Correcting for the free energy costs of bond or angle constraints in molecular dynamics simulations

Gerhard König*, Bernard R. Brooks

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Background: Free energy simulations are an important tool in the arsenal of computational biophysics, allowing the calculation of thermodynamic properties of binding or enzymatic reactions. This paper introduces methods to increase the accuracy and precision of free energy calculations by calculating the free energy costs of constraints during post-processing. The primary purpose of employing constraints for these free energy methods is to increase the phase space overlap between ensembles, which is required for accuracy and convergence.

Methods: The free energy costs of applying or removing constraints are calculated as additional explicit steps in the free energy cycle. The new techniques focus on hard degrees of freedom and use both gradients and Hessian estimation. Enthalpy, vibrational entropy, and Jacobian free energy terms are considered. 

Results: We demonstrate the utility of this method with simple classical systems involving harmonic and anharmonic oscillators, four-atomic benchmark systems, an alchemical mutation of ethane to methanol, and free energy simulations between alanine and serine. The errors for the analytical test cases are all below 0.0007 kcal/mol, and the accuracy of the free energy results of ethane to methanol is improved from 0.15 to 0.04 kcal/mol. For the alanine to serine case, the phase space overlaps of the unconstrained simulations range between 0.15 and 0.9%. The introduction of constraints increases the overlap up to 2.05%. On average, the overlap increases by 94% relative to the unconstrained value and precision is doubled.

Conclusions: The approach reduces errors arising from constraints by about an order of magnitude. Free energy simulations benefit from the use of constraints through enhanced convergence and higher precision. 

General significance: The primary utility of this approach is to calculate free energies for systems with disparate energy surfaces and bonded terms, especially in multi-scale molecular mechanics/quantum mechanics simulations. This article is part of a Special Issue entitled Recent developments of molecular dynamics.

Original languageEnglish
Pages (from-to)932-943
Number of pages12
JournalBiochimica et Biophysica Acta - General Subjects
Issue number5
Early online date16 Sept 2014
Publication statusPublished - 1 May 2015


  • Bennett's acceptance ratio
  • Constraint correction
  • Free energy calculation
  • Molecular dynamics simulation
  • Normal mode analysis


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