of 9
Article
https://doi.org/10.1038/s41467-024-51733-8
Data-driven
fi
ngerprint
nanoelectromechanical mass spectrometry
John E. Sader
1,2
,AlfredoGomez
3,5
,AdamP.Neumann
3
,AlexNunn
3
&
Michael L. Roukes
2,3,4
Fingerprint analysis is a ubiquitous tool for pattern recognition with applica-
tions spanning from geolocation and DNA analysis to facial recognition and
forensic identi
fi
cation. Central to its utility is
the ability to provide accurate
identi
fi
cation without an a priori mathem
atical model for the pattern. We
report a data-driven
fi
ngerprint approach for nanoelectromechanical systems
mass spectrometry that enables mass m
easurements of particles and mole-
cules using complex, uncharacterized
nanoelectromechanical devices of
arbitrary speci
fi
cation. Nanoelectromechanica
l systems mass spectrometry is
based on the frequency shifts of the nanoelectromechanical device vibrational
modesthatareinducedbyanalyteadso
rption. The sequence of frequency
shifts constitutes a
fi
ngerprint of this adsorption, which is directly amenable to
pattern matching. Two current requireme
nts of nanoelectromechanical-based
mass spectrometry are: (1) a priori kn
owledge or measurement of the device
mode-shapes, and (2) a mode-shape-ba
sed model that connects the induced
modal frequency shifts to mass adso
rption. This may not be possible for
advanced nanoelectromechanical devices with three-dimensional mode-
shapes and nanometer-sized features. The advance reported here eliminates
this impediment, thereby allowing
device designs of arbitrary speci
fi
cation
and size to be employed. This enab
les the use of advanced nanoelec-
tromechanical devices with complex vib
rational modes, which offer unpre-
cedented prospects for attaining t
he ultimate detection limits of
nanoelectromechanical mass spectrometry.
Mass spectrometry is used across a broad spectrum of applica-
tions, ranging from the chemical identi
fi
cation of compounds to
the sequencing of proteins and its use in drug discovery
1
7
.These
applications discriminate between analytes based on their mass-
to-charge (
m
=
z
) ratio, often in combination with an initial chro-
matographic separation.
In proteomic analysis,
m
=
z
analysis is
often complemented with information about the fragmentation
patterns of proteins and protein complexes from which
bioinformatics can enable the identi
fi
cation of the original, intact
analyte
6
,
8
11
.
In recent years, a new form of mass spectrometry has emerged
utilizing nanoelectromechanical systems (NEMS)
that circumvents
these requirements by enabling direct mass measurement of the
analyte, without the need for fragmentation or ionization
12
15
. Tre-
mendous advances in this new technology have been reported over
the past two decades, with the realization of routine single-molecule
Received: 14 March 2023
Accepted: 15 August 2024
Check for updates
1
Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, California 91125, USA.
2
Department of Applied Physics, California Institute of
Technology, Pasadena, California 91125, USA.
3
Department of Physics, California Institute of Technology, Pasadena, California 91125, USA.
4
Department of
Bioengineering, California Institute of Technology, Pasadena, California 91125, USA.
5
Present address: Language Technologies Institute, Carnegie Mellon
University, Pittsburgh, Pennsylvania 15213, USA.
e-mail:
jsader@caltech.edu
;
roukes@caltech.edu
Nature Communications
| (2024) 15:8800
1
1234567890():,;
1234567890():,;
mass measurements and the promise of single-Dalton mass
resolution.
NEMS mass spectrometry currently requires a mode-shape-based
mathematical model that connects the resonant response of the sen-
sing device to the mass of an adsorbate
13
19
. This has limited the scope
of NEMS devices used to those readily amenable to mathematical
modeling
e.g., using Euler-Bernoulli (E-B) elastic beam theory and thin
plate/membrane theory
leading to a proliferation of studies involving
cantilevered elastic beams, doubly-clamped beams, thin plate mem-
branes and similar devices
13
16
,
20
,
21
.
