# Entropy and Free Energy of a Mobile Loop Based on the Crystal Structures of the Free and Bound Proteins

^{*}

## Abstract

**:**

_{loop}between these states sheds light on the mechanism of binding. With our “hypothetical scanning molecular dynamics” (HSMD-TI) method ΔF

_{loop}= F

_{free}− F

_{bound}where F

_{free}and F

_{bound}are calculated from two MD samples of the free and bound loop states; the contribution of water is obtained by a thermodynamic integration (TI) procedure. In previous work the free and bound loop structures were both attached to the same “template” which was “cut” from the crystal structure of the free protein. Our results for loop 287−290 of AcetylCholineEsterase agree with the experiment, ΔF

_{loop}~ −4 kcal/mol if the density of the TIP3P water molecules capping the loop is close to that of bulk water, i.e., N

_{water}= 140 − 180 waters in a sphere of a 18 Å radius. Here we calculate ΔF

_{loop}for the more realistic case, where two templates are “cut” from the crystal structures, 2dfp.pdb (bound) and 2ace.pdb (free), where N

_{water}= 40 − 160; this requires adding a computationally more demanding (second) TI procedure. While the results for N

_{water}≤ 140 are computationally sound, ΔF

_{loop}is always positive (18 ± 2 kcal/mol for N

_{water}= 140). These (disagreeing) results are attributed to the large average B-factor, 41.6 of 2dfp (23.4 Å

^{2}for 2ace). While this conformational uncertainty is an inherent difficulty, the (unstable) results for N

_{water}= 160 suggest that it might be alleviated by applying different (initial) structural optimizations to each template.

## 1. Introduction

_{loop}between these two loop conformations in the bound and free proteins, rather than on the absolute values themselves. (As discussed later, ΔF

_{loop}not only sheds light on the mechanism of ligand binding, but in some cases it is an (ignored) component of the absolute free energy of binding.) In the present work we develop HSMD further, extending its applicability to more complex models, in particular to more complex models of loops, as explained below.

**x**) is rugged (

**x**is the 3N-dimensional vector of the Cartesian coordinates of the molecule’s N atoms), i.e., this surface is “decorated” by a tremendous number of localized wells and “wider” ones, defined over regions, Ω

_{m}(called microstates) each consisting of many localized wells. A microstate Ω

_{m}, (e.g., the α-helical region of a peptide) which typically constitutes only a tiny part of the entire conformational space, Ω, can be represented by a sample (trajectory) generated by a local MD [24,25] simulation. A molecule will visit a localized well only for a very short time [several femtoseconds (fs)] while staying much longer within a microstate [26,27] which is thus of a greater physical significance (for further discussions about microstates and their problematic definition in simulations, see [8,9]).

_{m}in thermodynamic equilibrium, which should be identified and their populations, p

_{m}= exp[–F

_{m}/k

_{B}T] calculated (k

_{B}is Boltzmann constant and T is the absolute temperature). In particular, identifying the global free energy minimum of a protein is the daunting task of protein folding.

_{m,n}= ∫dF between two microstates m and n with significant structural variance the integration from m to n becomes difficult and for large molecules unfeasible. This drawback of TI can be overcome to a large extent by methods that provide the absolute F

_{m}(and S

_{m}) from a given sample; thus, one is required to carry out (only) two separate local MD simulations of microstates m and n, calculating directly the absolute F

**and F**

_{m}**hence their difference ΔF**

_{n}_{mn}= F

_{m}– F

_{n}, where the complex TI process is avoided. HSMC(D) (and other techniques [28,29,30,31] such as the quasi-harmonic approximation [32,33,34]) enable one to calculate the absolute F (and S) (for a more extensive discussion on TI and other techniques for calculating differences, ΔF (and ΔS) see [17]).

_{i}

^{B}from which various free energy functionals can be defined. While the TPs of HSMC(D) are stochastic in nature (calculated by MD or MC simulations), all the system interactions are taken into account; in this respect HSMC(D) can be viewed as exact [3], where the only approximation involved is due to insufficient MC (MD) sampling for calculating the TPs. HSMC(D) has unique features: it provides rigorous lower and upper bounds for F, which enable one to determine the accuracy from HSMD results alone without the need to know the correct answer. Furthermore, F can be obtained from a very small sample and in principle even from any single conformation (e.g., see results for argon in [3]).

_{m,n}(and ΔS

_{m,n}) between the open and closed microstates. First, we treated the 7-residue mobile loop, 304–310 (Gly-His-Gly-Ala-Gly-Gly-Ser) of the enzyme porcine pancreatic α-amylase [8,9], where the system is modeled by the AMBER96 force field [35] alone and by AMBER96 with the GB/SA implicit solvent of Still and coworkers [36]. In this work only protein atoms close to the loop (the template) within a sphere of radius, R

_{tmpl}were considered, where they were kept fixed in their x-ray crystallography positions, while only the loop’s atoms were moved by MD. Later the implicit solvent was replaced by explicit solvent, i.e., the α-amylase loop was capped with 70 TIP3P [37] water molecules (within a sphere of radius R

_{water}= 14 Å), which (together with the loop’s atoms) were moved in the MD simulation. Because the application of HSMD to water has not been optimized yet, the contribution of water to the free energy was calculated by a TI procedure incorporated within the framework of HSMD. This HSMD-TI process leads to a relatively small statistical error because only the water-loop interactions are involved, where their full values are gradually decreased to zero, for a fixed loop structure.

_{loop}= F

_{free}– F

_{bound}~ −4 kcal/mol) [40,42,46]. Therefore AChE has been an ideal system for checking the performance of HSMD-TI, and in particular for examining the minimal number of water molecules needed to cap the loop. Thus we carried out two sets of calculations for systems of increasing size defined by R

_{t}

_{mpl}= 12 and R

_{water}= 13 Å, and by R

_{tmpl}= 13 and R

_{water}= 14 Å, for an increasing number of water molecules, N

_{water}= 80, 100, 120, 180 and 80, 100, 120, …, 220, respectively. We have found that to recover the experimental free energy difference the water density should be close to that of bulk water and our results for the first set, ΔF

_{loop}= −3.1 ± 2.5 and −3.6 ± 4 kcal/mol for a sphere containing 160 and 180 waters are equal within error bars to the experimental value [10].

_{loop}is known. Therefore, the loop of AChE is treated here again, where the simulations are carried out with respect to two different templates which are cut from the free and bound crystal structures; this requires applying the (relatively) elaborate HSMD-TI procedure mentioned above. While the basic theory is similar to that described in [10], we derive it here in a new and shorter way for better clarity.

_{loop}to ligand binding. One question of interest in this area is whether the conformational change adopted by a mobile loop upon ligand binding is induced by the ligand (induced fit [50,51]) or alternatively, the free loop (in the apo protein) interconverts among different microstates, one of which is selected upon binding (selected fit [52]). While some recent studies favor the latter mechanism, there is no general consensus, and the adopted mechanism is probably system dependent. The present study is not aimed at solving this problem, which would require to apply the longest simulation possible to the entire AChE protein soaked in explicit water, as done, for example, in [53] for studying loop 6 of TIM. Notice, however, that in the case of induced fit ΔF

_{loop}(in the apo protein) should be added to the total absolute free energy of binding of the protein-ligand complex. Thus, if the template remains unchanged (due to binding), calculation of ΔF

_{loop}carried out in [10] is expected to be adequate, whereas for a changed template the present calculation should be used. Indeed, in our recent study of the avidin-biotin complex ΔF

_{loop}is taken into account (for the first time) in the calculation of the absolute free energy of binding [54].

