Free Energy Fundamentals

Free Energy Fundamentals
Theory Definitions Exponential Averaging / Zwanzig Thermodynamic Integration Bennett Acceptance Ratio WHAM MBAR
Methods of Free Energy Simulations Thermodynamic Cycle Intermediate States (Constructing) Soft Core Potentials Running Simulations Extracting Simulation Information Analysis / Bootstrap Sampling
Improving Sampling and Accelerating Simulations Umbrella Sampling Hamiltonian Replica Exchange [math]\displaystyle{ \lambda }[/math]-dynamics / Expanded Ensemble

Free Energy How-to's
Example Calculations Absolute Solvation Free Energy Relative Binding Affinity Absolute Binding Affinity
Tutorials Ethanol Solvation - GROMACS Expanded Ensemble - GROMACS Absolute Binding Free Energy - GROMACS

Before one can carry out free energy calculations, one must first understand the underlying principals behind them. This page is dedicated to the most fundamental concepts of free energy calculations and should be an excellent place to start for those wanting to start doing their own calculations. This page also contains some of the common nomenclature and symbols that are seen throughout the rest of the free energy fundamentals pages.

This page is not meant to be an end-all repository of the background mathematics and principals required for free energy calculations, but it will serve as a good start point and hopefully a quick reference.

Assumptions for the Fundamentals

The assumptions listed here are carried out through the rest of the fundamentals sections. It should be noted that free energy calculations do not need these assumptions to be carried out, but to establish a baseline set of simulation conditions and to make conveying the fundamentals easier, they are made. It is encouraged for researchers to explore the literature and develop methods to overcome these assumptions as doing so will advance the field even further.

• A Classic molecular mechanics model will be assumed; this includes:
  • Harmonic bond angle terms
  • Periodic dihedral terms
  • Non-bonded terms made up of point charges and Lennard-Jones repulsion/dispersion terms
• Temperature will be held constant (discussed below)
• Masses do not alchemically change. If one wishes to do this, substitute all potential energies, [math]\displaystyle{ U }[/math], with the more general Hamiltonian, [math]\displaystyle{ \mathcal{H} }[/math].
• QM/MM will not be considered for fundamentals because of the field has yet to be sufficiently developed.

Core Free Energy Equation

Most free energy calculations and methods started from a single core equation derived from statistical mechanics. The free energy difference between two states is directly related to the ratio of probabilities of those states through the partition functions. This free energy difference for an NVT ensemble is

[math]\displaystyle{ \displaystyle \Delta A_{ij} = -k_B T \ln \frac{Q_j}{Q_i} = -k_B T \ln \frac{\int_{V_j} e^{-\frac{U_j(\vec{q})}{k_B T}} d\vec{q}}{\int_{V_i} e^{-\frac{U_i(\vec{q})}{k_B T}}d\vec{q}} }[/math]

where [math]\displaystyle{ \Delta A_{ij} }[/math] is the Helmholtz free energy difference between state [math]\displaystyle{ j }[/math] and state [math]\displaystyle{ i }[/math], [math]\displaystyle{ k_B }[/math] the Boltzmann constant, [math]\displaystyle{ Q }[/math] the canonical partition function, [math]\displaystyle{ T }[/math] is the temperature, [math]\displaystyle{ U_i }[/math] and [math]\displaystyle{ U_j }[/math] are the potential energies as a function of the coordinates and momenta [math]\displaystyle{ \vec{q} }[/math] for two states, and [math]\displaystyle{ V_i }[/math] and [math]\displaystyle{ V_j }[/math] are the phase space volumes of [math]\displaystyle{ \vec{q} }[/math] over which we sample.

From this equation, all free energy calculations are derived.

Facts from the Core Equation

There are a number of key items we can learn from the core equation for free energy calculations. Each of these are critical in interpreting simulation results and should be committed to memory.

