The dissociate rate (i.e. off rate) koffcan be determined by the strength of the bonds which affects the activation energy barrier for dissociation and by the thermal energy kBT, where kBis the Boltzmann constant. The activation energy for the dissociation is the difference in energy between the initial state and the transition state of the highest energy to which the system must be raised before dissociation can occur. It is important that we understand the binding of a ligand to its macromolecular protein receptor, not only for its fundamental importance but also as an aid in designing new potential drug candidates. Recent data of dissociation rates from thermodynamic and kinetic measure-ments have been supplemented by single-molecule experimeasure-ments using AFM, in which an external force is applied to dissociate the protein–ligand complex (Table 2). Moreover, molecular dynamic simulations (Grubmuller et al., 1996) that complement the original AFM analysis of the protein–ligand also give some insights into the interactions along the dissociation path (Florin et al., 1994; Moy et al., 1994).
3.2. Theory
It is known that a bound state can be idealized by confiding a single energy barrier at a given position along a specific reaction path (Fig. 5). Assuming that the kinetics of the forced unbinding can be analyzed in the context of the Kramer’s theory for activated processes (Kramers, 1940), the zero kinetic off rate (dissociation rate) for unbinding can be written as
koffð0Þ ¼ w eDE=kBT (4)
where kBis the Boltzmann constant, T the absolute temperature, kBT represents the thermal energy (4.14 1021J), w the frequency prefactor, and DE is the activation energy.
In AFM measurements, an external force is usually applied to a pair of atoms that are then pulled apart. Since the timescale
Fig. 5. A schematic diagram of the energy landscape shows the dissociation without an external force (solid line) and with an applied force (dashed line).
The intermolecular dissociation without an external force and with an applied force corresponds to the dynamic parameters, k(0) and k(F).
Fig. 6. The schematic diagram of the dissociation over a sharp energy barrier without an external force (solid line) and with an applied force (dashed line).
The dissociation over a sharp energy is characterized by an exponential increase of the barrier which is located at a distance, projected along the direction of the applied force.
Fig. 4. pH dependence of the unbinding force for glucagons–anti-glucagon pairs with different pulling velocities (16.7–166.7 nm/s). Reprinted with per-mission fromLin et al. (in press). Copyright 2006 Elsevier Ltd.
of the thermal motion and the time scale of the force variation in the AFM experiment are separated by at least several orders of magnitude, the assumption is usually made that the effect of the external force on the microscope kinetics is that of a constant force. As was first argued byBell (1978), a constant force F applied to x modifies the energy (and the free energy) by a factor Fx. Thus, if F and x are parallel (Fig. 6), then
DEðFÞ ¼ DE Fx (5)
which establishes a rate for unbinding that depends on the applied force such that
where it can be assumed that koff(Eq.(4)) and the prefactor w is not affected by the force.
3.3. Measurements
Kinetic parameters can be extracted from AFM data using theories developed above to understand bond dissociation, including protein–ligand dissociation under applied external forces. The kinetic rate-off koff( f) (Eq. (6)) is an essential determinant of the molecular interaction as it can provide insight into the strength of bonds, the occurrence of bonds, and the bond relaxation time. We can determine the koff( f) value from the unbinding force by looking at the range of the loading rate for protein–ligand unbinding events. Although an AFM is capable of applying a loading rate ranging between 10 and
Table 2
Protein–ligand off rate and binding energy
Molecular partners Loading rate
Not given 6 6.7 104 Not given Hinterdorfer et al.
(1996) Concanavalin A (ConA)
receptor/ConA
415–4980 0.27 0.17 Not given Chen and Moy
(2000)
Lactose/bovine heart (BHL) 20–10,000 1.02 0.17 0.09 Not given Dettmann et al.
