9. Non-linear elasticity of individual molecules
The AFM has been used to measure the linear elastic properties of biomolecules by selective indentation (Tao, Lindsay et al. 1992; A-Hassan, Heinz et al. 1998). The study of the non-linear elastic properties is a more recent development. Reif et. al (Rief, Gautel et al. 1997; Rief, Oesterhelt et al. 1997) demonstrated that remarkable structure appeared in the force versus distance curve when the AFM probe was pulled back after being pushed into a ‘sticky’ biopolymer attached to a substrate. We use the unfolding of the giant muscle protein, titin, as an example here (Rief, Gautel et al. 1997). Titin consists of a series of interconnected globular domains and may play a role as a ‘shock adsorber’ in muscle tissue. It adsorbs readily onto a gold substrate. Attached to an AFM tip by means of non-specific interactions (the tip is pushed hard into the adsorbed protein) it adheres well enough so that single molecules can be unfolded by pulling the tip away from the surface (illustrated in Fig 20a). The corresponding force-distance curve has a sawtooth structure with peaks of about 100pN separated by about 30nm distance. An example is shown in Fig. 20b (data courtesy of Wenhai Han using samples provided by John Trinick and Larissa Tskhovrebova). Rief et al. (Rief, Gautel et al. 1997) attributed each rapid drop to the unfolding of a single globular domain, and proved this to be the case by examining a series of constructs of a known small number of domains. The peaks increase in force from left to right, the weaker domains unwinding first. The same group used this method to pull on polysaccharide molecules, observing a bump in the force curve that they attributed to a conformational change within the interconnected sugar rings (Rief, Oesterhelt et al. 1997).
Figure 20: (a) showing a molecular unfolding experiment schematically. (b) Force vs. distance pulling on a titin molecule. (c) Image of the titin layer on the surface prior to pulling. (Image courtesy of Wenhai Han.)
Figure 21: (a) shows force (heavy line) and DFM amplitude (light dashed line) for pulling a chromatin molecule off a glass substrate. Regions labeled C are conservative. (b) Shows agreement between force calculated from the amplitude curve in the conservative regions (solid lines) and the measured force (dots). Arrows mark the points at which the integrations of stiffness data were started. (From Liu et al. (Liu, Leuba et al. 1999) with permission.)
None of the early work showed images of the substrate for the very good reason that the sample sticks to the tip. Images can be important, in, for example, ensuring that a single molecule (and not a bundle) sticks to the tip. It may also be important to check the conformation of the molecule on the substrate before pulling. It may be denatured with multiple attachment points to the substrate, so that peaks in the force curve reflect the process of pulling off the substrate rather than unfolding of internal domains. DFM can be useful here because it permits non-contact imaging. Thus, the sample can be surveyed prior to pulling, the tip then pushed into a molecule hard and the pulling done in the normal way. Fig. 20c shows an image of the area used to obtain the force curve shown in Fig. 20b. It was taken with magnetically-excited DFM.
The exquisite sensitivity of magnetic DFM for interfacial forces (O'Shea, Welland et al. 1994; Han and Lindsay 1998; O'Shea and Welland 1998) leads one to wonder if additional information might be obtained in non-linear experiments such as this. For small oscillation- amplitude (see section 5.4)
so, for a conservative system for which the force is given by the integral of stiffness with respect to extension distance, z,
. (22)
Figure 21 shows a plot of the dynamic amplitude signal (in nm p/p) with simultaneously-acquired force data (in nN) for pulling on a chromatin sample (Liu, Leuba et al. 1999). The stiffness is approximately inversely proportional to the amplitude signal according to Eqn 21. However Eqn. 22 cannot apply because the corresponding integral would rise continuously. In regions where the stiffness falls with extension, Eqn. 22 would predict negative forces, and these are clearly not observed. The reason is that the interactions in the regions marked NC are not conservative so Eqn. 22 does not apply. Within the regions of monotonicallly increasing force, Eqn. 22 can be applied as is demonstrated in Fig. 21b. Here, the constant of integration has been set to match the experimentally measured force at the start of each integration and the stiffness integrated to produce the solid lines. The dots are experimental force data. There are no fitting parameters beyond the known spring constant of the cantilever and the initial force. Excellent agreement is obtained, verifying the utility of Eqn. 22 in regions where interactions are conservative. These results show that enhanced signal to noise is obtained from the stiffness data. These data clearly contain information beyond that encapsulated in the relation Eqn. 22. In the non-conservative regions, much would be gained by recording both the amplitude and phase of the DFM signal and using these to determine a complex modulus for the molecule. The dissipative part is of particular interest because it would include information about bond-breaking and viscosity.
Figure 22: (a) Showing the arrangement of a bond rupture measurement schematically. The bond rupture force is obtained from the adhesion portion of force curves as shown in (b).
