Multistate Bennett Acceptance Ratio Free Energy Calculator

Calculator

Input potential unit

Output free energy unit

Temperature K

Reference state

Maximum iterations

Convergence tolerance

Iteration damping

Bootstrap replicates

Random seed

Sample matrix

Use CSV rows. First column is sampled state. Remaining columns are potentials for states 0, 1, 2, and onward.

Example Data Table

Sampled state	u0	u1	u2
0	0.00	1.10	2.30
0	0.20	1.00	2.10
1	1.20	0.00	1.15
1	0.90	0.10	1.00
2	2.50	1.20	0.00
2	2.20	0.90	0.15

Formula Used

The calculator solves the MBAR self consistent equations for reduced free energies.

f_i = -ln sum_n [ exp(-u_i(x_n)) / sum_k N_k exp(f_k - u_k(x_n)) ]

Here, f_i is the reduced free energy of state i. The term u_i(x_n) is the reduced potential of sample n evaluated at state i. The value N_k is the number of samples drawn from state k.

If energies are pasted, the calculator first uses u = E / (R T). Then it reports Delta F = R T Delta f when an energy output unit is selected.

How to Use This Calculator

Paste a CSV matrix with one sampled state column.
Add one potential column for every target state.
Select whether pasted values are reduced potentials or energies.
Enter temperature, reference state, tolerance, and iterations.
Use bootstrap replicates when a quick uncertainty spread is needed.
Press Calculate and read the result above the form.
Check overlap and effective sample size before exporting.

About Multistate Free Energy Estimation

Why MBAR Helps

Multistate Bennett acceptance ratio estimates free energy differences from many sampled states at once. It is common in molecular simulation. Each row represents one sampled configuration. Each column stores the reduced potential of that configuration in a target state. The method uses all rows together. It avoids comparing only two neighboring states.

Core Idea

The calculator solves dimensionless free energy offsets. It uses self consistent MBAR equations. The denominator mixes sample counts with trial offsets and reduced potentials. Each iteration updates every state. A reference shift keeps one state equal to zero. Convergence is reached when the largest offset change becomes smaller than the tolerance.

Data Quality Matters

Good overlap is essential. Samples from one state should still have reasonable probability in nearby states. Poor overlap causes unstable gaps. The overlap table is a quick diagnostic. Effective sample size also helps. A low value means few configurations carry most of the weight. More simulations or added intermediate states may be needed.

Energy Units

MBAR naturally works with reduced potentials. These values are unitless. They usually equal beta times potential energy. When you paste energies, the calculator converts them using temperature and the gas constant. Reported gaps can be shown in reduced units, kilojoules per mole, or kilocalories per mole. The chosen reference state controls the zero point, not the physical differences.

Uncertainty Notes

The optional bootstrap repeats the estimate on resampled data. It gives a practical spread for each relative free energy. It is not a full replacement for careful statistical analysis. Correlated trajectories should be subsampled first. Equilibrated data should be used. Always compare convergence, overlap, and uncertainty before trusting one number.

Limits

This calculator does not remove bias from bad sampling. It cannot create missing overlap. It also assumes the pasted matrix is already cleaned. Use independent blocks when possible. Compare with known benchmarks. Treat unusual warnings as a reason to inspect trajectories, restraints, temperatures, and state definitions carefully.

Practical Use

Use this tool for quick checks, teaching, and reproducible notes. Start with the example table. Replace it with your simulation matrix. Keep state labels as integers. Keep the same number of potential columns in every row. Then calculate, review diagnostics, and export the results.

FAQs

What does MBAR estimate?

It estimates relative free energy differences between states. It uses samples from all states together, rather than relying only on pairwise comparisons.

What is a reduced potential?

A reduced potential is a unitless energy value. It is commonly beta times potential energy, where beta equals one divided by thermal energy.

Can I paste normal energy values?

Yes. Choose kJ/mol or kcal/mol as the input unit. The calculator converts energies to reduced potentials using the selected temperature.

Why is overlap important?

MBAR needs shared probability between states. Low overlap means few samples connect the states. Results may then become noisy or biased.

What does effective sample size mean?

It estimates how many samples strongly support each state. Low values suggest that only a few configurations dominate the weighted estimate.

What reference state should I choose?

Choose any state as zero. Relative differences stay consistent. The reference only changes the displayed origin of the free energy scale.

Is bootstrap uncertainty exact?

No. It is a practical spread from resampled rows. For correlated simulations, use decorrelated data and compare block based analyses.

Why did the solver fail to converge?

Possible causes include poor overlap, extreme potentials, too few iterations, or weak damping. Try more iterations and inspect the input matrix.