1. Only free energy differences are ever calculated. There is never a calculation where absolute free energies are needed (and rarely can the be calculated at all) as all of the biological or thermodynamical quantities of interest are based on a free energy difference. As such, there must always be a minimum of two defined states.
   • Even ``absolute free energies of binding are still free energy differences between two states: the ligand restricted to the binding site, and the ligand free to explore all other configurations.
2. Temperature does not change with state. If it did, you would get [math]\displaystyle{ \Delta A_{ij} = -k_B T_j \ln Q_j + k_B T_i \ln Q_i }[/math], which is no longer a ratio calculation and not needed for biological systems of interest. Temperature dependence on free energy is more likely to be what is [math]\displaystyle{ \Delta A_{ij} }[/math] change at two different temperatures.
3. There are two different phase space volumes. [math]\displaystyle{ V_I }[/math] and [math]\displaystyle{ V_j }[/math] are often the same, but they are not required to be. The methods presented in the fundamentals assume that the two phase spaces overlap. When the spaces do not overlap, these methods break down and it is difficult to identify this problem without in depth knowledge of your system.
   • The degree to which the phase spaces are shared is called the "phase space overlap" and there are ways to address is, but there are many aspects of it that still need addressed.
   • Consider the example of a hard sphere solute with radius [math]\displaystyle{ \sigma }[/math] at state [math]\displaystyle{ i }[/math] and a Lennard Jones repulsion/dispersion potential, with the same [math]\displaystyle{ \sigma }[/math] at state [math]\displaystyle{ j }[/math]. Since [math]\displaystyle{ V_i }[/math] will not have molecules at a distance less than [math]\displaystyle{ \sigma }[/math], but [math]\displaystyle{ V_j }[/math] will, the two phase spaces are not the same and these methods will either break down or return very error-prone results.
   • It should also be noted that "near zero probability" and "always zero probability" are two distinct things when considering phase space. So long as there is a chance for an observation to be made, no mater how small, it is considered part of the phase space.

Nomenclature and Variables

In an effort to keep things uniform, this section contains all the common constants and variables that are seen throughout the fundamentals sections. Although the Table itself is not in any particular order, the sorting buttons should help you find what you are looking for. The alternate listings of some variables are things one may encounter in literature, although this site will strive to be uniform in its naming process.

Variable, Acronym, Term	Definition	Notes
[math]\displaystyle{ U }[/math]	Potential Energy	This is a general potential energy and can be for either canonical ensemble where it is just the natural potential energy, or for grand canonical ensemble where [math]\displaystyle{ U=U_i + PV_i }[/math]. In ether case, the equations work out the same
[math]\displaystyle{ k_B }[/math]	Boltzmann's Constant
[math]\displaystyle{ T }[/math]	Temperature	This is often shown with [math]\displaystyle{ k_B }[/math] as well, and so common in fact that there is a common variable affiliated with it: [math]\displaystyle{ \beta }[/math].
[math]\displaystyle{ \beta }[/math]	Inverse Boltzmann Temperature [math]\displaystyle{ \beta=(k_BT)^{-1} }[/math]	Because it is so common in most the equations, one will see [math]\displaystyle{ \beta^{-1} }[/math] more often than [math]\displaystyle{ k_BT }[/math]
[math]\displaystyle{ P }[/math]	Pressure
[math]\displaystyle{ V }[/math]	Volume (traditional)	CAUTION: Volume in this sense is the kind you think of with box size. Not to be confused with the Phase Space Volume. This particular volume is not often considered by itself and should only show up on this site within [math]\displaystyle{ PV }[/math] terms, or NVT.
[math]\displaystyle{ V_i }[/math]	Phase Space Volume: Total set of coordinates and momenta where the system has a nonzero probability of being found	CAUTION: This is not volume in the traditional sense. It occasionally is seen as the variable [math]\displaystyle{ \Gamma }[/math].
[math]\displaystyle{ q }[/math] or [math]\displaystyle{ \vec{q} }[/math]	Collective variable for coordinates and momentum.	Can also be seen as [math]\displaystyle{ \mathbf{x} }[/math], [math]\displaystyle{ \mathbf{\vec{x}} }[/math], and a number of other terms in literature.
[math]\displaystyle{ \epsilon }[/math] and [math]\displaystyle{ \sigma }[/math]	Standard Lennard-Jones parameters
[math]\displaystyle{ Q }[/math]	Partition Function (Canonical, NVT), will also be the arbitrary partition function. [math]\displaystyle{ Q=\int_{V_i}\exp\left(-\beta U \right)\,\mathrm{d}\vec{q} }[/math]	In the examples here, this will represent the canonical partition function, although, the theories shown here work for an arbitrary partition function as well.
NVT	System held at constant Number of particles, Volume (traditional), and Temperature. Used for Helmholtz free energy calculations.
NPT	System held at constant Number of particles, Pressue, and Temperature. Used for Gibbs free energy calculations.
TI	Thermodynamic Integration
EXP	Exponential Averaging	Also called "free energy perturbation" and the "Zwanzig relationship."
BAR	Bennett Acceptance Ratio
MBAR	Multistate Bennett Acceptance Ratio
"Boltzmann Weight"	A phrase to reference [math]\displaystyle{ \exp\left(-\beta U \right) }[/math]	This is not technically a "weight" since it it not normalized.
State (variable [math]\displaystyle{ K }[/math]	A unique set of conditions and parameters that completely describe a thermodynamic system.	For all examples here, the upper case variable [math]\displaystyle{ K }[/math] will denote all states (or the count thereof), and the lower case variable [math]\displaystyle{ k }[/math] will refer to a specific state.