For advanced NEMS devices with more complex geometries, that
may exhibit three-dimensional mode shapes, numerical methods can
in principle be used to compute their modes. However, small varia-
tions due to unavoidable fabrication uncertainties can alter both the
shape and sequential ordering of the eigenmodes in the frequency
domain (which is often used to identify the modes)
22
. Miniaturization
to nanoscale dimensions also obviates the use of standard optical
techniques to measure these modes in situ. For example, recent pho-
nonic crystal resonators
23
,
24
illustrated in Fig.
1
, and more generally
advanced NEMS devices of sub-micron-sized dimensions with complex
modes shapes, are dif
fi
cult (perhaps not possible) to characterize
using available optical techniques
23
,
24
. This prevents the use of the
existing paradigm for NEMS mass spectrometry that relies on precise
knowledge of the mode shapes.
Consider a (one-dimensional) elastic beam that is currently and
often used to measure the mass of an adsorbed analyte. At least two
eigenmodes are needed whose fractional eigenfrequency shifts,
Δ
f
n
where
n
=1,2,
...
,arerelatedtothemass,
M
analyte
, and position,
x
,of
the adsorbed analyte, by
13
16
,
18
Δ
f
n
=

M
analyte
M
device
Φ
2
n
x
ðÞ
ð
1
Þ
where
M
device
is the device mass, it is assumed that
M
analyte
M
device
,
Φ
n
ð
x
Þ
is the displacement
fi
eld of eigenmode
n
,and
Δ
f
n
f
n

f
0
ðÞ
n
Þ
=
f
0
ðÞ
n
where
f
n
and
f
0
ðÞ
n
are the eigenfrequencies in the presence
and absence of the analyte, respectively. Knowledge of the eigen-
modes measured and their displacement
fi
elds, i.e., mode shapes, is
central to the use of such mathematical models. These mode shapes
are often approximated by theoretical idealizations, frequently leading
to systematic yet unknown errors
22
. This issue presents a signi
fi
cant
bottleneck to utilizing the full spectrum of NEMS devices that can be
fabricated today and into the future.
In this article, we report a data-driven approach that circumvents
these limitations by eliminating these central requirements. These are
replaced with a calibration-based algorithm
termed the
learning
phase
(described below)
which uses a frequency-shift
fi
ngerprint
of several modes, to perform the mass measurement. This
fi
ngerprint
approach opens the door to the use of nonconventional and unchar-
acterized NEMS devices
potentially with complex three-dimensional
mode-shapes that are not amenable to experimental character-
ization
enabling a sole focus on NEMS device optimization for
mass responsivity that is independent of device composition and
1
2
3
4
5

f
n
6
4. Measure mass of unknown analyte
3. Compare with
database.
analyte fingerprint
advanced NEMS
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
fingerprint database
1. Learning phase.
Repeated random sampling
using mass calibration
standard.
2. Measurement phase.
Frequency shifts upon physisorption
of unknown analyte.
1

m
3.61 GHz
3.10 GHz
2.84 GHz
2.36 GHz
2.29 GHz
2.71 GHz
LM
ab
Fig. 1 | Fingerprint approach for NEMS mass spectrometry. a
Schematic of
fi
ngerprint approach for NEMS mass spectrometry with an illustrative example
reporting six eigenmodes of an advanced NEMS device: the defect of a phononic
crystal resonator
23
,
24
is shown. In the learning phase, analytes of known mass are
adsorbed at random positions on the device to determine the
fi
ngerprint database
(red text: 1). In the measurement phase, the mass of an analyte is measured by
matching its
fi
ngerprint to the
fi
ngerprint database (black text: 2, 3, 4).
b
Symbols
to indicate the mass distributions used in the learning (L) and measurement (M)
phases. Delta and Gaussian-like symbols refer to mono- and polydisperse dis-
tributions, respectively, denoting the various embodiments of the
fi
ngerprint
approach. These symbols are used in subsequent
fi
gures.
