Ligand-Receptor bonds

The first step to understand interactions in cells and between cells is to understand interactions between the involved biomolecules. Besides DNA/RNA and antigen-antibody interactions, there are many proteins (receptors) and corresponding ligands that play a vital role inside and in-between cells.

Figure 1.4: Sketch for streptavidin-biotin bond-force measurements, from [119]. A force is applied to biotin (red) to pull it out of the streptavidin.

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Because of their exceptionally high binding affinity, two of the most prominent ligand-receptor pairs are
streptavidin-biotin and avidin-biotin. Both proteins have a tetrameric
structure, so they can bind up to four ligands. Although many properties are valid for other ligand-receptor pairs, only these two ligand-receptor pairs were used in this thesis and, therefore, this section will focus on them.

During the last two decades, the rupture force between ligand and receptor was investigated with several different methods, such as Atomic Force Microscopy (AFM) [44,97,86,122,28,134], Surface Force Apparatus (SFA) [63,137], Dynamic Force Spectroscopy (DFS) [39,91], Flow Chamber measurements [102] and relaxation experiments with magnetic nanoparticles [77]. Complementary to the measurements, computer-
simulations were done by several
groups [59,70,40,65]. Figure 1.4 shows a single streptavidin-biotin
pair and exemplarily the applied
force in bond-force measurements.

Numerous measurements of the rupture force between ligand-receptor pairs were made with the AFM. In AFM experiments, the tip is coated by one part of the ligand-receptor pair and the other is fixed to a surface (e.g. an agarose bead [44]). After the tip contacts the surface and the ligand-receptor pair is bound, the tip is retracted using the force spectroscopy mode. Under appropriate experimental conditions [123,122], hundreds of single bond breaking events can be measured within a short time. But AFM measurements are somewhat limited, as shown in 1999 by MERKEL et al.  [91]. They showed that the bond-force depends strongly on the rate of force increase $F'$^1.4. Although this dependency was found with AFM measurements as well [134], only the DFS has the possibility to measure with loading-rates from 0.05pN/sec to 60nN/sec. Dynamic Force Spectroscopy, developed by EVANS et al.  [39] in 1991, utilises biomembrane probes and is a potent method for the analysis of rupture events. DFS consists of two micropipettes that push two vesicles against each other. A vesicles contains low amounts of either receptors or ligands. The separation of the vesicles is analysed under a microscope, and video is recorded to calculate the rupture force.

In contrast to the AFM measurements, where the measured bond-forces where around 100 to 300pN, it was possible to measure bond-forces of only 5pN with the DFS [91]. Experiments with other techniques, also got results in between those values, which support the loading-rate dependency. In addition, this thesis presents experiments that confirm the dependency in the range of extremely low loading-rates (see chapter 4).

In a different approach multi-wavelength x-ray diffraction methods were used to obtain the specific structure of ligand-receptor pairs with å ngstrøm precision [132,64,88,46,24]. Results of such x-ray diffraction methods can be seen in figures 1.2, 1.3 and 1.4.

In 1987 WEBER et al. [132] were the first group who fabricated streptavidin crystals and characterised the streptavidin-biotin complex with this method. Two years later HENDRICKSON et al. [64] made a comprehensive assay using a multi-wavelength anomalous diffraction method at an x-ray energy of 11921eV. They found that the biotin is buried deeply inside the $\beta$-barrel of the streptavidin protomer. Only the carboxylate oxygens and the ureido-ring nitrogen protrude to the outside. Multitudinous hydrogen-bonds and van der Waals interactions are involved in the biotin binding. There are three hydrogen bonds to the carbonyl group buried within the barrel and also hydrogen bonds to the ureido nitrogens and carboxyl oxygens. Four tryptophan amino acids are in contact with each biotin molecule. Most of the interactions result from the residues of a given subunit, although one tryptophan is supplied by a subunit related to the R diad axis, which is vital for the tetramer integrity. This is an explanation for the reduced affinity with less than four ligands that was also found elsewhere [76,113]. WEBER et al. [131] confirmed the results for streptavidin and apostreptavidin in 1992 and LIVNAH et al. [88] got similar results for the avidin-biotin complex in 1993. In 1998, CHU et al. [24] presented a very convincing experiment, where they removed the polypeptide loop that undergoes an open to closed conformational change when biotin is bound. They showed that the deletion caused a large decrease of the affinity for the full ligand-receptor bond.

