Chemistry Reference
In-Depth Information
ubiquitin. In order to investigate the source of the outliers in the obtained distribu-
tion, we next devised a maximum likelihood method at the single-molecule level.
If the unfolding process is Markovian, implying that the unfolding events are
independent of one another, yet involve different unfolding pathways (i.e. non-
homogeneous), the diversity in the pathways can be estimated using the maximum-
likelihood method (MLM) [65] to allocate a rate constant,
, to each single molecule
chain (Figure 13.4). The probability of a singlemodule unfolding in time
a
t
is assumed
e
(at)
as supported by the observation that each unfolding event takes
place in a single step on the timescale of the experiment. Therefore, for a given
to be p
¼
1
a
,
from the experimental trajectory of a sequence of
t
k
(Figure 13.4) and the appropriate binomial counting shown in Equation 13.1, the
unfolding probability is given by the product of probabilities,
k
events unfolding at times
e
aðNk
max
1Þt
det
Y
k
max
N
!
k
max
e
at
k
P
ð
t
1
; ...;
t
k
max
ja;
N
Þ¼a
:
ð
13
:
2
Þ
ð
N
k
max
Þ!
k
¼
1
The probability map for each unfolding trajectory is calculated over a wide range of
a
,
from 0.01 to 100 s
1
, as well as over the possible range of
(Figure 13.4) for the
particular trace shown. Since molecules detach from the cantilever at random times,
N
N
must lie between the number of observed steps and the engineered protein length.
The maximum likelihood method uses the unfolding probability,
P
(
t
1
...t
kmax
)in
Figure 13.4 Maximum Likelihood Method for
analyzing protein unfolding. A typical polyprotein
unfolding trajectory that exhibits six consecutive
unfolding events under a constant force of
110 pN is shown here. From the measured dwell
times, t
k
, and the detachment time, t
det
,we
estimate the most likely rate constant,
, and
length of the chain, N, using the equation for the
probability of unfolding as a function of time. The
inset shows the shape of the probability function
and the peak isolated for N
a
0.5 s
1
¼
10 and
a¼
for this trajectory.