(2000) Lactose/lactose-binding
immunoglobulin G (IgG)
20–10,000 0.72 0.09 0.9 Not given
Lactose/Viscum album (VAA) 20–10,000 0.75 0.16 0.09 Not given
Lactose/Ricinus communis (RCA) 20–10,000 0.41 0.07 0.8 Not given
Asialofetuin (ASF)/BHL 20–10,000 0.62 0.06 1.3 Not given
ASF/VAA 20–10,000 0.60 0.17 0.9 Not given
ASF/IgG 20–10,000 0.48 0.07 1.6 Not given
ASF/RCA 20–10,000 0.41 0.07 0.4 Not given
Streptavidin/biotin 100–1000
Not given Yuan et al. (2000)
Streptavidin mutant
Not given Lo et al. (2001)
Ryanodine receptor 1 (RYR1)/anti-RYR1
42–2000 0.2 12.7 Not given Kada et al.
(2001) Ligand (RGD)/human platelet
aIIbb3receptor
10,000–50,000 0.1 22.6 6.37 Lee and Marchant
(2001)
Not given 0.25 Not given 4.7 Cai et al. (2003)
Fibrinogen (aIIbb3)/RGD 100–1,000,000 0.103 1.5 6.4 (2.64 1020J) Lee and Marchant
(2003) Fibrinogen (aIIbb3)/
HHLGGAKQAGDV
100–1,000,000 0.109 47.58 6.5 (2.67 1020J)
Human a5b1integrins/GRGDSP Above 59,000 Below 59,000
0.09 2.77
787 0.015
Not given Kokkoli et al.
(2004) Sendai-purple membrane
(Sendai-PM)/anti-Sendai antibody
2000–50,000 0.12 6 Not given Kienberger et al.
(2005)
akB: Boltzmann constant = 1.381 1023J; T: room temperature = 278 K.
1000 nN/s, instability at rates >100 nN/s due to hydrodynamic effect is often experienced for protein–ligand measurements (Lee and Marchant, 2003).
Experimentally, the plot of unbinding force versus the logarithm of the loading rate (i.e. the product of the force constant of the AFM cantilever and its retraction speed) (Schwesinger et al., 2000) can be employed to estimate the dissociation rate (zero kinetic off-rate koff(0)) at zero force.
3.4. Examples
3.4.1. Concanavalin A (ConA) receptor–ConA interaction In 2000, Chen and Moy used an AFM to determine whether receptor cross-linking can increase cell adhesion through enhanced cooperative binding. They used the AFM to obtain force measurements before and after chemically cross-linking adhesion receptors on the surface of fibroblast cells. They measured the adhesive strength between concanavalin A (ConA) coupled to an AFM tip and ConA receptors on the surface of the NIH3T3 fibroblast cells. Their results showed that cross-linking of the receptors with either glutaraldehyde or 3,30-dithio-bis(sulfosuccinimidylproprionate) led to an increase in adhesion that can be attributed to enhanced cooperativity among the adhesion complexes. Moreover, to determine if changes in the loading rate contributed to the increased ConA/
ConA receptor adhesion, they obtained force measurements for the breakage of individual ConA and D-mannose complexes over a range encompassing the observed loading rates of unfixed and fixed cell measurements. For these previous
experiments, the adhesive strength was determined at different force loading rates, and the force histograms were acquired from measurements carried out using a ConA-functionalized tip on a D-mannose-linked agarose bead. In addition, the rupture force f* was dependent on the loading rate rfobtained and the dissociation rate constant was calculated from the Bell’s model. According to the Bell’s and Evans model, the dissociation rate can be obtained from the plot of f* versus log (rf) to be 0.17 s1, more information as described in Section 3.2. This investigation demonstrated that the cross-linking of surface receptors caused a shift towards cooperative binding and hence results in an increase in the receptor binding strength.
