Dissipation at the angstrom scale: Probing the surface and interior of an enzyme

Pursuing a materials science approach to understanding the deformability of enzymes, we introduce measurements of the phase of the mechanical response function within the nanorheology paradigm. Driven conformational motion of the enzyme is dissipative as characterized by the phase measurements. The dissipation originates both from the surface hydration layer and the interior of the molecule, probed by examining the effect of point mutations on the mechanics. We also document changes in the mechanics of the enzyme examined, guanylate kinase, upon binding its four substrates. GMP binding stiffens the molecule, ATP and ADP binding softens it, while there is no clear mechanical signature of GDP binding. A hyperactive two-Gly mutant is found to possibly trade specificity for speed. Global deformations of enzymes are shown to be dependent on both hydration layer and polypeptide chain dynamics.

Figure 1

Nanorheology setup, showing the flow chamber with enzyme-tethered GNPs, the parallel plates capacitor geometry used for mechanical excitation, and the evanescent wave scattering optics used for read out.

Figure 2

Frequency scan showing the mechanical response of the wild type (WT), under our standard conditions (in SSC/3, $50\text{mM}$ total ionic strength, $\text{pH}=7.0$; circles) and with the addition of $0.5\%$ DMSO (a kosmotropic agent affecting the hydration layer; squares). The data are obtained from the same sample. (a) RMS amplitude $x_0$ of the response (in $\AA$) vs. frequency $\nu =\omega /2\pi$ (in $\text{cycles}/\text{s}$). The lines are fits with Eq. (5), returning the values $F_0/\gamma =20\AA /\mathrm{s}$, ${\omega }_c=163\mathrm{rad}/\mathrm{s}$ (circles) and $F_0/\gamma =14\AA /s$, ${\omega }_c=153\text{rad}/\text{s}$ (squares). (b) Phase $\phi$ of the response [defined operationally in Eq. (2)] vs. frequency. The lines are one-parameter fits with Eq. (5) and show that the Maxwell model Eq. (3) does not quite describe the system. (c) This plot is a measure of dissipation. For the same data as (a) and (b), the quantity $\pi x_0\text{sin}(-\phi )$ is plotted (in $\AA$) vs. frequency. For a linear system this quantity would be equal to $W/F_0$ [Eq. (7)], where $W$ is the energy dissipated per cycle and $F_0$ is the amplitude of the applied force. For the Maxwell model, this quantity is proportional to $1/\omega$ [Eq. (6)]; the lines are one-parameter fits with the form $\text{const}./\omega$.

Figure 3

(a) Map of the relative distance change of $\mathrm{C}\alpha$ atoms of GK from the open to the closed states; the distance change is averaged over $\mathrm{C}\alpha$ atoms along the chain within a cutoff $X$ ($X=15\AA$ for this graph). Two major peaks appear in this strain map: one for residues 29–35, which is the p-loop, and another for residues 175–176, the region often called “hinge.” The same features appear when varying the cutoff $X$ from 8 to $18\AA$. (b) Color map of the graph in (a) painted on the GK structure (only $\mathrm{C}\alpha$ atoms are shown), with increasing “strain” from blue to red. The structures used for this map were PDB 1ZNX (closed state) and 1ZNW (open state).

Figure 4

Comparison of the enzymatic activity (plotted on a log scale) of the WT and the two mutants. The quantity plotted is the initial speed of the enzymatic reaction, measured with a pyruvate-NADH coupled enzymatic assay (see Sec. 4). Conditions were the same for all measurements (optimal conditions for the WT, [GMP] = 1 mM, [ATP] = 2 mM), the nominal enzyme concentration being determined with the Bradford assay.

Figure 5

Representative nanorheology frequency scans for the mutant C1. Panels (a), (b), and (c) show, respectively, the root-mean-square amplitude of the mechanical response, the phase, and the dissipation [the quantity $\pi x_0\text{sin}(-\phi )$]. The frequency is $\nu =\omega /2\pi$, in $\text{cycles/s}$. The lines are fits with the Maxwell model predictions Eqs. (5) and (9).

Figure 6

Representative frequency scans for the mutant B1, arranged as in Fig. 5. The lines are fits with the Maxwell model [Eqs. (5) and (9)]. Mechanical differences between the mutants are not evident, but can be observed with sufficient averaging, as shown in Fig. 7.

Figure 7

Amplitude of the response for the WT (circles), B1 mutant (squares) and C1 mutant (triangles), averaged over several samples. The WT is averaged over five different samples, B1 over four, and C1 over three. There are systematic differences in the mechanics, especially between the WT and C1. Namely, in the region $\omega >{\omega }_c$ the amplitude for C1 decreases faster with frequency compared to the WT. Because this region is not captured well by the Maxwell model, we use a different form to fit the data (see text). Overall, the figure indicates that internal friction (as opposed to surface friction) is increased for C1.

Figure 8

Binding isotherms for GDP, ADP, ATP, measured by nanorheology for the “fast” mutant C1. Displayed are the amplitude and phase signals vs ligand concentration, measured at the the fixed frequency $\nu =12\text{Hz}$. For GDP, there is no signal above the scatter of the data, both for the amplitude (a) and the phase (b). For ADP, the amplitude signal (c) is unclear, whereas the phase (d) shows a clear signature of binding. The line is a fit with the two-states binding isotherm (13), yielding the dissociation constant $K_d^{\text{ADP}}\left(C1\right)=230\mu \text{M}$.

(e) ATP binding curve obtained from the signal amplitude; the line is a fit with Eq. (13), yielding $K_d^{\text{ATP}}\left(C1\right)=1.2\text{mM}$.

Figure 9

Binding isotherms obtained by nanorheology for the WT. Similar to C1, there is no clear signature of GDP binding either in the amplitude (a) or phase (b) for concentrations $\left[\mathrm{GDP}\right]<1\text{mM}$. The signature of ADP binding is not clear in the amplitude (c), but is visible in the phase (d). The line is a fit with Eq. (13), giving $K_d^{\text{ADP}}\left(\text{WT}\right)=240\mu \text{M}$.

Figure 10

GMP binding curves for B1 [(a) and (b)] and C1 [(c) and (d)]. Both mutants become stiffer upon binding GMP, as does the WT [29]. For B1, both the amplitude (a) and phase (b) carry a signature of GMP binding; fitting with Eq. (13) returns the values $K_d^{\text{GMP}}\left(B1\right)=5.7\mu \text{M}$ from (a) and $K_d^{\text{GMP}}\left(B1\right)=4.6\mu \text{M}$ from (b). (c) For C1, the GMP binding curve based on the amplitude shows two binding events, the first with midpoint $\left[\text{GMP}\right]\approx 5\mu \text{M}$ and the second at $\left[\text{GMP}\right]\approx 600\mu \text{M}$. See text for more discussion. The line is a fit with Eq. (14).

Figure 11

Crystal structure of GK in open state from PDB 1ZNW. The GMP binding domain is shown in red, ATP binding domain in violet, residues 075, 171, and 042 in cyan, residue 176 in blue, and residue 175 in green. The orange part is a water channel as identified in Ref. [37].