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**|**HHS Author Manuscripts**|**PMC2716075

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- Abstract
- I.Introduction
- II. General theoretical considerations
- III. Methods for calculating the free energy
- IV. Conclusions
- References

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Curr Protein Pept Sci. Author manuscript; available in PMC 2010 April 1.

Published in final edited form as:

PMCID: PMC2716075

NIHMSID: NIHMS97861

Department of Computational Biology, University of Pittsburgh School of Medicine, 3059 BST3, Pittsburgh, PA 15260

The publisher's final edited version of this article is available at Curr Protein Pept Sci

See other articles in PMC that cite the published article.

The Helmholtz free energy, *F* and the entropy, *S* are related thermodynamic quantities with a special importance in structural biology. We describe the difficulties in calculating these quantities and review recent methodological developments. Because protein flexibility is essential for function and ligand binding, we discuss the related problems involved in the definition, simulation, and free energy calculation of microstates (such as the α-helical region of a peptide). While the review is broad, a special emphasize is given to methods for calculating the absolute *F* (*S*), where our HSMC(D) method is described in some detail.

The *absolute* entropy, *S* and the *absolute* Helmholtz free energy, *F* (or *G* – Gibbs free energy) are fundamental quantities in statistical mechanics with a special importance in structural biology. *S* is a measure of order where changes in the *S* of water lead to the hydrophobic interaction – the main driving force in protein folding. *F* constitutes the criterion of stability, which is essential for studying the structure and function of peptides, proteins, nucleic acids, and other biological macromolecules. The free energy defines the binding affinities of protein-protein and protein-ligand interactions, it also quantifies many other important processes such as enzymatic reactions, electron transfer, ion transport through membranes, and the solvation of small molecules.

However, calculation of *F*(*S*) by computer simulation is extremely difficult, and considerable attention has thus been devoted in the last 50 years to this subject. While significant progress has been made (see reviews in [1-9]), in many cases the efficiency (or accuracy) of existing methods is unsatisfactory and the need for new ideas has kept this field highly active. We summarize here mainly recent developments in this area of research where the emphasis is on methodology issues and less on applications. The present article constitutes a substantial extension of a concise review appeared recently [7].

In this section we define various thermodynamic quantities and discuss the problems involved in estimating them by computer simulation.

The commonly used simulation techniques, Metropolis Monte Carlo (MC) [10] and molecular dynamics (MD) [11,12] are *exact* methods which enable one to generate samples of system configurations *i* distributed according to the Boltzmann probability, *P _{i}*

$${P}_{i}^{\text{B}}=\frac{exp[-{E}_{i}/{k}_{\text{B}}T]}{Z},$$

(1)

where *T* is the absolute temperature, *k*_{B} is the Boltzmann constant and *E _{i}* is the potential energy of configuration

$$Z=\sum _{i}exp[-{E}_{i}/{k}_{\text{B}}T],$$

(2)

where the summation (integration for a continuum system) is carried out over the *entire* ensemble of configurations. The ensemble averages of the energy, <*E*>, and the absolute entropy, *S*, are given by

$$\langle E\rangle =\sum _{i}{P}_{i}^{\text{B}}{E}_{i}$$

(3)

and

$$S=\langle S\rangle =-{k}_{\text{B}}\sum _{i}{P}_{i}^{\text{B}}ln{P}_{i}^{\text{B}}.$$

(4)

where the free energy, *F*, can also be expressed (formally) as an ensemble average,

$$F=\langle F\rangle =-{k}_{\text{B}}TlnZ=\sum _{i}{P}_{i}^{\text{B}}[{E}_{i}+{k}_{\text{B}}Tln{P}_{i}^{\text{B}}]=\langle E\rangle -TS.$$

(5)

An extremely important property of this representation of *F* (but not of other representations) is that its variance vanishes, σ^{2}(*F*)=0; indeed, substituting the expression for *P _{i}*

$$\overline{E}={n}^{-1}\sum _{t=1}^{n}{E}_{t}$$

(6)

where the values of *E _{t}* are easily measured for each of the sampled configurations (e.g, the sum of the Lennard-Jones interactions for argon). (One has to distinguish between a summation over the entire ensemble which is denoted by the index

$$\overline{S}={n}^{-1}\sum _{t=1}^{n}ln{P}_{t}^{\text{B}},$$

(7)

is not straightforward because (unlike the energy) the *value* of ln *P _{t}*

The above discussion in terms of a discrete system also holds for a continuum system where the potential energy is *E*(**x**), **x** Ω is a 3*N*-dimensional vector of the Cartesian coordinates of the *N* atoms, and Ω is the entire configurational space. Thus, the summations (over the entire ensemble) are replaced by integrations over Ω. Moreover, this theory also applies to any *partial region* Ω* _{m}* of Ω, where a corresponding partial free energy,

$${F}_{m}=-{k}_{\text{B}}Tln{Z}_{m}=-{k}_{\text{B}}Tln\underset{{\Omega}_{m}}{\int}exp[-E(x)/{k}_{\text{B}}T]dx={E}_{m}-T{S}_{m}.$$

