1
SUPPLEMENTARY
INFORMATION
Data
-
driven fingerprint nanoelectromechanical mass spectrometry
John E. Sader
1
*
, Alfredo Gomez
2
, Adam P. Neumann
2
, Alexander R. Nunn
2
and Michael L. Roukes
2
*
1
Graduate Aerospace Laboratories
and Department of Applied Physics
, California Institute of Technology,
Pasadena, California 91125, USA
2
Department of Physics, California Institute of Technology, Pasadena, California 91125, USA
CONTENTS
A
.
Mass uncertainty due to database
discreteness
B.
Variance in the position discrepancy of the analyte
C.
Database filtering using noise threshold
D.
Numerical validation using a cantilevered beam
E
.
Experimental validation using polystyrene particles measured with a
suspended microchannel
resonator
* e
-
mail: jsader@caltech.edu; roukes@caltech.edu
2
Section
A
–
Mass uncertainty due to
database
discrete
ness
Th
e d
iscrete nature of the fingerprint database introduces
uncertainty
into mass measurement
s
because the
most
parallel
fingerprint
v
ector
extracted
from the database
may not be
truly parallel
to
the
measurement
.
Here, we quantify this
error
for 1D
mass
placement
and
derive
formulas
for the
resulting
uncertainty
in the measured mass.
A
fingerprint database is said to be
“single
-
valued”
if
each
of its
fingerprint
s
has
only one
direction
and magnitude
, as
its
number of elements,
푁
!"#"$"%&
→
∞
; barring
fingerprint
set
s
of zero measure.
Such a
database has the following properties:
i.
A
single
-
valued
database
can have
fingerprints
that
originate from multiple
spatial
positions on
the device, e.g., due to device symmetry.
This
does not affect mass measurement accuracy
or the
developed theory
.
ii.
Finite
푁
!"#"$"%&
can introduce
ambiguity
in selection of the most parallel
vector if
the
measured
fingerprint
for
the
analyte
does not belong to the
database
,
as we
shall
discuss
.
The
scenario
in (ii)
can generate large mass anomalies and is excluded from the theory
, which
is
derived in the asymptotic limit of large
푁
!"#"$"%&
.
1. Theory for mass uncertainty
C
onsider a one
-
dimensional
device
,
whose
governing
equation
for
the
fractional
frequency shift
,
Δ
푓
'
,
of
its
eigen
mode
,
푛
,
is
Δ
푓
'
=
−
푀
"(")*#&
푀
!&+,-&
Φ
'
.
(
푥
)
,
(
1
)
where
Φ
'
(
푥
)
is
the
corresponding
displacement field of
the
eigenmode
;
푛
=
1
,
2
,
...
,
푁
;
the
analyte
mass
is
푀
"(")*#&
; the device mass is
푀
!&+,-&
; and
푥
is the
1D
spatial
position
of the
analyte
mass
on the
device
.
The fingerprint
is
defined as
the sequence of
fractional frequency shifts, i.e.,
{
Δ
푓
/
,
Δ
푓
.
,
...
,
Δ
푓
0
}
.
The algorithm chooses a fingerprint in the discrete database that is
most parallel
to the
measured
fingerprint
of
the
analyte.
This
is equivalent to choosing
the
spatial
position
on the device
from which
th
at
specific
most parallel
vector
arises.
Inevitably, there
can be
a
(small)
difference between
that
spatial
position and
the actual analyte position,
푥
,
due to database discretization
; this difference
is
denoted
Δ
푥
.
The
components of the
most parallel
vector
are
then
Δ
푓
'
|
!"#"$"%&
=
−
푀
!"#"$"%&
푀
!&+,-&
Φ
'
.
(
푥
+
Δ
푥
)
,
(
2
)
where
푀
!"#"$"%&
is the
standardized
mass used to generate the database
.
We assume the database has sufficient discretization
so
that
Δ
푥
can be considered small
,
i.e.,
푁
!"#"$"%&
≫
1
,
and Eq.
(
2
) is expanded
accordingly
,
Δ
푓
'
|
!"#"$"%&
=
−
푀
!"#"$"%&
푀
!&+,-&
8
Φ
'
.
(
푥
)
+
Δ
푥
Φ
'
.
1
(
푥
)
+
⋯
:
,
(
3
)
3
where
′
denotes the spatial derivative.
