Alchemical Free Energy

Exploits the state function property of free energy by constructing a closed thermodynamic cycle. The non-physical alchemical transformation is combined with physical binding or solvation processes, allowing the computationally intractable direct physical path to be replaced by a calculable alchemical one. This is the foundational logic enabling relative binding free energy calculations.

Introduces a continuous coupling parameter λ ∈ [0,1] that scales the Hamiltonian between the initial state (λ=0) and final state (λ=1). At intermediate λ values, the system samples a mixed potential that represents a non-physical superposition of the two end states. The free energy difference is obtained by integrating the derivative of the Hamiltonian with respect to λ along this path.

Addresses the end-point singularity problem that occurs when atoms are created or annihilated during the alchemical transformation. Standard Lennard-Jones potentials diverge as interatomic distances approach zero, causing numerical instability. Soft-core potentials modify the interaction function to remain finite at zero distance, ensuring smooth and stable sampling near the end states.

A statistically optimal estimator that computes free energy differences by using all samples from all λ windows simultaneously. Unlike simpler methods like Thermodynamic Integration or the Bennett Acceptance Ratio, MBAR minimizes statistical variance by solving a set of self-consistent equations that reweight data across the entire ensemble of intermediate states, maximizing the information extracted from the simulation data.

Distinguishes between two calculation types:

• Relative Binding Free Energy (RBFE): Calculates the difference in binding affinity between two similar ligands by morphing one into the other within the binding pocket and in solution. Cancellation of errors improves accuracy.
• Absolute Binding Free Energy (ABFE): Calculates the binding affinity of a single ligand by physically separating it from the receptor along a defined path, requiring more extensive sampling but providing a direct thermodynamic measurement.

A hybrid Hamiltonian method where both the initial and final molecular topologies are present simultaneously throughout the simulation, but their interactions are scaled by λ. Atoms unique to the initial state have their interactions scaled to zero as λ → 1, while atoms unique to the final state have their interactions scaled from zero as λ → 0. This avoids the need to define a physical morphing path and handles ring-breaking transformations naturally.

The standard free energy change when a ligand binds to a receptor from an unbound state in solution. Unlike relative free energy, this involves physically separating the ligand from the binding pocket along a defined path, requiring careful treatment of translational and rotational restraints. Double-decoupling is the most common approach, where the ligand is alchemically annihilated in both the bound and unbound states.

A statistically optimal estimator for calculating free energy differences by combining data from all intermediate alchemical states simultaneously. MBAR minimizes statistical variance by solving a set of self-consistent equations that weight configurations from every lambda window. It supersedes the older Weighted Histogram Analysis Method (WHAM) and is the standard analysis tool in modern alchemical free energy workflows.

A method that computes free energy differences by integrating the ensemble average of the derivative of the Hamiltonian with respect to the coupling parameter lambda. The formula is:

• ΔG = ∫⟨∂H/∂λ⟩ dλ
• Requires smooth, continuous lambda schedules
• Often used alongside soft-core potentials to avoid endpoint singularities when atoms appear or disappear

An exponential averaging method that calculates free energy differences directly from the overlap of energy distributions between adjacent states. FEP relies on the Zwanzig equation:

• ΔG = -kBT ln⟨exp(-ΔU/kBT)⟩
• Sensitive to phase space overlap between states
• Often combined with BAR for bidirectional estimates when both forward and reverse perturbations are available

An extended ensemble approach where the alchemical coupling parameter λ is treated as a dynamic variable that evolves during the simulation. This allows the system to visit multiple alchemical states in a single continuous trajectory, automatically focusing sampling on thermodynamically relevant intermediates. λ-dynamics can efficiently screen multiple ligands simultaneously by using a multidimensional λ-space.

A non-equilibrium work theorem that relates the exponential average of work performed during fast, irreversible switching processes to the equilibrium free energy difference:

• exp(-ΔG/kBT) = ⟨exp(-W/kBT)⟩
• Enables free energy estimation from fast-growth or steered MD simulations
• Convergence requires sufficient sampling of rare low-work trajectories, making it challenging for complex biomolecular transformations