The conformational states of Mg·ATP in water

Abstract

The conformational equilibria of Mg·ATP in solution is studied using molecular dynamics (MD) augmented with umbrella sampling methods. Free energy comparisons show that the Mg²⁺ ion is equally likely to coordinate the oxygens of the two end phosphates, or of all three phosphates. The MD trajectories reveal two major degrees of freedom of the Mg·ATP molecule in solution, and we compute the free energy as a function of these variables, and determine its elastic properties. Comparing the free energy function with several crystallographic structures of ATP analogs, we find that the crystal structures correspond to states where ATP would be elastically strained. The average water density around Mg·ATP is investigated to show the average number of hydrogen bonds and the hydrophobicity.

Appendices

Appendix A: atom numbers

The atom numbers used in this report are shown in Fig. 5.

Fig. 5.

The chemical structure of Mg·ATP. Atom numbers are the same as in CHARMM

Appendix B: methods

Simulations of Mg·ATP in water were carried out using the molecular dynamics program CHARMM (Brooks et al. 1983; MacKerell et al. 1998). We performed two different sets of simulations, one using the original partial charge distributions provided in CHARMM (Pavelites et al. 1997), and the other using the partial charge distributions derived by Minehardt (Minehardt et al. 2002). Both simulations used the same Mg²⁺ parameters from CHARMM version 27. The simulations were carried out in a cube 31.1 Å on a side, containing 1,000 TIP3P water molecules. Periodic boundary conditions were imposed using Ewald sums (Billing et al. 1996; Frenkel et al. 2002). Atom O5' (as shown in Fig. 5) was harmonically constrained and the size of the water box was chosen so that the ATP molecule is free to rotate without contacting the boundary of the box. The number of water molecules was such that the bulk density of water is 1 g cm⁻³. Using a shifted potential and the SHAKE algorithm (Ryckaert et al. 1977), the long-range non-bonded interactions were limited to 13 Å. The Verlet leapfrog algorithm was used for numerical calculations using a time step of 1 fs (Hockney et al. 1988). The system was first minimized by the adopted basis Newton–Raphson method for 100 steps (Brooks et al. 1983). After the energy minimization, the system was heated from 0 K to 300 K over the course of 15 ps, and then equilibrated at 300 K for 30 ps. Simulation runs of 1 ns or more were then carried out after equilibration.

To obtain free energy functions, umbrella sampling was employed and the overall free energy function was obtained by combining statistics from different regions (Ferrenberg et al. 1989). The θ–Φ plane was partitioned into 21 regions to apply the umbrella sampling method with different potentials. The quadratic potential function

$${E = {1 \over 2}{\left[ {k_{\theta } {\left( {\theta - \theta _{0} } \right)}^{2} + k_{\phi } {\left( {\phi - \phi _{0} ^{2} } \right)}} \right]}}$$

was used to fit the potentials. The centers (θ₀, Φ₀) of the potential functions in these 21 regions are listed in Table 2.

Table 2.

The potential coefficients k _θ=6 kcal mol⁻¹, k _Φ=6 kcal mol⁻¹ were used for the first two rows of potentials. For the last row, Φ is close to π/2, where the circumferential angle θ is not defined, so we added a steep exponential function to avoid sampling too close to Φ=π/2. The potential functions for umbrella sampling of the bottom row were

$$ {E = {1 \over 2}{\left[ {k_{\theta } {\left( {\theta - \theta _{0} } \right)}^{2} + k_{\phi } {\left( {\phi - \phi _{0} ^{2} } \right)}} \right]} + {\mathop{\rm e}\nolimits} ^{{ - \alpha {\left( {\phi - \phi _{0} } \right)}}} } $$

where α=8 kcal mol⁻¹ was chosen to bias the potential. Stronger potentials of k _θ=16 kcal mol⁻¹ and k _Φ=10 kcal mol⁻¹ were also applied to enhance the avoidance of divergent sampling at Φ=π/2. In the Φ direction Φ₀ was set as −1.4 rad so that the effective minimum after adding the exponential term was at Φ₀=−1.22, as proposed in the table. The procedure of the umbrella sampling simulation, including the algorithm, periodic boundary conditions with Ewald sums, minimization, and the equilibration process, is the same as described for standard MD simulations. The simulation runs of 1 ns were carried out for each sampling region.

Appendix C: coordination of the Mg²⁺ for ADP

We carried out simulations of Mg·ADP in water, using the Minehardt parameter set. The results show that Mg²⁺ always coordinates one oxygen atom from the β-phosphate, one oxygen atom from the α-phosphate, and four water molecules (Fig. 6A). This coordination agrees with the experimental results of Mg·ADP in water (Bridger et al. 1983). That is, in order to hydrolyze Mg·ATP, the original coordination to the γ-phosphate must be removed. The crystal structures of Mg·ADP bound enzymes demonstrate that, in many cases, Mg²⁺ only coordinates one oxygen atom from β-phosphate (Fig. 6B), although in some cases it also coordinates with oxygen atoms of α and β-phosphates (Fig. 6A).

