Chemistry Reference
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minimal model [91] the second stage in the folding mechanism involves the kinetic
ordering of the foldable chain, describing the proteins search among the space of
unfolded structures, such that the sequence directs the folding reaction towards
structures that are native-like. The search among the compact structures leading to
one of the minimumenergy structures is thought to proceed by a diffusive process in
the rugged energy landscape. The timescale for diffusion,
t
KO, and the subsequent
kinetic ordering is given by
D N z
t
t
ð
13
:
5
Þ
KO
where
z
is a dynamical folding exponent and
t D is a time constant. Numerical analysis
has found that a good estimate for
z
is 3 for heteropolymers while the exponent is
much higher, up to
6 for model proteins with native interactions [96]. The time
constant
t D is thought to correspond roughly to the timescale for local dihedral angle
transitions, which is obtained from simulations to be
10 8 s [91]. The search for
a structure close to the native state corresponds to combined stages 2 and 3
(Figure 13.6), during which the protein attains the end-to-end length of the folded
protein under a low stretching force. The timescale for folding of ubiquitin for this
stage of the folding trajectory is then predicted to be 0.4ms from Equation 13.5.
Clearly this timescale is vastly different from the timescales observed in the force-
clamp trajectory (Figure 13.6) in which a diffusional search for a native-like
conformation can take of the order of seconds and is highly dependent on the
magnitude of the applied force (Figure 13.7A).
The final stage in this minimal model is the activated transition from one of the
many native-like minimum energy structures to a native-state conformation [91].
The timescale for the barrier activated transition,
t
AT , is given by
p
corr e 0:6
t
t
ð
13
:
6
Þ
AT
where
corr is de ned as the correlation time for harmonic fluctuations in the
measured reaction coordinate [97]. This equation is based on a model in which the
protein diffuses through a rough energy landscape with multiple energy minima.
These transitions, involving the formation and rupture of native and non-native
contacts to establish the folded transition state, are presumed to take place during
stage 4, below the length resolution of our force-clamp technique. We therefore turn
to force-extension experiments to directly investigate the N dependence on the
folding times at zero force in the same proteinmolecule since the scaling laws for the
multipathway mechanism have been derived in the absence of force [91].
We have utilized protein engineering to control the contour length and therefore
the number of amino acids unraveled in a single protein [98]. Force extension curves
were obtained for a polyprotein for repeated unfolded and refolding cycles. In each
cycle the protein was first unfolded, resulting in a sawtooth pattern of unfolding
peaks. The force was then dropped to zero and the protein was given a period of time
to refold. Subsequently, the protein was againmechanically unfolded. The fraction of
refolded proteins was obtained for a particular delay period, for a wide range of
waiting times. From this experimental data the average refolding time at zero force
t
 
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