Although JONES and KURZBAN [72] presented good evidence that the streptavidin-biotin binding is not cooperative^1.5 in the sense of the MWC-model^1.6, many other publications clearly indicate that the bond strength makes a step for the full bond. This can be seen as a positive cooperativity, as proposed by WILLIAMS et al. [133] and SANO et al. [113]. Besides the already mentioned x-ray diffraction experiments, these results were also supported by fourier-transform infrared spectroscopy and fluorescence spectroscopy [49].

Figure 1.5: Asymmetric two-well potential U(x), used in KRAMERS' model. Escape occurs via the forward rate k$^+$ and the backward rate k$^-$. The corresponding activation energies are E$_b^+$ and E$_b^-$. Taken from [61]

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All theoretical descriptions of ligand-receptor bonds and their breaking is based on the Transition-State-
Theory (TST) and KRAMERS' model [15]. The TST was developed by
POLANYI and WIGNER in 1928 [103] and expanded by EYRING in 1935 [41]. Generally the classical TST describes any two physical states that are separated by a bottleneck in
phase space. Two assumptions have to be made to apply the TST. First, the strong-coupling assumption, i.e. that all effects from a divergence of the thermal equilibrium are neglected, and second, the point of no return, i.e. that separated states do not reconnect. Even when these assumptions are applicable, TST can only give an upper bound to the true rate for any dividing surface [61]. KRAMERS' model describes a chemical reaction as a classical particle that moves in a one dimensional asymmetric double-well potential (confer to figure 1.5). A thorough description of KRAMERS' model can be found in [61].

The new experimental results about ligand-receptor bonds stimulated several theoretical works that extend the TST and KRAMERS' model. In 1996 GRUBMÜLLER et al. [59] presented computer-simulations of the streptavidin-biotin bond that matched their AFM measurements. The simulations supported their experimental results, that the measured bond-force increased with an increasing loading rate, and that the bond-force is around 280pN for a loading-rate of over 100nN/sec. But they simulated an extremely stiff cantilever (the spring constant was nearly 20 times higher than in the experiments), the time-scale of force increase was nanoseconds rather than milliseconds (as in experiments) and only a streptavidin monomer has been simulated. In 1997 IZRAILEV et al. simulated the avidin-biotin bond and found very high rupture forces of up to 450pN. They also presented a theoretical study that demonstrates that the nanosecond simulations can not reproduce thermally activated bond rupture that requires milliseconds. Also in 1997 EVANS and RITCHIE [40] published a thorough extension of KRAMERS' model to simulate force-activated bond rupture, and tested their model using smart Monte Carlo simulations. Additionally, they proposed a law for the exponential loading-rate dependency of the avidin-biotin rupture between 1 and 10$^{20}$pN/sec and 100 to 400pN.

Figure 1.6: Conceptual energy landscape of a ligand-receptor bond. The dashed line represents an applied force that lowers the potential barriers and, therefore, the total width of the potential narrows ($x_2 < x_1$). Remade after [91]

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As shown 2000 by STRUNZ et al. 
[121] the simplest possible model for ligand-receptor pairs has at least one intermediate state. Figure 1.6 illustrates such a model. Without any outer force the width of the potential ($x_1$) is wide. An applied force (represented by the dashed line) lowers the outer barrier and therefore the inner barrier with a narrower potential width ($x_2$) is relevant for the bond rupture properties. For the narrowing of the potential width a linear lever rule is valid: The smaller the potential width, the higher the rupture force. These models can only be applied if there is a single preferred path for the reaction [15],
which was shown experimentally by FREITAG et al. [45]. Such a model was introduced in 1978 by BELL [8] and used e.g. by MERKEL et al. to explain the loading-rate dependency of the ligand-receptor bond in 1999. Experimental data from bond-force measurements can be fitted to a linear function [121]:

$$F = \frac{k_{\rm B} T}{x_\beta} \ln\frac{x_\beta\cdot r}{k_{\rm B} T\cdot k_{\rm off}}$$ (1.1)

where $k_{\rm B} T = 4.114$pNnm is a Boltzmann factor at 298K, $r$ is the loading-rate, $x_\beta$ is the potential width (see above) and $k_{\rm off}$ is the natural off rate.