Article
https://doi.org/10.1038/s41467-024-51733-8
Nature Communications
| (2024) 15:8800
2
geometry. This dramatically magni
fi
es the spectrum of devices
that can be employed in NEMS mass sp
ectrometry, facilitating the
realization of the long-standing goal of single-Dalton mass reso-
lution using advanced NEMS.
Results
Fingerprint approach for mass measurement
The response of a NEMS device to an adsorbed analyte can be char-
acterized using the sequence of fractional eigenfrequency shifts,
Δ
f
n
(
n
=1,2,
...
,
N
), of the eigenmodes chosen for measurement, forming
the
fi
ngerprint vector
:
Ω

Δ
f
1
,
Δ
f
2
,
...
,
Δ
f
N

ð
2
Þ
For the practical case of a small and rigid analyte with
M
analyte
M
device
, the analyte mass affects only the magnitude of
Ω
(not its direction in the
N
-dimensional
con
fi
guration space
). The
overriding assumption is that the direction of the
fi
ngerprint vector is
unique for mass adsorption at any single position on the device. As we
shall discuss, this assumption is automatically satis
fi
ed by choosing a
minimum number of eigenmodes,
N
. This eliminates the need to
measure the analyte position.
The sequence of chosen eigenmodes (and hence eigen-
frequencies) in Eq. (
2
) need not be in any particular order, and char-
acterization of the eigenmodes is immaterial. As per traditional
fi
ngerprint analysis, a
fi
ngerprint database
denoted by the set
F
consisting of individual
fi
ngerprints,
Ω
database
must be sourced for
the NEMS device in question. This
learning phase
is performed
experimentally through sequential adsorption of individual
particles of known and identical mass,
M
database
,atrandomand
unspeci
fi
ed positions on the entire device; see Fig.
1
. There is no need
to remove the particles between adsorption events, provided
N
particle
M
analyte
M
device
,where
N
particle
is the total number of
adsorbed particles. The resulting locus of points from this set
F
,in
their
N
-dimensional con
fi
guration space, de
fi
nes the required data-
base upon which
fi
ngerprint NEMS mass spectrometry can be
performed.
Importantly, the learning phase must be applied to the device of
interest and cannot be used between two nominally identical devices,
because of slight fabrication differences. If the device is substantially
altered after performing the learning phase, then the learning phase
must be repeated; adding small masses does not constitute a sub-
stantial change.
In the subsequent
measurement phase
, the mass of an individual
analyte,
M
analyte
, is determined following its random adsorption onto
the surface of the NEMS device. This is achieved by measuring the
analyte
s
fi
ngerprint vector,
Ω
analyte
, and identifying the database
fi
n-
gerprint vector,
Ω
database
2
F
, which is most parallel to its direction.
Because the direction of each
fi
ngerprint vector uniquely de
fi
nes an
adsorption position (see above), the unknown analyte mass can then
be determined by comparing the magnitude of these
fi
ngerprint vec-
tors, namely:
M
analyte
=
jj
Ω
analyte
jj
jj
Ω
parallel
jj
M
database
ð
3a
Þ
where
Ω
parallel
= arg max
Ω
database
Ω
analyte

Ω
database
jj
Ω
analyte
jjjj
Ω
database
jj
ð
3b
Þ
This
fi
ngerprint approach does not concern itself with the ana-
lyte
s position on the NEMS device, which is an inconsequential hidden
variable. It only assumes that the adsorbate weakly perturbs the
dynamics of the NEMS device.
