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Steady-state simulations using weighted ensemble path sampling

Divesh Bhatt, Bin W. Zhang, Daniel M. Zuckerman

arXiv:0910.5255v3physics.bio-phphysics.comp-ph

TL;DR

The paper addresses slow steady-state attainment by ordinary weighted ensemble sampling when metastable intermediates are present. It develops an enhanced WE scheme using probability adjustment, and reports that the enhanced method becomes more efficient than regular WE and brute force as landscape ruggedness increases.

  • Problem

    Ordinary WE reaches steady state slowly when significant metastable intermediates must become substantially populated.

    Source: §1 · p. 3

  • Method

    The paper extends weighted ensemble path sampling to steady state and uses probability adjustment to initialize simulations closer to steady-state bin probabilities.

    Source: §1 · p. 3, §2.5 · p. 6

  • Results

    As energy-landscape ruggedness increases, enhanced WE becomes more efficient than both regular WE and brute-force simulations.

    Source: §6 · p. 22

  • Takeaways & Limitations

    With minor changes to probability adjustment, enhanced WE can also sample equilibrium, and the adjustment can reduce equilibration time.

    Source: §6 · p. 22

    Rate-matrix estimates require τ to exceed intrabin relaxation time and multiple observed transitions for good statistics, which takes longer to estimate.

    Source: §2.5 · p. 7

Abstract

from arXiv · show

We extend the weighted ensemble (WE) path sampling method to perform rigorous statistical sampling for systems at steady state. The straightforward steady-state implementation of WE is directly practical for simple landscapes, but not when significant metastable intermediates states are present. We therefore develop an enhanced WE scheme, building on existing ideas, which accelerates attainment of steady state in complex systems. We apply both WE approaches to several model systems confirming their correctness and efficiency by comparison with brute-force results. The enhanced version is significantly faster than the brute force and straightforward WE for systems with WE bins that accurately reflect the reaction coordinate(s). The new WE methods can also be applied to equilibrium sampling, since equilibrium is a steady state.

1 Introduction

Steady-state sampling connects to first-passage kinetics, but existing path-sampling methods had not been applied to steady state. The paper extends weighted ensemble sampling and adds probability adjustment to speed convergence when metastable intermediates are present.

1 Introduction

  • Steady-state phenomena occur in biological and chemical systems, including homeostatic protein production and enzymatic reactions with constant reactant concentration and product removal.The paper also notes steady-state conditions in motor-protein movement and some in vitro enzymatic studies described by Michaelis–Menten kinetics [2,3].Source: §1 · pp. 1–2
  • Hill showed that steady-state flux from A to B with immediate feedback to A equals the inverse mean-first-passage time from A to B [4].The paper presents a path-sampling procedure for efficiently establishing steady state and calculating steady-state and first-passage rates.Source: §1 · p. 2
  • Path-sampling methods had been developed to compute rate constants for slow kinetics, but the authors knew of no application of these methods to steady state.Earlier approaches established steady state by enforcing net zero flux across configurational regions [20,21]; vanden–Eijnden and Venturoli extended this to calculate steady-state rate and flux [22].Source: §1 · p. 2
  • Enhanced WE extends weighted ensemble path sampling to rigorous statistical sampling at steady state, where feedback and statistical net zero flux establish the steady state.Unlike Dinner and coworkers’ approach, it does not enforce strict net zero flux in a region [20,21].Source: §1 · pp. 2–3
  • Probability adjustment speeds steady-state attainment when significant intermediates make ordinary WE slow because their bin probabilities take time to become stationary.The authors note that intermediate dwell times can also challenge methods that do not shortcut them.Source: §1 · p. 3

2 Method

The method establishes steady state through feedback and net probability-flux balance, then uses weighted ensemble sampling to reach it. Enhanced WE estimates near-steady bin populations to reduce relaxation, and the paper outlines how it evaluates efficiency against brute force.

2.1 General points

  • For region i, steady state means its probability remains constant as incoming and outgoing probability fluxes balance; equilibrium is the special case with no flux between regions.A more general steady state can have nonzero flows between regions, and feedback from the final state to the start is one way to establish it.Source: §2.1 · pp. 3–4

2.2 Steady state with brute force

  • Brute-force trajectories with feedback from the final state to the initial state eventually establish steady state, but large barriers make transitions inefficient to study.Source: §2.2 · p. 4

2.3 Brief review of generic Weighted Ensemble

  • Ordinary WE evolves trajectories for intervals τ, then splits or combines those in occupied bins while reallocating probabilities rigorously, without biasing system dynamics.Source: §2.3 · p. 5

2.4 ‘Regular’ weighted ensemble attainment of steady state

  • Regular WE attains steady state by feeding trajectories that enter the target back into the start; the feedback loop is part of the steady-state definition.The start can be a finite region or a single configuration, while the target must be a finite region to permit finite flux.Source: §2.4 · pp. 5–6

