Dependence of Work on the Pulling Speed in Mechanical Ligand Unbinding

In single-molecule force spectroscopy, the rupture force Fmax required for mechanical unfolding of a biomolecule or for pulling a ligand out of a binding site depends on the pulling speed V and, in the linear Bell–Evans regime, Fmax ∼ ln(V). Recently, it has been found that non-equilibrium work W is better than Fmax in describing relative ligand binding affinity, but the dependence of W on V remains unknown. In this paper, we developed an analytical theory showing that in the linear regime, W ∼ c1 ln(V) + c2 ln2(V), where c1 and c2 are constants. This quadratic dependence was also confirmed by all-atom steered molecular dynamics simulations of protein–ligand complexes. Although our theory was developed for ligand unbinding, it is also applicable to other processes, such as mechanical unfolding of proteins and other biomolecules, due to its universality.

Then taking average over the random force we have: Using this equation and rupture time ≈ we obtain

Region 2: ‡ < ≤
After rupture, the viscosity term should be taken into account, because the ligand speed becomes greater than in the first region. Then we have: By using Fourier transformation obtain: Similar to the region 1 case, the work was calculated by using Equations S8-S10: Substituting x(ꞷ) and v(ꞷ) from Equations S18 and S19 into this equation we arrive at Averaging over the random force, we obtain .
Coefficients a 1 -a 6 in Eq. 8 in the main text Substituting these parameters into Eq. (S25) we obtained the coefficients a 1a 2 in Table S1.

Choice of protein-ligand complexes
To support our theory we performed molecular simulations for two receptor-ligand complexes.
The first complex, which has the PDB ID 1FKH [2] (Figure S2), contains a small SBX compound located at the binding site of the FKBP12 protein. Another complex (PDB ID: 4KZ6 [3]) consists the AmpC beta-lactamase protein and the ZB6 carboxylic acid, a small compound ( Figure S2). The FKBP12 protein [4] has 107 amino acids; it is a 12 kDa cytosolic protein that is abundantly expressed in all tissues. It binds to FK506 and rapamycin, which mediates the immunosuppressive effect of drugs [5]. Binding to FKBP12 allows drugs to subsequently interact with mechanistic targets of their action in immunosuppression [6].
Beta-lactamase [7], an enzyme produced by bacteria [8], is responsible for the resistance of bacteria to many beta-lactam antibiotics like penicillins, cephalosporins and carbapenems. A typical AmpC protein consists of 716 amino acids with approximately 6310 atoms and a mass of 79.88 kDa. Suppression of AmpC activity can prevent bacterial resistance [9,10].

Preparation of initial conformations for molecular dynamics simulations
From the PDB protein data bank, the ligandprotein conformation was downloaded and then the pdb format file was loaded by Pymol [11] software version 2.0 and separately saved into two files for a protein and a ligand. For the ligand conformation, an auto geometry optimization and an atomH adding were performed by Avogadro [12]

Steered molecular dynamic simulations
In the steered molecular dynamics (SMD) simulations an external force F=k s (∆x-vt) was applied to a dummy atom that connects to the ligand through a spring with the spring constant k s , v is the pulling speed and Δx is the displacement of the pulled atom of the ligand from its initial position.
To simulate the AFM experiment [17], we chose the cantilever stiffness = 600 . 2 . The pulling direction of the ligand from the binding site was determined using the minimal steric hindrance method [18] (Figure S2). All Cα atoms of the receptor were restrained to avoid its drift under the influence of an external force. To obtain reliable results we carried out many SMD trajectories and their number depends on the system and the pulling speed (Table S2). With a slow pull, the errors are smaller and fewer runs are required.
From the force-displacement/time profile we collected the rupture force for the calculation of <F max >. The pulling work for a given trajectory was calculated using Equation (1) in the main text and the trapezoid rule: .
Results obtained for the average rupture force, rupture time and pulling works are shown in Table S3.   Table S3: Average rupture force and pulling work obtained by using SMD simulations for two complexes. Error bars are standard deviations.