The most-parallel-vector condition arises from the discrete nature
of the
fi
ngerprint database. This introduces uncertainty into the mass
measurement, in addition to any other measurement noise. For analyte
adsorption along a one-dimensional cantilevered beam of length,
L
,
which is often used in current NEMS mass spectrometry, the resulting
relative standard deviation in the measured mass is (Supplementary
Information Sections A and B):
SD
M
analyte
=
2
N
database
ffiffiffiffiffiffiffiffiffi
2
L
x
min
s
ð
4
Þ
where
N
database
is the number of
fi
ngerprints in the database and
x
min
is
the analyte
s closest distance to the cantilever
s clamped end, which is
assumed to be small, i.e., the analyte position spans most of the can-
tilever length. For two-dimensional adsorption on an advanced NEMS
device, e.g., onto the surface of a phononic crystal device
23
,
24
, a scaling
law with respect to
N
database
that varies between that of Eq. (
4
)andthe
reciprocal of
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
N
database
p
is expected, depending on the nature of the
eigenmodes. These scaling laws establish that the
database discretiza-
tion error
, characterized by SD
M
analyte
, is systematically reduced by
increasing the number of measurements,
N
database
, in the learning
phase. Equation (
4
) shows that SD
M
analyte
diverges if mass adsorption
occursneartheclamp,andthus,suchasituationmustbeavoided.This
is not a signal-to-noise issue but a phenomenon that arises from
discreteness of the
fi
ngerprint database and quasi-linear dependence
of the eigenmodes near the clamp.
This phenomenon can be avoided
in practice by rejecting
fi
ngerprints within the measurement noise; see
Supplementary Information Section C.
Phononic crystal device
To demonstrate the
fi
ngerprint approach, we
fi
rst report synthetic
numerical simulations using a Monte Carlo algorithm that implements
the above-described procedure on an advanced NEMS device: a pho-
nonic crystal resonator
23
,
24
. Analogous results for a cantilevered beam
are given in Supplementary Information Section D. The resonating
adsorption platform (the
defect site
23
,
24
) of these gigahertz frequency
phononic crystal devices are approximately one micron in size which
prohibits conventional optical characterization of their eigenmodes
22
.
Nonidealities in fabrication can distort these eigenmodes and their
ordering in frequency, given these resonant frequencies are closely
spaced; see Fig.
1
. Moreover, the eigenmodes undergo shear (in-plane)
deformation which further complicates mode-shape measurement;
see Fig.
1
.
The Monte Carlo algorithm places particles, in both the learning
and measurement phases, randomly onto the device surface while
calculating the resulting
fi
ngerprints from the mode-shapes. This uses
the two-dimensional generalization of Eq. (
1
)
because the analyte
adsorbs to a surface of the device
and simulates an actual experiment
where adsorption positions are not recorded. As per Fig.
1
,the
fi
n-
gerprint database is generated in the learning phase by adsorbing
(calibrated) particles of a single speci
fi
ed mass,
M
database
, numerous
times at random positions on the device surface; 1000 particles are
used in this demonstration. The resulting
fi
ngerprint databases, gen-
erated using the
fi
rst 3 and 4 eigenmodes within the band gap of the
device
23
,
24
, are shown in Fig.
2
a, d This learning phase primes the
fi
n-
gerprint approach, which can then be used in the measurement phase
to determine the unknown mass of an analyte particle.
Figure
2
b, e gives the measurement phase of the
fi
ngerprint
approach using (additional) 1000 analyte particles. The analyte parti-
cles are of identical mass to those used in the learning phase. Changing
the analyte mass gives identical plots in the measurement phase,
except that they are rescaled by the relative mass change due to
Article
https://doi.org/10.1038/s41467-024-51733-8
Nature Communications
| (2024) 15:8800
3
linearity of Eq. (
3a
). The particle mass, averaged over all measure-
ments, is accurately determined in the measurement phase. This is
regardless of the number of eigenmodes used, though using 4 eigen-
modes gives substantially less variance. The displayed sorted density
plots highlight this feature: using 3 eigenmodes gives the correct mass
with a standard deviation of ~40% over all measurements, while the use
of 4 eigenmodes exhibits a standard deviation of only ~4%.
The poorer accuracy in the 3-eigenmode measurement phase data
is due to the multivalued nature of its
fi
ngerprint database. While some
fi
ngerprints are uniquely de
fi
ned by their direction in the con
fi
gura-
tion space, others are not, i.e., there exist multiple branches of solu-
tion. This violates the overriding assumption of uniqueness in the
fi
ngerprint vector (see above), which is guaranteed when 4 or more
eigenmodes are used for two-dimensional adsorption on the physical
surfaces of NEMS devices; see Supplementary Information Section D
and for adsorption on a one-dimensional device.