2.5 ‘Enhanced’ weighted ensemble attainment of steady state

  • In enhanced WE, unbiased trajectory dynamics estimate interbin transition rates, which yield near-steady bin probabilities for initialization before regular WE relaxation.The target-bin probability is set to zero and the remaining bin probabilities are normalized; accuracy of the estimates determines how much relaxation time is reduced.Source: §2.5 · pp. 6–7
  • Rate-matrix estimates for enhanced WE require dynamics after intrabin relaxation and multiple transitions for reliable statistics, trading longer sampling for more accurate rates.The authors’ implementation uses a single WE τ interval for rate estimates, though they note that multiple intervals could be supported with better bookkeeping.Source: §2.5 · p. 7

2.6 Equilibrium sampling

  • For equilibrium sampling, the procedure solves the steady-state equations for all bins, and accurate bin populations can reduce relaxation time to equilibrium.Source: §2.6 · p. 8

2.7 Analysis of WE efficiency versus brute force to obtain steady state

  • Efficiency comparisons focus on total simulation time to reach steady-state flux within one standard deviation, estimating brute-force time from the mean first-passage time.Enhanced WE flux is block-averaged after transients, while the brute-force estimate assumes Poissonian hopping between states separated by a high barrier.Source: §2.7 · pp. 9–10

3 Model systems and dynamics

The paper tests steady-state WE on one- and two-dimensional toy landscapes and all-atom alanine dipeptide, using systems with barriers, intermediates, and multiple pathways. The models also compare binning choices and specify the dynamics used for each system.

3.1 One dimensional toy model

  • The one-dimensional toy landscape contains three wells separated by 6kBT barriers, with state A at 0.98 < x < 1.17 and state B at x > 4.5.WE uses 25 bins, and each WE increment comprises 10 integration steps of overdamped Langevin dynamics.Source: §3.1 · pp. 10–11
  • The one-dimensional model uses an overdamped Langevin equation with Gaussian random displacements, and the WE target-state trajectories are fed back into state A.The random displacement has zero mean and variance 2kT∆t/(mγ).Source: §3.1 · p. 11

3.2 Two dimensional toy model

  • The two-dimensional toy landscapes have two reaction channels separated by significant barriers, allowing tests of multiple pathways and bins that do not optimally match the reaction coordinate.Two-dimensional bins isolate wells, whereas one-dimensional bins can contain separated wells and require longer relaxation to estimate populations accurately.Source: §3.2 · pp. 11–12
  • The two-dimensional study uses both one- and two-dimensional binnings on the same surfaces, with trajectories entering state B fed back to a single point.Bin indices identify discretized regions along x and y; the two-dimensional dynamics use the same parameters as the one-dimensional model.Source: §3.2 · pp. 11–12

3.3 Alanine Dipeptide

  • The alanine dipeptide model has significant intermediates and multiple pathways, with its implicit-solvent configurational space represented by the ψ and φ torsional angles.The initial state is C7eq, the target is C7ax, and periodicity of the angles permits multiple paths between states.Source: §3.3 · pp. 12–13
  • Alanine dipeptide is simulated atomistically with the CHARMM 19 force field and implicit solvent, using 30° × 30° WE bins and Langevin dynamics.Enhanced WE resampling occurs every 100 fs, while transitions are evaluated every 1000 fs at 500 K and every 10000 fs at 300 K.Source: §3.3 · p. 13

4 Results

Across the tested systems, enhanced WE reaches steady state faster when intermediates or rugged landscapes make brute-force and regular WE sampling slow. Its advantage is not universal: brute force is more efficient on a relatively smooth two-dimensional landscape, while the WE methods also agree with reference fluxes and can be used for equilibrium sampling.

4.1 One–dimensional system

  • On the one-dimensional landscape, enhanced and regular WE agree on steady-state flux, but enhanced WE reaches that flux significantly sooner.The authors report that probability adjustment does not disturb the natural system dynamics.Source: §4.1 · p. 14

4.2 Two–dimensional system

  • For two-dimensional surfaces, both WE versions match the brute-force-derived flux on the smoother landscape, and both binning schemes give correct fluxes.One-dimensional binning produces noisier estimates, but enhanced WE still reaches the correct flux faster with that binning.Source: §4.2 · p. 15
  • Enhanced WE reaches steady state in approximately the same time on smooth and rough 2D landscapes, whereas regular WE and brute force slow by more than three orders of magnitude on the rough landscape.The enhanced approach requires only a few transition events for probability adjustment, while brute-force time scales with the mean first-passage time.Source: §4.2 · p. 15, Efficiency comparison with brute force · p. 16
  • For a rugged 2D landscape, an intermediate-bin population plateaus quickly with enhanced WE but remains two orders of magnitude below that plateau after 10^5 τ increments with regular WE.The bin examined is at index (10,4).Source: §4.2 · p. 15

4.3 Alanine dipeptide

  • At 500 K, both alanine-dipeptide WE methods give the correct steady-state flux, reaching it in approximately the same number of τ increments.At 300 K, enhanced WE agrees with brute force while regular WE has not reached the correct flux after 1400 increments; the intermediate-bin population plateaus with enhanced but not regular WE.Source: §4.3 · pp. 16–17
  • At 300 K, enhanced WE reaches within one standard deviation of mean steady-state flux in 150 ns versus about 5 µs for brute force; on rougher landscapes, it progressively outperforms brute force.On the smoother surface, enhanced WE takes 2×10^7τ, versus 4×10^5τ for brute force, making brute force more efficient.Source: Efficiency comparison with brute force · pp. 16–17