(8)

Notice that the integral defining *Z _{m}* has the dimension of

$${p}_{m}/{p}_{n}={Z}_{m}/{Z}_{n}.$$

(9)

While the difficulty in calculating the absolute *S* (*F*) discussed above is common to all systems, biological macromolecules such as peptides and proteins, are particularly challenging due to their rugged potential energy surface, *E*(**x**). More specifically, this surface is “decorated” by a tremendous number of localized energy wells and “wider” ones that are defined over microstates (regions Ω* _{m}*), each consisting of many localized wells (Fig. 1); a microstate can be represented by a sample (trajectory) generated by a

Schematic one-dimensional representation of part of the energy surface of a peptide or a protein, as a function of a coordinate X. The two large potential energy wells are defined over the corresponding microstates denoted Ω_{1} and Ω_{2}. Each **...**

Free energy calculations are also required in problems which are less challenging than protein folding, i.e., in cases of *intermediate flexibility*, where a flexible protein segment (e.g., a side chain or a surface loop), a cyclic peptide, or a ligand bound to an enzyme populates significantly several microstates in thermodynamic equilibrium. It is of interest, for example, to know whether the conformational change adopted by a loop (a side chain, ligand, etc.) upon protein binding has been induced by the other protein (induced fit [17,18]) or alternatively the free loop already interconverts among different microstates where one of them is selected upon binding (selected fit [19]). This analysis requires calculating the relative populations, *p _{m}*/

Calculating populations, *p _{m}* or ratios

However, this definition should be used with caution. Thus, a short simulation will span only a small part of Ω_{h} and this part will grow constantly as the simulation continues; correspondingly, the calculated average potential energy, *E*_{h} and the entropy *S*_{h} (obtained by any method) will both increase and the free energy, *F*_{h} is expected to change as well. As the simulation time is increased further, side chain dihedrals will “jump” to different rotamers, which according to our definition should also be included within Ω_{h}; for a long enough simulation the peptide is expected to “leave” the α-helical region and move to a different microstate. Thus, *in practice*, the microstate size and the corresponding thermodynamic quantities can depend on the simulation time used to define the microstate. In some cases, one can better define Ω_{h} by discarding structures with dihedral angles beyond predefined Δ* _{k}* and Δψ

The various methods are divided into three main categories, the “counting approach”, thermodynamic integration/perturbation, and methods for calculating the absolute *F* and *S*. For brevity in what follows we denote microstates by *m* and *n* rather than by Ω* _{m}* and Ω

As has been already pointed out, in many cases one is interested in differences Δ*S _{mn}* and Δ

$$\Delta {F}_{mn}={k}_{\text{B}}Tln[(\#m)/(\#n)]$$

(10)

and #*m* (#*n*) is the number of times the molecule visited *m* (*n*) during the simulation. However, because of high energy barriers, the transition between microstates at room temperature might require long times, nanoseconds or more even for side chain rotamers, meaning that reliable sampling of #*m* (#*n*) might become prohibitive. This problem can be alleviated by applying enhanced sampling techniques such as replica exchange [33] or the multicanonical method [34,35]; however, the conformational search capability of these methods is also limited and microstates of interest might be visited poorly (or not at all). The common analysis is based on projecting MD (MC) trajectories onto a small number of coordinates using principal component analysis, PCA (to help define/identify microstates), or in simpler cases calculating the populations along one or two physically significant reaction coordinates [36,37].

Differences Δ*F* and Δ*S* are commonly calculated by thermodynamic integration (TI) over physical quantities such as the energy, temperature, pressure, specific heat, etc. [38,39], as well as non-physical parameters, for instance, using a coupling parameter to act on the interaction potential to effect an “alchemical mutation”. In addition to TI, free energy perturbation (FEP) [1-9,40-47] and histogram analysis methods [48-50] can also be applied and will be included in this category. These are robust and highly versatile approaches, which have been reviewed extensively [1-9] and therefore only recent developments will be discussed here, some of them in detail.

An important application of TI is calculating the difference in the binding free energy of two ligands **a** and **b** bound to a protein (or a single ligand bound to a protein before or after a mutation in the protein). In this case two different simulations (integrations) are carried out in which **a** is mutated to **b** in water (**aw**→**bw**) and in the protein environment (**Pa**→**Pb**), and the corresponding differences in free energy, Δ*F*_{aw}_{→}** _{bw}** and Δ

A thermodynamic cycle for the binding of two ligands **a** and **b** to a protein **P**. In the experiment the ligands are transferred from the solvent to the active site where one measures the difference ΔΔ*F* = Δ*F*_{aw} _{→}_{Pa} - Δ **...**

$$\Delta {F}_{\text{aw}\to \text{Pa}}-\Delta {F}_{\text{bw}\to \text{Pb}}=\Delta {F}_{\text{aw}\to \text{bw}}-\Delta {F}_{\text{Pa}\to \text{Pb}}.$$

(11)

