Functional Modes and Residue Flexibility Control the Anisotropic Response of Guanylate Kinase to Mechanical Stress

Abstract

The coupling between the mechanical properties of enzymes and their biological activity is a well-established feature that has been the object of numerous experimental and theoretical works. In particular, recent experiments show that enzymatic function can be modulated anisotropically by mechanical stress. We study such phenomena using a method for investigating local flexibility on the residue scale that combines a reduced protein representation with Brownian dynamics simulations. We performed calculations on the enzyme guanylate kinase to study its mechanical response when submitted to anisotropic deformations. The resulting modifications of the protein's rigidity profile can be related to the changes in substrate binding affinity observed experimentally. Further analysis of the principal components of motion of the trajectories shows how the application of a mechanical constraint on the protein can disrupt its dynamics, thus leading to a decrease of the enzyme's catalytic rate. Eventually, a systematic probe of the protein surface led to the prediction of potential hotspots where the application of an external constraint would produce a large functional response both from the mechanical and dynamical points of view. Such enzyme-engineering approaches open the possibility to tune catalytic function by varying selected external forces.

Introduction

The importance of protein flexibility and dynamics for the understanding of protein function has now been clearly established (1–4). During the execution of their biological function, proteins can be subjected to forces and their mechanical properties evolved in response to fit this selection pressure. Experimentally, many techniques, such as optical and magnetic tweezers (5–7) or atomic force microscopy (8–10), make it possible to probe biomolecular mechanics directly on the single-molecule level (11). In particular, experiments with linkages other than the usual N-to-C-terminal have shown how these mechanical properties strongly depend on the loading geometry (12–15).

Although the first experiments mostly investigated the mechanical response of proteins and the sequence of unfolding events that would result from the application of a force (8), recent setups have focused more on their functional response, thus leading to the field of mechanoenzymatics (16,17). In this perspective, the mechanism of allosteric control (18,19) of an enzyme, which plays a crucial part in signaling pathways in the cell (20), can now be studied experimentally via the building of protein-DNA chimeras where a DNA molecular spring is coupled to the protein at specific locations on its surface (21–23). Through this allosteric spring probe (ASP), one can affect the static and dynamic conformations of the protein and follow its functional response to the application of an external stress (24).

From the theoretical point of view, atomic coordinate-based methods, such as constrained molecular dynamics simulations, can mimic force-extension experiments but operate on much shorter timescales and remain computationally expensive (25–29). Therefore, lower-resolution models have been widely used in recent years to study protein dynamics (30–35). These coarse-grained representations comprise the elastic network model (ENM) (36,37), which reduces the protein to a set of pseudoatoms with pairs below a given cutoff distance being linked by Gaussian springs. Despite their simplicity, these models led to many results concerning protein mechanics and dynamics (38–44). Recently, coarse-grained approaches were used to successfully model the anisotropy of the mechanical response of proteins subjected to an external force (45–47).

In this work, we used a method combining a coarse-grained protein representation and Brownian dynamics simulations. This approach was previously successfully applied to model the mechanical response of the green fluorescent protein to understand the single-molecule experiments carried out by Dietz et al. (14,47). Here, we investigated the enzyme guanylate kinase (GK), which was studied by Tseng et al. via the ASP approach (48). GK is an essential enzyme that catalyzes the transfer of a phosphate group from adenosine triphosphate (ATP) to guanosine monophosphate (GMP) (49). Upon substrate binding, GK undergoes a structural transition from the open to the closed state through a movement of the two lobes formed by the LID and GMP domains (see Fig. 1), leading to an ∼1 nm conformational change (50,51). With DNA springs anchored on three different locations on the protein surface, Tseng et al. determined for each case the changes in substrate binding affinities and catalytic rate constant resulting from the directional stress exerted on the protein. They showed that the functional response strongly depends on the direction of load. Using our molecular modeling approach, we investigated the mechanics and dynamics of GK when subjected to an external constraint and related our results to the variations of the enzymatic activity observed experimentally.

Figure 1.

A cartoon representation of GK with the pulling directions tested experimentally by Tseng et al. (48) The color coding of the protein is according to the domain definitions of Hible et al. (67) The GMP and ATP binding sites are located at the GMP/CORE and CORE/LID interfaces, respectively. The arrow indicates the direction of the opening/closing transition, which constitutes the first mode of motion of the protein. The images in this figure and in the upper part of Fig. 5 were prepared using visual molecular dynamics (71).

