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* Bijvoet Center for Biomolecular Research, Utrecht University, Utrecht, The Netherlands; and Laboratory of Physical Chemistry, Swiss Federal Institute of Technology, Zürich, Switzerland
Correspondence: Address reprint requests to Alexandre M. J. J. Bonvin, Tel.: 31-30-253-3859; Fax: 31-30-253-7623; E-mail: a.m.j.j.bonvin{at}chem.uu.nl.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONREFERENCES |
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; Sattentau and Moore, 1991). This is followed by the binding of the chemokine co-receptor to gp120 (Berger et al., 1999; Rizzuto et al., 1998; Trkola et al., 1996; Wu et al., 1996) and the extension or stretching of the coiled-coil gp41, leading to the insertion of the membrane fusion peptide into the host membrane, which induces fusion of the viral and host membranes (Chan and Kim, 1998). Specific recognition of CD4 by gp120 is evidently the linchpin of the CD4-mediated viral infection pathway. Recent extensive biochemical and biophysical studies have provided ample evidence of a substantial structural rearrangement during the gp120-CD4 recognition process (Wyatt and Sodroski, 1998). Large enthalpic (–259 ± 13 kJ mol–1) and entropic changes (220 ± 13 kJ mol–1 at 310 K) upon binding were observed experimentally (Kwong et al., 2002; Myszka et al., 2000). The free energy of binding, however, is small due to a remarkable entropy/enthalpy compensation. This means that the affinity of gp120 for CD4 and CD4-induced monoclonal antibodies (CD4MAb) is relatively small, and can be fine-tuned by small variations in those two rather large contributions. The gp120/CD4 interaction is predominantly determined by the core of gp120: removal of the hypervariable loops corresponding to nearly 40% of the total sequence shows little effect on the thermodynamic properties of binding (Myszka et al., 2000).
Comprehensive structural insight is, however, limited by the intrinsic flexibility of the system. The structure of the ternary complex of the core of gp120, CD4, and a CD4-induced antibody is the only available atomic information to date (Kwong et al., 2000, 1998). This truncated form of gp120 (core gp120) does not compromise much the biophysical and biological properties compared to the full-length wild-type (Kwong et al., 2002; Myszka et al., 2000; Pollard et al., 1992; Wyatt et al., 1993). It provides a suitable structural template for molecular dynamics (MD) studies allowing a significant reduction in the size for the simulation system as compared to full-length gp120. Using MD simulations we have identified concerted loop motion around the CD4 binding site in gp120 upon binding (Hsu and Bonvin, 2004). The enthalpic change upon gp120/CD4 complex formation, calculated as the difference in protein-protein plus protein-solvent potential energy between the simulations of the complex and of the separate proteins (–127 kJ mol–1 at 300 K with a standard error, SE, of 11 kJ mol–1), is of the same order of magnitude as the experimentally determined value (–259 ± 13 kJ mol–1 at 310 K; Myszka et al., 2000). (Note that SE is defined as the standard deviation SD divided by the square-root of the number of sampling point N. The energies were calculated every 5 ps as reported earlier by Hsu and Bonvin, 2004, and therefore N is 16,001 when analyzing the 2–10 ns simulation segments.) Encouraged by the rough agreement between the theoretical and experimental binding enthalpies, we aimed at extracting the binding entropy from our simulations and at gaining insight into CD4 binding-induced conformational changes in gp120 that are related to entropy changes.
The estimation of configurational entropy from molecular dynamics trajectories was first proposed by Karplus and Kushick (1981) using a quasiharmonic method. The difference in configurational entropy between two molecular conformations a and b can be estimated as 
S = kB/2 ln(det
a/detb), where a and b are the covariance matrices of atomic positional fluctuations of the two conformers and kB is Boltzmann's constant. The method was formulated in terms of internal (non-Cartesian) coordinates, which made it less easily applicable. This approach was extended and applied to various biomolecular systems (Di Nola et al., 1984; Edholm and Berendsen, 1984; Levy et al., 1984). A decade later, Schlitter (1993) introduced a heuristic formula, based on Cartesian coordinates, to compute an upper bound to the absolute entropy of a molecule from a simulation trajectory. Calculation of the absolute molecular entropy would require a complete translational and rotational sampling. This is not yet reachable for proteins with the current computation timescale of nanoseconds. Instead, an assessment of the configurational entropy can be obtained from a MD trajectory based on the covariance matrix of the Cartesian atom-positional fluctuations after elimination of translational and overall rotational motion by atom-positional least-squares fitting of molecular trajectory structures onto each other. This method was successfully tested for biomolecular simulations of peptide folding (Schäfer et al., 2001, 2000) and applied to simulations of protein molten globule states (Schäfer et al., 2002). Recently Andricioaei and Karplus (2001) revised the quasiharmonic approach to allow for the use of Cartesian coordinates.