Figure
2
c, f gives mass measurements (using the same
fi
ngerprint
databases shown in Fig.
2
a, d) for 10,000 analyte particles obeying a
log-normal mass distribution in the measurement phase only. More
analyte particles are used here in the measurement phase to reduce
uncertainty in the sorted density plots and ensure an accurate com-
parison. The above-mentioned multivalued nature of the 3-eigenmode
database in the learning phase is apparent in Fig.
2
c, with a visible
difference in the measured and true mass distributions. Increasing the
number of eigenmodes to four immediately eliminates any visible
difference, see Fig.
2
f, which is restricted to the above-mentioned
database discretization error.
Results and discussion
The
fi
ngerprint approach is validated using experimental measure-
ments on two current NEMS mass spectrometry devices: (1) a doubly-
clamped elastic beam operating in vacuum, and (2) a suspended
microchannel resonator (SMR) for buoyant mass measurements in
liquid. The purpose is to demonstrate and benchmark the
fi
ngerprint
approach on actual experimental data. Nominally identical particles
are used in both the learning and measurement phases so that the
measured masses are implicitly calibrated; the use of different parti-
cles would require independent calibration of their masses and gen-
erate additional uncertainty in this benchmark study.
Doubly-clamped beam measurements in vacuum
We
fi
rst apply the
fi
ngerprint approach to measure the mass of indi-
vidual proteins physisorbed onto the surface of a doubly-clamped
NEMS beam in a high vacuum. These macromolecules are of identical
(
fi
xed) mass to within a degree of high precision; modulated only
by the binding of hydrogen, water molecules etc. A hybrid Orbitrap-
NEMS system illustrated in Fig.
3
a is used to perform single-molecule
nanomechanical mass measurements of E. coli GroEL chaperonin, a
noncovalent complex consisting of 14 identical subunits; its theore-
tical molecular mass is 800.7664 kDa for which independent
M
particle
M
ref
Particle number (sorted)
M
particle
M
ref
Particle number (sorted)
-Δf1
-Δf2
-Δf3
-Δf4
-Δf2
-Δf3
0
500
1000
0.5
1
2
0
500
1000
0.5
1
2
0
5000
10000
0.5
1
2
0
5000
10000
0.5
1
2
M
LM
3 modes
4 modes
abc
de
f
Fig. 2 | Numerical simulations of a phononic crystal device
23
,
24
.
Fingerprint
approach using the
fi
rst three (2.29, 2.36, and 2.71 GHz) and four (2.29, 2.36, 2.71,
and 2.84 GHz) eigenmodes of a phononic crystal resonator (in its measurement
band gap) illustrated in Fig.
1
.
a
,
d
Learning phase showing
fi
ngerprint databases
with
N
database
= 1000
fi
ngerprints, which are used in all measurement phases.
b
,
e
Measurement phases for 1000 identical analyte particles, shown as an ordered
density plot.
c
,
f
Measurement phases for 10,000 different analyte particles, whose
masses obey a log-normal distribution. Orange curve is the true mass distribution.
Blue dots (which appear as a single curve) are the measured masses using the
fi
ngerprint approach.
Article
https://doi.org/10.1038/s41467-024-51733-8
Nature Communications
| (2024) 15:8800
4
measurements have con
fi
rmed a value of 800.7822 ± 0.0236 (SD)
kDa
25
27
. GroEL is pre-selected using the quadrupoles of the orbitrap
system, ensuring only intact GroEL molecules are delivered to the
NEMS. A 20-device NEMS array of doubly-clamped beams (Fig.
3
b) is
used to localize the focal point of the ion beam. Subsequently, the
smallest NEMS device in the array with the best mass resolution (length
of 7
μ
m) is operated in isolation. Details concerning the operation and
fabrication of this array are provided in refs.
25
28
.