_{loop}, which measures the difference in stability between the two (possibly) meta-stable microstates is of interest. For this purpose, generating the longest possible MD trajectories might be a disadvantage because the loop is expected to escape from the original microstates during the simulations. Therefore, ΔF

_{loop}is calculated from relatively short trajectories (0.5 ns) where the loop remains in its original microstates (bound and free).

_{loop}(t), ΔS

_{loop}(t) and ΔF

_{loop}(t) are stable during a long enough time, t; for more details see [8,9].

_{loop}, ΔS

_{loop}and ΔF

_{loop}between two loop microstates might be small, the fluctuations in E and S increase as N

^{1/2}, where N is the number of atoms simulated. Therefore, simulating the entire protein might be prohibitive and to keep the fluctuations manageable the coordinates of the template in our calculations are held fixed. Notice, however, that this is not an inherent limitation as in our present application of HSMD-TI to the avidin-biotin complex [54] part of the template is also moved in the MD simulation since the size of the fluctuations remains under control.

## 2. Theory and Methodology

#### 2.1. The Loop and the Protein’s Template

^{2}). Afterwards the loop and (TIP3P) water atoms were allowed to relax in the presence of a fixed template. These minimizations eliminate bad atomic overlaps and strains in the original structures, while keeping the atoms reasonably close to the PDB coordinates.

^{2}) (as compared to 100 kcal/Å

^{2}used in [10] for 2ace) was chosen to allow more extensive changes in the bound structure due to its relatively large B-factors and the chemical change imposed on it by removing the phosphate group; correspondingly, larger structural changes were also allowed to occur in the free structure. To assess the change in the structure during minimization we list the various RMSD values (Å) in pairs, for the beginning and the end of the process: RMSD for all atoms, (0.98, 0.99), backbone atoms, (0.43, 0.41), sidechains, (1.20, 1.21), whole template, (0.85, 0.86), template backbone, (0.35, 032), template sidechains, (1.05, 1.06), whole loop (2.71, 2.87), loop backbone (1.39, 1.45), loop sidechains (2.92, 3.10). Thus, the strongest changes occurred for the loops.

_{tmpl}atoms closest to the loop, where the rest of the protein atoms are ignored. More specifically, the center of mass of the backbone atoms of the free loop (in the 2ace structure) is calculated as a (3D) reference point denoted

**x**

_{cmb}and a distance (R

_{tmpl}) is chosen. If the distance of any atom of a residue from

**x**

_{cmb}is less than R

_{tmpl}, the entire residue is included in the template; otherwise, the residue is eliminated. We use R

_{tmpl}= 12 Å which was found in [10] to lead to a large enough template consisting of N

_{tmpl}= 944 atoms; however, for the present (somewhat different) structure N

_{tmpl}= 961, and exactly all these atoms were included in the template which was “cut” from the crystal structure of the bound protein (2dfp). We added to each template 33 crystal water molecules selected from those which are closest to the loop (30 in [10]).

#### 2.2. Addition of Water

**x**

_{cmb}with a radius, R

_{water}(R

_{water}= R

_{tmpl}+ 1 Å) where waters are added at random to the hemisphere oriented towards the exterior of the template. To hold these waters around the loop they are restrained with a flat-welled half-harmonic potential (with a force constant of 10 kcal mol

^{-1}Å

^{-2}) based on their distance from

**x**

_{cmb}. That is, if the distance of a water oxygen from

**x**

_{cmb}is greater than R

_{water}a harmonic restoring force is applied, otherwise the restraining force is zero.

_{tmpl}and R

_{water}, N

_{water}should be 140–180, it is of interest to check this conclusion again for the present case of two different templates; therefore, we carry out calculations for N

_{water}= 40, 60, 80, 100, 120, 140, and 160.

_{water}. Again, at the end of this optimization the energy is minimized where the loop’s atoms and the water molecules are free to move. The change in the loops’ conformations from their crystal structures measured by the RMSD are relatively small; for the free loop we obtain 0.31, 1.03, and 0.94 Å for backbone, side chains and all loop atoms, respectively, where the corresponding results for the bound loop are 0.37, 1.06, and 0.98 Å (see the discussion related to Table 1).

_{total}is the sum of partial energies related to the loop and water (the template-template energy is constant and thus is ignored):

_{total}= [E

_{loop-loop}+E

_{loop-tmpl}]+[E

_{water-water}+E

_{water-tmpl}+E

_{water-loop}] = E

_{loop}+E

_{water}

_{loop-loop}is the intra loop energy, E

_{loop-tmpl}is the energy due to loop-template interactions; these energies define the total loop energy E

_{loop}, and the interactions related to water are defined in a similar way, where their total is defined as E

_{water}. The reconstruction of the loop structure is carried out in internal coordinates; therefore, the loop conformations simulated by MD are transferred from Cartesians to the dihedral angles φ

_{i}, ψ

_{i}, and ω

_{i}(i = 1,N

_{res}= 4), the bond angles θ

_{i},

_{l}(i = 1,N

_{res}, l = 1,3), the side chain angles χ, and the corresponding bond angles. For convenience, all these angles (ordered along the backbone) are denoted by α

_{k}, k = 1,37 = K. We have argued in [7,8] that to a good approximation bond stretching can be ignored.

#### 2.3. Statistical Mechanics of a Loop in Internal Coordinates

_{res}= 4 residues) is carried out in internal coordinates; therefore, the loop conformations simulated by MD are transferred from Cartesians to the dihedral angles φ

_{i}, ψ

_{i}, and ω

_{i}(i = 1, N

_{res}), the bond angles θ

_{i},

_{l}(i = 1, N

_{res}, l = 1, 3), the side chain angles χ, and the corresponding bond angles. For convenience, all these angles (ordered along the backbone) are denoted by α

_{k}, k = 1,K; We have argued in [7,8] that to a good approximation bond stretching can be ignored, thus the bond lengths are considered to be constant.

**x**

_{loop},

**x**

^{N}) = E

_{total}is defined in Equation (1),

**x**

_{loop}is the Cartesian coordinates of the loop in microstate m and ${x}^{N}$ is the 9N Cartesian coordinates of the water molecules; E

_{total}also depends on the coordinates of the “frozen” template which are omitted for simplicity. For the same reason the letter m will be omitted in most of the equations and N

_{water}will be replaced in the theoretical section by N (N = N

_{water}). After changing the variables of integration from

**x**

_{loop}to internal coordinates, the integral becomes a function of the K dihedral and bond angles, α

_{k}, k = 1,K and a Jacobian, cos(θ

_{i},

_{l}) that depends (only) on each of the bond angles θ

_{i},

_{l}[27,28,32]:

#### 2.4. Exact Future Scanning Procedure

_{i}

^{B}, while the value of P

_{i}

^{B}is not provided (due to the dynamical character of these methods). [To simplify the discussion we use the discrete probability, P

_{i}

^{B}rather than the probability density ${\text{\rho}}^{\text{B}}([{\alpha}_{k}],{x}^{N})$ defined in Equation (4).] Thus, properties such as the energy that are “written” on i can easily be calculated, while a direct calculation of the absolute S is difficult because lnP

_{i}

^{B}is unknown (it depend not only on i but on the entire ensemble through the partition function Z, which cannot be obtained from a finite sample).