One advantage of approaches based on the covariance matrix of atomic fluctuation is the possibility to compute this quantity for different subsets of atoms or degrees of freedom. One can, for example, resolve the entropic contributions of hydrophilic and hydrophobic residues to protein-protein binding, which is experimentally impossible. It should be noted, though, that entropies originating from different degrees of freedom need not be additive and that a decomposition will neglect correlation of motions along different degrees of freedom. Yet, the neglect of correlation between a particular subset of atoms and the rest of the molecule is a reasonable first-order approximation. Such analysis nonetheless enables us in the current study to estimate the entropic contribution of subsets of atoms, or degrees of freedom of interest, down to residue and atomic levels. Comparison of the configurational entropies of gp120 and the CD4 D1 domain (denoted CD4 for simplicity in the following) in the free and bound states suggests that the formation of intermolecular hydrogen bonds and hydrophobic contacts contributes most to the entropy changes. In line with previous postulates based on thermodynamic data (Kwong et al., 2002; Myszka et al., 2000; Xiang et al., 2002), the conformational rearrangement in the bridging sheet of gp120 upon CD4 binding is accompanied by a significant loss of entropy. The large-amplitude relocalization of the V3 loop of gp120 upon binding is, however, free from any substantial entropic cost.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONREFERENCES |
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).
The GROMACS program package (Lindahl et al., 2001) was used for the MD simulations with the GROMOS96 43A1 force field (Daura et al., 1998; van Gunsteren et al., 1996), and the simple point-charge water model (Berendsen et al., 1981) with rectangular periodic boxes with a 1.4-nm solute-wall minimum distance. Nonbonded interactions were calculated using twin-range cutoffs of 0.8 and 1.4 nm. Long-range electrostatic interactions beyond the cutoff were treated with a generalized reaction field model (Tironi et al., 1995), using a dielectric constant of 54. For further simulation details, we refer to Hsu and Bonvin (2004). Owing to the large system sizes, 
2 ns was required to reach equilibrium. The entropy analysis was therefore performed on the last 8 ns of each simulation, using molecular configurations that are two picoseconds apart.
Schlitter's formula was used for the configurational entropy calculation, which yields an upper bound to the true entropy Strue,
 | (1) |
Plank's constant divided by 2
, M the mass matrix that holds on the diagonal the masses belonging to the atomic Cartesian degrees of freedom, and the covariance matrix of atom-positional fluctuations. The elements of are given by
 | (2) |
Two sets of atoms were used in the superposition of molecular configurations:
, and C') of the most stable structural elements of both proteins, the first -helix of gp120 (residues 17–31), and part of the ß-sheet of CD4 consisting of residues 26–30 and 82–86 (see Fig. 1). The set of atoms is indicated by the code 2nd. , and C') atoms of individual residues were used for superposition of trajectory configurations when calculating the entropy per residue (code: fir).
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, and C') atoms (code: bb); see Fig. 4.
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 | (3) |
 | (4) |
To assess the degree of overlap between the configurational spaces sampled in two simulations, one may append one trajectory to the other trajectory and compute the development of the configurational entropy S with time; see Fig. 2. Denoting the trajectories by I and II we then obtain
 | (5) |
 | (6) |
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The coverage of the conformational space of the second trajectory segment with respect to the first segment of a combined trajectory can be quantitatively measured by the difference between S at the end of the second segment of the combined trajectory buildup curve and S at the end of the first segment. For example, the coverage of the free trajectory with respect to the complex trajectory is
 | (7) |
 | (8) |
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONREFERENCES |
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. Unlike other structural parameters such as the atom-positional root mean-square deviations (RMSD) of the trajectory structures from the starting structure, which seem to reach equilibrium after 2 ns for CD4 and 4 ns for gp120 (Fig. 3; for details see Hsu and Bonvin, 2004), only the backbone configurational entropy starts to level off after 4 ns (Fig. 4, A and B), and the all-atom values are still in the buildup phase throughout the 8-ns trajectories (Fig. 4, C and D). Unexpectedly, the configurational entropies of both backbone and all atoms of gp120 (Fig. 4, A and C) in the complex formed (dashed lines until 8 ns) toward the 8-ns time point are slightly higher (
and 
) than those in the free form (solid lines until 8 ns). This increase, however, only amounts to <2% of the absolute value.