3.4.2. Ryanodine receptor 1 (RYR1)–anti-RYR1 interaction In 2000, Kada et al. adsorbed the skeletal muscle Ca2+
released channel (ryanodine receptor 1, RYR1) to a mica substrate with the cytoplasmic side facing up. They found that a single specific RYR1–anti-RYR1 recognition event can be detected at the single-molecule level with AFM tips to which the anti-RYR1 was tethered. In linear scans, the occurrence of the RYR1–anti-RYR1 binding showed a significant lateral depen-dence, which allowed for the localization of the binding sites with a nm resolution. Variation of the loading rate in the force spectroscopy experiments revealed a logarithmic dependence of the unbinding forces, ranging from 42 pN at 2 nN/s to 73 pN at 9 nN/s. As we know from theoretical predictions (Grubmuller et al., 1996; Evans and Ritchie, 1997), the unbinding force f will rise linearly with the loading rate on a half-logarithmic scale, which is characteristic for a single-energy barrier in the thermally
Fig. 7. (A) Scheme depicting an AFM tip to which an antibody is covalently tethered via a cross-linker. (B) A force–distance cycle where a single-molecule unbinding event results in an unbinding force of 120 pN and an unbinding length of 15 nm. (C) A probability density function which shows a pdf for the antibody–antigen interaction at a loading rate of 9 nN/s. (D) An unbinding force with dependence of the loading rate (force spectroscopy). Reprinted with permission fromKienberger et al. (2005). Copyright 2005 Elsevier Ltd.
activated regime (Merkel et al., 1999). The bond length of L = 0.2 nm and the dissociation rate constant of koff= 12.7 s1 were determined using Eq.(3)where the slope and intercept is at zero force.
3.4.3. Human a5b1integrins–GRGDSP dissociation
In 2004, with an AFM, Kokkoli et al. engineered a novel bio-mimetic system to study the mechanistic details of the unbinding processes of a5b1–GRGDSP pairs at collective and the single-molecule level. Their investigation demonstrated for the first time that at a collective level, the separation of multiple identical bonds within a5b1–GRGDSP, is a combination of multiple unbinding events as the pair does not break at once but in multiple steps, and in a stretching process. Force histograms were collected at a loading rate of 1–305 nN/s. The bond strength at a specific loading rate can be estimated by inspecting the force histogram and by calculating the autocorrelation function for the histogram.
Moreover, the unbinding force versus the logarithm of the loading rate for a single a5b1–GRGDSP bond can be plotted.
The force spectrum revealed two linear regimes with different slopes within the range of the loading rates that were examined, and the Bell parameters (xB and k0off) were also estimated from the slope and the intercept of force versus the logarithm of the loading rate (Evans and Ritchie, 1997; Tees et al., 2001). The best fit to the data revealed an inner barrier for the loading rates which was seen at a value above 59 nN/s at xB= 0.09 nm that is characterized by a very rapid unstressed transition rate of 1/toff= 787 s1. For a value below 59 nN/s, a second regime appeared in the force spectrum that mapped an outer barrier at xB= 2.77 nm. This outer barrier is characterized by a much slower transition rate that defines the dissociation rate constant k0off in the absence of force, k0off ¼ 0:015 s1. This value is in good agreement with the dissociation rate constant of 0.01 s1 as reported between the fibronectin and fibronectin receptor (a5b1) on the fibroblast cells in the solution.
3.4.4. Sendai-purple membrane (Sendai-PM)–anti-Sendai interaction
Kienberger et al. (2005) investigated the molecular recognition of antibodies to membrane-antigens and the extraction of the antigens out of the membranes at a single-molecular level. Dynamic force microscopy imaging and enzyme immunoassay was used to detect the binding of the anti-Sendai antibodies to Sendai-epitopes genetically fused into bacteriorhodopsin molecules from the purple membranes under physiological conditions. Moreover, they measured the unbinding force fuas a function of the loading rate r (Fig. 7).
The dissociation rate constant (k0off) was determined from the slope and the intercept of force versus the logarithm of the loading rate (Evans and Ritchie, 1997, 1999; Kerkel et al., 1999). The Sendai–anti-Sendai interaction strength of 70–
170 pN at loading rates of 2–50 nN/s yielded a barrier width of x = 0.12 nm and a kinetic dissociation rate (off-rate) (corresponding to the barrier height) of koff= 6 s1, respec-tively.
4. Determination of energy barriers