_{i}

^{B}hence to S and F. Practically, a loop/water/template configuration would be generated by initially building a loop structure (in the presence of moving water) followed by the construction of a configuration of the surrounding water molecules (in the presence of a fixed loop conformation). In this way a sample of statistically independent system configurations can be obtained.

_{k}, 1 ≤ α

_{k}≤ 6M = K with values within microstate m; the loop is surrounded by N

_{water}water molecules moving within the volume defined by a sphere of radius, R

_{water}, the template, and the loop. We seek to generate a configuration of the entire system by first generating a loop conformation and then a configuration of the water molecules.

**x**

_{loop}) is determined step-by step. Thus, the position of the first atom k’ = 1 is defined by the simultaneous determination of the first pair of dihedral and bond angles α

_{1,}and α

_{2}. The maximum range Δα

_{1}Δα

_{2}which will keep the loop within m is defined, and each of Δα

_{1}and Δα

_{2}is divided into n

_{b}small bins (of sizes Δα

_{1}/n

_{b}and Δα

_{2}/n

_{b}) denoted j

_{1}and j

_{2}, j

_{1}= 1…n

_{b}, j

_{2}= 1…n

_{b}, respectively. A long MD simulation of the whole system (loop + water) is carried out within microstate m, where a conformation is retained every l fs leading to a huge sample of size n; then, the number of conformations n

_{j}

_{1j2}that visit simultaneously the (double) bin j

_{1}j

_{2}is calculated from which the corresponding TP is obtained, p

_{j}

_{1j2}= n

_{j}

_{1j2}/n (or ρ

_{j1j2}= [n

_{j}

_{1j2}/n]/[Δα

_{1}Δα

_{2}/n

_{b}

^{2}]). A double bin is then selected by a random number according to the probabilities p

_{j}

_{1j2}which defines the position of atom k’ = 1 (and its hydrogen or oxygen). The position of this atom is not changed in the next steps of the build-up process, i.e., it becomes part of the “past”. The position of the second atom (k’ = 2) is determined in the same manner from a long MD simulation of the future part of the system (i.e., atoms k’ = 2…3M and water) where α

_{3}and α

_{4}are considered, bins Δα

_{3}/n

_{b}and Δα

_{4}/n

_{b}are defined, probabilities are calculated and a “lottery” (like above) determines the values of α

_{3}and α

_{4}which define the position of atom k’ = 2; the process continues until the positions of all the loop’s atoms (and their hydrogens or oxygens) have been determined. A configuration of the N

_{water}molecules is then determined in a similar way step-by-step in the presence of the fixed loop structure previously constructed (for details see [3]). Obviously, the smaller are the bins the higher is the accuracy of the construction process, provided that the statistics is adequate, i.e., that the (future) MD simulations are long enough; this stochastic scanning method becomes exact as the bin size → 0 (n

_{b}→ ∞) and n→ ∞. Notice that in applications of the (deterministic) scanning method to lattice models, only part of the future has been considered, (i.e., only f steps ahead), where this part has been scanned completely; therefore, the corresponding TPs are approximate but deterministic (rather than stochastic), and accurate results were obtained by using an additional importance sampling procedure [58].

_{1}, …, α

_{k-2}) and they are kept fixed (defining the “past”); α

_{k-1}and α

_{k}(which will determine the position of atom k’) are defined with the exact TP density $\text{\rho}({\text{\alpha}}_{k-1}{\text{\alpha}}_{k}{|\text{\alpha}}_{k-2},\cdots ,{\text{\alpha}}_{1})$:

_{future}is carried out over the positions of atoms k’ = k/2 + 1…K/2 (which affect angles, ${\alpha}_{k+1},\cdots ,{\alpha}_{K}$) and the 9N coordinates ${x}^{N}$ of the water molecules (which will be determined in future steps of the build-up process). Notice that this integration is carried out in a restrictive way where the corresponding conformations (of the loop) remain within microstate m. Also, in this integration the atoms treated in the past (1… k’ − 1) (which were determined by ${\alpha}_{1}\cdots {\alpha}_{k-2}$) are held fixed in their coordinates. For simplicity the integrations below are written over the angles rather than the Cartesian coordinates (

**x**

_{loop}) of the loop atoms, k’ = k/2+1…K/2. Thus:

_{loop}is defined up to an additive constant. Extending the exact scanning procedure to side chains is straightforward, where again the position of a side chain atom is defined by two angles as described above.

_{j}) required with the scanning method), as described below; also, because we are mainly interested in entropy differences, approximations (e.g., ignoring the Jacobians and bond stretching) can be applied without compromising the accuracy of the results.

#### 2.5. Loop Reconstruction by HSMC(D)

_{i}

^{B}whereas with HSMD an already generated structure (by MD) is reconstructed in order to obtain its probability. Thus, one starts by generating an MD sample of the loop in water in microstate m; the configurations of this sample will be reconstructed by HSMD (application of HSMC is similar). First, the conformations are represented in terms of the dihedral and bond angles α

_{k}, 1 ≤ α

_{k}≤ 6M = K, and the variability range Δα

_{k}is calculated:

_{k}(max) and α

_{k}(min) are the maximum and minimum values of α

_{k}found in the sample, respectively. Δα

_{k}, α

_{k}(max), and α

_{k}(min) enable one to verify that the sample has not “escaped” from microstate m. Notice that in the discussion below we define the loop conformation by the set of angles [α

_{k}] rather than by the positions of the loop atoms,

**x**

_{loop}, which are determined by [α

_{k}].

**x**

^{N}. Because the position of atom k’ is defined by a dihedral and a bond angle one has to calculate their TP simultaneously. Thus, at step k’ (k = 2k’) of stage 1, the k−2 angles ${\alpha}_{k-2}\cdots {\alpha}_{1}$ have already been reconstructed and the TP density of ${\alpha}_{k-1}{\alpha}_{k}$, $\text{\rho}({\alpha}_{k-1}{\alpha}_{k}{|\alpha}_{k-2},\cdots ,{\alpha}_{1})$, is calculated from an MD run, where the entire future of the loop and water is moved [i.e., the loop’s atoms k’,k’ + 1,…K/2 and their connected hydrogens, or oxygens and the water coordinates

**x**

^{N}] while the past (loop’s atoms 1,2,…,k’−1 and their connected hydrogens or oxygens) are held fixed at their values in conformation i. By considering a future conformation every 10 fs, a sample of size n

_{f}is generated. Two small segments (bins) δα

_{k}

_{-1}and δα

_{k}are centered at α

_{k}

_{-1}(i) and α

_{k}(i), respectively, and the number of simultaneous visits, n

_{visit}, of the future chain to these two bins during the simulation is calculated; one obtains (see Figure 1):

