Leveraging isotopologues as a general strategy to image neurotransmitters with
vibrational microscopy.

Gabriel F. Dorlhiac1, Markita P. Landry1,2,3,4,5∗, Aaron Streets1,4,5,6,7∗

1Biophysics Graduate Group, University of California Berkeley, Berkeley, CA,
94720, USA
2Department of Chemical and Biomolecular Engineering, University of
California, Berkeley, Berkeley, CA, 94720, USA
3Innovative Geneomics Institute, University of California Berkeley, Berkeley,
CA, 94720, USA
4California Institute for Quantitative Biosciences, QB3, University of
California, Berkeley, Berkeley, CA, 94720, USA
5Chan-Zuckerberg Biohub, San Francisco, CA, 94158, USA
6Department of Bioengineering, University of California, Berkeley, Berkeley,
CA, 94720, USA
7UC Berkeley-UC San Francisco Graduate Program in Bioengineering, University
of California, Berkeley, Berkeley, CA, 94720, USA

## Abstract

####

Chemical neurotransmission is central to neurotypical brain function but also
implicated in a variety of psychiatric and neurodegenerative diseases. The
release dynamics of neurotransmitters is correlated with but distinct from
electrical signal propagation. It is therefore necessary to track
neurotransmitter modulation separately from electrical activity. Here, we
present a new approach for imaging deuterated neurotransmitter molecules with
vibrational microscopy in the cell-silent window. Using stimulated Raman
scattering microscopy, we perform direct imaging of the neurotransmitters
dopamine and GABA in PC12 chromafin cells, and primary hippocampal neurons,
respectively, based on their C-D vibrational frequencies. We find that isotope
labelling does not perturb neurochemical activity. We further show that
stimulation of neurotransmitter release results in an overall 20-50% signal
reduction, with the ability to observe inter- and intracellular variation in
vesicular release potential. Taken together, our data suggest that
neurotransmitter isotopologues can serve as a generic, non-perturbative,
method to image neurotransmitters.

## Introduction

####

Information flow in the nervous system is mediated by both electrical and
chemical transmission processes, an understanding of both of which is
necessary to study neurotypical activity, and any deviations which can result
in neuropathology. Neuronal stimulation results in intracellular electrical
signal propagation, in the form of action potentials, on the millisecond
timescale. Interneuronal communication, however, is most often mediated by
small molecule, i.e. neurotransmitter, release at the neuronal synapse.
Canonically these neurotransmitters are locally confined to the sub-micron
sized synapse, leading to rapid communication shortly following action
potential arrival. One class of neurotransmitters, known as neuromodulators,
are thought to exhibit additional extrasynaptic diffusive action, which may
modulate broader neuronal networks on the order of microns and seconds.
Frequently, the release of these compounds is also not well correlated with
action potential arrival, as in the case of dopamine, where estimates of the
correlation are as low as 6-10% (1, 2). Abnormal regulation of neuromodulator
activity is associated with a wide variety of neuropathologies, including
schizophrenia (3), epilepsy (4), Parkinson’s (5), and Huntington’s disease
(6), and addiction (7), among others. The study of chemical neurotransmission
is, therefore, of fundamental importance to our understanding of
neuropathologies.

Owing to the high spatiotemporal resolution of fluorescence imaging, its
adaptability to image whole organs and organisms, and the broad abundance of
fluorescence-based probes to image neurons and neuronal activity,
fluorescence-based techniques have been widely adopted to study the brain and
the process of neurotransmission.

