Biomedical Engineering Reference
In-Depth Information
probability of finding
n
bonds formed when the sample mean is
N
is given by the
Poisson distribution:
e
−
N
N
n
n
!
P
(
n
;
N
)=
(3.96)
The sum over all probabilities for which an adhesion event occurs (
n
0) gives the
fraction of all possible adhesion events for any bond valency. Therefore, the frac-
tion of occurrences of adhesion events
N
e
within the total sample of attempted force
measurements
N
tot
is equal to the sum of probabilities with
n
=
≥
1:
n
=
1
e
−
N
N
n
∞
N
e
N
tot
e
−
N
n
!
=
−
=
1
(3.97)
which gives the mean number of bonds formed in terms of the fraction of adhesion
events:
ln
1
N
=
(3.98)
1
−
N
e
/
N
tot
With this solution for
N
, one can solve for
P
in Equation 3.96 to find the
fraction of single-bond events out of the total number of samples:
(
n
=
1;
N
)
1
ln
N
1
N
tot
=
N
e
N
tot
1
−
(3.99)
1
−
N
e
/
N
tot
Likewise, the fraction of single bonds out of the number of successful adhesion
events follows from
P
(
n
=
1;
N
)
/
(
N
e
/
N
tot
)
as
N
e
N
tot
1
ln
−
1
N
1
N
e
=
1
−
(3.100)
1
−
N
e
/
N
tot
Rewriting Equation 3.98 in terms of
N
, we can solve for the frequency of successful
adhesion events that give single bonds on average:
N
e
N
tot
1
=
0
.
63
(3.101)
N
=
However, inserting this frequency in Equation 3.100, we find that only 53% of the
successful events will contain single bonds, while the remaining 47% will consist of
a mixture of doubles, triples, and so on. Therefore, aiming for a target of single bonds
on average will still leave the measurements containing a significant portion (nearly
half!) of higher bond numbers. Alternatively, one can look to filter out the higher
bond numbers by decreasing the frequency of interactions further. For example, to
achieve adhesion events containing 90% single bonds, one would aim for an adhesion
event frequency of about 18%. This translates to 16% of single bonds out of the
total number of attempted measurements. To put this in perspective, to measure 100
single-bond events, one would need to make more than 600 measurements, resulting
in an excess of 500 “empty” measurements.
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