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6 ns (Fig. 4, A and C) is due to a leveling off of the curve for the free form, which suggests that convergence is near for the free molecule. Appending the free and complex trajectories of CD4 gives rise to a small stepwise increase of the entropy buildup of the backbone atoms (Fig. 4 B at 8 ns), regardless of the order of appending. This suggests that two slightly different conformational spaces are sampled in the free and complex forms of the molecule, representing a situation of case A (Fig. 2 A). For the all-atom analysis of CD4 (Fig. 4 D), on the other hand, a smooth continuation at 8 ns of the configurational entropy buildup is observed when the complex trajectory is appended to the free one (solid line), whereas a slight stepwise increase is visible when the trajectories are appended in the reverse order (dashed line). This illustrates case B (Fig. 2 B), where a subspace of the conformational space of the free molecule is sampled in the complex state.
Configurational entropy of interacting elements and CD4 binding-induced conformational changes in gp120
The gp120/CD4 interface involves two major interaction modes:
The loss of entropy stemming from the rigidification of the residues in gp120 forming stable hydrophobic contacts to CD4-F43 (
; see Fig. 5 A) is of the same size as the corresponding loss of entropy in CD4-F43 (
see Fig. 5 B). Appending the complex trajectory of gp120 to the free one gives rise to a small increase (
), indicating that the complex form samples slightly different parts of conformational space (Fig. 5 A). Interestingly, for CD4-F43, the corresponding value is negative, –0.01 kJ K–1 mol–1 (solid line in Fig. 5 B). This would only occur when the averaged atom-positional fluctuations, i.e., the determinant of the covariance matrix, are reduced due to the addition of the second trajectory segment (case C, Fig. 2 C). In other words, the second MD trajectory segment covers a small subspace of the conformational space sampled by the first segment and hence reduces the overall size of the elements of the covariance matrix through averaging over the combined trajectories. As an illustration of this, the root mean-square fluctuations of the phenyl ring of CD4 Phe43 after least-squares fitting on all heavy atoms of CD4 decreases from 0.30 ± 0.07 nm in the free form to 0.08 ± 0.01 nm in the complex form. Moreover, all Phe43 side-chain conformations in the complex are contained within the ensemble of conformations of the free form.
In addition to the changes directly related to the intermolecular contact, CD4 binding also induces conformational changes away from the binding interface that are crucial for the subsequent events of viral entry into the host cell. It is generally accepted from thermodynamic and biochemical data that the ß-sheet of gp120 that bridges the inner and outer domain of the gp120 core, the bridging sheet, is only fully formed and stabilized upon complexation with CD4 (Fig. 1), and that this stabilization contributes a sizeable entropic loss and/or enthalpic gain (Jardetzky, 2002; Kwong et al., 2002; Xiang et al., 2002). Concomitantly, the third hypervariable loop (V3) undergoes a rearrangement to a somewhat different conformation leading to accessibility of the epitope for co-receptor binding. Our previous MD study revealed that lid-closure motions upon CD4 binding are accompanied by concerted structural changes leading to a substantial increase of rigidity of the bridging sheet and a large-amplitude translocation of the V3 loop (Hsu and Bonvin, 2004). The corresponding changes in entropy are shown in Fig. 6. The bridging sheet shows a clear entropy difference between the free and complex forms of gp120 with a loss of entropy of 0.23 kJ K–1 mol–1. Not only is its flexibility reduced, but its configuration is also altered. This can be concluded from the subsequent change after appending the trajectories of the complex and of the free form to each other, which is consistent with the previously defined case A (Fig. 2 A). The entropy of the V3 loop in the CD4-bound form initially builds up more slowly than the entropy in the free form, but crosses the latter curve in the last 1.5 ns, resulting in a final increase of 0.24 kJ K–1 mol–1 (Fig. 6 B). In addition, the V3 loop shows an increase in entropy when the two trajectories are combined, irrespective of the order of appending. These observations suggest that the V3 loop undergoes major changes in conformation and/or localization upon CD4 binding, although its intrinsic flexibility persists, or even possibly increases.