_{f}(n

_{f}→∞) and very small bins (δαk-1, δαk→0). This means that in practice ${\text{\rho}}^{\text{HS}}({\alpha}_{k-1}{\alpha}_{k}{|\alpha}_{k-2},\cdots ,{\alpha}_{1})$ will be somewhat approximate due to insufficient future sampling (finite n

_{f}) and relatively large bin sizes. The same treatment can be applied to side chain atoms; however, notice that the sidechain of Arg is treated here approximately, where at each step three consecutive atoms are considered and their positions are defined only by the corresponding dihedral angles. In [8] we have shown that δα

_{k}and δα

_{k}

_{+1}can be optimized. Notice that with HSMD the future loop conformations generated by MD at each step k‘ remain in general within the limits of m, which is represented by the analyzed MD sample. The corresponding probability density related to the loop is:

**Figure 1.**An illustration of the HSMD reconstruction process of conformation i of a peptide consisting of three glycine residues, where for simplicity the oxygens and most of the hydrogens are discarded. At step k’ = 6 the TPs related to the “past” atoms k’ = 1…5 (depicted by full spheres) have already been determined and these atoms are kept fixed in their positions at conformation i. At this step (6) one calculates the TPs of bond angle α

_{k}(defined by C’-Cα-N) and dihedral angle α

_{k−1}(defined by C’-Cα-N-C’) which are related to C’, where k = 2k’ and thus k = 12. The TPs are obtained from an MD simulation where the as yet unreconstructed atoms k’ = 6…10 (the “future” atoms) are moved (depicted by empty spheres connected by dashed lines) while the past atoms k’ = 1…5 are kept fixed; notice that the future part should remain within the limits of the microstate and future-past interactions are taken into account. Small bins δα

_{k−1}and δα

_{k}are centered at the values α

_{k−1}and α

_{k}in i. The TP is calculated from the number of visits (n

_{visit}Equation 11) of the future part to δα

_{k−1}and δα

_{k}simultaneously during the simulation. After ${\text{\rho}}^{\text{HS}}({\alpha}_{11}{\alpha}_{12}{|\alpha}_{10},\cdots ,{\alpha}_{1})$ (Equation 11) has been determined the coordinates of C’ (and O) are fixed at their positions in i, i.e., they become “past” atoms and the process continues.

_{s}) generated by MD using the arithmetic average:

_{s}and should appear with a bar as well. However, from now on only estimations will be considered and for simplicity, all of them will appear without the bar, like the energies defined in Equation (1). ${S}_{\text{loop}}^{\text{A}}$ (Equations (13) and (14)) constitutes a measure of a pure geometrical character for the loop flexibility, i.e., with no direct dependence on the interaction energy. When the converged or the best value of ${S}_{\text{loop}}^{\text{A}}$ is considered, it will be denoted by S

_{loop}; thus, F

_{loop}= E

_{loop}−TS

_{loop}is defined as the loop’s contribution to the total free energy, where E

_{loop}is defined in Equation (1). In the same way, the difference in the loop entropies between the free and bound microstates obtained for a specific set of parameters is denoted by $\text{\Delta}{S}_{\text{loop}}^{\text{A}}$ while the converged difference is denoted by $\text{\Delta}{S}_{\text{loop}}$:

_{loop}= ΔE

_{loop}−TΔS

_{loop}, where $\text{\Delta}{E}_{\text{loop}}$ is obtained from Equation (1). Notice that (unlike in [10]) ΔS

_{loop}and ΔF

_{loop}are obtained for different templates; for simplicity their dependence on the templates is omitted.

#### 2.6. Thermodynamic Integration of Water

_{water}(our main interest) can be obtained by a TI procedure (applied to each system) where these interactions are gradually increased (from zero) during an MD simulation of water (while the loop structure and template remain fixed). (Notice that while the templates are structurally different, the template-template interactions are ignored, therefore the templates are considered to be in the same state.)

_{s}times for each of the loop structures $[{\alpha}_{k}]$ of the sample of size n

_{s}. Denoting these $[{\alpha}_{k}]$ of the sample by t and keeping for brevity the letter m (“free” or “bound”) only on the left side of the equation, leads to:

_{s}’ integrations (denoted by t) are carried out, each starts from a different water configuration and a set of velocities (denoted t) thus:

#### 2.7. The Reconstruction Procedure with HSMD

_{init}initial considered conformations are discarded for equilibration. For each of the next n

_{f}(considered) future conformations the pair (α

_{k-}

_{1},α

_{k}) (k = 2k’) are calculated and their contribution to n

_{visit}(Equation (11)) is calculated. An essential issue is how to guarantee an adequate coverage of microstate m, i.e., that the future chains will span its entire region (in particular the side chain rotamers) while avoiding their “overflow” to neighboring microstates, conditions that will occur for a too small and a too large n

_{f}, respectively. [Note that even at step k’, where the “past” of the loop (atoms 1…k’−1) is kept fixed, the (future) unfixed part (atoms k’, k’ + 1…) can leave the microstate during long MD simulations. Such an “overflow” is more likely to happen for small residues such as Gly and for small k’.]

_{f}values used are not large and the microstates are concentrated (i.e., the Δα

_{k}values of Equation (10) are relatively small; see discussions in 3.2).

## 3. Results and Discussion

#### 3.1. Simulation Details

_{init}= 250 initial structures (2.5 ps) were discarded for equilibration. The future samples were generated for several bin sizes, δ, where results are presented for δ = Δα

_{k}/l, l = 5, 10, 20, 30, 40, and 50, centered at α

_{k}(i.e., α

_{k}± δ/2) (Equation (11)). If the counts of the smallest bin are smaller than 50 the bin size is increased to the next size, and if necessary to the next one, etc. In the case of zero counts, n

_{visit}is taken to be 1; however, an event of zero counts is very rare.

#### 3.2. Dihedral Angles for Different Microstates

_{k}(min), α

_{k}(max) and Δα

_{k}(Equation (10)) for the backbone dihedral angles φ and ψ and the sidechains, χ. These values are based on the samples of 1,000 conformations generated for the free and bound microstates for N

_{water}= 140 (rather than the results for N

_{water}= 160, which show some inconsistencies, see 3.4). The table reveals that the Δα

_{k}values for the backbone of the free and bound microstates are relatively small (in most cases smaller than 80

^{o}) and they are comparable to those obtained by other samples (based on different N

_{water}values). Somewhat larger Δα

_{k}values were obtained for χ, where all these results are also comparable to those obtained in [10]; this suggests that ΔS

_{loop}(Equation (15b)) would be small as well. These small Δα

_{k}values stem from the constraints imposed by the template on the inner loop.

_{k}(crystal). These angles enable one to determine whether the samples have escaped from their original microstates. While exact definition of a microstate is practically unfeasible (see discussion in section 1.3 of [8]), we have accepted an “escape” criterion for a dihedral angle when α

_{k}(crystal) + 60

^{o}is smaller than α

_{k}(max) or α

_{k}(crystal)−60

^{o}is larger than α

_{k}(min), i.e., if some angle values fall beyond the range α

_{k}(crystal) ± 60

^{o}; these angles are bold-faced in the table. The table reveals that only one backbone angle, φ(Arg) of both the free and bound microstates has been escaped but with small deviations of 18

^{o}and 10

^{o}, respectively. The number of escaped sidechain angles is seven for the free microstate and five for the bound one. Thus, the original microstates are retained for the backbone but only partially for the side chains, which might be expected since we model the loops in solution rather in the crystal environment.

