Rigorous free energy simulations

Thermodynamic integration (TI) and thermodynamic perturbation (TP) have been the most widely used methodologies in the above mentioned free 

The efficiency of a TI calculation can be improved using the interpolation method of Brooks [246]. According to the method, an ensemble average in a hybrid state A, can be calculated using conformations sampled at another hybrid state A, which results in the following equation for evaluating the particular 

P = /kgT, where kp is the Boltzman constant and T is the absolute temperature. The important consequence is that the number of windows in a TI calculation can be significantly reduced by performing only a limited number of simulations with the potentials and interpolating the ensemble averages (dYp /dX) for 

The other referred option, the TP formalism [247], is usually associated with the definition of a series of M hybrid states A for which M-1 free energy 

The number of windows can be significantly reduced when a double-wide sampling [248] is adopted a molecular simulation is carried out for each hybrid state A, and the free energy variations to the previous A,./ and to the next A,+/ states are evaluated using the sam.e produced trajectory. AF is then calculated as  

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In the method of constraints, a force of the form AV is applied at each step such that remains constant throughout the simulation. To see how A can be related to dA/d , we recall that the free energy A is the effective or average potential acting on . From physical intuition, it should be true that -dA/d is the average of the force acting on . In a constraint simulation, this force is equal to -A. Therefore we can expect to have d/l/d - - (A). We now make this statement more rigorous. 

We present and analyze the most important simplified free energy methods, emphasizing their connection to more-rigorous methods and the underlying theoretical framework. The simplified methods can all be superficially defined by their use of just one or two simulations to compare two systems, as opposed to many simulations along a complete connecting pathway. More importantly, the use of just one or two simulations implies a common approximation of a near-linear response of the system to a perturbation. Another important theme for simplified methods is the use, in many cases, of an implicit description of solvent usually a continuum dielectric model, often supplemented by a simple description of hydrophobic effects [11]. 

There now exist several methods for predicting the free energy associated with a compositional or conformational change.7 These can be crudely classified into two types "exact" and "approximate" free energy calculations. The former type, which we shall discuss in the following sections, is based directly on rigorous equations from classical statistical mechanics. The latter type, to be discussed later in this chapter, starts with statistical mechanics, but then combines these equations with assumptions and approximations to allow simulations to be carried out more rapidly. 

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