim for 8 to 12 points spanning low signal to near plateau, because narrow ranges can make Ka or C50 poorly identified. Optional weights let you emphasize high precision measurements, such as averaged replicates. The parser accepts commas, tabs, or spaces.

Search strategy and numerical stability

Fitting uses a coarse grid in log10 space for Ka, K, or C50, then refines the best region with a bracketed optimizer. For advanced n mer mode, n is scanned across a user range with a chosen step size. Each prediction solves the mass balance by bisection, keeping the monomer solution between 0 and Ct. For isodesmic polymerization the solver finds x=K[M] from x=K Ct(1−x)^2, which avoids divergence.

Quality metrics and model comparison

RMSE summarizes average deviation between observed and predicted fractions on the 0 to 1 scale; an RMSE of 0.03 means about three percentage points typical error. An AIC style score balances fit against parameter count, helping compare a one parameter curve to a two parameter alternative on the same dataset. Inspect residual trends across Ct; systematic curvature often signals baseline bias or a mismatched mechanism. If the curve saturates early, widen log bounds and check whether f approaches one; incomplete saturation limits identifiability of strong constants often here.

Exports and reproducible reporting

CSV exports Ct in the entered unit and in molar, predicted f, residuals, and weights so you can recreate plots elsewhere. The PDF provides a compact snapshot of model choice, parameters, RMSE, AIC, and the first rows for quick review. For reporting, record the unit, temperature used for the optional ΔG° estimate, and the fitting bounds for reproducibility.

FAQs

What does Ct represent in this tool?

Ct is the total analytical concentration of all species, in your chosen unit. The calculator converts it to molar internally to apply mass balance equations.

When should I choose the n‑mer model versus isodesmic?

Use n‑mer when a dominant oligomer size forms in one step. Use isodesmic when growth is stepwise with similar binding energy per added monomer, producing a distribution of lengths.

Why are Ka or K reported with unusual units?

Association constants depend on reaction order. Ka for n‑mer assembly scales as M^(1−n), while isodesmic K scales as 1/M. The units follow directly from the equilibrium expressions and keep predictions dimensionally consistent.

How do weights influence the fit?

Weights multiply each squared residual in the objective. Larger weights force the fitted curve to pass closer to those points, which is useful when some measurements have smaller uncertainty or more reliable normalization.

What if my observed fraction is below 0 or above 1?

Values outside 0–1 usually indicate baseline drift or scaling issues. The calculator clamps inputs to protect the solver, but you should correct the raw signal and re‑normalize so the fitted parameters remain meaningful.

How is the PDF download created?

The PDF is generated directly on the server as a simple one‑page report. It includes the selected model, fitted parameters, RMSE, AIC, and a preview of the first data rows for documentation.