**Table 1.**Minimum and maximum values of dihedral angles, α

_{k}(min) and α

_{k}(max) and their differences Δα

_{k}(in degrees) for the 140 free and bound samples*.

Free | Bound | |||||||

Residue angle | α
_{k}(crystal) | α
_{k}(min) | α
_{k}(max) | Δα
_{k} | α
_{k}(crystal) | α
_{k}(min) | α
_{k}(max) | Δα
_{k} |

Ser ψ | 165 | 135 | 162 | 27 | 172 | 159 | 186 | 27 |

ω | 179 | 156 | 179 | 23 | 177 | 142 | 172 | 30 |

Ile φ | −120 | −150 | −109 | 41 | −57 | −60 | −8 | 52 |

ψ | 138 | 136 | 185 | 49 | −44 | −93 | −31 | 62 |

Phe φ | 70 | 29 | 79 | 50 | −98 | −131 | −69 | 62 |

ψ | 43 | 32 | 88 | 56 | 130 | 73 | 139 | 66 |

Arg φ | −139 | −217 | −133 | 84 | −113 | −126 | −43 | 83 |

ψ | 136 | 84 | 160 | 76 | 13 | −37 | 16 | 53 |

Phe φ | −122 | −155 | −82 | 73 | −88 | −93 | −41 | 52 |

Side Chains | ||||||||

Ile χ^{1} | −118 | −93 | −38 | 55 | −82 | −65 | −25 | 40 |

χ^{2} | 38 | −98 | −34 | 64 | −47 | −62 | −17 | 45 |

χ^{3} | 180 | −180 | 180 | 360 | 180 | −179 | 180 | 359 |

χ^{2’} | 180 | 31 | 91 | 60 | 180 | 14 | 77 | 63 |

Phe χ^{1} | −139 | −174 | −119 | 55 | −62 | −86 | −40 | 46 |

χ^{2} | −36 | −58 | 8 | 66 | −51 | −94 | −31 | 63 |

Arg χ^{1} | −68 | −129 | −51 | 78 | −72 | −88 | −39 | 49 |

χ^{2} | −55 | −223 | −148 | 75 | −71 | −228 | −150 | 78 |

χ ^{3} | −170 | 30 | 104 | 74 | −50 | −36 | 90 | 126 |

χ^{4} | −97 | 61 | 176 | 115 | 176 | −183 | −51 | 132 |

χ^{5} | 0 | −37 | 35 | 72 | 0.4 | −29 | 37 | 66 |

χ^{6} | 0 | −42 | 42 | 84 | 0 | −39 | 34 | 73 |

χ^{6’} | 0 | −37 | 35 | 72 | 0 | −48 | 44 | 92 |

Phe χ^{1} | −43 | −62 | −13 | 49 | 40 | 36 | 76 | 40 |

χ^{2} | −80 | −104 | −41 | 63 | −88 | −132 | −79 | 53 |

_{k}(min), α

_{k}(max), and Δα

_{k}are defined in Equation (10); their values were calculated from samples of 1,000 loop conformations (0.5 ps) generated for the free and bound microstates. The values of α

_{k}(crystal) were calculated from the PDB crystal structures, 2ace [47] and 2dfp [40] of the free and bound protein, respectively.

**Table 2.**HSMD results (in kcal/mol) for the loop entropy, $T{S}_{\text{loop}}^{\text{A}}$ and for entropydifferences $T{S}_{\text{loop}}^{\text{A}}$ between the free and bound microstates at T = 300*.

N_{water} = 140 | ||||

Bin size | nf | $T{S}_{\text{loop}}^{\text{A}}$ | $T\text{\Delta}{S}_{\text{loop}}^{\text{A}}$ | |

Free | Bound | |||

Δα_{k}/5 | 2,000 | 63.4 | 63.3 | 0.1 |

Δα_{k}/10 | 2,000 | 61.6 | 61.5. | 0.1 |

Δα_{k}/20 | 2,000 | 61.3 | 61.4 | −0.1 |

Δα_{k}/30 | 200 | 60.8 | 60.9 | −0.1 |

“ | 400 | 61.0 | 61.1 | −0.1 |

“ | 1,200 | 61.3 | 61.3 | 0.0 |

“ | 1,600 | 61.3 | 61.3 | 0.0 |

2,000 | 61.3 | 61.4 | −0.1 | |

Δα_{k}/40 | 200 | 60.8 | 60.9 | −0.1 |

“ | 400 | 61.0 | 61.1 | −0.1 |

“ | 1,200 | 61.2 | 61.3 | −0.1 |

" | 1,600 | 61.3 | 61.4 | −0.1 |

2,000 | 61.3 | 61.4 | −0.1 | |

Δα_{k}/50 | 200 | 60.8 | 60.9 | −0.1 |

" | 400 | 61.0 | 61.1 | −0.1 |

" | 1,200 | 61.3 | 61.3 | 0.0 |

" | 1,600 | 61.3 | 61.3 | 0.0 |

2,000 | 61.3 | 61.3 | 0.0 | |

Errors ≤ | ±0.3 | ±0.3 | 0.0 | |

Converged | 61.3 ± 0.3 | 61.3 ± 0.3 | 0.0 ± 0.2 | |

QH | 10^{4} | 72.1 ± 1.0 | 71.5 ± 1.1 | 0.6 ± 2.0 |

_{water}= 140. The results are calculated as a function of the bin size δ = Δα

_{k}/l (Equation (10)) and n

_{f}(Equation (11)) the sample size of the future chains used in the reconstruction process.

#### 3.3. Results for the Loop Entropy

_{water}= 140 for the free and bound microstates and for their difference $T[{S}_{\text{loop}}^{\text{A}}(\text{free})-{S}_{\text{loop}}^{\text{A}}(\text{bound})]=T\text{\Delta}{S}_{\text{loop}}^{\text{A}}$ [see the discussion preceding Equation (15a)]. These results were obtained by reconstructing n

_{s}= 80 loop structures, distributed homogeneously along the entire sample of 1,000 system configurations. The simulated future consists of the future part of the loop including all the surrounding water molecules. The results are presented for several values of n

_{f}- the sample size of the future chains [Equation (11)], where n

_{f}= 200, 400, 1,200, 1,600, and 2,000; these values of n

_{f}are used for pairs of angles, such as a backbone dihedral and the successive bond angle. However, for the sidechains we also reconstruct a single χ angle and triplets of successive χ angles (e.g., for Arg) for which the maximal n

_{f}is 1,000 and 4,000 (rather than 2,000), respectively (see Equation (11)). The results are also presented as a function of bin size, δ = Δα

_{k}/l (Equations (10) and (11)) where l = 30, 40 and 50, while for n

_{f}= 2,000 we also provide results for larger bin sizes defined by l = 5, 10 and 20. The statistical errors were obtained from the fluctuations (standard deviation).

