Chemistry Reference
In-Depth Information
ing rate,
, as a function of the pulling force, F (Figure 13.3D) directly demonstrates
that the unfolding rate of polyubiquitin is exponentially dependent on the pulling force.
An Arrhenius termcan be
fit by the data to quantify the time to rupture a bond under a
mechanical stretching force [63]. This approximationmodels a two-state process with a
single unfolding energy barrier in the reaction coordinate of the end-to-end length.
This model then yields a measure of the distance to the transition state along the force
induced reaction coordinate,
a
0.17 nm, beyond which the mechanical stability of
the protein is lost. The distance is comparable to the length of a hydrogen bond in
water, which is in excellent agreement with the result of computer simulations that
identify the crucial hydrogen bonds that stabilize the proteins transition state under
force [62]. It should be noted that the Arrhenius term [63], while currently used widely,
assumes a force-independent distance to the transition state. Other models have
recently been proposed which offer a more detailed analysis of the force-induced
transitions [24, 64].
In rare instances in the unfolding trajectory we are able to capture mechanically
stable intermediate structures that add up to the expected step size of 20 nm,
suggesting some diversity in the unfolding energy barriers. Interestingly, collecting
a much larger data set also begins to show important deviations from the
ts to the
two-state behavior, as the less-traveled unfolding pathways become statistically
signi
cant. We investigate the stochastic dynamics of unfolding and the success of
the two-state model in the next section.
D
x
¼
13.3
Order Statistics in Unfolding
Using the force-clamp technique we are able to test the diversity in the pathways and
the correlation between the protein modules in each chain by investigating the
kinetics of unfolding in further detail.
We
first develop an analysis method for analyzing the unfolding dwell times
from the force-clamp technique. Since the number of modules in each chain varies
up to the engineered protein length (
N¼
12) (Figure 13.3B) and the dwell times
depend on the order of the event
we need to apply order statistics to the data
to investigate correlations between the modules and the heterogeneity in the
pathways.
Even without correlations (Markov process) and given a single reaction pathway,
statistics deems that the dwell times,
k
t
, to the unfolding events depend on the order
number of the event,
k
, and the chain length,
N
. Indeed, the probability of observing
k
unfolding events out of
N
folded protein modules in time
t
should follow a binomial
distribution,
N
!
ð
k
1
Þ
ð
N
k
þ
1
Þ
e
at
e
at
P
ð
t
;
N
;
k
Þ¼a
Þ!
ð
1
Þ
ð
Þ
ð
13
:
1
Þ
ð
k
1
Þ!ð
N
k
where
a
is the unfolding rate constant for the ensemble. With an extensive pool of
data we showed that this distribution could not account for all the unfolding events of