This procedure is extremely valuable because it enables one to calculate *small* free energy differences, Δ*F*_{aw}_{→}** _{bw}** and Δ

However, TI has weaknesses that should be emphasized. Thus, if one seeks to calculate Δ*F***_{mn}** between microstates

As discussed below, these drawbacks can be overcome to a large extent with methods that calculate the absolute free energy. In this context it should be pointed out that the absolute *F* can also be obtained with TI provided that a reference state *R* with known *F* is available and an efficient integration path *R*→*m* can be defined. A classic example is the calculation of *F* of liquid argon or water by integrating the free energy from an ideal gas reference state. However, for non-homogeneous systems such integration might not be trivial, and in models of peptides and proteins defining adequate reference states is not straightforward (see later discussions in III.3.b.7). However, in spite of these problems, the TI approach is applied regularly for calculating the free energy of binding (and other properties) and the required computer programs are implemented in the commonly used molecular mechanics/molecular dynamics software packages, such as AMBER [56], CHARMM [57], NAMD [58], BOSS [59], GROMOS [60], GROMACS [61], TINKER [62], and others.

An interesting development in the TI category is the Adaptive Integration Method (AIM) for computing free energies, radial distribution functions, and potentials of mean force [63]. A general TI process is based on the integral

$$\Delta F=\underset{0}{\overset{1}{\int}}{\langle \frac{dH(\lambda ,x)}{d\lambda}\rangle}_{\lambda}d\lambda ,$$

(12)

where 0 ≤ λ ≤ 1 defines a hybrid Hamiltonian, *H*(λ) = (1- λ)*U*_{0} + λ*U*_{1}, that is varied between two energy functions *U*_{0} and *U*_{1}. (*H*(λ) can also be defined by more general nonlinear scalings.) This integral is commonly evaluated by carrying out *l separate* MD (or MC) simulations at *l* intermediate λ values, where the *l* corresponding averages (in conformational space **x**) of the derivative <*dH*(λ,**x**)/*d*λ>_{λ} = <*U*_{1}-*U*_{0}>_{λ} are calculated. With AIM, on the other hand, the sampling is performed within an MC procedure that allows transitions between coordinates as well as between different λ values. The parameter, λ, is therefore treated as an additional coordinate thus defining an expanded (λ,**x**) “super-system”. Thus, if the (a-priori unknown) partition function at λ is *Z*_{λ} (*Z*_{λ} = ∫ exp[−*H*(*λ*,**x**)/*k*_{B}*T*]*d***x**), a normalized (Boltzmann) probability for the super-system to be at (λ,**x**) can be defined as

$${P}^{\text{B}}(\lambda ,\text{x})=\frac{exp[-H(\lambda ,\text{x})/{k}_{\text{B}}T]/{Z}_{\lambda}}{\text{\u2211}_{{\lambda}^{\prime}}\int exp[-H({\lambda}^{\prime},\text{x})/{k}_{\text{B}}T]d\text{x}/{Z}_{{\lambda}^{\prime}}}.$$

(13)

Note that each λ value has been weighted (by 1/*Z*_{λ}) to give the same probability, *P*^{B}(λ)=1/{Σ_{λ′}∫exp[-*H*(λ′,**x**)/*k*_{B}*T*]*d***x**/Z_{λ′}}. The MC transition probabilities should satisfy the detailed balance condition, *p*(λ_{1}→ λ_{2}) / *p*(λ_{2}→ λ_{1})=*P*^{B}(λ_{2},**x**)/ *P*^{B}(λ_{1},**x**), which leads to

$$p({\lambda}_{1}\to {\lambda}_{2})=min\lfloor 1,exp\{-[H({\lambda}_{2},x)-H({\lambda}_{1},x)+{\overline{F}}_{{\lambda}_{1}}-{\overline{F}}_{{\lambda}_{2}}]/{k}_{\text{B}}T\}\rfloor $$

(14)

Because *Z*_{λ} is not known a-priori, the free energy, * _{λ}* = −

The authors claim that a larger number of bins (λ values) can be treated with AIM than with TI (for the same amount of computer time) which leads to a much finer resolution. Another potential advantage of AIM lies in the fact that a bin might be visited many times during the simulation, each visit starts from a different structure (seed) leading to an adequate sampling of the contributing microstate(s) for this λ. With TI, on the other hand, only a single simulation (starting from one seed) is typically performed and the coverage of the contributing microstates is expected to be more limited.

It should be pointed out that simulation techniques based on an adaptive calculation of (relative) free energies and entropies have been suggested before, starting with the multicanonical technique of Berg and Neuhaus [34], the method of expanded ensembles of Lyubartsev et al. [64], and the simulated tempering method of Marinari and Parisi [65]. The more recent (and relatively simple) random walk algorithm of Wang and Landau [66] has been used extensively, and has become the basis for more sophisticated techniques developed, for example, by de Pablo's group [67-69]. Also, to enhance efficiency, Escobedo and collaborators have devised methods [70-74], which combine the expanded ensembles idea with other known procedures (e.g., Bennett's method [75]). However, unlike AIM, which is aimed at calculating the free energy, most of these methods are designed primarily as simulation tools that enable a system with a rugged energy surface to cross energy barriers efficiently, while differences in free energy (or entropy) are obtained (like other properties) as byproducts of the simulation. A detailed discussion of these methods is beyond the scope of this review and extensive relevant literature can be found in the references cited above. Finally, it should be pointed out that further development of multicanonical ideas has also been pursued by the groups of Okamoto and Nakamura (also for MD simulations) and this approach has been applied extensively to peptides and proteins in explicit water (see for example [76-82] and references cited therein).