_{f}= 2,000, the $T{S}_{\text{loop}}^{\text{A}}(\text{free})$ values are 63.4, 61.6, 61.3, 61.3, 61.3, and 61.3 (kcal/mol + constant), i.e., they decrease for l = 5, 10, and 20 and converge to 61.3 (±0.3) kcal/mol for l = 30, 40 and 50; a similar behavior is observed for $T{S}_{\text{loop}}^{\text{A}}(\text{bound})$. One would also expect $T{S}_{\text{loop}}^{\text{A}}$ to decrease as n

_{f}increases in each bin. On the other hand, the table reveals that the central values of $T{S}_{\text{loop}}^{\text{A}}$ slightly increase in going from n

_{f}= 200 to 1,200 and then become constant. While most of these changes are within the error bars, they reflect an insufficient equilibration of 2.5 ps (see discussion following Equation (13)). Thus, eliminating the results for n

_{f}= 200 (i.e., increasing the equilibration to 4.5 ps) would lead to the expected decrease among the (new) $T{S}_{\text{loop}}^{\text{A}}$ results due to the expected increase in the (new) results for n

_{f}= 200 and 1,000 (however, to be consistent with [10] the results in the table are based on a 2.5 ps equilibration). In any case, within the error bars the $T{S}_{\text{loop}}^{\text{A}}$ values in Table 2 can be considered as converged; they are only slightly larger (by ~1 kcal/mol) than those of Table 4 of [10] and the errors in both tables are comparable.

**σ**, is obtained from a local MD sample and N is the number of internal coordinates used in the HSMD procedure. It should be pointed out that in a detailed study of QH Chang et al. [61] have found that the method might be unreliable when used in Cartesian coordinates or applied (in internal coordinates) to several microstates. On the other hand, QH was found suitable for treating a single microstate, while the convergence of the results is slow and large samples are typically needed; also, because QH takes into account only second order correlations ${S}_{\text{loop}}^{\text{QH}}(m)$ constitutes an upper bound for the correct S. Still, entropy differences $\text{\Delta}{S}_{\text{loop}}^{\text{QH}}(m)$ are expected to be reliable (see also [21]). Thus, to obtain reasonable precision, results for ${S}_{\text{loop}}^{\text{QH}}(m)$ (m is bound or free) were obtained from MD samples of 10,000 loop-water configurations. To avoid the “escape” of a sample from the original microstate, it consists of 10 separate samples of 1,000 configurations (0.5 ns), each started from the same structure with a different set of initial velocities, where the initial trajectory of 0.5 ns is used for equilibration and is thus discarded. The values of $T{S}_{\text{loop}}^{\text{QH}}$ exceed the HSMD results for $T{S}_{\text{loop}}$ (for n

_{s}= 2,000) by 11–10 kcal/mol for the free and bound microstates. These elevated results are in accord with ${S}_{\text{loop}}^{\text{QH}}$ being an upper bound and are comparable to the overestimation values found in our previous studies [6,7,8,9,10]. On the other hand, ${S}_{\text{loop}}^{\text{QH}}(\text{free)-}{S}_{\text{loop}}^{\text{QH}}(\text{bound)}$ = 0.6 ± 2.0 is in accord (within the error bars) with the HSMD result ΔS

_{loop}= 0.

_{water}= 160 is 8.1 hours CPU on a 2.1 GHz Athlon processor, meaning that the entire reconstructions required 1,296 and 1,472 h CPU; this time is smaller for N

_{water}= 140. However, we have shown that considering only 10% (n

_{f}= 200) of the maximal reconstruction samples and using smaller samples of n

_{s}= 40 (rather than 80) have led to sufficiently accurate entropy differences, meaning that the total computer time can be reduced to 65 h CPU, respectively. We have generated the relatively large reconstruction samples to verify the convergence of the results.

#### 3.4. The Effect of Water

_{loop}(Equation (1)), for the loop-water free energies (calculated by TI), ${F}_{\text{water}}^{\text{TI}}$(ch,loop), ${F}_{\text{water}}^{\text{TI}}$(LJ,loop), and their sum ${F}_{\text{water}}^{\text{TI}}$(loop) [Equation (16)], for the water-template free energies ${F}_{\text{water}}^{\text{TI}}(\text{ch,tmpl})$, ${F}_{\text{water}}^{\text{TI}}(\text{LJ,tmpl})$ and their sum ${F}_{\text{water}}^{\text{TI}}(\text{tmpl})$ (Equation (17)); we also provide results for F

_{sum}= E

_{loop}+ ${F}_{\text{water}}^{\text{TI}}$(loop) + ${F}_{\text{water}}^{\text{TI}}(\text{tmpl})$ (which is the total free energy without the contribution of the loop entropy, S

_{loop}, Equations (13) and (14)), and for the total energy, E

_{total}(Equation (1)). For simplicity we have omitted the letter “m” from these quantities; however, each one of them is calculated for the free and bound microstates, and their difference, Δ (free bound) is also provided. These results are obtained for 40 ≤ N

_{water}≤ 160. Details about the integrations are described in the Appendix.

_{s}is 80 (N

_{water}≤ 140) and 40 for N

_{water}= 160. Statistical errors, s/(n

_{s})

^{1/2}, where s is the standard deviation, are provided only for N

_{water}= 140 and 160, which are larger than those obtained for N

_{water}< 140. The largest error of 2.5 kcal/mol is for E

_{total}(N

_{water}= 160), where the other errors are significantly smaller. The errors in the differences, Δ are again very stable, and smaller than 2.6 kcal/mol for ΔF

_{sum}(N

_{water}= 160); these errors are comparable to those obtained in [10]. It is in particular encouraging that small errors are found for the water-template free energies, ${F}_{\text{water}}^{\text{TI}}(\text{ch,tmpl})$, ${F}_{\text{water}}^{\text{TI}}(\text{LJ,tmpl})$ and their sum ${F}_{\text{water}}^{\text{TI}}(\text{tmpl})$, which are based on an integration of a significant number of atoms and are calculated for the first time in this paper.

_{loop}= F

_{free}– F

_{bound}~ −4 kcal/mol for water densities close to that of bulk water, which for the present system occur for N

_{water}ranging from 140 to 180. Still, it is of interest to check this expectation by calculating results also for smaller N

_{water}. We have seen in 3.2 that for N

_{water}= 140 ΔS

_{loop}~0, where small ΔS

_{loop}was found also in [10]. We therefore assume that ΔS

_{loop}~0 for other sets of results (N

_{water}≠ 140) meaning that ΔF

_{sum}actually constitutes the difference in free energy between the two microstates. However, the results for ΔF

_{sum}in the table are positive for all N

_{water}, even for N

_{water}= 140 and 160, i.e., they differ significantly from the experimental value; in particular, for N

_{water}= 140 ΔF

_{sum}= 18 ± 2 kcal/mol.

_{loop}to be close to zero for all values of N

_{water}, also means that the total entropy, TΔS

_{total}= ΔE

_{total}– ΔF

_{sum}is mainly due to water. The table shows that in all cases ΔE

_{total}is positive, i.e., the total energy of the free loop is larger than that of the bound one. Correspondingly, in all cases (besides for N

_{water}= 100), ΔE

_{total}is larger than ΔF

_{sum}, i.e., the entropy of the free loop (as expected) is also the largest, where TΔS

_{total}ranging from 5 (N

_{water}= 140) to 42 kcal/mol for N

_{water}= 160. Thus the entropy contributes significantly to the free energy where it constitutes between 20 and 64% of the total energy.

