1 \section{Mechanical unfolding experiments}
4 % AFM unfolding procedure
5 In a mechanical unfolding experiment, a protein polymer is tethered
6 between two surfaces: a flat substrate and an AFM tip. The polymer is
7 stretched by increasing the separation between the two surfaces
8 (\cref{fig:unfolding-schematic}). The most common mode is the
9 constant speed experiment in which the substrate surface is moved away
10 from the tip at a uniform rate. The tethering surfaces, \ie, the AFM
11 tip and the substrate, have much larger radii of curvature than the
12 dimensions of single domain globular proteins that are normally used
13 for folding studies. This causes difficulties in manipulating
14 individual protein molecules because nonspecific interactions between
15 the AFM tip and the substrate may be stronger than the forces required
16 to unfold the protein when the surfaces are a few nanometers apart.
17 To circumvent these difficulties, globular protein molecules are
18 linked into polymers, which are then used in the AFM
19 studies\citep{carrion-vazquez99a,chyan04,carrion-vazquez03}. When
20 such a polymer is pulled from its ends, each protein molecule feels
21 the externally applied force, which increases the probability of
22 unfolding by reducing the free energy barrier between the native and
23 unfolded states. The unfolding of one molecule in the polymer causes
24 a sudden lengthening of the polymer chain, which reduces the force on
25 each protein molecule and prevents another unfolding event from
26 occurring immediately. The force versus extension relationship, or
27 \emph{force curve}, shows a typical sawtooth pattern
28 (\cref{fig:expt-sawtooth}), where each peak corresponds to the
29 unfolding of a single protein domain in the polymer. Therefore, the
30 individual unfolding events are separated from each other in space and
31 time, allowing single molecule resolution despite the use of
32 multi-domain test proteins.
34 \nomenclature{force curve}{Or force--distance curve. Cantilever-force
35 versus piezo extension data aquired during a force spectroscopy
36 experiment (\cref{fig:expt-sawtooth}).}
40 \subfloat[][]{\asyinclude{figures/schematic/unfolding}%
41 \label{fig:unfolding-schematic}}
43 \subfloat[][]{\asyinclude{figures/expt-sawtooth/expt-sawtooth}%
44 \label{fig:expt-sawtooth}}
45 % Possibly use carrion-vazquez00 figure 2 to show scale of afm tip
46 \caption{\protect\subref{fig:unfolding-schematic} Schematic of the
47 experimental setup for mechanical unfolding of proteins using an
48 AFM (not to scale). An experiment starts with the tip in contact
49 with the substrate surface, which is then moved away from the tip
50 at a constant speed. $x_t$ is the distance traveled by the
51 substrate, $x_c$ is the cantilever deflection, $x_u$ is the
52 extension of the unfolded polymer, and $x_f=x_{f1}+x_{f2}$ is the
53 extension of the folded polymer.
54 \protect\subref{fig:expt-sawtooth} An experimental force curve
55 from stretching a ubiquitin polymer with the rising parts of the
56 peaks fitted to the WLC\index{WLC} model
57 (\cref{sec:sawsim:tension:wlc})\citep{chyan04}. The pulling speed
58 used was $1\U{$\mu$m/s}$. The irregular features at the beginning
59 of the curve are due to nonspecific interactions between the tip
60 and the substrate surface, and the last high force peak is caused
61 by the detachment of the polymer from the tip or the substrate
62 surface. Note that the abscissa is the extension of the protein
63 chain $x_t-x_c$.\label{fig:procedure}}