Conformational free energy differences

We have previously calculated conformational free energy differences for a well-suited model system, the catalytic subunit of cAMP-dependent protein kinase (cAPK), which is the best characterized member of the protein kinase family. It has been crystallized in three different conformations and our main focus was on how ligand binding shifts the equilibrium among these ([Helms and McCammon 1997]). As an example using state-of-the-art computational techniques, we summarize the main conclusions of this study and discuss a variety of methods that may be used to extend this study into the dynamic regime of protein domain motion. 

Conformational free energy simulations are being widely used in modeling of complex molecular systems [1]. Recent examples of applications include study of torsions in n-butane [2] and peptide sidechains [3, 4], as well as aggregation of methane [5] and a helix bundle protein in water [6]. Calculating free energy differences between molecular states is valuable because they are observable thermodynamic quantities, related to equilibrium constants and... 

The second application of the CFTI approach described here involves calculations of the free energy differences between conformers of the linear form of the opioid pentapeptide DPDPE in aqueous solution [9, 10]. DPDPE (Tyr-D-Pen-Gly-Phe-D-Pen, where D-Pen is the D isomer of /3,/3-dimethylcysteine) and other opioids are an interesting class of biologically active peptides which exhibit a strong correlation between conformation and affinity and selectivity for different receptors. The cyclic form of DPDPE contains a disulfide bond constraint, and is a highly specific S opioid [llj. Our simulations provide information on the cost of pre-organizing the linear peptide from its stable solution structure to a cyclic-like precursor for disulfide bond formation. Such... 

The Cyc conformer represents the structure adopted by the linear peptide prior to disulfide bond formation, while the two /3-turns are representative stable structures of linear DPDPE. The free energy differences of 4.0 kcal/mol between pc and Cyc, and 6.3 kcal/mol between pE and Cyc, reflect the cost of pre-organizing the linear peptide into a conformation conducive for disulfide bond formation. Such a conformational change is a pre-requisite for the chemical reaction of S-S bond formation to proceed. 

The free energy differences obtained from our constrained simulations refer to strictly specified states, defined by single points in the 14-dimensional dihedral space. Standard concepts of a molecular conformation include some region, or volume in that space, explored by thermal fluctuations around a transient equilibrium structure. To obtain the free energy differences between conformers of the unconstrained peptide, a correction for the thermodynamic state is needed. The volume of explored conformational space may be estimated from the covariance matrix of the coordinates of interest, = ((Ci [13, lOj. For each of the four selected conform-... 

The free energy difference between two stable conformations can be obtained by a thermodynamic integration approach [38,39]. Let q and represent the centers of the two corresponding energy wells. The free energy derivative is seen to be... 

A prototype of such phenomena can be seen in even the simplest carboxylic acid, acetic acid (CH3CHOOH). Acidity is determined by the energy or free energy difference between the dissociated and nondissociated forms, whose energetics usually depend significantly on their conformation, e.g., the syn/anti conformational change of the carboxyl-ate group in the compound substantially affects the acid-base equilibrium. The coupled conformation and solvent effects on acidity is treated in Ref. 20. 

The free-energy difference between conformers is referred to as the conformational free energy. For substituted cyclohexanes, it is conventional to specify the value of — AC ° for the equilibrium... 

The steric parameters for the estimation of reactant state effects were chosen to be the conformational free energy differences for cyclohexane axial-equatorial equilibria (A-values) (8). In order to establish the methyl group as the standard size group, modified A-values (A ) for the various groups were used, by simply subtracting the A value for the methyl group (1.70) from the A values of the various substituents ... 

This equation means that when there is a free energy difference of a few fcs T the probability P( ) is reduced considerably, that is, those conformations with large A( ) are sampled very rarely. This is a very important observation in terms of numerical efficiency. At the transition region for example, the free energy is maximum and typically very few sample points are obtained during the course of molecular dynamics simulation. In turn this results in very large statistical errors. Those errors can only be reduced by increasing the simulation time, sometimes beyond what is practically feasible. 

The same technique can be used in some cases to obtain accurate estimates of binding free energy differences for a set of ligands of interest [25, 31-34]. The molecule taken as the reference need not be a real molecule. Indeed, the reference molecule could be intermediate between a large set of molecules of interest, so that conformations that are sufficiently representative of them all are sampled in the reference simulation. The justification for this approach is discussed in detail in Chap. 6. To achieve this for a variety of substituted phenols, Liu et al. [25] added dummy atoms to the ring at the sites they wished to substitute. Such dummy atoms can be softer ... 

Protein stability is just the difference in free energy between the correctly folded structure of a protein and the unfolded, denatured form. In the denatured form, the protein is unfolded, side chains and the peptide backbone are exposed to water, and the protein is conformationally mobile (moving around between a lot of different, random structures). The more stable the protein, the larger the free energy difference between the unfolded form and the native structure. 

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