Wanhao Cai,Miriam Jäger,Jakob T. Bullerjahn,Thorsten Hugel,Steffen Wolf,Bizan N. Balzer
Anisotropic Friction in a Ligand-Protein Complex
atomic force microscopy,single molecule,friction,targeted molecular dynamics simulation,steered molecular dynamics simulation,ligand-protein complex
The effect of an externally applied directional force on molecular friction is so far poorly understood. Here, we study the force-driven dissociation of the ligand-protein complex biotin-streptavidin and identify anisotropic friction as a not yet described type of molecular friction. Using AFM-based stereographic single molecule force spectroscopy and targeted molecular dynamics simulations, we find that the rupture force and friction for biotin-streptavidin vary with the pulling angle. This observation holds true for friction extracted from Kramers’ rate expression and by dissipation-corrected targeted molecular dynamics simulations based on Jarzynski’s identity. We rule out ligand solvation and protein-internal friction as sources of the angle-dependent friction. Instead, we observe a heterogeneity in free energy barriers along an experimentally uncontrolled orientation parameter, which increases the rupture force variance and therefore the overall friction. We anticipate that anisotropic friction needs to be accounted for in a complete understanding of friction in biomolecular dynamics and anisotropic mechanical environments.
Anisotropic Friction in a Ligand-Protein Complex

The effect of an externally applied directional force on molecular friction is so far poorly understood. Here, we study the force-driven dissociation of the ligand-protein complex biotin-streptavidin and identify anisotropic friction as a not yet described type of molecular friction. Using AFM-based stereographic single molecule force spectroscopy and targeted molecular dynamics simulations, we find that the rupture force and friction for biotin-streptavidin vary with the pulling angle. This observation holds true for friction extracted from Kramers’ rate expression and by dissipation-corrected targeted molecular dynamics simulations based on Jarzynski’s identity. We rule out ligand solvation and protein-internal friction as sources of the angle-dependent friction. Instead, we observe a heterogeneity in free energy barriers along an experimentally uncontrolled orientation parameter, which increases the rupture force variance and therefore the overall friction. We anticipate that anisotropic friction needs to be accounted for in a complete understanding of friction in biomolecular dynamics and anisotropic mechanical environments.

The dynamics of biomolecules, such as protein folding and ligand-protein binding, underlies a functional hierarchy spanning from single molecule functions to cell activities. While a full microscopic description of these processes is usually impossible, they can be coarse-grained as diffusion along a one-dimensional free energy surface. The system is partitioned into slow “system” degrees of freedom in the form of reaction coordinates and free energies and into the remaining fast “bath” degrees of freedom, which form a source of friction. Employing a Langevin equation framework or Kramers’ rate theory, process rates can then be recovered based on free energy and friction alone. Methods for the derivation of free energy surfaces are available for both experiments and simulations. However, obtaining friction factors and elucidating their physical origins is less established. A well-known source of molecular friction is the ubiquitous solvent friction that is related to solvent viscosity. Furthermore, fluctuations within a biomolecule, based on intramolecular interactions and configurational rearrangements, result in internal friction. Notably, internal friction plays a significant role when ligands or amino acid segments are buried inside the protein, which has been validated by different methods such as Förster resonance energy transfer (FRET) and molecular dynamics (MD) simulations. However, it is still a challenge to study sources of friction where anisotropic environments are involved. For example, in atomic force microscopy (AFM), the main obstacle is that the applied external forces bias the diffusion of the molecular system on its free energy landscape, resulting in changes in friction and the degrees of freedom accessed by the system. In this study, we reveal a new type of friction, anisotropic friction, and its molecular origin in a biomolecular host-guest system, combining AFM-based stereographic single molecule force spectroscopy (SMFS), steered MD (SMD) simulations, and targeted MD (TMD) simulations. As a simple test system, we focus on the well-established biotin-streptavidin (SA) complex. Defining the pulling direction in spherical coordinates, we perform stereographic pulling experiments at different polar angles θ, with unknown azimuthal angle φ, i.e., orientation of the protein. Complementarily, we carry out SMD simulations using a harmonic restraint with well-defined angles θ and φ. We find that both the rupture force and its variance keep increasing with θ, indicating a rise of molecular friction for pulling biotin out of SA. By further employing dissipation-corrected targeted MD (dcTMD) and a nonequilibrium principal component analysis (PCA) on both ligand-protein contacts and protein backbone dihedral angles, we relate the rise in friction to a heterogeneity in free energy barriers present along different φ. This can be understood along the lines of Zwanzig’s seminal paper on diffusion in a rough energy landscape, where the roughness results in an effective diffusion coefficient. Here, the anisotropy in free energy barriers for different φ has a growing influence with increasing θ, which gives rise to the increase of anisotropic friction when φ is projected out.