Computational Details

Brownian Dynamics simulations

Rigidity profile of a protein

Coarse-grained Brownian Dynamics (BD) simulations were run using a modified version of the ProPHet (probing protein heterogeneity) program (41,42), where an external mechanical constraint can be applied between two residues. In this approach, the protein is represented using an ENM. Diverging from most common coarse-grained models, where each residue is described by a single pseudoatom (52), we chose a more detailed representation (53) that involves up to three pseudoatoms per residue and enables different amino acids to be distinguished. Pseudoatoms closer than the cutoff parameter, R_c = 9 Å, are joined by Gaussian springs that all have identical spring constants of γ = 0.42 N m⁻¹ (0.6 kcal mol⁻¹ Å⁻²). The springs are taken to be relaxed for the experimentally observed conformation of the protein, in this case the crystallographic structure of guanylate kinase from Mycobacterium tuberculosis in its open conformation available in the protein data bank with the code 1S4Q.

Mechanical properties are obtained from 200,000 BD steps at 300 K. The simulations are analyzed in terms of the fluctuations of the mean distance between each pseudoatom belonging to a given amino acid and the pseudoatoms belonging to the remaining residues of the protein. The inverse of these fluctuations yields an effective force constant k_i that describes the ease of moving a pseudoatom with respect to the overall protein structure:

$$k_i=\frac{3k_{\text{B}}T}{\langle {\left(d_i-\langle d_i\rangle \right)}^2\rangle },$$

where 〈〉 denotes an average taken over the whole simulation and d_i = 〈d_ij〉_j∗ is the average distance from particle i to the other particles j in the protein (the sum over j^∗ implies the exclusion of the pseudoatoms belonging to residue i). The distances between the C_α pseudoatom of residue i and the C_α pseudoatoms of the adjacent residues i − 1 and i + 1 are excluded, since the corresponding distances are virtually constant. The force constant for each residue k is the average of the force constants for all its constituent pseudoatoms i. We will use the term rigidity profile to describe the ordered set of force constants for all the residues of the protein.

Applying an external constraint on the protein

Whereas in our previous work on the green fluorescent protein (47), the mechanical stress was simply modeled by applying a constant force between the C_α pseudoatoms of the corresponding residues, in this study, we chose to model the external constraint by adding to the ENM representation a supplementary spring termed the constraint spring in opposition to the structural springs resulting from the original conformation of the protein,. This way we could model more accurately the experiment of Tseng et al. (48), where DNA molecular springs of identical length (60 bp) were used at three different locations on the surface of the protein to apply a controlled mechanical stress (see Fig. 1). From the available experimental data regarding the contour length of the DNA spring (200 Å) and the amount of elastic energy that resides in the protein (54) (∼1 kT), we derived for our constraint spring the parameters equilibrium length (L_C = 150 Å) and spring constant (γ_C = 0.84 N m⁻¹) (1.2 kcal mol⁻¹ Å⁻²). This constraint spring was added between the C_α pseudoatoms of the anchor residues of the three locations tested in the experiment, Thr⁷⁵/Arg¹⁷¹, Cys⁴⁰/Arg¹⁷¹, and Cys⁴⁰/Lys¹³⁰ (see Fig. 1).

Principal component analysis of the coarse-grained trajectories

The BD trajectories for the protein without (relaxed protein) and with the application of an external force (protein under stress) were investigated using principal component analysis (PCA) (55–58) with tools from the Gromacs (59–61) software package. In particular, we calculated the inner product matrices of the ten first eigenvectors, which always cover >89% of the total variance of the protein (see Fig. S1, Fig. S2, and Table S1 in the Supporting Material) of each constrained trajectory with the 10 first eigenvectors of the relaxed trajectory to assess how the mechanical constraint affects the leading modes of motion of the protein.

Systematic scan of the protein surface

To determine whether the deformations that were studied experimentally are representative of the full heterogeneity of the GK structure, we performed a more systematic study of residue-pair deformations. The selection of representative pairs is first narrowed by limiting our choice to surface residues, that is, residues with at least 5% solvent accessibility (62) (as calculated by the NACCESS program (63)), which are thus amenable to experimental study. Second, we chose residue pairs separated by at least 20 Å and 30 amino acids in the primary sequence. Last, we eliminated residue pairs that differed from already-selected pairs by fewer than five residues in the primary sequence in either of the constituent residues. This method led to the selection of 236 residue pairs, which were all tested with the constraint spring described earlier (see Fig. S3).