Individual molecular adsorption events of intact GroEL mole-
cules abruptly shift the resonant frequencies of the
fi
rst two
fl
exural
eigenmodes of the smallest NEMS device; see experimental data in
Fig.
3
c. The tracked frequency data collected for each
fl
exural
eigenmode are time series, with jump events due to molecular
adsorption. These jump events are detected and analyzed using a
statistical algorithm described in ref.
29
. This produces a dataset
consisting of pairs of frequency shifts (for
fl
exural eigenmodes 1 and
2 of the device) for each molecular adsorption event, each of which is
a
fi
ngerprint.
The
fi
ngerprint approach is used to analyze this measured
fi
n-
gerprint dataset to recover the mass of each molecule. The result is
compared to a conventional mode-shape-based method reported by
Dohn et al.
18
(termed, the
Dohn method
), which
fi
ts the relative
frequency shifts of multiple eigenmodes
measured at a single particle
position
to E-B theory using a weighted least-squares approach.
Uncertainty in the mass measurements is reported, allowing for a
robust comparison. In the learning phase, the
fi
ngerprint database is
generated from the
fi
rst 24 entries of the overall frequency shift
dataset. The remaining 48 entries in the frequency shift dataset are
then analyzed in the measurement phase of the
fi
ngerprint approach,
to determine the masses of their corresponding GroEL molecules. The
Dohn method is applied directly to each entry of the
fi
ngerprint
database, with the median of the resulting 24 measured masses used as
the mass reference in the measurement phase.
Figure
3
dshowsaplotofall
fi
ngerprints, for the learning and
measurement phases, in their con
fi
guration space. Figure
3
d also
shows the exclusion zone of E-B theory which is not accessible by the
Dohn method, i.e., no particle mass can be recovered from these
positions in the con
fi
guration space. In contrast, the
fi
ngerprint
approach has no such limitation and naturally handles this situation
(which may naturally arise due to frequency noise). Comparison of the
measured masses of each of the above-described 48 GroEL molecules
using the
fi
ngerprint approach and the Dohn method is reported in
Fig.
3
e. Excellent agreement is observed using these independent
approaches. This constitutes an experimental validation of the pro-
posed
fi
ngerprint approach for a widely-used NEMS mass spectro-
metry set up: physisorption of a small analyte onto the surface of a
NEMS device, showing that the
fi
ngerprint approach can be used with
con
fi
dence.
14
15
16
17
Time (s)
-12
-8
-4
0
4
Relative frequency change (
10
-6
)
Mode 1
Mode 2
Fingerprint
Dohn
Dohn outliers
b
e
d
c
a
M
L
Fig. 3 | Mass measurements of GroEL molecules with a doubly-clamped NEMS
beam in high vacuum. a
Architecture of the Hybrid Q Exactive-NEMS System that
delivers intact proteins to the Orbitrap chamber for analysis of mass-to-charge
ratio and then onto the NEMS for single molecule analysis.
b
SEM image of a 20-
device array of doubly-clamped beams showing their metallization layers, AlSi
(colorized in yellow), used to interconnect the electrical connections of each
resonator.
c
As GroEL molecules physisorb to a single NEMS resonator, the reso-
nant frequency of each tracked
fl
exural eigenmode abruptly shifts. Sub-
fi
gures
a
c
taken from ref.
26
.
d
All 72 measured
fi
ngerprints of the
fi
rst two eigenmodes in
their con
fi
guration space: [open circles] 24
fi
ngerprints in the learning phase;
[closed circles] 48
fi
ngerprints in the measurement phase; [triangles] 2
fi
ngerprints
that are outliers of the Dohn method; [solid line] best
fi
t to the E-B model; [shaded
region] exclusion zone of E-B model where the Dohn method does not apply.
e
Measured mass of 48 particles using: [blue dots] the measurement phase of the
fi
ngerprint approach; [orange dots] the Dohn method; [triangles] the Dohn
method which does not give a converged solution. Error bars refer to 95% con-
fi
dence intervals.
Article
https://doi.org/10.1038/s41467-024-51733-8
Nature Communications
| (2024) 15:8800
5