First, we study the friction of biotin-SA in HEPES (50 mM, pH = 7.9) via AFM-based SMFS in vertical pulling mode (θ = 0°, i.e., perpendicular to the surface, Figure 1a). Note that the indicated angles have an upper limit of uncertainty of 7° (section Uncertainty of pulling angle, SI). A single biotin molecule is attached to a polyethylene glycol (PEG) linker and covalently functionalized onto a cantilever tip, while SA is covalently anchored to a glass substrate with multiple anchor points via PEG linkers, which largely restrict the free rotation of SA (section Materials and sample preparation, SI). AFM imaging (Figure 1b) shows that the SAs are well-distributed on the substrate with a mean height of (5.5 ± 1.1) nm, in good agreement with the diameter of a single SA tetramer of 4 to 5 nm. We use vertical force-extension (Fz vs Rz) curves of biotin-SA (Figure 1c) to determine the vertical rupture (unbinding) forces Fr,z and extract the associated loading rates Ḟr,z from the slope of the corresponding force-time curves just prior to rupture (Figure S1). The specificity of biotin-SA unbinding is confirmed by a series of control experiments (Figure S2). Therefore, the resulting dynamic force spectrum (Fr,z vs Ḟr,z, Figure 1d) enables us to extract the dynamics parameters for biotin-SA as well as the friction coefficient. In the following, we go beyond the conventional Bell-Evans analysis and instead make use of the Dudko-Hummer-Szabo (DHS) model, which is based on Kramers’ theory (see also section Comparison to previous studies in SI for a comparison to previous studies based on the Bell-Evans model). To realize this approach, we have developed an open-source analytical protocol for fitting the results precisely with the DHS model (section Rupture force data analysis, SI), which predicts the force-dependent unbinding rate k(F) to be: The model is valid for rupture forces below the critical value Fc = ΔGu/(vxu), where ΔGu is the height of the free energy barrier separating the bound and the unbound state, koff is the spontaneous unbinding rate, xu is the distance between the bound state and barrier, β = 1/kBT is the inverse thermal energy, and v is a shape-dependent parameter (1/2 or 2/3, depending on whether the free energy landscape is cusp-like or linear-cubic). Fitting the measured rupture forces to the rupture force distribution associated with eq 1 gives ΔGu = (13.1 ± 0.3) kBT, xu = (0.47 ± 0.02) nm, and koff = (0.038 ± 0.01) s–1 for v = 2/3 at θ = 0°, while similar results are also obtained for v = 1/2 (see Tables S1 and S2 for detailed parameters). For diffusion on the free energy surface, the Kramers’ rate expression offers an analytical approach to a uniform friction coefficient γ as a function of the free energy landscape G(x) (section Friction analysis, SI):where G(x) takes different expressions for different surface shapes, and G″(x) denotes the curvatures of the free energy landscape at its minimum (x = xb) and maximum (x = xt) (Figure 1e). Here, γ is 0.01 to 0.02 pN·s/nm (for θ = 0°) depending on the free energy landscape. Note that this γ reflects the friction around the barrier top, while γ along the full pulling distance can be position-dependent for the projected dynamics on a one-dimensional surface, which will be detailed by our simulations further below.

The influence of the pulling angle on the friction is measured by applying stereographic SMFS onto biotin-SA, varying the polar angle θ from 0 to 75°. The polar coordinate θ of the pulling angle is well-defined, by controlling the relative motion of the cantilever tip and substrate with nanometer accuracy (Figure 2a). The azimuthal angle φ however depends on the SA orientation on the surface, which cannot be controlled in our AFM experiments. During pulling, the vertical rupture force Fr,z (last peak of the respective force-extension curve) is probed and recorded according to the vertical deflection of the cantilever (see Figure S4 for exemplary curves). Then, we recover the rupture force along the pulling direction Fr,p (as well as the associated loading rate Ḟr,p) from its vertical component Fr,z (Ḟr,z) to construct angle-dependent dynamic force spectra using: As shown in Figure 2b, different rupture forces are observed for different θ. We relate this difference to the inherent asymmetry of the biotin-SA complex. Furthermore, the azimuthal angle φ and the tilt of the tetramer due to the SA immobilization constitute additional degrees of freedom for pulling at a defined θ. As the experimentally measured force values for a given θ do not resolve these degrees of freedom, they are lumped together into a one-dimensional representation, resulting in so-called memory kernels and ultimately friction. Furthermore, because the angle φ and the exact SA immobilization points cannot be controlled, dynamics along φ and the tetramer tilt in respect to the surface constitute dissipation channels. By extracting the unbinding dynamics for each θ using the DHS model (Figure S5), we observe that both Fr,p and γ increase with increasing θ (Figure 2b,c), resulting in a higher friction than in the vertical pulling case. Besides, we also observe an increase in the rupture force variance ⟨δFr,p2⟩ with increasing θ that might be indicative of a more heterogeneous free energy landscape at larger θ (Figure 2d). Contrary to this effect, the transition state barrier height stays approximately constant (Tables S1 and S2). Our attachment chemistry using NHS esters to anchor SA to several PEG linkers on the substrate restricts any rotation of the SA. This enables us to pull biotin-SA at defined polar angles θ. If SA would be immobilized through a single anchor point, SA would reorient upon directional pulling, allowing biotin to detach along the same pathway for every angle analogous to vertical pulling.