Results

Mechanical properties of guanylate kinase

The rigidity profile of GK is represented in Fig. 2 a. It is worthy of remark that most of the force-constant peaks from the first half of the protein sequence correspond to residues belonging to ligand-binding sites, such as Lys³⁴ and Glu¹¹⁹ for the ATP/Mg²⁺-binding site, or Ser⁵³, Glu⁸⁸, and Thr¹⁰¹ for the GMP-binding site. In particular, note the peaks corresponding to Ser²⁷ and Lys³⁴, two residues surrounding the flexible P-loop, a highly conserved motif that binds the β-phosphate of the ATP donor in nucleoside monophosphate kinases (64,65); and the highly rigid area on the β7-sheet, around Glu¹¹⁹, which corresponds to residues interacting with GMP in the closed conformations of GK from Saccharomyces cerevisiae (50) and from Mus musculus (66).

Figure 2.

(a) Rigidity profile of GK when no mechanical stress is applied on the protein. (b) Variation of the force-constant profile upon applying an external constraint on the protein in pulling directions 75/171 (upper), 40/171 (middle), and 40/130 (lower). The black horizontal bars at the upper left of Fig. 2, a and b,indicate the position of the P-loop and the β7-sheet along the sequence.

The variation of the force-constant profile of the protein upon mechanical stress is represented in Fig. 2 b. The P-loop base and the β7-sheet are the protein segments whose mechanical properties are the most sensitive to the application of an external stress, independent of the pulling direction. The mechanical response of GK is nevertheless markedly anisotropic. Although the 40/171 and 40/130 pulling directions only lead to weak (<20 kcal mol⁻¹ Å⁻²) variations in the force constant of the residues, stressing the protein along the 75/171 direction results in a strong rigidity decrease of Ser²⁷ (−57 kcal mol⁻¹ Å⁻²), Lys³⁴ (−31 kcal mol⁻¹ Å⁻²), Glu¹¹⁹ (−64 kcal mol⁻¹ Å⁻²), and Val¹²⁰ (-78 kcal mol⁻¹ Å^-2). The initial distances between the C_α atoms from the 75/171, 40/171, and 40/130 residue pairs are 35.7 Å, 27 Å, and 28 Å, respectively. This indicates that the stress exerted by the constraint spring, which has an equilibrium length of 150 Å, should intrinsically be less pronounced for the 75/171 direction than for the 40/171 and 40/130 directions of load. However, the highly anisotropic architecture of the protein leads to the opposite effect, with the 75/171 direction inducing the most important changes in the enzyme's mechanics.

Dynamics of the constrained protein

In a second step, we used the PCA approach to compute the inner product of the 10 first eigenvectors of a constrained trajectory with the eigenvectors of the relaxed trajectory. The resulting matrices for the 75/171, 40/171, and 40/130 pulling directions are plotted in Fig. 3, a–c, respectively. Table 1 summarizes the overlap between the covariance matrices (CMOs) of the relaxed and constrained trajectories. We can see how the application of an external stress along the 40/130 direction feebly modifies the modes of motion of the protein (CMO of 0.83), thus resulting in an almost diagonal matrix. On the other hand, applying an external constraint along the 75/171 and 40/171 directions induces some important disruptions of the protein's dynamics, but in different ways. Although pulling the protein along the 75/171 direction leads to a generally more important perturbation of GK movements compared with the 40/171 direction, with CMOs of 0.65 and 0.80, respectively, it turns out that the main functional mode of motion of the enzyme, corresponding to its opening and closing around the GMP and ATP binding sites, is more preserved for the 75/171 than for the 40/171 direction. This is shown by the projections of the first eigenvector of the constrained trajectory on the first eigenvector of the relaxed trajectory, which amount to overlaps of 0.83 and 0.77, respectively. This variation in the disruption of the enzyme dynamics is easily understandable, since the 75/171 direction actually coincides with opening and closing motions of GK, whereas pulling the protein via a constraint spring anchored on Cys⁴⁰ and Arg¹⁷¹ introduces a new direction of motion with a component orthogonal to the main enzymatic movement.

Figure 3.

Inner-product matrices of the 10 first eigenvectors of the constrained trajectories over the relaxed trajectory. The pulling directions are (a) 75/171, (b) 40/171, (c) 40/130, and (d) 65/122. The color scale ranges from 0 to 1.

Table 1.