_{loop}, it is of interest to discuss the results for N

_{water}= 160. These results (in particular those for the differences Δ) show an unexpected “jump” as compared to their counterparts for N

_{water}< 160. Thus, while the results for ΔF

_{sum}for 60 ≤ N

_{water}≤ 140 are comparable (ranging between 15 and 26 kcal/mol), ΔF

_{sum}(N

_{water}= 160) = 95 kcal/mol. Correspondingly, ΔE

_{total}increases from 23 (N

_{water}= 140) to 137 kcal/mol (N

_{water}= 160). The main contributors for these results are the relatively large values of the sum of the loop energies, ΔE

_{loop}= 98 and $\text{\Delta}{F}_{\text{water}}^{\text{TI}}(\text{LJ,tmpl})$ = 55 kcal/mol; these values are significantly larger than their counterparts for N

_{water}< 160. To check this point further we integrated the water-template interactions in our sample for the free microstate generated in [10] for N

_{water}= 160 to find the still relatively large results, ΔF

_{sum}= 43 kcal/mol and ΔE

_{total}= 134 kcal/mol. In these calculations (which do not appear in the table) ΔE

_{loop}= 117 contributes significantly to the large ΔF

_{sum}; on the other hand, $\text{\Delta}{F}_{\text{water}}^{\text{TI}}(\text{LJ,tmpl})$ = −64 kcal/mol is negative! It has been difficult to identify the configurationl elements that lead to these energetic changes even after examining the individual configurations numerically and by computer graphics. However, since the template is fixed, these results suggest that as the number of waters increase to N

_{water}= 160 their configuration undergoes a significant change, probably due to the increase of pressure in the sphere. Thus, water molecules that stay in the bulk for N

_{water}<160 enter the template already in the equilibration stage (i.e., when the water-template interactions are still intact) affecting thereby the results for $\text{\Delta}{F}_{\text{water}}^{\text{TI}}(\text{LJ,tmpl})$ and $\text{\Delta}{F}_{\text{water}}^{\text{TI}}(\text{ch,tmpl})$.

_{water}= 160 are not expected to disappear to a large extent when a single template is used, because changes in the arrangement of water will affect approximately equally the results for the free and bound microstates and the corresponding deviations will mostly be cancelled in differences.

_{water}= 160 discussed above, the results for N

_{water}< 160 show a consistent behavior, and based on [10] one would expect the simulations with N

_{water}= 140 to lead to the experimental, ΔF

_{loop}= −4 rather than ΔF

_{loop}= +18 kcal/mol. This disagreement might stem from the uncertainty in the crystal structure of the bound protein, 2dfp, which is characterized by large B-factors (with an average of 41.6 Å

^{2}) whereas for many atoms the values are much larger; for comparison, the average B-factors in 2ace, the crystal structure of free AChE, is 23.4 Å

^{2}. The large cavity formed in the active site of the bound template mentioned above might also affect the results for the bound protein. These are inherent problems which might not be easy to overcome completely; however, one might alleviated them by applying different initial optimizations to the two templates. Thus, in this work we have used the same harmonic force constants of 1 kcal/(molÅ

^{2}) to the free and bound x-ray structures, while it is plausible to anticipate that the force constant should be commensurate with the quality of the crystal structure, i.e., stronger for well defined structures. The results for N

_{water}= 160 provide some support for this assumption; thus, F

_{sum}= 43 is better (i.e., smaller) than F

_{sum}= 95 kcal/mol where these values are based on templates optimized by the force constants, 100 ([10]) and 1 kcal/(molÅ

^{2}) (here), respectively. However, the optimal force constants are unknown a-priori and their determination probably would require quite a few trials.

_{loop}= −4 is known makes AChE a suitable system for checking the calculation of ΔF

_{loop}based on two different templates. As previously discussed, such calculations are of interest, as they mimic better the experimental conditions, and for an enzyme with a mobile loop, ΔF

_{loop}might be an important ingredient of the total absolute free energy of binding. Our calculations have revealed some instability in the results for N

_{water}= 160 but have led to consistent results for N

_{water}≤ 140; obviously, it is somewhat disappointing that ΔF

_{loop}= −4 kcal/mol has not been recovered for N

_{water}= 140. However, this is not a failure of HSMD-TI per se but reflects the uncertainty in the crystal structure of the bound protein. One should also bear in mind that no other studies of a loop attached to two templates have been carried out thus far and besides our HSMD-TI papers [8,9,10,62] only few calculations of ΔF

_{loop}(based on a single template) are available [48,49]. Thus, the present work should be considered as the first step and a basis for future more extensive studies, which can be pursued in several avenues: for example, HSMD-TI can be applied to a mobile loop with known ΔF

_{loop}where the quality of the crystal structures of both, the bound and the free protein is high. As is already stated above, one can try optimizing the templates by changing the force constants in a systematic way for N

_{water}= 140, and more work can be done to elucidate the problematic behavior of the results for N

_{water}= 160, which again reflects problems in the simulation itself rather than in the application of HSMD-TI.

**Table 3.**Results in kcal/mol for energy and free energy components and their difference (Δ) between the free and bound microstates*.

E loop energy | F loop-water (TI) | F template-water (TI) | |||||||||

template-loop | loop-loop | sum | CH | LJ | sum | CH | LJ | sum | F sum | E total | |

40 waters n_{s} = 40 | |||||||||||

Free | −93 | −48 | −140 | −60 | 8 | −52 | −242 | −12 | −254 | −446 | −432 |

Bound | −173 | −49 | −222 | −13 | −1 | −14 | −246 | −11 | −257 | −493 | −494 |

∆ | 81 | 1 | 82 | −47 | 9 | −38 | 3 | 0 | 3 | 47 | 61 |

60 waters n_{s} = 40 | |||||||||||

Free | −91 | −51 | −142 | −60 | 12 | −48 | −313 | −2 | −315 | −505 | −675 |

Bound | −174 | −47 | −221 | −8 | 15 | 7 | −305 | −4 | −309 | −523 | −725 |

∆ | 84 | −4 | 79 | −52 | −2 | −55 | −8 | 2 | −7 | 18 | 50 |

80 waters n_{s} = 40 | |||||||||||

Free | −90 | −49 | −139 | −67 | 45 | −23 | −350 | 11 | −339 | −500 | −874 |

Bound | −168 | −47 | −215 | −11 | 30 | 19 | −354 | 29 | −326 | −521 | −928 |

∆ | 78 | −2 | 76 | −56 | 15 | −41 | 5 | −18 | −13 | 21 | 54 |

100 waters n_{s} = 80 | |||||||||||

Free | −108 | −39 | −147 | −66 | 51 | −15 | −379 | 37 | −342 | −504 | −1075 |

Bound | −168 | −45 | −213 | −6 | 46 | 40 | −395 | 38 | −357 | −530 | −1134 |

∆ | 60 | 6 | 66 | −60 | 5 | −55 | 16 | −1 | 15 | 26 | 5 |

120 waters n_{s} = 80 | |||||||||||

Free | −105 | −39 | −143 | −64 | 59 | −5 | −426 | 73 | −353 | −500 | −1296 |

Bound | −152 | −40 | −192 | −24 | 63 | 40 | −442 | 79 | −363 | −516 | −1339 |

∆ | 47 | 1 | 49 | −40 | −4 | −44 | 16 | −6 | 11 | 15 | 38 |

140 waters n_{s} = 80 | |||||||||||

Free | −89 (0.5) | −32 (0.7) | −121 (0.9) | −75 (0.3) | 82 (0.8) | 8 (0.8) | −471 (0.4) | 126 (0.6) | −345 (0.6) | −459 (1.3) | −1491 (1.4) |

