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

To understand the advantages and disadvantages of different free energy methods, it's important to review the underlying principles first. This page is dedicated to the most fundamental concepts of free energy calculations and is designed to give an in-depth view of the approaches, but starting from the basics. This page also contains some of the common nomenclature and symbols that are seen throughout the rest of the free energy fundamentals pages and in the literature as a whole.

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.

Why the name "Alchemical"?

One of the first questions, and often most confusing points, about a number of free energy calculations why we refer to them as "alchemical" changes in a large number of computational methods. When people first hear the word "alchemy," they usually think of the medieval alchemists who were trying to turn lead into gold, or other such transformations. How computational free energy adopted the name takes a bit of understanding of simulation limitations and the properties of what we are calculating.

Considering that the natural evolution of some of the processes we try to model are well beyond reasonable simulation times, we must come up with very efficient ways to compute the free energy differences. Because free energy is a state variable (path independent), we can design simulations that provide a convenient pathways to computing free energy. Furthermore, since we are doing simulations, we are not limited by experimentally observable conditions so long as we are meticulous with our calculations.

With these observations in mind, it was found that we can simulate modifications to atoms which can change their properties to reflect non-physical or entirely different species. This is roughly the same definition of alchemy from old in that we are changing the properties of the atoms to be something else, although we do it in a mathematically sound and rigorous manner; hence, the term "alchemical" was coined as many free energy calculations are "like alchemy" in their pathways and methods.

There is of course many factors and checks that must be done to ensure accuracy and robustness of the calculations, many of which will be shown in the rest of the free energy fundamentals pages.

Assumptions for the Fundamentals

The assumptions listed here are carried out through the rest of the fundamentals sections. Free energy calculations can usually be set up without these assumptions, but we'll make these assumptions to simplify the explanations.

• A standard 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
• constant temperature is maintained: (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.

Free Energy Difference 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_{\Gamma_j} e^{-\frac{U_j(\vec{q})}{k_B T}} d\vec{q}}{\int_{\Gamma_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{ \Gamma_i }[/math] and [math]\displaystyle{ \Gamma_j }[/math] are the phase space volumes of [math]\displaystyle{ \vec{q} }[/math] over which we sample, or the total set of all allowed positions and momenta of the system.

From this equation, all free energy calculations are derived.

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{ k_B }[/math]	Boltzmann's Constant
[math]\displaystyle{ U }[/math]	Potential Energy	This is the total internal energy calculated a classical simulation, the sum of the potential and kinetic energies.
[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{ u }[/math]	Reduced potential, expressed as [math]\displaystyle{ u=\beta(U+PV-\sum \mu N_i) }[/math]. For many equations, it is simpler to work with the reduced potential.	In the case of NVT ensembles, the PV and [math]\displaystyle{ \mu N_i }[/math] terms are zero. For NPT ensembles, the [math]\displaystyle{ \sum \mu N_i }[/math] is zero, and for [math]\displaystyle{ NV\mu }[/math] ensembles, the PV term is zero.
[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{ \Gamma }[/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 standard physical sense, meaning the volume of the box, but the volume space times the momentum space.
[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]	Lennard-Jones parameters for the well depth and the radius of the point particle respectively.
[math]\displaystyle{ Q }[/math]	Partition function for the canonical or NVT ensemble, but sometimes also signifies an arbitrary partition function of any ensemble. [math]\displaystyle{ Q=\int_{\Gamma}e^{-\beta U (\vec{q})}\,\mathrm{d}\vec{q} }[/math]	In the examples here, this will represent the canonical partition function although most of the theory shown here works for an arbitrary partition function as well, though sometimes with slight modifications, where [math]\displaystyle{ Q=\int_{\Gamma}e^{-u(\vec{q})}\,\mathrm{d}\vec{q} }[/math]
[math]\displaystyle{ \tau }[/math]	Autocorrelation time
NVT	System held at constant Number of particles, Volume, and Temperature. Free energy calculations performed in this ensemble yield Helmholtz free energies.
NPT	System held at constant Number of particles, Pressue, and Temperature. Free energy calculations performed in this ensemble yield Gibbs free energies.
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	The probability of a microstate [math]\displaystyle{ \vec{q} }[/math], defined [math]\displaystyle{ e^{-u(\vec{q})}/Q }[/math], where [math]\displaystyle{ u(\vec{q}) }[/math] is the reduced potential. It can also refer to simply [math]\displaystyle{ e^{-u(\vec{q})} }[/math], although this is not a full weight because it is not normalized by the partition function.
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.
[math]\displaystyle{ \lambda }[/math]	Alchemical variable, thermodynamic path coordinate	The most common symbol for denoting/tracking the progress along some alchemical pathway. Since most free energy calculations involve an alchemical thermodynamic path, this site was named accordingly.
Annihilating/Annihilated	The act of removing ALL of an atoms interactions with the system, this includes the molecule's interactions with itself.	Intermolecular force = OFF, Intramolecular forces = OFF. Also known as disappeared or destroyed. Not to be confused with "decoupling."
Decoupling	The act of removing an atom or molecules interactions with its surroundings ONLY; the interactions between the molecule itself remain active	Intermolecular force = OFF, Intramolecular forces = ON. Not to be confused with "annihilation."

Facts from the Free Energy Difference Definition

There are a number of key notes we can learn from the definition of free energy differences. Each of these can be important in interpreting simulation results.

1. Only free energy differences are ever calculated. There is never a calculation where absolute free energies are needed (and rarely can they 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 thermodynamic 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. Free energy differences between states at different temperatures are usually not what you want to be calculating for problems of interest. 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{ \Gamma_i }[/math] and [math]\displaystyle{ \Gamma_j }[/math] are often the same, but they are not required to be. The methods presented here almost always assumes that the two phase spaces overlap. However, 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. 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{ \Gamma _i }[/math] will not have molecules at a distance less than [math]\displaystyle{ \sigma }[/math], but [math]\displaystyle{ \Gamma_j }[/math] will, the two phase spaces are not the same and these methods will either break down or return very error-prone results.
   • The degree to which the phase spaces are shared is called the "phase space overlap". Efficient free energy calculations require significant phase space overlap. There are a number of strategies to address lack of overlap between target spaces, but determining the best way for any given situation is still a research question.
   • 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.