Another approach for calculating the (*reversible*) Δ*F* is based on Jarzynski's identity [47],

$$\Delta F=-{k}_{\text{B}}T{\langle exp[-{W}_{f}/{k}_{\text{B}}T\rangle}_{0}$$

(15)

where <>_{0} represents an average over *non-reversible* forward-directed work values, *W _{f}*, generated by starting an equilibrium simulation at

Shirts and Pande [91] reviewed and developed theoretical estimates for the bias and variance of Jarzynski's identity, TI, and Bennett's method [75]. They applied these methods to toy models but could not define a preferred method for calculating Δ*F*; however, in applications to simple atomistic models the lowest variance and bias were obtained with Bennett's method. Pande's group also developed efficient methods for calculating the absolute *F* of binding [92]. In a recent study [93] the accuracy and precision of nine free energy methods have been compared, where among them are, TI, AIM, FEP, Bennett's method, and single-ensemble path sampling [84]. Δ*F* was calculated for growing a (neutral) Lennard-Jones sphere in water and for charging a Lennard-Jones sphere in water. The efficiency was found to depend on the system and extent of accuracy sought, where overall AIM is the most efficient. Jarzynski's identity was also applied to realistic systems of proteins [94], where steered MD was used for calculating potential of mean force for unbinding acetylcholine from the alpha7 nicotinic acetylcholine receptor ligand-binding domain; four different procedures were checked in this study (see also [95]).

Problems associated with the free energy difference-based approaches discussed earlier (e.g., TI) can be remedied to a large extent by calculating the absolute free energy; then, *F***_{m}** and

A commonly used approach for estimating the absolute *S* is based on the harmonic approximation which was introduced to biomolecules by Gō and Scheraga [96,97]. They obtained

$$S=\text{}-\left({k}_{\text{B}}/\text{2}\right)\text{ln}\left[\text{Det}\left(\text{Hessian}\right)\right],$$

(16)

where Hessian is the matrix of second derivatives of the force field with respect to internal coordinates around an energy minimized structure; the quantum mechanical version (Einstein oscillators) was applied later for peptides by Hagler's group [98]. A related approach, “the second generation mining minima” method (M2) [99,100] has been developed by Gilson's group. With M2, low energy minimized structures (within a microstate) are initially identified, the free energies of the corresponding local potential wells are calculated with a method that considers both harmonic and an-harmonic effects, and the contribution of the individual wells is then accumulated.

An important development has been the introduction of the quasiharmonic (QH) method by Karplus and Kushick [101], where the Boltzmann probability density of structures defining a microstate is approximated by a multivariate Gaussian. Thus,

$${S}_{\text{QH}}=\frac{{k}_{\text{B}}}{2}\{N+ln[{(2\pi )}^{N}\text{Det(}\sigma )]\}$$

(17)

where the covariance matrix, **σ**, is obtained from a local MD (MC) sample and *N* is (usually) the number of internal coordinates. Clearly, *S*_{QH} constitutes an upper bound for *S* since correlations higher than quadratic are neglected; also, an-harmonic contributions are ignored, and QH is not suitable for diffusive systems such as water.

While QH has been used extensively during the years (see [102-104] and references cited therein), a systematic study of its performance has been carried out only recently by Gilson's group [105]. They studied linear alkanes and a host-guest system (urea receptor with the ethylenurea ligand) comparing the QH results to those obtained by the M2 method mentioned above. The conclusions of this study are that QH can be accurate for a highly populated single energy well, where the calculation is based on internal coordinates; the use of Cartesians, however, leads to errors of several kcal/mol. When the simulation covers several energy wells the errors of QH (in internal coordinates) can increase to tens of kcal/mol and are significantly larger with QH(Cartesians). Also, while errors sometimes get cancelled in entropy differences, the host-guest studies have shown that the errors in Δ*S*_{QH} are substantial. Finally, the convergence of the QH results is slow and in the host-guest system convergence has not been obtained even with 6 ns MD runs, which is in accord with previous studies. These conclusions probably apply to other versions of QH where **σ** is defined in Cartesian coordinates, such as the ad-hoc quantum mechanical approximation of Schlitter [106,103] and the exact derivation of quantum mechanical QH [107]; the performance of these two methods has been compared [108].