Retaining the leading
-
order terms
only
and using Eq. (
1
)
, Eq.
(
3
)
becomes
Δ
푓
'
|
!"#"$"%&
=
푀
!"#"$"%&
푀
"(")*#&
Δ
푓
'
|
#23&
−
푀
!"#"$"%&
푀
!&+,-&
Δ
푥
Φ
'
.
1
(
푥
)
.
(
4
)
Here,
Δ
푓
'
|
#23&
are
components of the
analyte’s
fingerprint at its true position
,
푥
,
i.e., the
measured
fingerprint
from Eq. (
1
)
.
The
measured
mass of the analyte
,
using
the
most parallel vector
extracted from the database
in
Eq. (
4
)
,
is
푀
4&"%32&!
=
‖
{
Δ
푓
'
|
#23&
}
‖
‖
{
Δ
푓
'
|
!"#"$"%&
}
‖
푀
!"#"$"%&
,
(
5
)
where {...} defines
a
sequence, i.e.,
a
fingerprint.
Substituting Eq.
(
4
) into Eq.
(
5
) gives
푀
4&"%32&!
=
‖
{
Δ
푓
'
|
#23&
}
‖
A
B
푀
!"#"$"%&
푀
"(")*#&
Δ
푓
'
|
#23&
−
푀
!"#"$"%&
푀
!&+,-&
Δ
푥
Φ
'
.
1
(
푥
)
C
A
푀
!"#"$"%&
,
(
6
)
showing
that
푀
4&"%32&!
can differ from the true analyte mass
,
푀
"(")*#&
.
Expanding
Eq.
(
6
)
to leading order
in
Δ
푥
,
and using Eq.
(
1
)
, gives
the relative error in the measured
mass
,
Δ
푀
≡
푀
4&"%32&!
−
푀
"(")*#&
푀
"(")*#&
=
−
∑
Φ
'
.
(
푥
)
Φ
'
.
1
(
푥
)
0
'
5
/
∑
Φ
'
6
(
푥
)
0
'
5
/
Δ
푥
.
(
7
)
It
then follows
from Eq. (
7
)
that
the relative variance in the measured mass
of an analyte at
a
fixed
position,
푥
—
derived
by selecting the most parallel
database
vector to
its
fixed
fingerprint
—
is
Var
Δ
푀
=
1
4
K
1
퐹
푑퐹
푑푥
N
.
Var
Δ
푥
,
(
8
)
w
here
퐹
(
푥
)
=
P
Φ
'
6
(
푥
)
0
'
5
/
,
(
9
)
which is the
squared
magnitude of
a
dimensionless fingerprint. Thus,
Var
Δ
푀
in Eq
.
(
8
)
is proportional
to the relative rate
-
of
-
change in the
squared
magnitude of the fingerprint
.
Var
Δ
푥
is
the value over all
realizations of the database.
An expression for
Var
Δ
푥
is required to
complete the solution
. The
fingerprint
database is
constructed
in the learning phase
by randomly placing the mass standard
at
discrete
spatial positions
4
in
the interval,
푥
∈
[
푥
4,(
,
푥
4"7
]
.
As shown in Section B,
the
analyte’s
position discrepancy,
Δ
푥
,
relative
to the position corresponding to the most parallel vector,
satisfies
Var
Δ
푥
=
훿푥
.
2
,
(
10
)
where
훿푥
≡
푥
4"7
−
푥
4,(
푁
!"#"$"%&
.
(
11
)
Substituting Eqs. (
1
0)
and
(
1
1
) into Eq. (
8
), gives the variance in the measured mass placed at a
single
(known)
position,
푥
,
due to selection of the most parallel vector over all
realizations of the
database,
Var
Δ
푀
|
8
=
1
8
K
푥
4"7
−
푥
4,(
푁
!"#"$"%&
N
.
K
1
퐹
푑퐹
푑푥
N
.
.
(
12
)
Next,
we
consider an ensemble of measurements of the s
a
me
analyte
placed randomly
over
a
finite
spatial region of the device
,
푥
∈
[
푥
4,(
,
푥
4"7
]
.
A
veraging
Eq. (
1
2
)
over all
analyte
positions
—
because
the
se
position
s
are
distributed
randomly
with uniform probability
along
the device
during
mass
measurement
(see above)
—
gives the
average
relative mass
uncertainty,
Var
Δ
푀
=
푥
4"7
−
푥
4,(
8
푁
!"#"$"%&
.