Configurational entropy changes per residue
Changes in conformational entropy upon binding can also be calculated per residue. For this, the atom-positional least-squares fitting of trajectory structures was performed using backbone atoms (N, C, and C') of the individual residues to exclude from the entropy contributions of collective motions of larger structural segments. The intraresidue conformational entropy obtained in this way provides, therefore, only information about side-chain motions. The intraresidue configurational entropy per residue was calculated using Eq. 1 and then normalized by dividing it by the number of atoms of each residue. Intraresidue entropy differences between the free and complex forms 
(Eq. 4) are plotted as a function of residue number in Fig. 7.
On average, CD4 shows an intraresidue entropy decrease of –0.5 kJ K–1 mol–1 per atom, although gp120 shows an unexpected increase of 0.3 kJ K–1 mol–1. Most of the residues of CD4 and gp120 that are involved in the intermolecular hydrogen-bond network (solid bars in Fig. 7) show reductions in intraresidue configurational entropy upon complexation, except for K22 and D53 in CD4, and C126, S365, and E429 in gp120. The relative large entropy of these five residues illustrates their intrinsic flexibility that is reflected in their large B-factors in the crystal structure (Kwong et al., 1998). In addition, our previous MD analysis of the intermolecular hydrogen bonding shows that all these residues, except CD4-K22, form only marginally stable intermolecular hydrogen bonds. Despite its high occurrence (78%), the salt bridge between CD4-K22(N
) and gp120-E429(C
) located at the edge of the interface of the molecular complex might be subject to fewer structural limitations than those in the center of the interface. Furthermore, the relatively long side chains of both residues might also tolerate a higher degree of flexibility despite the presence of a salt bridge.
In line with the previous analysis of the functional elements, CD4-F43 shows a large intraresidue entropy reduction upon complexation (Fig. 7) due to the steric restriction imposed by the receptive cavity of gp120. The neighboring residue Q40, which lies at the center of the complex interface, shows the largest reduction among all residues (top row in Fig. 7). Although less confined than CD4-F43, it is in close contact (<0.5 nm interatomic distance) with T283, G473, and D474 of gp120, of which the latter two were found to form intermolecular hydrogen bonds with F43 and Q25 of CD4, respectively (Hsu and Bonvin, 2004). In gp120, regions that show concerted lid-closure motions upon CD4 binding, namely LD, C3, C4, and V5, display substantial intraresidue entropy reductions upon complexation especially for those residues that are involved in intermolecular hydrogen bonding (Fig. 7). Note that the outer half of the bridging sheet (ß3 and ß4) and the tip of the hairpin structure of the V3 loop (residues 297–330) do not exhibit significant entropy changes, most likely due to the lack of direct contact with CD4. This illustrates that changes in conformation do not necessarily involve a change in entropy or extent of motion.
In addition to the residues that are directly involved in intermolecular contacts or are in close proximity to the intermolecular interface, there are a few residues in gp120 that show large intraresidue entropy changes upon complexation (see Fig. 7). Some residues show positive intraresidue entropy change upon complexation. Several are located in the putative trimerization interface, S243, C378, and I439, and two of them, V255 and L260, are located in the ß-turn that connects the inner and outer domains of gp120. This indicates that the intermolecular interactions between gp120 and CD4, although decreasing the intraresidue entropy of residues in direct contact, can induce intraresidue entropy changes at the interdomain interface within the gp120 monomer and at the putative trimeric gp120 interface.
Error analysis of configurational entropies
The configurational entropy differences between complex and free forms reported above were obtained from the entropy differences at 8 ns (the full trajectory length used for analysis). To estimate errors, several independent simulations could be performed and compared. This is, however, rather prohibitive, considering the system size in this particular case. An alternative is to divide a trajectory into N blocks of equal size over which sub-averages and standard errors can be calculated; the standard error is given by the standard deviation divided by squared root of the number of sampling points. An upper limit to the standard error at equilibrium can then be estimated by fitting the block average standard errors as a function of increasing trajectory block sizes to a single-exponential function. This so-called block average procedure has been applied, for example, to obtain standard errors of the average area per lipid molecule in membrane simulations (Anezo et al., 2003).