A new version of QH has been suggested recently by Wang and Brüschweiler [109], which enables one to estimate the contribution of different potential wells, e.g. rotameric states. Thus, defining a peptide conformation by the dihedral angles θ* _{j}*, a PCA analysis is carried out for a sample of conformations with respect to the complex variables
${e}^{i{\theta}_{j}}$ (rather than θ

Another approach for calculating the absolute *S* (*F*) has been suggested by Meirovitch and has been implemented initially in two *approximate* techniques of general applicability (i.e., not restricted to harmonic conditions), the local states (LS) [111,22-28] and the hypothetical scanning (HS) methods [112,115]. With both methods each conformation *i* of a sample [generated by MC or MD] is *reconstructed* step-by-step (from nothing) using transition probabilities (TPs); the product of these TPs leads to an approximation *P _{i}* for the correct

The philosophy of this approach is based on the ideas of the *exact* scanning method, which is thus described first [122,123]. While these methods are applicable to a wide range of systems, they are described here as applied to a simple peptide – a polyglycine molecule of *N* residues where its conformations are defined by the dihedral angles * _{i}*,ψ

The *exact* scanning method [122,123] is a step-by-step construction procedure for a peptide conformation based on calculating (consecutively) TPs for the α* _{k}*, and determining their values and the positions of the related atoms [124]. For example, the angle defines the coordinates of the two hydrogens connected to C

$$\rho ({\alpha}_{k}{|\alpha}_{k-1},\cdots ,{\alpha}_{1})={Z}_{\text{future}}({\alpha}_{k},\cdots ,{\alpha}_{1})/[{Z}_{\text{future}}({\alpha}_{k-1},\cdots ,{\alpha}_{1})].$$

(18)

That is, *ρ*(*α _{k}*|

$${Z}_{\text{future}}({\alpha}_{k},\cdots ,{\alpha}_{1})=\underset{{\Omega}_{\text{h}}}{\int}exp[-E({\alpha}_{6N},\cdots ,{\alpha}_{1})/{k}_{\text{B}}T]d{\alpha}_{k+1}\cdots d{\alpha}_{6N}.$$

(19)

The product of the TPs [equation (18)] leads to the (Boltzmann) probability density of the entire conformation [equation (1)],

$${\rho}^{\text{B}}({\alpha}_{6N},\cdots ,{\alpha}_{1})=\prod _{k=1}^{6N}\rho ({\alpha}_{k}{|\alpha}_{k-1},\cdots ,{\alpha}_{1}).$$

(20)

This construction procedure is not feasible for a large molecule because the scanning range grows exponentially and the helical region is not known, as discussed in II.4; therefore this method was used as a conformational search technique, where only a limited number of future angles were scanned [124]. However, the ideas of the exact scanning method constitute the basis for the three methods, HS, HSMC(D), and LS, as discussed below.

The *exact* scanning method is equivalent to any other exact simulation technique (in particular MC and MD) in the sense that large samples generated by such methods lead to the same averages and fluctuations within the statistical errors. Therefore, one can assume that a given MC or MD sample has rather been generated by the exact scanning method, which enables one to reconstruct each conformation *i* by calculating the TP densities that *hypothetically* were used to create it step-by-step. This idea has been implemented initially in two different ways in the LS and HS methods. However, an exact reconstruction of the TPs [equation (18)] is feasible only for a very small peptide. Therefore, calculation of future partition functions [equation (19)] by these methods has been carried out only approximately, by considering a partial future (or a limited past in the case of LS) as discussed in III.3.b.5. On the other hand, with HSMC(D) the *entire* future is considered and in this respect HSMC(D) can be considered to be exact.

Because HSMC and HSMD are based on the same theoretical grounds, we denote the related probability functions by ‘HS’, where the theory is described for HSMD, which for peptides has been found to be the more practical and efficient method among the two. One starts by generating an MD sample of the helical microstate; the conformations are then represented in terms dihedral and bond angles,1≤ α* _{k}* ≤ 6

$$\Delta {\alpha}_{k}={\alpha}_{k}(max)-{\alpha}_{k}(min),$$

(21)

where α* _{k}*(max) and α

As mentioned in III.3.b.2, with our approach a sample conformation *i* is reconstructed step-by-step by calculating the TP density of each α* _{k}* [equation (18)] from the future partition functions Z

$$\rho ({\alpha}_{k}{|\alpha}_{k-1},\cdots ,{\alpha}_{1})\approx {\rho}^{\text{HS}}({\alpha}_{k}{|\alpha}_{k-1},\cdots ,{\alpha}_{1})={n}_{\text{visit}}/[{n}_{f}\delta {\alpha}_{k}]$$

(22)

where the relation becomes exact for very large *n _{f}* (

Illustration of the HSMC(D) reconstruction process of conformation *i* of a peptide consisting of three glycine residues. At each step the transition probability (TP) of a dihedral angle and the successive bond angle is determined and the related atoms **...**

$${\rho}^{\text{HS}}({\alpha}_{k+1},{\alpha}_{k}{|\alpha}_{k-1},\cdots ,{\alpha}_{1})={n}_{\text{visit}}/[{n}_{f}\delta {\alpha}_{k}\delta {\alpha}_{k+1}].$$

(23)

Notice that in the deterministic calculation of *Z*_{future}, [equation (19)] the limits of Ω_{h} are in practice unknown. On the other hand, with HSMD the future structures generated by MD at each step *k* remain in general within the limits of the microstate Ω_{h} defined by the analyzed MD sample due to the microstate's (meta) stability.

