Crossing a barrier with overlapping umbrella windows

Hold a rarely visited position with a spring

Choose collective variable $s(x,y)$ . Window $k$ adds a spring centered at $s_k$ with stiffness $\kappa_k$ . The original potential remains $U$ ; sampling uses $U+w_k$ . Dimensional stiffness has units of energy per squared coordinate; our values are dimensionless.

The windows divide the work. Each trajectory explores a local region without repeatedly completing the entire rare transition. That region still has to equilibrate, including directions the spring does not drive.

w_k(s)=\frac{\kappa_k}{2}(s-s_k)^2,\qquad p_k(\mathbf r)=\frac{e^{-\beta[U(\mathbf r)+w_k(s(\mathbf r))]}}{Z_k}

Each window has its own partition function. The bias changes which configurations are common during sampling; the target remains the original equilibrium distribution.

Source: Giovanni Bussi · Michael R. Shirts and John D. Chodera

Undo the spring’s preference

Multiplying the biased density by $e^{\beta w_k}$ cancels the spring in the exponent. An unknown normalization disappears in a ratio of weighted averages. The identity can still be a poor finite-sample estimator when a few large weights represent an important region.

Worked example: two classes appear $80$ and $20$ times with spring energies zero and $k_{\mathrm B}T\ln 4$ . Each observation of the second class receives weight four. Total weights become $80$ and $80$ , giving equal unbiased populations. Raw counts would incorrectly favor the first class fourfold.

\langle A\rangle_0=\frac{\langle A e^{\beta w_k}\rangle_k}{\langle e^{\beta w_k}\rangle_k}

The denominator normalizes the result. Distinct windows have distinct normalization constants; combining them requires their relative values.

Estimate a width, then inspect actual overlap

If the local free energy is flat, a spring gives Gaussian width $\sigma_s=\sqrt{k_{\mathrm B}T/\kappa}$ . At unit thermal energy and stiffness nine the width is about $0.333$ . Centers separated by one third share substantial probability. A separation of two is six standard deviations in this approximation.

The landscape can shift and narrow the actual distributions. Thermal widths start the design; observed distributions, weights and exchanges test it. Evenly spaced spring centers alone establish little.

Original Gaussian design schematic. Top: neighboring spring-only distributions share appreciable support. Bottom: widely separated windows leave a gap. Actual overlap must be checked from sampled full configurations.

Projected overlap can hide incompatible configurations

A spring restrains $x$ , but $y$ has two valleys separated by a high barrier. Neighboring $x$ histograms may overlap even when one trajectory stays in the upper valley and another in the lower one. Their full configurations may share almost no support.

Start each window in both hidden valleys, compare conditional distributions and extend the runs. Persistent disagreement calls for more hidden-coordinate sampling, a suitable second variable or another method. More springs along the same coordinate may not solve it.

A disconnected chain cannot reliably determine the relative free-energy zero of its two parts. Report supported within-part shapes and retain the gap. Smooth interpolation across it would imply information absent from the data.

Try the reasoning in an experiment

Design a connected chain before sampling

Estimate thermal widths and choose the overlapping preset. Compare raw pooled counts with the reweighted shape.
Drag a window away or use the gapped preset. Predict which relative height loses support, then inspect the overlap warning.
Choose hidden-coordinate trapping. Inspect the hidden distribution at equal projected position, export the seeded trajectories and propose a change for the next run.

Exercises and explanations

Increase spring stiffness fourfold. How does its width change in a flat region, and does precision necessarily improve?
Read the explanation
The standard deviation halves. A window localizes more tightly, but neighboring windows can share fewer configurations. Global precision depends on coverage and overlap as well as local counts.
Every projected histogram has large counts. Why inspect conditional distributions?
Read the explanation
Large counts establish repeated visits to the projection. An uncontrolled slow coordinate can retain its initial state and give incorrect conditional weights. Different starts and hidden-coordinate distributions help expose that error.

Sources and reproduced scope

Worked examples and schematic figures are original teaching material. The numerical research slice compares specified estimator equations on our seeded two-coordinate records with independent quadrature and PyMBAR. The final chapter reads Figure 1 of the MBAR paper and states which experimental data remain outside that calculation.

Model definitions, scripts and numerical records

Giovanni Bussi, PLUMED Masterclass 21.3, Umbrella sampling (2021; documentation v2.9)
Exercises on static restraints, combining windows with binless WHAM, reweighting a different coordinate and uncertainty. Its alanine-dipeptide trajectories are not used here.
Michael R. Shirts and John D. Chodera, Statistically optimal analysis of samples from multiple equilibrium states, J. Chem. Phys. 129, 124105 (2008)
Frozen arXiv v3: Sections II–IV, Eqs. (9), (11)–(14), Section V and Figure 1, Appendices A–C. We implement the equilibrium estimating equations on our synthetic model. Figure 1 is read as an experimental example; its DNA-hairpin traces and plotted values are not reproduced.