To reveal the underlying mechanism of the angle-dependent friction based on a SA-biotin complex with well-defined pulling along both polar angle θ and azimuthal angle φ, we have further performed SMD and TMD simulations based on the SA-biotin MD simulation system by Rico et al. The former simulation approach is used to ideally perform the angular pulling process of stereographic SMFS and the latter, utilizing the dcTMD framework, is used to obtain a quantitative overview of the free energy barriers and friction parameters present during the unbinding process. For each SMD trajectory, we determine the maximum force peak as the rupture force for further analysis and evaluate the results for each θ and φ (Figure 3b,e, see Figure S6 for exemplary curves). The other peaks in the trajectory are associated with intermediate states and cannot be observed with the resolution of AFM. Even if all force peaks are taken into account, the results do not significantly change (Figure S7). As for experiments, we find an increase in the mean rupture force ⟨Fr,p⟩ as well as the associated rupture force variance ⟨δFr,p2⟩, with increasing θ and loading rate Ḟr,p (Figure 3c,f). Combining stereographic SMFS and SMD results (Figure 3d,g) we observe a nonlinear increase of the ⟨Fr,p⟩ and ⟨δFr,p2⟩ in this broad range of log(Ḟr,p). These observations are consistent with previous experimental reports and theoretical predictions on SA. To relate the angle-dependent rupture forces observed in both experiments and simulations with friction, eqs 1 and 2 can be used, as done above for the experimental data. However, it is unclear whether the DHS model still holds at the extreme pulling velocities employed in the MD simulation data. On the basis of dcTMD, we are able to calculate an alternative nonequilibrium friction coefficient ΓNEQ (section Friction analysis, SI). The friction coefficient γ from eq 2 and ΓNEQ both need to relate to the same molecular sources of friction and thus should follow similar trends. Assuming that friction is connected to the fluctuations that drive biotin over the transition barrier, we propose the following relations to hold (see section Friction analysis, SI, for details):among the friction coefficient γ entering the Kramers’ rate, the dcTMD friction coefficient ΓNEQ, and the empirically observed rupture force variance ⟨δFr,p2⟩. The increase in ⟨δFr,p2⟩ with polar angle θ can therefore be qualitatively associated with an increase in γ. Indeed, both experiment and simulation show that an increasing pulling angle θ correlates with an increase in the rupture force variance. The ⟨δFr,p2⟩ for a single φ may increase or decrease with θ (Figure 3e) but increases significantly with θ when multiple φ are combined (Figure 3f), i.e., φ is integrated out as coordinate. To assess the source of the different rupture forces along φ, we investigate possible differences in unbinding pathways and in protein conformational dynamics via a nonequilibrium PCA on amino acid-biotin contacts and on protein dihedral angles.Figures 4a and S8 display the corresponding contact PCA for all SMD simulations at 0.5 m/s with data projected onto the first few eigenvectors xi. We display the position probability P(xi) analogous to free energies in form of a “biased free energy landscape”: (further projections and details in Figures S8 to S10). In of the different pulling directions, different population regions appear that are summarized in Table 1, taking into account that the variation of θ only reveals gradual changes in contact domains but does not result in distinct pathways. In contrast, varying φ displays clearly distinguishable distributions, which correspond to different domains being accessed. In the dihedral analysis in Figures 4b and S11 to S13, we observe an additional conformational state γ′ that is exclusive for φ = 0° and gradually accessed with increasing θ. Along this direction, we find the L3/4 loop (Figure 3a in yellow), and the main contributions to the first three dihedral eigenvectors (Figure S13) contain residues 47 to 50 found in this loop. Likewise, a root-mean-square fluctuation analysis in Figure 4c displays increased fluctuations of this loop at φ = 0°. While conformational changes of L3/4 have been linked with biotin unbinding before, we observe that the extent of this conformational change prominently depends on φ. In summary, we observe that the increase in rupture force variance ⟨δFr,p2⟩ is accompanied by differences in unbinding paths and changes in conformational dynamics of the protein. To obtain a quantitative overview of the free energy barriers and friction parameters present during unbinding, we have performed dcTMD in combination with temperature-boosted Langevin equation simulations along θ = 0° as well as θ = 45°. In the latter case, we have performed calculations separately for the azimuthal angles φ = 0, 90, and 