Ligand-induced changes of the apparent transition-state position in mechanical protein unfolding

Abstract

Force-spectroscopic measurements of ligand-receptor systems and the unfolding/folding of nucleic acids or proteins reveal information on the underlying energy landscape along the pulling coordinate. The slope Δx(‡) of the force-dependent unfolding/unbinding rates is interpreted as the distance from the folded/bound state to the transition state for unfolding/unbinding and, hence, often related to the mechanical compliance of the sample molecule. Here we show that in ligand-binding proteins, the experimentally inferred Δx(‡) can depend on the ligand concentration, unrelated to changes in mechanical compliance. We describe the effect in single-molecule, force-spectroscopy experiments of the calcium-binding protein calmodulin and explain it in a simple model where mechanical unfolding and ligand binding occur on orthogonal reaction coordinates. This model predicts changes in the experimentally inferred Δx(‡), depending on ligand concentration and the associated shift of the dominant barrier between the two reaction coordinates. We demonstrate quantitative agreement between experiments and simulations using a realistic six-state kinetic scheme using literature values for calcium-binding kinetics and affinities. Our results have important consequences for the interpretation of force-spectroscopic data of ligand-binding proteins.

Figure 1

Kinetic network model for the folding/unfolding of CaM₃₄ as presented in Stigler and Rief (24). The domain can exist in its calcium-free form, with one calcium ion bound or with two calcium ions bound. In addition, each form can exist in a folded or unfolded conformation. Note that due to cooperative binding, the one-calcium bound state is only sparsely populated.

Figure 2

Typical passive-mode equilibrium traces of CaM₃₄ at three different calcium concentrations. The protein fluctuates between the holo-folded state (green) and the unfolded state (red) (24). (Asterisks) Transient populations of the apo-state that unfold before calcium binds (24). The population of individual apo-states decreases with increasing calcium concentration. Due to calcium-dependent stabilization of the holo-state, the forces at which the protein is in equilibrium also increase with calcium. (Black lines) Downsampled traces; (shaded lines) level identifications from a hidden Markov model (29). To see this figure in color, go online.

Figure 3

Folding (open circles) and unfolding (solid circles) rate constants for the transition between the unfolded and the long-lived folded holo-state in CaM₃₄. (A) Experimental values. (B) Simulated values. (Left three columns) At varying calcium concentrations; (right column) folding (open squares) and unfolding (solid squares) rate constants between the unfolded and folded apo-states at 10 mM EDTA. (Dashed lines) Fits to the unfolding rates according to Eq. 1 (see text).

Figure 4

Experimental values (data points) and simulation results (lines) for the force-dependent kinetics of the long-lived folded state N in CaM₃₄ at varying calcium concentrations. Values at [0]mM Ca²⁺ (squares) are for the short-lived apo-state N^∗. (Continuous lines) Simulated values for a realistic calcium on-rate constant of k_on = 0.6 × 10⁸ M⁻¹ s⁻¹. (Dashed line) Simulation for an assumed quasi-equilibrium with k_on = 0.6 × 10¹⁰ M⁻¹ s⁻¹. (A and B) Slope Δx^‡ value and zero-force extrapolated rate constant k⁰_u for unfolding. (C and D) Transition-state distance ΔL^‡_f and zero-force extrapolated rate constant k⁰_f for folding. (Shaded area) Calcium concentration is ≲ $\sqrt{K_1{K}_2}$, so that the long-lived state is not populated. The rate-constant data points in (B) and (D) were taken from Stigler and Rief (24).

Figure 5

Energy-landscape representations for the kinetic network $U\rightleftharpoons N^{\ast }\rightleftharpoons N$ at (A) low and (B) high ligand concentrations and a force F that induces unfolding of the apo-state. U represents the unfolded state, N^∗ is the apo-state, and N is the holo-state. The Δx^‡ value that is measured in force spectroscopy depends on the ligand concentration (see text). (C) At effectively zero ligand concentration ([L] << K_D), N will no longer be populated and the unfolding pathway starts from N^∗. The measured Δx^‡ will hence be Δx^‡_apo.

Figure 6

(A) Analytical solutions for the effective unfolding rates in the one-binding-site model according to Eq. 5 (lines). In this model, the unfolding rates saturate at k_off for high forces. Note that in experiments, especially in passive mode, only a small force range is accessible, and the predicted bending of rate-versus-force plots is therefore masked. To make the plot comparable with kinetic simulations of CaM₃₄, we chose the parameters k_on = 0.6 × 10⁸ M⁻¹ s⁻¹ and k_off = k_on ⋅ K, with K = 10^−6.21 M and k_u,apo = 305 s⁻¹. Data points are simulation results from the full six-state model (see Fig. 3B). (B) Extracted slopes of Δx^‡ values according to Eq. 6, calculated at a fixed force of 7 pN (dashed line). Note that the experimentally accessible range is close to the ligand-concentration-dependent midpoint force F_1/2([L]), where folding and unfolding rates are equal. (Continuous line) Δx^‡ at F_1/2([L]).

Figure 7

Simulated trajectories of transitions between folded and unfolded states according to the model of Fig. 5 at forces where the folded holo-state and the unfolded state are approximately equally populated. (Asterisks) Short-lived apo-states (see Fig. 2).

Figure 8

Results for calmodulin’s N-terminal domain CaM₁₂. (A) Shift of Δx^‡ values according to the simple model (see Fig. 6B). (Dashed line) Δx^‡ measured at a fixed force of 7 pN. However, because the experimentally accessible range is close to the calcium-dependent equilibrium force F_1/2([Ca²⁺]) for folding and unfolding, we also calculate Δx^‡ at F_1/2([Ca²⁺]) (continuous line). (B) Experimental values (data points) and simulation results (lines) according to the six-state model for CaM₁₂. (Shaded areas) Concentrations <K_D ≈ $\sqrt{K_1{K}_2}$, where the holo-state does not get populated and the depicted solutions are invalid.