We followed here a similar procedure to estimate errors on the conformational entropy changes upon complex formation for the sets of atoms that are involved in intermolecular interactions and/or show conformational changes (Table 1). Each 8-ns trajectory was divided into 16, 8, and 4 blocks of non-overlapping trajectory segments of 0.5-, 1-, and 2-ns length, respectively, and five and three partially overlapping blocks of 4- and 6-ns length, respectively, by shifting the time origin by 1-ns increments. As an illustration, conformational entropy buildup curves as a function of block size with error bars derived from the block averaging procedure are presented for gp120 atom sets in Fig. 8. Since overlap between blocks can reduce the standard deviation and hence the standard error (Fig. 8 B), only the first three data points derived from non-overlapping blocks (0.5-, 1-, and 2-ns block sizes) were used for fitting the standard errors to an exponential function using the xcrvfit software (http://www.pence.ca/software/xcrvfit/) and extrapolating the standard error to infinite block size. The V3 loop of gp120 shows a monotonous increase of the average configurational entropy difference (circles in Fig. 8 B) with markedly larger standard errors than the other structural elements. This may reflect the large structural change and the crossing of the free and bound buildup curves (Fig. 6 B). The intermolecular hydrogen-bonded residues in gp120 and CD4 both show fast equilibration and small standard errors with increasing block size (triangles in Fig. 8). The same is observed for the bridging sheet of gp120 (squares in Fig. 8), for F43 of CD4 (F43), and the F43-cavity of gp120 (F43cav) (not shown). This suggests that, for these sets, sufficient sampling of conformational space has been achieved and that relatively precise conformational entropy differences between the two states can be obtained, although the values for the V3 loop should be taken with caution. The corresponding conformational entropy differences with their respective standard error estimates from the block averaging analysis are summarized in Table 2.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONREFERENCES |
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The conformational entropy change of gp120/CD4 complexation roughly agrees with the experimental value despite the fact that the contribution from bulk solvent was neglected in our analysis. Assessment of the complete system was not yet feasible due to insufficient sampling, especially for gp120. Nevertheless, by considering functionally important sets of atoms, a localized, rather than global, analysis provided insights into the entropic contribution of various degrees of freedom. The intermolecular hydrogen bond network and the insertion of CD4-F43 into its receptive cavity seem to predominantly determine the large entropy loss upon complexation. The bridging sheet of gp120 also plays a key role in the entropy change, as has been proposed experimentally (Myszka et al., 2000; Xiang et al., 2002). Although in some cases, conventional structural parameters such as atom-positional fluctuations may fail to identify thermodynamic differences between two states, the conformational entropy analysis of combined trajectories can provide a complementary way of evaluating spatial distributions and their statistical weight. This is essentially equivalent to the clustering approach, which was previously proposed to assess equilibration and convergence of biomolecules simulations (Smith et al., 2002). With the rapid advance in computing power and methodology, we are hopeful that a thorough description of the thermodynamics of such complex systems can be ultimately achieved via computer simulations and will meet experimental data in the near future.
Submitted on April 23, 2004; accepted for publication September 20, 2004.
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(
and 
) d < 0.5 nm; (•) 0.5 nm < d < 1 nm; (
) 1 nm < d < 2 nm; and (x) d > 2 nm. The intermolecular distance is defined as the minimum interatomic distance from any atom of the residue of interest to any atom of the other molecule in the structure taken at 6 ns, which is well-equilibrated and is representative of the most populated V3 conformation (see 
with standard errors, SE, for various sets of atoms in the free (open symbols and dashed lines) and complex (solid symbols and solid lines) states as a function of the trajectory block size. (Circles) gp120 V3 loop; (squares) gp120 bridging sheet; and (triangles) gp120 intermolecular hydrogen-bounded residues (for definition, see 
between bound and free forms and standard errors SEdiff as a function of the trajectory block size. The standard error SEdiff of the entropy difference is calculated as 