However, fluorescence-based approaches face a major obstacle when imaging
neurochemical communication, in that neurotransmitters are small molecules
that do not intrinsically fluoresce in the visible range. Therefore, several
approaches have been taken to enable fluorescence-based measurements of
neurotransmitter activity, including synthesis of fluorescent chemical
analogues, the use of genetically encoded receptors with affinity for the
neurotransmitter of interest, and readout of neuron activity by fluorescent
reporters as a proxy for neurotransmitter release. For instance, monitoring of
vesicular fusion can be accomplished through introduction of exogenous
amphipathic FM dyes (8), or by using the acidic vesicular environment to
quench genetically encoded pHluorin sensors (9), which report on fusion when
exposed to the neutral synaptic environment. Yet neither technique reports on
the neurotransmitter populations themselves inside the vesicles, nor whether
they are released. Other genetically encoded fluorescent receptors report
binding of the neurotransmitter itself. For example, cell-based
neurotransmitter fluorescent engineered reporters (CNiFERS), linking
neurotransmitter-binding G-protein coupled receptors to fluorescent readout,
have been used to study a number of different neurotransmitters (10, 11). This
technique, however, requires the transplantation of reporter cells into
neuronal tissue, exhibiting low spatiotemporal resolution relative to
neurochemical signaling. To address this issue GRABDA was developed by
engineering dopamine receptors with circularly permuted eGFP, to be expressed
in cell membranes, and is used to image exocytosed volume transmission of
dopamine (12). Patriarchi et al concurrently produced a suite of genetic
encoded sensors, operating on similar principles, called dLight (13). Both
probes require extensive protein engineering which is not easily translated to
other neurotransmitters, and can only measure neurotransmitters post-
exocytosis. Another class of sensors based on near-infrared fluorescent
nanoparticles (14)can be localized to the extracellular space in the brain and
produce fluorescence changes upon neurotransmitter binding, allowing imaging
of synaptic-scale neurotransmitter volume transmission (15). These sensors
have recently been used to image dopamine release from somatodendritic
processes and to characterize dopamine release from individual synapses (16).
All aforementioned fluorescence-based probes face similar research and
development challenges, requiring new approaches to their design, development,
and deployment with each new neurotransmitter target. Furthermore, their use
is limited to the measurement of extracellular analytes, making it difficult
to study neurotransmitters in the intracellular environment.

To address the challenge of intracellular neurotransmitter imaging, direct
chemical modification of neurotransmitters has been used to produce visible
fluorescence, allowing tracking of the molecule itself. False
neurotransmitters analogs have been under development for many years as
artificial alternatives to their endogenous counterparts. To this end,
fluorescent false neurotransmitters (FFN) enable intracellular tracking and
subsequent release of the neuromodulators dopamine (17, 18)and serotonin (19).
However, relatively large structural changes to the native neurotransmitter
are necessary to produce a visibly fluorescent analog, requiring extensive
validation and complex syntheses. Furthermore, it remains unclear to what
extent the structural modifications present in FFNs affects their
biodstribution and function. A number of investigations have therefore used
the natural autofluorescence of certain neurotransmitters, such as dopamine
and serotonin (20–22). While this approach provides a direct readout of the
compound, the use of UV fluorescence limits penetration depth, and limits
application to other neurotransmitters.

Fluorescence, however, is not the only physical property that can be used to
gain chemical specificity. Molecular vibrations are specific to the types of
atoms and bonds in a molecule, and can be probed through a number of different
optical interactions, many of which are well suited to implementation as
imaging techniques. The most common probes of molecular vibrations are
infrared (IR) and Raman spectroscopy, although many others exist including
surface-enhanced Raman scattering (SERS), photoacoustic imaging, and nonlinear
techniques such as coherent anti-Stokes Raman scattering (CARS), and
stimulated Raman scattering (SRS). However, vibrational imaging faces its own
difficulty in isolating the vibrational signature of a single molecule of
interest in situ in a biological system. A typical vibrational spectrum of a
cell contains a substantial amount of information about cellular composition
but, even with high spectral resolution, is too complex to deconvolve into
individual molecular contributions. To date, only a single neurotransmitter,
acetylcholine, has been imaged using vibrational microscopy in the frog
neuromuscular junction (23). The lack of subsequent development of vibrational
probes for neuroscience illustrates the challenge in specifically imaging a
single type of molecule in a complex cellular context.

Nonetheless, in the animal systems of interest, a gap appears in the
vibrational spectrum from about 1800 - 2800 cm-1, often called the cell-silent
window. This corresponds to the frequency range of alkyne and nitrile
vibrations, functional groups that are largely absent from biomoleules in
animals. By modifying molecules to have vibrational frequencies in this
window, it is possible to image their biodistribution with minimal background
from other cellular components. One could directly introduce a triple bond to
produce this effect. This strategy has been used to study a number of
processes including imaging of newly synthesized DNA by incorporation of
5-ethynyl-deoxyuridine (EdU) (24, 25), to study mitochondria (26), membrane
proteins (27), and drug uptake (25, 28). Such a modification was recently
proposed to study dopamine; however, imaging based on the alkyne vibration has
not yet been demonstrated (29).