Fig. 6A, B.

Coordinations of Mg²⁺. A ADP: in simulation, experimental results, and some crystal structures. B ADP: in several crystal structures

Appendix D: bowl shape trajectory of the γ-phosphorus

In order to describe the various conformations of the Mg·ATP molecule, a body-fixed coordinate system was used. MD trajectories show that the major conformational changes of the molecule involve rotations about the phosphoanhydride bonds that form the long axis of the molecule. The adenine ring does not experience significant distortions. Therefore, we place our reference frame on the adenine ring, with the origin located at nitrogen atom N9 in the adenine linking the carbon atom in the ribose, as shown in Fig. 7. The x-axis lies along the direction from atom N9 to atom C4, with the adenine ring lying in the x–y plane, and the z-axis perpendicular to it.

Fig. 7.

Stick representation showing the coordinate system used to interpret our simulations

Figure 8 shows a stereo view of points visited during a 1-ns trajectory within the body-fixed coordinate frame using the Minehardt parameter set. Each dot represents the location of the γ-phosphorus in a Cartesian coordinate system with the origin at the center of the adenine, and x–y plane parallel to that of the adenine ring. Configurations shown are sampled every 0.1 ps. Five regions (labeled ABCDE) are assigned in the figure with higher populations of atomic configurations. The configurations of the ATP molecule in these five regions are also shown. Figure 8 also shows that the trajectory points form a "bowl" shape, suggesting that the motion can be described by a simple spherical coordinate system.

Fig. 8.

A stereo view of points visited during a 1 ns trajectory shown in the Cartesian coordinate system adapted to the nucleotide. Using the adenine ring as a reference frame, each dot represents the location of the γ-phosphorus of Mg·ATP at 0.1-ps intervals. Five different configurations of Mg·ATP corresponding to the higher populations of atomic configurations labeled ABCDE are also shown. The orientation is transformed so that the adenine ring is nearly horizontal for all the configurations

Appendix E: free energy contours from molecular dynamics

The free energy contours for Mg·ATP and Mg·ADP obtained from straightforward 1-ns MD simulations in the θ–Φ plane are shown in Fig. 9. For Mg·ADP, the trajectory of the β-phosphorus, instead of the γ-phosphorus for Mg·ATP, is used to define the two collective coordinates. The overall free energy surfaces of Mg·ATP in water using different sets of parameters are different, while the regions of the minimum free energy and the pattern around the free energy wells of both cases are similar. The free energy surfaces of Mg·ADP in water and Mg·ATP in water are very similar, indicating that, in terms of the θ–Φ coordinates, the properties of the molecules are close to each other. Because the results were obtained by direct MD simulations, the sampling may be stuck at a certain region, and thus umbrella sampling reflects better statistics. For the simulation of Mg·ADP, the locations of crystal structures mapped to θ–Φ plane are also shown in Fig. 9.

Fig. 9A–C.

The free energy (in units of k _B T) contours over the θ–Φ plane. A The free energy contour for Mg·ATP in water using CHARMM parameters. B The free energy contour for Mg·ATP in water using Minehardt parameters. C The free energy contour for Mg·ADP in water using Minehardt parameters. Locations of the β-phosphorus for several ADP molecules in different crystal structures are marked

Appendix F: estimation of elastic properties

We locate the minimum free energy in Fig. 3, and estimate the surface shape around the minimum by cutting cross-sections of constant θ and Φ. Figure 10A shows the cross-section of the free energy surface at θ=36° , and Fig. 10B shows the cross-section of the free energy surface at Φ=−12°, for Mg·ATP with CHARMM parameters. Although the curves are rough and slightly asymmetric, they can be approximated by a parabola up to 1.5–2 k _B T above the free energy minimum, as shown in the figure. From this we obtain that the effective elastic constant in the Φ direction (keeping θ constant) is about 4.69 k _B T rad⁻². That is, with the radius of motion of about 8.8 Å for Mg·ATP, the elastic constant k _Φ for Mg·ATP in solution around the minimum free energy regions are about 25 pN nm⁻¹. Similarly, we fit a parabolic curve to estimate the effective elastic constant k _θ in the θ direction as 35 pN nm⁻¹ for Mg·ATP, showing that the molecule is stiffer in the θ direction than in the Φ direction. The same procedure can be used to estimate the elastic properties of Mg·ADP. From Fig. 9C, we obtain that k _Φ of Mg·ADP is 7.6 k _B T rad⁻², or 74 pN nm⁻¹, and k _θ of Mg·ADP is 24.6 k _B T rad⁻², or 239 pN nm⁻¹.

Fig. 10A, B.

Cross-sections of the free energy surface for Mg·ATP and the fitted parabolic curves used to determine the effective elastic constants. A The cross-section of the free energy at constant θ for Mg·ATP. B The cross-section of free energy at a constant Φ