Bound | −128 (0.4) | −36 (0.7) | −164 (0.8) | −44 (0.4) | 76 (0.6) | 32 (0.8) | −479 (0.6) | 135 (0.8) | −344 (0.9) | −477 (1.4) | −1515 (1.6) |

∆ | 39 (0.6) | 4 (1.1) | 43 (1.2) | −30 (1.5) | 6 (1.0) | −24 (1.1) | 8 (0.6) | −9 (1.0) | −1 (1.1) | 18 (1.9) | 23 (2.2) |

160 waters n_{s} = 40 | |||||||||||

Free | −82 (0.7) | −28 (0.9) | −111 (1.0) | −73 (0.6) | 106 (1.0) | 33 (1.3) | −523 (0.5) | 332 (1.4) | −191 (1.5) | −269 (2.1) | −1524 (2.5) |

Bound | −167 (0.8) | −42 (1.2) | −209 (1.2) | −8 (0.7) | 96 (0.9) | 88 (0.9) | −520 (0.5) | 276 (1.6) | −244 (1.5) | −364 (1.6) | −1661 (2.4) |

∆ | 84 (0.7) | 13 (1.2) | 98 (1.2) | −65 (0.7) | 10 (0.9) | −54 (1.0) | −2 (0.5) | 55 (1.7) | 53 (1.7) | 95 (2.1) | 137 (2.6) |

_{loop}, and the total energy, E

_{total}are defined in Equation (1). F

_{loop-water}(TI) are ${F}_{\text{water}}^{\text{TI}}$(ch) and ${F}_{\text{water}}^{\text{TI}}$(LJ) (Equation (16)) which are free energies calculated by TI, where the loop-water charges and Lennard-Jones interactions, respectively, are gradually eliminated; their sum is ${F}_{\text{water}}^{\text{TI}}(\text{loop})$. Correspondingly, F

_{tmpl-water}(TI) are ${F}_{\text{water}}^{\text{TI}}(\text{ch,tmpl})$ and ${F}_{\text{water}}^{\text{TI}}(\text{LJ,tmpl})$ [Equation (17)] which are free energies calculated by TI, where the template-water charges and Lennard-Jones interactions, respectively, are gradually eliminated; their sum is ${F}_{\text{water}}^{\text{TI}}(\text{tmpl})$. F

_{sum}= Eloop + ${F}_{\text{water}}^{\text{TI}}(\text{loop)}+{F}_{\text{water}}^{\text{TI}}(\text{tmpl)}$ is the total free energy (without the contribution of the loop entropy, S

_{loop}). Δ is the difference, free bound. n

_{s}is the sample size. Statistical errors, s/(n

_{s})

^{1/2}, where s is the standard deviation, are given only for results based of a number of water molecules, N

_{water}= 140 and 160; the errors for N

_{water}< 140 are smaller than those obtained for N

_{water}= 140 and 160.

## 4. Summary and Conclusions

^{2}) and the energy was minimized. Afterwards the loop and (TIP3P) water atoms were allowed to relax in the presence of a fixed template. The relatively weak force constant (1 kcal/Å

^{2}) (as compared to 100 kcal/Å

^{2}used in [10] for the pdb structure 2ace) was chosen to allow more extensive changes in the bound structure due to its relatively large B-factors and the chemical change imposed on it by removing the phosphate group; correspondingly, larger structural changes were also allowed to occur in the free structure. Then two cuts from each crystal structures were performed, which contain exactly the same atoms; these templates and the attached loops were subjected to further optimization steps as described earlier.

_{water}≤ 140 the errors are relatively small and the results for the free-bound differences in the total free energy and energy, ΔF

_{sum}and ΔE

_{total}are comparable, while for N

_{water}= 160 the errors become larger (in particular for the Lennard-Jones part of the water-template TI) and a significant increase in ΔF

_{sum}and ΔE

_{total}occurs. We attribute this jump in the results to changes in the configuration of water stemming from the increase in water pressure in the sphere as the number of waters grows.

_{loop}based on separate templates which are cut from the free and bound crystal structures is more realistic than using a single common template for both loop conformations. ΔF

_{loop}should also be considered in the calculation of the absolute free energy of binding when the conformational change of the loop is due to ligand binding is of an induced fit type. Thus, while the present calculations show that recovering the experimental result, ΔF

_{loop}= F

_{free}− F

_{bound}~ −4 kcal/mol is not straightforward, to achieve improvement more research is needed, which will be based on the conclusions of the initial work carried out here, as specified in the summary of the previous section.

## 5. Appendix

#### 5.1. Thermodynamic Integration of Water

^{2}, prevents the divergence of the potential (and its derivative) at small pair separations; a similar scaling function is used for the Coulomb interactions. The free energy derivatives with respect to λ, ∂F/∂λ is:

_{i}. The (λ = 1→λ = 0) integration of the electrostatic interactions (i.e., charge elimination) is carried out first (in the presence of intact LJ interactions) followed by a ε = λ = 1→0 integration of the LJ interactions. Thus, the entire two-stage process is based on 40 ∂F/∂λ

_{i}integration steps.

_{i}) step (window) starts with energy minimization (based on λ

_{i}) of the last structure obtained in the simulation of the i-1 step, followed by 5 ps MD simulation for equilibration, which is discarded. The next step the production MD simulation, is of 20 ps in the loop-water TI, where a configuration is retained every 0.02 ps, i.e., altogether 1,000 configurations are used for evaluating <∂F/∂λ

_{i}>. For the template-water TI, the production simulation is longer, of 40 ps (2,000 configurations), due to the larger size of the template. It should be pointed out again that these two integrations are carried out for two different templates. The computer time is discussed in the Appendix of [10].

## Acknowledgments

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**MDPI and ACS Style**

Mihailescu, M.; Meirovitch, H.
Entropy and Free Energy of a Mobile Loop Based on the Crystal Structures of the Free and Bound Proteins. *Entropy* **2010**, *12*, 1946-1974.
https://doi.org/10.3390/e12081946

**AMA Style**

Mihailescu M, Meirovitch H.
Entropy and Free Energy of a Mobile Loop Based on the Crystal Structures of the Free and Bound Proteins. *Entropy*. 2010; 12(8):1946-1974.
https://doi.org/10.3390/e12081946

**Chicago/Turabian Style**

Mihailescu, Mihail, and Hagai Meirovitch.
2010. "Entropy and Free Energy of a Mobile Loop Based on the Crystal Structures of the Free and Bound Proteins" *Entropy* 12, no. 8: 1946-1974.
https://doi.org/10.3390/e12081946