Fig. 1 Overview of isotopologue imaging with vibrational microscopy.
(a) Deuterated neurotransmitters are incubated with the sample of interest.
Endogenous neurotransmitter transporters uptake the deuterated isotopologues,
facilitating their packaging into vesicles and allowing their visualization
using vibrational microscopy techniques such as stimulated Raman scattering
(SRS). (b) The commercially available, four deuterium substituted dopamine
isotopologue used for tracking dopamine uptake and release. (c) The SRS
spectrum of the dopamine isotopologue.

An alternative strategy to the incorporation of alkyne functional groups is to
use isotope labeling, as the frequency of a molecular vibration is inversely
proportional to the reduced mass of the atoms involved. Increasing the mass of
the involved atoms will therefore reduce the vibrational frequency, for
example by employing one of two common isotopic substitutions used in mass
spectrometry and nuclear magnetic resonance (NMR): 1H to 2H (D) or 12C to 13C.
These substitutions move the C-H stretches that are normally at 2800-3000 cm-1
down into the cell-silent window. Isotope labeled compounds have been used in
conjugation with vibrational imaging to study lipid and cholesterol dynamics
(30), proteome degradation in steady-state as well as with Huntington protein
aggregation (31), in addition to being used as a general readout of metabolic
activity (32). Furthermore, due to the application of isotope labeling in mass
spectrometry and NMR, many neurotransmitters are already commercially
available in isotope-labeled form, as analytical standards. This availability
provides a rapid source of neurotransmitters with vibrational signatures that
can be easily separated from the cellular background, obviating the need for
special syntheses, or genetic engineering. We hypothesize these compounds can
be introduced into neurobiological systems or cell cultures, and uptaken by
endogenous neurotransmitter transporters (Figure 1a), where vesicular
populations of the neurotransmitter of interest can then be monitored under
native conditions.

In this report, we demonstrate the application of SRS microscopy with this
labeling strategy to image two such commercially-available deuterated
neurotransmitters, d4-dopamine (d4-DA) and d6-$\gamma$-aminobutyric acid
(d6-GABA), in cultured PC12 cells and primary hippocampal neurons,
respectively. SRS is a coherent 3rd order nonlinear process which makes use of
two pulsed lasers, and is implemented in a point-scanning configuration for
microscopy, like with multiphoton excited fluorescence microscopy. The two
laser frequencies are chosen such that the difference in their frequency is
resonant with a vibrational mode of interest. As such, for any single image
only a single spectral channel is acquired, in contrast to confocal
spontaneous Raman microscopy. However, through coherent driving of the
vibration, signal enhancements of up to 108 over spontaneous Raman scattering
can be obtained (33, 34). We show that leveraging this signal enhancement
allows for visualization of d4-DA and d6-GABA neurotransmitter isotopologues
at speeds comparable to confocal fluorescence microscopy, with probe
structures that are functionally identical to their native counterparts.

## Results

### SRS microscopy allows observation of uptake and release of d4-DA

####

As a proof of principle we investigated the capability to image the uptake and
intracellular distribution of d4-DA in cultured PC12 cells. The PC12 cell
line, derived from rat chromaffin cells in the 1970s, is a popular in vitro
culture system for studying catecholamines and neurite development (35).
Figure 1b shows the hydrogen-to-deuterium substitutions made for this
particular isotopologue, while its corresponding SRS spectrum acquired in the
cell-silent window is displayed in panel c. We first tested the ability to
detect uptake of the labeled compound d4-DA, by adding it to a final
concentration of 50 $\mu$M to the cell culture medium. The PC12 cells were
incubated for one hour, after which they were washed and then imaged in a PBS
buffer.

To verify uptake, images were acquired at 2135 cm-1, the maximum of the d4-DA
spectrum in this window (figure 1c), and compared to the images of cells which
were not pre-incubated with the compound. To provide a reference for the
location and structure of the entire cell, images were also acquired at 2950
cm-1, corresponding to CH3 stretches, a signal largely dominated by proteins.
d4-DA internalization was observed for the d4-DA-incubated cells, largely
localized to the cell periphery, and in punctate structures, with clear
nuclear exclusion (figure 2a, middle row). Under identical imaging and culture
conditions, unincubated cells displayed no such signal, consistent with
detection of a signal specific to the d4-DA neurotransmitter (figure 2a, top
row). We observe a significant separation between the average intracellular
signal intensity in incubated vs unincubated cells (figure 3a; p $<$ 0.001,
$\Delta$ $>$ 15).