Overlap of constrained trajectories with relaxed trajectory along the 10 first modes of motions of the protein.

Pulling direction	Covariance matrix overlap (CMO)	Overlap of first eigenvectors
Thr75-Arg171∗	0.65	0.83
Cys40-Arg171∗	0.80	0.77
Cys40-Lys130∗	0.83	0.97
Asp65-Leu122	0.66	0.55

^∗Asterisks identify the pulling directions experimentally tested by Tseng et al. (48)

Prediction of hotspots on the protein surface

We performed a systematic search of the protein mechanical response to the application of an external constraint by probing 236 new nonredundant pulling directions via residues anchored all over the surface of the protein. The resulting variations of the force constant of each residue are plotted in Fig. 4. From the qualitative point of view, most directions of load result in variations of the rigidity profile that are similar to those previously observed for the experimentally tested directions. Once again, the most important changes in the force constants of the residues occur in the P-loop and β7-sheet areas, which usually undergo a strong increase in flexibility.

Figure 4.

Distribution of the force-constant variation over all the pulling directions that have been modeled. Δk is expressed in kcal mol⁻¹ Å⁻². The black horizontal bars above the figure indicate the position of the P-loop and the β7-sheet along the sequence. The extended pairs have been ordered starting with the direction leading to the largest average perturbation, 〈abs(Δk)〉, of the GK rigidity profile. A negative/positive, value of Δk denotes a decrease/increase, of the rigidity of the residue.

The pulling direction that led to the most important perturbation of the rigidity profile (in terms of both the maximum force constant variation and the average perturbation of the profile) was formed by Asp⁶⁵ and Leu¹²² (see Fig. 5). As can be seen in Fig. 5 (lower), the application of a constraint spring between these two residues induces an important disruption of the GMP binding site, with force-constant decreases beyond 100 kcal mol⁻¹ Å⁻² for Glu¹¹⁹ and Val¹²⁰, which is not surprising since the anchor residues actually surround the catalytic site. It is of interest that this new pulling direction also strongly disturbs the enzyme's dynamics. As we can see from Fig. 3 d and Table 1, the 65/122 direction yields a CMO between the constrained and the relaxed trajectories of 0.66, whereas the projection of the first eigenvector is now reduced to 0.55, a much lower value than previously obtained for the experimental directions of load.

Figure 5.

(Upper) Cartoon representation of GK with the 65/122 pulling direction. The ellipse indicates the location of the GMP-binding site. (Lower) Variations in force constant (kcal mol⁻¹ Å⁻²) as the protein moves along the 65/122 pulling direction. The black horizontal bars above the figure indicate the position of the P-loop and the β7-sheet along the sequence.

Discussion and Conclusion

In the ASP experiment, protein-DNA chimeras are used to strain the conformation of a protein, thus potentially providing further insight into the mechanism of allosteric control of biological function. In their work, Tseng et al. (48) applied a mechanical constraint at three different locations on the surface of GK, Thr⁷⁵/Arg¹⁷¹, Cys⁴⁰/Arg¹⁷¹, and Cys⁴⁰/Lys¹³⁰. These experiments yielded different results in terms of changes of the enzymatic activity: The 75/171 pulling direction induced a decrease in binding affinity for GMP, thus increasing the Michaelis-Menten constant, K_G, which was measured by GMP titration experiments, but having little or no effect on K_A (the binding affinity for ATP) and k_cat (the catalytic rate of the enzymatic reaction). For the 40/171 direction, Tseng et al. (48) observed a decrease of k_cat, whereas here, K_A and K_G were not affected. Finally, for the 40/130 pulling direction, no noticeable effect was observed for K_G, K_A, or k_cat. In this study, we combined a coarse-grained protein representation and BD simulations to investigate the mechanical and dynamical response of GK when an external stress is applied on the protein. During the simulations, the spring network oscillates around its equilibrium state within a limited range, with the deformations amounting to an ∼1-Å root-mean-square deviation from the average conformation of the protein. Our model is therefore well adapted to describe the experiment of Tseng et al., where the protein's structure undergoes very few changes.