270° (simulations at 180° produced artifacts, see SI). Our results are displayed in Figure 4d: For the reference angle θ = 0°, we observe a single binding minimum with a transition state of ΔGu ≈ 24 kBT at a pulling distance of x ≈ 0.3 nm. After the transition state, a single additional minimum follows around x ≈ 0.4 nm, but the free energy landscape levels off quickly to a free energy ΔG ≈ 20 kBT. The main peaks in friction around x ≈ 0.5 nm coincide roughly with biotin and ligand binding site hydration (Figure S14). Temperature-boosted Langevin simulations with this free energy and friction profile predict a binding rate kon = (5.0 ± 1.0)·106 M–1 s–1 and an unbinding rate koff = (0.02 ± 0.01) s–1. Note that this koff estimate excellently agrees with the koff values obtained for the same θ in our AFM experiments (Tables S1 and S2). While the resulting dissociation constant KD = koff/kon = (4.0 ± 3.0) nM is significantly larger than the picomolar affinity known from experiments, it is a reasonable estimate in light of the peculiar quantum mechanical interactions within the binding site that cannot be reproduced with classical force fields. The shape of the free energy landscape is markedly different for a θ = 45° pulling angle (Figure 4d). For the three azimuthal angles φ = 0, 90, and 270°, we observe a shift of the transition state to x ≈ (0.5-0.9) nm and an increase of the barrier height to ΔGu ≈ (22-32) kBT. In particular for φ = 90°, we observe two additional free energy barriers at x ≈ 0.3 and 0.8 nm. In all three cases displayed, the dcTMD-derived friction coefficient ΓNEQ at the barrier is comparable or smaller at θ = 45° than for 0° (Table S3). It is therefore unlikely that protein-internal or solvent friction causes the apparent increase in friction. Altogether, our dcTMD calculations demonstrate that pulling at angles θ ≠ 0° gives rise to varying free energy barriers for different φ. The more θ deviates from the symmetric orientation at θ = 0°, the more the heterogeneity of the free energy barriers along φ increases its impact on ⟨δFr,p2⟩ and thus friction. How can the increase of the two employed friction coefficient definitions γ and ΓNEQ be combined to one common molecular source of angle-dependent friction? In the case of γ and eq 2, the fraction in front of the Boltzmann factor constitutes the attempt frequency of biotin to cross the unbinding transition state. Considering SA’s molecular structure in Figure 1a, the optimal unbinding angle should be θ = 0°. Pulling at larger angles causes a motion of the ligand toward the unbinding path to become less likely and therefore lowering the attempt frequency to optimal unbinding. Based on eq 2, this is observed as an increase in friction. The dcTMD analysis reveals that for the same θ, there exists a heterogeneous free energy surface that varies with the azimuthal angle φ. Owing to different ligand-protein contacts and protein conformations, φ constitutes an uncontrolled parameter in the experiments. We note that the simulations cannot take into account a possible denaturation of SA, but the consistency with the experiments renders such a denaturation unlikely. Hence, the dynamics along φ and the protein tilt constitute dissipation channels and thus sources of friction. An increase of θ leads to a stronger influence of uncontrolled pulling over different barriers, leading to an increase of friction with increasing θ. Combining the findings from our stereographic SMFS experiments, the SMD and the dcTMD approach, we propose a general interpretation: when biotin is pulled at an increasing polar angle θ, the conventional unbinding path becomes less likely. Under the influence of the external driving force, higher free energy barriers are crossed as well. The unbinding process becomes irreversible, as it is performed along the uncontrolled parameter φ of the system, where work is dissipated and perceived as an increase in friction.

We use AFM-based stereographic SMFS and TMD simulations to study the effect of pulling direction on the friction of biotin-SA unbinding. We observe an increase in rupture force variance with an increasing pulling polar angle θ, which we relate to an increase in anisotropic friction. While friction in biomolecular systems so far has been described in the form of internal and solvent friction, we here extend this list of friction sources by anisotropic friction. Friction anisotropy is known to occur in surface friction. This effect however is based on a well-ordered surface shape anisotropy, while in our case, anisotropic friction stems from an additional degree of freedom that cannot be controlled in experiment or mechanical environments. Anisotropic friction may therefore occur in any situation where directional forces act on biomolecules, such as folding, mechanics of motor proteins and possibly to bioassemblies or biomaterials at larger scales.