Similar to equation (20), the corresponding *overall* probability density for HSMD is

$${\rho}^{\text{HS}}({\alpha}_{6N},\cdots ,{\alpha}_{1})=\prod _{k=1}^{6N}{\rho}^{\text{HS}}({\alpha}_{k+1},{\alpha}_{k}{|\alpha}_{k-1},\cdots ,{\alpha}_{1}),$$

(24)

where in the product only odd values of *k* are used. *ρ*^{HS}(*α*_{6}* _{N}*,,

$${S}^{\text{A}}=-{k}_{\text{B}}\underset{{\Omega}_{\text{h}}}{\int}{\rho}^{\text{B}}ln{\rho}^{\text{HS}}d{\alpha}_{1}\cdots d{\alpha}_{6N}$$

(25)

$${F}^{A}=\langle E\rangle -T{S}^{\text{A}}=\langle E\rangle +{k}_{\text{B}}T\underset{{\Omega}_{\text{h}}}{\int}{\rho}^{\text{B}}ln{\rho}^{\text{HS}}d{\alpha}_{1}\cdots d{\alpha}_{6N}.$$

(26)

In these equations *ρ*^{HS} = *ρ*^{HS}(*α*_{6}* _{N}*,,

$$\overline{{\sigma}_{\text{A}}}={\left[\frac{1}{n}\sum _{t=1}^{n}{[{\overline{F}}^{\text{A}}-{E}_{t}-{k}_{\text{B}}Tln{\rho}_{t}^{\text{HS}}]}^{2}\right]}^{1/2}.$$

(27)

One can also define a free energy functional, *F*^{B} which constitutes a rigorous upper bound for the correct *F* [116,117]. Thus, by increasing computer time (and/or decreasing δα* _{k)}*) a set of improving bounds can be obtained which enable one to determine the accuracy from HSMC(D) results alone without the need to know the correct answer (a “self checking” property). Furthermore,

Unlike the limited applicability of methods that are based on harmonic approximations, HSMC(D) is applicable to fluids, random coil polymers, as well as microstates of a peptide. Thus, results for liquid argon, TIP3P water [117,125], and self-avoiding walks on a square lattice [118] were found to agree within error bars to TI results. Also, for polyglycine molecules, differences Δ*F***_{mn}** and Δ

It is important to understand the basis for the cancellation of errors discussed above. We examine first two one-dimensional harmonic microstates (oscillators) with the same mass defined by different spring constants *f*_{1} and *f*_{2}. The *exact* entropy difference, Δ*S _{mn}* (here written Δ

$$\Delta {S}_{2,1}=(1/2){k}_{\text{B}}ln({f}_{1}/{f}_{2})={k}_{\text{B}}[ln(<{x}^{2}{>}^{1/2})-ln(<{y}^{2}{>}^{1/2})]$$

(28)

One can estimate Δ*S*_{2,1} from two separate MD simulations, where the corresponding variances are calculated. If *f*_{1} is significantly smaller than *f*_{2} (i.e., *f*_{1} defines a flatter parabola) and the same step size is used in both simulations a longer simulation will be required for *f*_{1} than for *f*_{2} to gain the same statistical precision. Therefore, if the same sample size is used for both microstates the statistical precision of Δ*S*_{2,1} will be determined mostly by that of *S*_{1}.

We now examine the entropy contributed by a backbone dihedral angle, α* _{k}* (denoted α for simplicity) in the course of the reconstruction process. α varies in microstates 1 and 2 within the ranges Δα

$$\Delta {S}_{0}(\alpha )={k}_{\text{B}}[ln{\Delta}_{2}-ln{\Delta}_{1}]$$

(29)

which is similar to that of equation (28) above (for brevity we shall omit α from the equations below). For better HSMD approximations,
$\Delta {S}_{0}^{{n}_{f}}(l)$ we define the bins δ_{1}=Δ_{1}/*l* and δ_{2}=Δ_{2}/*l*, where *l* is an increasing integer; the corresponding probabilities are
${p}_{1}^{{n}_{f}}(l)$ and
${p}_{2}^{{n}_{f}}(l)$ which are defined by *n*_{visit}/*n _{f}* [equations (22) and (23)]. One obtains,

$$\begin{array}{l}\Delta {S}_{0}^{{n}_{f}}(l)={k}_{\text{B}}[ln({p}_{1}^{{n}_{f}}(l)/{\delta}_{1})-ln({p}_{2}^{{n}_{f}}(l)/{\delta}_{2})]={k}_{\text{B}}\{ln[{p}_{1}^{{n}_{f}}(l)/{p}_{2}^{{n}_{f}}(l)]+ln({\Delta}_{2}/{\Delta}_{1})\}\\ \text{or}\phantom{\rule{6.5em}{0ex}}\Delta {S}_{0}^{{n}_{f}}(l)=\Delta {S}^{{n}_{f}}(l)+\Delta {S}_{0}\end{array}$$