W
K
1
퐹
푑퐹
푑푥
N
.
푑푥
8
!"#
8
!$%
.
(
13
)
2
.
Mass placement
near a stationary
support
of the device
We consider two
practical
supports
: (A) clamped
, i.e.,
zero displacement and slope
;
and (B) simply
supported
,
zero displacement and moment.
Case A:
Near a clamp (at
푥
=
0
),
Φ
'
(
푥
)
~
푎
'
푥
.
where
푎
'
is a
constant
, and Eq.
(
1
3
) gives
Var
Δ
푀
~
2
푥
4"7
−
푥
4,(
푥
4,(
푁
!"#"$"%&
.
as
푥
4,(
→
0
.
(
14
)
This
establishes that uncertainty in the measured mass diverges if
any
mass
position approaches a
clamp
ed support
.
Case B:
F
or
a device that is
simply
supported at
푥
=
0
, we have
Φ
'
(
푥
)
~
푎
'
푥
,
and Eq.
(
1
3
) becomes
Var
Δ
푀
~
푥
4"7
−
푥
4,(
2
푥
4,(
푁
!"#"$"%&
.
as
푥
4,(
→
0
,
(
15
)
which also diverges
in the same fashion
as for a clamped support
.
This is because the eigenmodes have
similar
functional form
s
near the supports,
which leads to deterioration of the
level of
uniqueness
in
the fingerprint database
. Namely
, the fingerprint
s
generated near
a
support are nearly parallel.
5
This shows that mass
placement
near the
supports
of a
device
should be avoided, not only because
of
inevitably
poor signal
-
to
-
noise
in the measured fingerprint
s
, but because
mass
uncertainty
diverges
.
An algorithm for the rejection of these
fingerprints is detailed in Section C.
(a)
(b)
(c)
(d)
Supplementary
Figure
1
:
Normalized standard deviation of the measured mass,
푁
&'(')'*+
SD
Δ
푀
,
due to the
discrete nature of the fingerprint database. Analyte position range is
푥
,-.
<
푥
<
1
: (a)
푥
,-.
=
0
.
03
; (b)
푥
,-.
=
0
.
1
;
(c)
푥
,-.
=
0
.
5
; (d)
푥
,-.
=
0
.
9
. Results using (i) near
-
support formula, Eq. (
1
4) (green line), (ii) exact formula, Eq.
(
1
3) (red line), and (iii) Monte Carlo simulations with error bars specifying 95% C.I.
Number of fingerprints in the
database is indicated.
푁
&'(')'*+
=
100
; similar results for
푁
&'(')'*+
=
300
,
1000
(
data
not shown).
M
ass
measurements are performed 100 times, which are repeated 100 times to estimate 95% confidence intervals.
Outliers discussed in
the “Results and discussion” section
are rejected by only accepting simulations where the
mean and median of the repeat 100 samples (to compute the 95% C.I.) differ by no more 50%.
3
. Comparison to Monte Carlo simulations
Supplementary Figure
1
compares
Eqs. (
1
3
) and (
1
4
) to Monte Carlo simulations
for ensemble
measurements of a single analyte over a finite region of
a cantilevered beam
,
employ
ing
at least 3
eigenmodes
—
satisfying the database
single
-
valued
requirement
(see definition above)
. The distance
of the minimum analyte position to the clamped end,
푥
4,(
,
is varied to illustrate the effect of mass
measurement near
a
support (clamp). The results in
Supplementary Figure
1
validate Eq. (
1
3
) and
show that increasing the number of eigenmodes from
푁
=
3
to 10 has a weak effect on the mass
measurement uncertainty. Interestingly, the near support formula, Eq. (
1
4
), gives reasonable
performance for all
푥
4,(
, estimating the true value to within a factor of approximately two.
4
6
8
10
8
10
12
14
16
Max mode number, N
N
database
SD
Δ
M
4
6
8
10
3
4
5
6
7
8
9
Max mode number, N
N
database
SD
Δ
M
4
6
8
10
1.5
2.0
2.5
3.0
Max mode number, N
N
database
SD
Δ
M
4
6
8
10
0.6
0.8
1.0
1.2
Max mode number, N
N
database
SD
Δ
M