Fig. 2 SRS and two-photon fluorescence imaging of a deuterated dopamine
isotopologue and a false fluorescent analogue in PC12 cells.
(a) SRS images of PC12 cells without incubation with the deuterated dopamine
isotopologue (top), with incubation with the dopamine isotopologue (middle),
and with incubation with the dopamine isotopologue, but two minutes post
stimulation to release neurotransmitter with 50 mM potassium (bottom). The
left column displays images at 2950 cm-1, the C-H stretching frequency range,
broadly corresponding to protein signal and allowing cell localization. The
right column displays images at 2135 cm-1, the C-D stretching frequency range,
corresponding to the signal originating from the deuterated dopamine compound.
(b) Two-photon excited fluorescence images of PC12 cells incubated with a
false fluorescent neurotransmitter, FFN102 (top), and cells incubated with
FFN102 imaged two minutes post stimulation to release neurotransmitter with 50
mM potassium.

The observed subcellular distribution of d4-DA reveals signal along the cell
periphery and in distinct puncta as predicted, suggesting a biologically-
relevant internalization mechanism, with the isotopologue confined to
intracellular vesicles. Comparatively little to no signal is observed outside
these bright puncta, consistent with a much lower concentration of
neurotransmitter present in the cytoplasm, estimated to be 0.5 – 5 $\mu$M (36,
37), compared to 110 – 190 mM in the vesicles (38, 39). Furthermore, we
observe the expected lack of internalization into the nucleus. To test whether
this d4-DA signal did arise from an active vesicular population of d4-DA, we
repeated the incubation experiments, stimulated d4-DA release with 50 mM
potassium, and imaged cells two minutes post-stimulation. In line with the
expected response to potassium stimulation, there was an observable decrease
in the measured d4-DA signal for the population of cells imaged post
stimulation (figure 2a, bottom row), which is significant compared to the
unstimulated cells (figure 3a; p $<$ 0.001, $\Delta$ $>$ 0.35). This 22%
decrease is commensurate with prior reports that quantal release of dopamine
in PC12 cells is fractional.

We compared the response to potassium stimulation in PC12 cells incubated with
d4-DA to PC12 cells incubated with a dopamine FFN, FFN102 ([FFN102]=10
$\mu$M). A strong signal was visible within cells incubated with the
fluorescent analogue when imaged by two-photon excited fluorescence (TPEF)
microscopy (figure 2b, middle row). Notably, there was stronger readout from
the entire cell, perhaps due to the higher sensitivity of TPEF to measure
lower cytoplasmic concentrations of the analogue; however, unlike d4-DA-
labeled cells, some FFN signal was observed in the nucleus suggesting some
non-specific internalization mechanisms of the dopamine FFN analogue.
FFN102-incubated cells imaged post potassium stimulation also displayed a
lower overall signal (figure 2b, bottom row). This 21% decrease for FFN-
incubated cells post stimulation was observed to be significant over a large
sample of cells (figure 3b; p $<$ 0.001, $\Delta$ $>$ 0.48). The two
distributions for d4-DA and FFN102, in fact, show very similar trends, and
suggest d4-DA can report on relevant dopamine uptake and release in cultured
cells.

Fig. 3 Cell-averaged SRS or fluorescence intensity distributions for cells
incubated with deuterated dopamine analogue, or FFN102.
(a) The cell-averaged SRS intensity for cells that were not incubated with
deuterated dopamine (left column), cells that were incubated with deuterated
dopamine (middle column), and cells that were incubated with deuterated
dopamine, stimulated to release neurotransmitter with 50 mM potassium, and
imaged two minutes later (right column). * p $<$ 0.001, $\Delta$ $>$ 15
(incubated vs negative control), $\Delta$ $>$ 0.35 (incubated vs incubated
post stimulation). N(negative control) = 1273, N(Incubated, pre-stimulation) =
1152, N(Incubated, post-stimulation) = 1145. (b) The cell-averaged two-photon
excited fluorescence intensity for cells that were not incubated with FFN102
(left column), cells that were incubated with FFN102 (middle column), and
cells that were incubated with FFN102, stimulated to release neurotransmitter
with 50 mM potassium, and imaged two minutes later (right column). *p $<$
0.001, $\Delta$ $>$ 0.48 (incubated vs incubated post-stimulation). N(Negative
control) = 1802, N(Incubated, pre-stimulation) = 1904, N(Incubated, post-
stimulation) = 2920.