(30)

where
$\Delta {S}^{{n}_{f}}(l)$ can be viewed as an an-harmonic term. One can write,
${p}_{\text{i}}^{\text{exact}}(l)={p}_{i}^{{n}_{f}}(l){x}_{i}^{{n}_{f}}(l)$ for *i*=1,2, where
${p}_{\text{i}}^{\text{exact}}(l)={p}_{i}^{{n}_{f}=\infty}(l)$ and
${x}_{i}^{{n}_{f}}(l)$ are thus factors (systematic errors) satisfying
${x}_{i}^{{n}_{f}}(l)\to 1$ for very large *n _{f}*; for a given

$$\Delta {S}^{{n}_{f}}(l)={k}_{\text{B}}\{ln{p}_{\text{1}}^{\text{exact}}(l)-ln{p}_{\text{2}}^{\text{exact}}(l)+ln[{x}_{2}^{{n}_{f}}(l)/{x}_{1}^{{n}_{f}}(l)]\}$$

(31)

However, for large bins, δ (small *l*), one would expect to obtain probabilities that are close to the exact ones,
${p}_{\text{1}}^{\text{exact}}(l)$ and
${p}_{\text{2}}^{\text{exact}}(l)$ [i.e.,
${x}_{1}^{{n}_{f}}(l)$ and
${x}_{2}^{{n}_{f}}(l)$ are ~1] for a relatively small *n _{f}* due to adequate statistics, i.e., relatively large

The relation
${x}_{2}^{{n}_{f}}(l)\approx {x}_{1}^{{n}_{f}}(l)$ stems from two reasons, where the first one is the fact that HSMD takes all interactions into account and thus for a given *n _{f}* the future part of the chain is treated with the same level of approximation in both microstates. Secondly, because with MD the atoms are moved along their potential gradients, the simulations are equally efficient in both microstates. For peptides [32] the condition
${x}_{2}^{{n}_{f}}(l)\approx {x}_{1}^{{n}_{f}}(l)$ occurs for much smaller

With the HS method one seeks to reconstruct a chain by a *deterministic* calculation of *Z*_{future} [equation (19)] for each α* _{k}*, based only on a partial future scanning defined by α

With the LS method applied to a peptide, [22-28] the conformations of a given sample (of a microstate) are initially expressed in terms of internal coordinates and then a three-stage analysis is carried out where the sample is visited three times. In the first visit the variability range Δα* _{k}* is calculated, [equation (21)]. Each range, Δα

$$\rho ({\alpha}_{k}{|\alpha}_{k-1}\cdots {\alpha}_{1})\approx n({\nu}_{k},\cdots ,{\nu}_{1})/\{n({\nu}_{k-1},\cdots ,{\nu}_{1})[\Delta {\alpha}_{k}/l]\}$$

(32)

where *n*(ν* _{k}*,,ν

$${\rho}_{i}(b,l)=\prod _{k=1}^{6N}p({\nu}_{k}|{\nu}_{k-1},\mathrm{\dots},{\nu}_{k-b})/(\Delta {\alpha}_{\text{k}}/l),$$

(33)

the larger are *b* and *l* the better the approximation (for enough statistics). ρ_{i}(*b*,*l*) allows one to define an approximate entropy and free energy functionals, *S*^{A} and *F*^{A}=<*E*>-*TS*^{A} [as in equations (25) and (26), where ρ_{i}(*b*,*l*) replaces *ρ*^{HS}(*α*_{6}* _{N}*,,

The above discussion demonstrates that LS (unlike HS) is of a “geometrical” character, i.e., calculation of the entropy does not depend directly on the interaction energy. Other methods that are based on calculating the distribution of local states (but not transition probabilities) have been suggested recently by Hnizdo et al. [128] and Killian et al. [129] who tested them on small molecules and peptides.

Finally, it should be pointed out that both LS, the above two methods, and the mining minima technique [99,100] can be applied to samples based on several microstates (where LS is also applicable to the random coil state), while QH lead to reliable results only for a single microstate [105]. However, QH, which considers the (quadratic) correlations among *all* variables, is expected to lead to better results than LS for a single microstate. Indeed, for peptides [30,32] and a surface loop of the protein α-amylase [120,121] the entropy results of QH were found to be better (i.e., smaller) than those of LS based on *b*=2 and *l*=10. However, the corresponding results for *S*[HSMC(D)] have always been the lowest (i.e., better).

