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Adaptively Biased Molecular Dynamics for Free Energy Calculations
Volodymyr Babin, Christopher Roland, Celeste Sagui
TL;DR
The paper addresses the need for efficient free-energy calculations with non-equilibrium dynamics, especially where metadynamics requires costly long simulations. It presents ABMD, an evolving-potential umbrella-sampling method with multiple-walker and replica-exchange extensions, and demonstrates its use in peptide folding studies. The method provides O(t) scaling, improves speed and accuracy through enhanced sampling, and supports reusing results for new collective variables.
Problem
Metadynamics may require long biomolecular simulations, has O(t^2) cost, and involves many parameters whose effects are entangled.
Method
ABMD computes free-energy surfaces with a time-dependent biasing potential and supports multiple walkers plus replica exchange across temperatures and collective variables.
Results
ABMD has O(t) scaling, while its multiple-walker and replica-exchange variants improve speed and accuracy through better sampling.
Takeaways & Limitations
Replica exchange can provide free-energy projections for additional collective variables and reuse previously obtained results to enhance new calculations.
Abstract
from arXiv · showhide
We present an Adaptively Biased Molecular Dynamics (ABMD) method for the computation of the free energy surface of a reaction coordinate using non-equilibrium dynamics. The ABMD method belongs to the general category of umbrella sampling methods with an evolving biasing potential, and is inspired by the metadynamics method. The ABMD method has several useful features, including a small number of control parameters, and an $O(t)$ numerical cost with molecular dynamics time $t$. The ABMD method naturally allows for extensions based on multiple walkers and replica exchange, where different replicas can have different temperatures and/or collective variables. This is beneficial not only in terms of the speed and accuracy of a calculation, but also in terms of the amount of useful information that may be obtained from a given simulation. The workings of the ABMD method are illustrated via a study of the folding of the Ace-GGPGGG-Nme peptide in a gaseous and solvated environment.
I. INTRODUCTION
ABMD is introduced as an efficient non-equilibrium free-energy method that uses an evolving biasing potential, with extensions for multiple walkers and replica exchange. It addresses metadynamics' costly history-dependent potential and large parameter set through O(t) scaling and two control parameters.
- ABMD computes a reaction-coordinate free-energy surface using non-equilibrium dynamics and an evolving umbrella-sampling potential.
- ABMD uses only two control parameters and has O(t) numerical cost with molecular-dynamics time.
- Metadynamics can require long runs, has O(t^2) cost, and uses many entangled control parameters.
- The evolving biasing potential is updated alongside ordinary molecular dynamics and converges toward the negative free energy under suitable flooding timescale and kernel-width conditions.The kernel is positive, symmetric, and interpreted as a smoothed Dirac delta function.
- The implementation uses discretized bias potentials and has constant evaluation cost over time, giving O(t) scaling and reasonable storage requirements.The bias is discretized with cubic B-splines, and sparse arrays can reduce storage for multidimensional variables.
- Multiple walkers share one evolving bias, while replica exchange can combine replicas with different temperatures, collective variables, or static and evolving biases.These extensions improve sampling and can enhance convergence to the free energy.
III. CASE STUDY: A SHORT PEPTIDE.
The peptide case study benchmarks ABMD free-energy estimation and extends it from single simulations to multiple walkers and replica exchange. These extensions improve sampling and enable higher-dimensional free-energy characterization in gas and solvated environments.
- Case-study setup: The Ace-GGPGGG-Nme peptide was simulated in the gas phase and in cyclohexane to illustrate ABMD free-energy calculations.The radius of gyration of the heavy atoms was used as the collective variable, with a reference profile for benchmarking.
- Single-trajectory benchmark: ABMD was more accurate than the corresponding metadynamics run and used about 25 times less memory for the biasing potential.The comparison used the same kernel width and flooding timescale; the reported memory advantage was based on roughly 10^8 ABMD hills versus 5 × 10^3 metadynamics Gaussians.
- Multiple walkers: Multiple walkers provided nearly linear speedup for moderate walker numbers while also improving accuracy through better sampling of the evolving canonical distribution.The multiple-walker comparison used one, two, four, and eight trajectories at T = 300 K with τF = 1 ns.
- Parallel tempering: Eight-replica parallel tempering maintained accuracy even when τF was reduced to 11.2 ps.The replicas spanned 300 K to 600 K, and simulations with τF = 90 ps reached an ERMS of approximately 1 kcal/mol or less as t →∞.
- General replica exchange: General replica exchange reused the radius-of-gyration free energy to accelerate a two-dimensional (Rg, NOH) calculation that distinguished at least two globular states.The eight replicas enhanced sampling for a ninth replica, and the resulting map added information beyond one-dimensional free-energy profiles.
- Solvated peptide: In cyclohexane, the folded β-turn was clearly favored over the globular structure.The solvated-peptide calculation used replica-exchange ABMD at T = 300 K, including coarse and finer nonequilibrium estimates and an equilibrium biased-replica-exchange correction.
IV. CONCLUSIONS AND OUTLOOK.
ABMD computes reaction-coordinate free energy surfaces with non-equilibrium dynamics using two control parameters and O(t) scaling. Its multiple-walker and replica-exchange extensions improve sampling and enable projections across collective variables, while the peptide study remains an initial demonstration.
- ABMD computes reaction-coordinate free energy surfaces using non-equilibrium dynamics, with two control parameters and O(t) scaling in molecular dynamics time.The control parameters are the flooding timescale and kernel width.
- Multiple walkers and replica exchange improve speed and accuracy through better sampling of the evolving canonical distribution.
- Replica exchange can use different temperatures or collective variables and produce free-energy projections for multiple variables.
- Figure 8 reports accurate gas-phase peptide free energies as functions of Rg across temperatures from 300 K to 600 K.
- ABMD was implemented in AMBER, and its application to more complicated biomolecular systems was reserved for future publications.
APPENDIX A: REFERENCE FREE ENERGY CURVE.
The appendix provides simulation details for constructing the reference free-energy curve.
- The appendix introduces the simulation details used for the reference free-energy curve.
reference free energy curve.
The reference curve was generated with staged multiple-walker and parallel-tempering ABMD simulations, followed by a long run for error estimation. The resulting free-energy curves span several temperatures and have a reported maximum error of about 0.15 kcal/mol over the stated Rg interval.
- A 5 ns, eight-walker ABMD run at 600 K reconstructed the global well before parallel-tempering runs used its bias as the initial bias across eight temperatures.
- ≈0.15 kcal/mol was the reported maximum error over 3.3 Å < Rg < 6.3 Å, while the RMS error was considered probably much smaller.
- The resulting accurate free-energy curves were reported as functions of temperature.