The force-constant profiles, which were obtained for trajectories with and without the application of a mechanical constraint, present rigidity peaks that correspond to residues belonging to the ligand-binding sites, thus stressing once more the importance of the catalytic site's stiffness for enzymatic activity (41,42). From the qualitative point of view, the force-constant variations observed for the protein under stress were mainly located around the P-loop and the β7-sheet. Quantitatively, however, only the 75/171 pulling direction led to an important decrease of the rigidity of residues Ile¹¹⁸ to Asp¹²¹, which belong to the GMP binding site and do normally form interactions with the ligand in the closed conformation of the protein (67). Since such catalytic residues require an enhanced rigidity for the execution of their biological function (68–70), this disruption of the mechanical properties of the GMP binding site provides a first explanation for the decrease of the GMP binding affinity observed experimentally with the 75/171 chimera.

We then used PCA to study the variation in protein dynamics induced by the external stress. The resulting inner-product matrices obtained for the first 10 eigenvectors indicate that the 75/171 pulling direction leads to the most important perturbations of the general enzyme dynamics. However, if we focus only on the first mode of motion of the protein, which corresponds to the opening and closing movement of the LID and GMP domains over the CORE domain, and which is essential for GK to perform its catalytic function, it turns out that this mode is most perturbed when load is applied along the 40/171 direction. Once again this result is in agreement with the experimental data, where the 40/171 mutant alone presented a decrease of its catalytic rate, k_cat. All in all, the variations in enzymatic activity observed via the ASP experiments can either be related to some local mechanical perturbation of a substrate binding site (in the case of the GMP binding affinity), or to more global changes in the protein large-amplitude movements (for the catalytic rate constant).

Eventually, since Tseng et al. raised the question of the prediction of hotspots at the surface of the protein where a mechanical perturbation would produce a large functional response, we scanned 236 nonredundant locations on the protein surface. From a mechanical point of view, all pairs led to similar changes in the protein properties, with a rigidity decrease in the P-loop and β7-sheet regions. We were yet able to single out a specific residue pair that would be interesting to study experimentally, Asp⁶⁵-Leu¹²². Since these two residues surround the GMP binding site, pulling in the 65/122 direction should yield an important disruption of its rigidity. It is also noteworthy that the neighboring Asp¹²¹ residue is implicated in GMP binding and initiation of the enzyme's closure (67). From a dynamical point of view, it appears that this direction of load also induces a strong perturbation of the protein's first mode of motion. This means that, were experiments performed on a 65/122 mutant, one should observe a decrease in both the enzyme's binding affinity for GMP and its catalytic rate. We furthermore tried to assess from the results of the systematic scan whether a particular pulling direction could specifically disrupt the ATP binding site while leaving the GMP binding site intact. This would result in a protein-DNA chimera where K_A, but not K_G, would be affected. In terms of mechanics, this meant finding a direction of load leading to a decrease in rigidity of the protein around the P-loop but not in the β7-sheet area. However, we could not find any residue pairs that would satisfy such a criterion, and it seems that the mechanical properties of these two elements of GK are tightly coupled and cannot be modulated independent of each other. To mechanically separate these subunits, one would probably have to disrupt the set of interactions that bind together the α1-helix and the β1- and β7-sheets via site-directed mutagenesis. In this perspective, residues Leu²⁶, Lys³⁴, Val³⁸, Leu¹¹⁷, and Glu¹¹⁹, whose side chains are directed toward the center of the CORE domain, appear to be potential candidates. It is interesting that all these residues also present important force constants (>35 kcal mol⁻¹ Å⁻²) in the rigidity profile of GK, which supports the idea that they might play an important part in the enzyme's structural stability. However, since Lys³⁴ and Glu¹¹⁹ are also involved in ATP- and GMP-binding, respectively, it is unlikely that one could perturb the protein's intrinsic mechanics without disturbing its biological activity as well.

Altogether, we showed how a simple protein representation combined with BD simulations can yield a molecular level picture of the way the application of an external constraint on the GK enzyme perturbs its mechanical properties and dynamics. These perturbations can then be related to the changes observed experimentally in parameters K_G, K_A, and k_cat of the enzymatic reaction, helping us to understand the origin of the anisotropic functional response of the protein to various pulling directions. It is interesting to note that the apparent origin of the variations in thermodynamic parameter K_G is a disturbance of the protein's mechanical properties around its GMP binding site, whereas the changes in kinetic parameter k_cat arise from a perturbation of the protein dynamics, more precisely from the disruption of its first mode of motion, which defines the opening and closing movement performed by the protein when it undergoes a catalytic cycle. Our model can also be used in a predictive outlook. From a general search on the protein surface, we could suggest a new direction of load that should lead to the simultaneous perturbation of the two parameters K_G and k_cat, an effect not observed with the protein-DNA chimeras produced thus far.