With QH, LS, and HSMC(D) calculation of Δ*S _{mn}*=

Obviously, if *m* is less stable than *n* the *t* values should be adjusted (i.e., decreased) to fit the stability of *m*. If *m* is significantly larger than *n, t _{m}* should be large enough to allow an adequate coverage of

Unlike QH and LS, HSMC(D) is not based on gathering statistics from the studied sample; therefore, the required sample size is relatively small; also, *F*[HSMC(D)] (but not necessarily *E* and *S*[HSMC(D)]) can be obtained from a very small sample (even from a single conformation) [117]. Therefore, in our studies of peptides and loops populating significantly different microstates [29,32,120,121] the sample size for HSMC(D) is small and has been determined by the range of *t* values for which the average of *E _{m}* (

As pointed out earlier, the absolute *S* (*F*) can be obtained in principle also by TI, provided that a convenient reference state with known *F* is defined. In the early work of Stoessel and Nowak [130] a harmonic reference state *U*_{H}=*k*Σ(*r** _{i}*-

In a recent paper [132] Ytreberg and Zuckerman define a simple numerically calculable reference state designed to overlap the particular microstate of interest. Here, the microstate (say, an α-helical state of a peptide) is first simulated locally by MD, where the range of each internal coordinate, *k*, is divided into bins, and (normalized) populations are obtained from the (MD) sample. Using these probabilities a large sample of reference system structures, *i*, is then generated with known probabilities,
${P}_{i}^{\text{ref}}=\Pi {p}_{k}\left(i\right)$, where *p _{k}*(

Expressions for Δ*F _{mn}* based on two separate simulations of

$$\frac{{Z}_{n}}{{Z}_{m}}=\frac{{\langle {M}_{T}[{U}_{n}({\text{R}}_{m}+\text{D})-{U}_{m}({\text{R}}_{m})]\rangle}_{n}}{{\langle {M}_{T}[{U}_{m}({\text{R}}_{n}-\text{D})-{U}_{n}({\text{R}}_{n})]\rangle}_{m}}$$

(34)

where *M _{T}* is the Metropolis function,

Finally, we provide a short list of other recent methods for calculating *F* (*S*). Two methods involve TI [135,136] and one is based on the identity 1/*Q*=<*w*(**p**,**x**)exp[*H*(**p**,**x**)/*k*_{B}*T*]>, where *Q* is the partition function, **p** is the momentum and *w*(**p**,**x**) is a weight function [137]; this identity is used in HSMC(D) [117] and was used before with various *w* functions. With another method the joint probability density is represented by a two-dimensional Fourier series [138], and in a fourth method energy decomposition approach is used for evaluating *S* [139]. We mention also two methods for calculating the absolute protein-ligand binding free energy [140,141] and a method for calculating free energy profiles of enzymatic reactions by the linear response approximation [142].

In this review we have discussed the difficulties in calculating the entropy *S* and the free energy *F* focusing on the related problems involved in the definition and simulation of microstates of peptides and proteins. While the review is broad, the emphasis is on efficiency issues related to recently developed techniques, in particular techniques for calculating the absolute *F* (*S*), where our HSMC(D) method is discussed in some detail. We describe equilibrium and non-equilibrium techniques for calculating the *relative* binding free energy of ligands to an active site; however, methods for calculating the standard *absolute* binding free energy have not been covered. Also, we do not elaborate on practical aspects of protein-ligand (DNA-ligand, etc.) interactions, such as modeling of the solvent and calculating its contribution to the free energy. These topics are dealt more extensively in other recent reviews [8,9].

In this context one should emphasize the strong effects of modeling (in particular of electrostatic interactions) on the results for *F* (*S*) and other thermodynamic and structural properties. In fact, incompatibility of theoretical results with experimental data due to unreliable modeling can be much more severe than method-related inaccuracies in the calculation of *F* (*S*). Therefore, to gain progress in computational structural biology, the existing force fields and solvation models should be improved, efficient techniques for simulation of biological macromolecules should be devised, as well as better techniques for calculating *F* (*S*).

TI is the most general methodology, which in many cases is also the easiest to implement. Furthermore, various versions of TI (in particular procedures for calculating the relative free energy of ligands bound to an active site) are already programmed in the commonly used molecular mechanics/molecular dynamics software packages (see II.2.a). Among the TI based techniques, AIM [63,93] appears to be very efficient (at least for the systems studied), but it has not been applied as yet to biological molecules. Simulation methods (e.g., the multicanonical method) that lead to an efficient conformational search and based (like AIM) on an adaptive buildup of the (relative) free energy, have been applied to small biological macromolecules - peptides and loops (see II.2.b); these simulations have been performed with in-house programs. Also, path-based limitations in TI have led to the development of techniques for computing the absolute *F* (*S*). Thus, calculation of Δ*F***_{mn}**=

We thank Dan Zuckerman, Marty Ytreberg, and Mihail Mihailescu for valuable discussions. This work was partially supported by National Science Foundation Information Technology Research Grant, NSF 0225636 and NIH grant 2 R01 GM066090-04 A2.

- MC
- Monte Carlo
- MD
- Molecular dynamics
- HS
- Hypothetical scanning method
- HSMC(D)
- Hypothetical scanning MC (MD) method
- LS
- Local states method
- AIM
- Adaptive integration method
- TI
- Thermodynamic integration
- FEP
- Free energy perturbation

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