4.4 Equilibrium sampling

  • Enhanced WE also establishes equilibrium faster than regular WE on the tested smooth and rugged 2D landscapes; regular WE does not equilibrate on the rugged landscape within the simulation timescale.The authors estimate enhanced-WE equilibrium times of approximately 5×10^7τ for the smooth surface and 5×10^8τ for the rugged surface.Source: §4.4 · p. 18, Efficiency comparison to brute–force… · p. 18

5 Discussion

Enhanced WE gains efficiency over regular WE as landscapes become more rugged. The discussion attributes this to early probability adjustment and outlines bin, population, repeated-adjustment, and implementation changes that could improve performance further.

5.1 Comparison with Markov models

  • Enhanced WE uses eq 1 only to estimate steady-state probabilities, so final rates remain exact and independent of a Markovian approximation.Unlike Markov models focused on hopping between states, WE produces continuous paths that include time spent within each state.Source: §5.1 · p. 19
  • Estimating interbin rates after intrabin relaxation improves eq 1 probability estimates, although this requires waiting for the Markovian picture to emerge.The needed relaxation time is within each bin.Source: §5.1 · p. 19

5.2 Comparing enhanced and regular WE simulation

  • Enhanced WE’s efficiency gain over regular WE becomes more pronounced as the energy landscape grows more rugged.The adjustment wait is tied to the arrival of fast trajectories, whose target-arrival time is affected less by ruggedness than the mean first-passage time.Source: §5.2 · pp. 19–20

5.3 Possible improvements

  • Bins closer to steady-state probabilities can minimize post-adjustment relaxation; smaller bins also reduce transverse relaxation within a bin.Initial paths or fast trajectories may help construct better bins.Source: §5.3 · p. 20
  • Allocating more trajectories to bins with high adjusted populations and slow population decreases may speed relaxation after probability adjustment.The current scheme gives every bin the same number of trajectories, which may be inefficient.Source: §5.3 · pp. 20–21
  • Repeated probability adjustments can use progressively shorter τ as within-bin distributions approach steady state, improving probability estimates per simulation time.Each rate calculation requires τ to exceed the intrabin relaxation time.Source: §5.3 · p. 21
  • For alanine dipeptide, CHARMM reinitialization overhead makes wallclock time scale nonlinearly with τ.Increasing τ from 100 fs to 1000 fs increased WE wallclock time by a factor of two; integrating WE with the dynamics code could improve efficiency.Source: §5.3 · p. 21

6 Conclusions

The paper presents WE with feedback into the starting state as a steady-state path-sampling method that generates continuous paths without relying on Markovian decomposition. Its enhanced probability adjustment is more efficient as landscapes become more rugged, and the approach also applies to equilibrium sampling.

6 Conclusions

  • A feedback loop into the starting state extends standard WE to establish steady state, generate continuous paths without Markovian decomposition, and calculate ordinary rates.The ordinary rate is the inverse mean-first-passage time.Source: §6 · pp. 21–22
  • As energy-landscape ruggedness increases, enhanced WE becomes more efficient than both regular WE and brute-force simulations.Source: §6 · p. 22
  • With minor probability-adjustment changes, enhanced WE applies to equilibrium sampling and allows rapid equilibration.Suggested further improvements include optimal bins and bin populations informed by initial paths, and integrating WE with the underlying dynamics code.Source: §6 · p. 22

One–dimensional system

The one-dimensional system uses a potential assembled from six complementary half-well pairs, with specified well parameters and steep boundary repulsions.

One–dimensional system

Two–dimensional system

The two-dimensional tests compare enhanced and regular WE across smooth and rugged landscapes, different binning schemes, alanine dipeptide, and equilibrium sampling. The figures assess flux or bin-probability convergence against brute-force results where shown.

Two–dimensional system

  • Across smooth and rugged 2D surfaces, enhanced WE reaches steady flux sooner than regular WE, while results agree across binning approaches and with long brute-force simulations.Source: Figure 6 · p. 33, Figure 7 · p. 34
  • For a single intermediate, enhanced WE reaches a steady bin probability fairly quickly, whereas regular WE does not.The plotted probability is for bin (10,4) and is on a log scale.Source: Figure 8 · p. 35
  • At 500 K, enhanced WE, regular WE, and long brute-force simulations converge to the same flux, but enhanced WE is not more efficient than regular WE.Source: Figure 9 · p. 36
  • For alanine dipeptide at 300 K, regular WE fails to reach steady state, whereas enhanced WE establishes it rapidly in both flux and intermediate-bin probability.Both panels are window averaged.Source: Figure 10 · p. 37
  • In equilibrium sampling, enhanced WE reaches bin probabilities faster than regular WE on both smooth and rugged 2D surfaces.Source: Figure 11 · p. 38, Figure 12 · p. 39
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