From free-energy profiles to activation free energies

Given a chemical reaction going from reactant (R) to the product (P) on a potential energy surface (PES) and a collective variable (CV) discriminating between R and P, we define the free-energy profile (FEP) as the logarithm of the marginal Boltzmann distribution of the CV. This FEP is not a true free energy. Nevertheless, it is common to treat the FEP as the “free-energy” analog of the minimum potential energy path and to take the activation free energy, ΔF‡ RP, as the difference between the maximum at the transition state and the minimum at R. We show that this approximation can result in large errors. The FEP depends on the CV and is, therefore, not unique. For the same reaction, different discriminating CVs can yield different ΔF‡ RP. We derive an exact expression for the activation free energy that avoids this ambiguity. We find ΔF‡ RP to be a combination of the probability of the system being in the reactant state, the probability density on the dividing surface, and the thermal de Broglie wavelength associated with the transition. We apply our formalism to simple analytic models and realistic chemical systems and show that the FEP-based approximation applies only at low temperatures for CVs with a small effective mass. Most chemical reactions occur on complex, high-dimensional PES that cannot be treated analytically and pose the added challenge of choosing a good CV. We study the influence of that choice and find that, while the reaction free energy is largely unaffected, ΔF‡ RP is quite sensitive

Improved Sampling of Adaptive Path Collective Variables by Stabilized Extended-System Dynamics

Because of the complicated multistep nature of many biocatalytic reactions, an a priori definition of reaction coordinates is difficult. Therefore, we apply enhanced sampling algorithms along with adaptive path collective variables (PCVs), which converge to the minimum free energy path (MFEP) during the simulation. We show how PCVs can be combined with the highly efficient well-tempered metadynamics extended-system adaptive biasing force (WTM-eABF) hybrid sampling algorithm, offering dramatically increased sampling efficiency due to its fast adaptation to path updates. For this purpose, we address discontinuities of PCVs that can arise due to path shortcutting or path updates with a novel stabilization algorithm for extended-system methods. In addition, we show how the convergence of simulations can be further accelerated by utilizing the multistate Bennett’s acceptance ratio (MBAR) estimator. These methods are applied to the first step of the enzymatic reaction mechanism of pseudouridine synthases, where the ability of path WTM-eABF to efficiently explore intricate molecular transitions is demonstrated

Improved Sampling of Adaptive Path Collective Variables by Stabilized Extended-System Dynamics

Because of the complicated multistep nature of many biocatalytic reactions, an a priori definition of reaction coordinates is difficult. Therefore, we apply enhanced sampling algorithms along with adaptive path collective variables (PCVs), which converge to the minimum free energy path (MFEP) during the simulation. We show how PCVs can be combined with the highly efficient well-tempered metadynamics extended-system adaptive biasing force (WTM-eABF) hybrid sampling algorithm, offering dramatically increased sampling efficiency due to its fast adaptation to path updates. For this purpose, we address discontinuities of PCVs that can arise due to path shortcutting or path updates with a novel stabilization algorithm for extended-system methods. In addition, we show how the convergence of simulations can be further accelerated by utilizing the multistate Bennett’s acceptance ratio (MBAR) estimator. These methods are applied to the first step of the enzymatic reaction mechanism of pseudouridine synthases, where the ability of path WTM-eABF to efficiently explore intricate molecular transitions is demonstrated

From free-energy profiles to activation free energies

Given a chemical reaction going from reactant (R) to the product (P) on a potential energy surface (PES) and a collective variable (CV) discriminating between R and P, we define the free-energy profile (FEP) as the logarithm of the marginal Boltzmann distribution of the CV. This FEP is not a true free energy. Nevertheless, it is common to treat the FEP as the “free-energy” analog of the minimum potential energy path and to take the activation free energy, [Formula: see text], as the difference between the maximum at the transition state and the minimum at R. We show that this approximation can result in large errors. The FEP depends on the CV and is, therefore, not unique. For the same reaction, different discriminating CVs can yield different [Formula: see text]. We derive an exact expression for the activation free energy that avoids this ambiguity. We find [Formula: see text] to be a combination of the probability of the system being in the reactant state, the probability density on the dividing surface, and the thermal de Broglie wavelength associated with the transition. We apply our formalism to simple analytic models and realistic chemical systems and show that the FEP-based approximation applies only at low temperatures for CVs with a small effective mass. Most chemical reactions occur on complex, high-dimensional PES that cannot be treated analytically and pose the added challenge of choosing a good CV. We study the influence of that choice and find that, while the reaction free energy is largely unaffected, [Formula: see text] is quite sensitive. </jats:p