### PC12 cells display cell-to-cell heterogeneity of d4-DA uptake and release

####

We next sought to take pre- and post-stimulation measurements on individual
unincubated, d4-DA-incubated, and FFN102-incubated cells. While population
level measurements have relevance for several important biological questions,
they often obscure cell-to-cell heterogeneity. It is, therefore, frequently of
interest to make repeat measurements on the same individual cells. Single-cell
measurements are particular relevance for neurons, where measuring the effect
of stimulation and the relevant kinetics of neurotransmitter release are of
fundamental importance to understand neuro-communication. Furthermore, in PC12
cells, it is known that multiple vesicular populations (40)can exist
heterogeneously across individual cells, producing cell-to-cell variability in
dopamine concentration that is often not apparent when making bulk population
level measurements (35).

To measure d4-DA concentration within the same PC12 cells pre- and post-
stimulation, we built a simple microfluidic flow cell, with inlets for the
injection of stimulating potassium solution. Cells were grown inside these
devices, composed of coverslips on the top and bottom, and PDMS around the
sides.

Fig. 4 SRS and two-photon excited fluorescence imaging on the same single
cells pre- and post-potassium stimulation.
(a) Workflow for calculation of relative SRS intensity changes due to
potassium stimulation. 1. A single image is acquired in the 2950 cm-1 (C-H)
channel, broadly corresponding to protein distribution, which is used as a
mask. 2. A mask is created from this image using adaptive thresholding and
binarization. 3. Contour tracing allows single cells to be retrieved from the
mask. 4. A single image is acquired in the 2135 cm-1 (C-D) channel,
corresponding to the deuterated dopamine signal. The average signal is
determined for each masked cell. 5. The cells are stimulated to release
neurotransmitter with 50 mM potassium solution, and a second image in the 2135
cm-1 channel is acquired two minutes later. 6. The relative change in SRS
signal is calculated for each masked cell. (b) The distribution of relative
SRS intensity change in the 2135 cm-1 channel on a cell-averaged level due to
stimulation with 50 mM potassium. (c) The relative change in FFN102
fluorescence intensity due to stimulation with 50 mM potassium.

Using these flow cells, we first incubated PC12 cells with 50 $\mu$M d4-DA, as
previously. Prior to imaging the cells were again rinsed; however, imaging was
performed in clean culture medium as opposed to PBS. We next acquired acquired
an image at 2950 cm-1, to localize cell boundaries, and which served as a mask
to identify and separate cells computationally. Subsequently an image was
acquired in the 2135 cm-1 to quantify d4-DA concentration prior to
stimulation. Next the cells were stimulated to release the isotopologue by
exposure to a 50 mM potassium solution, before a second image was acquired at
2135 cm-1 to quantify d4-DA concentration post stimulation (figure 4a). We
next quantified the relative change in d4-DA intensity, i.e. the change in
d4-DA intensity normalized to the initial intensity, in response to potassium
stimulation. This analysis revealed a bimodal distribution for PC12 cells
incubated with d4-DA, with one population exhibiting a strong 64% release of
d4-DA in response to potassium stimulation, and another population exhibiting
a weaker 15% release of d4-DA (figure 4b). This bimodal distribution is
similar to the distribution observed when the experiment is repeated for PC12
cells incubated with FFN102. We next compared the relative change in intensity
due to potassium stimulation versus the initial pre-stimulation intensity in
d4-DA incubated cells (figure S1). We observed a general trend where the
greater the initial pre-stimulation intensity, the greater the relative change
in intensity in response to potassium stimulation, which was split roughly
into two populations. Interestingly, while we observe these two populations
for FFN102-incubated cells, a third, high initial intensity, but weakly-
responsive, population is also present in FFN102-incubated cell. This result
may potentially point to an additional, non-specific, internalization
mechanism of the FFN, that is not active for d4-DA uptake.

### d6-GABA uptake is also observable with SRS microscopy

####

Fig. 5 A deuterated GABA isotopologue and its spectrum.
(a) The H to D substitutions for the commercially available GABA isotopologue
used in this study. (b) The corresponding SRS spectrum for the deuterated GABA
istopologue in the cell-silent window.

We next assessed whether the strategy of deuteration and SRS microscopy could
be applied more broadly to different commercially-available deuterated
neurotransmitters. GABA plays a crucial role in the nervous system as the main
inhibitory neurotransmitter, and is consequently centrally implicated in many
neurochemical processes throughout the brain. Furthermore, while dopamine as a
neuromodulator exhibits volume transmission-based activity, GABA is thought
not to, representing a class of neurotransmitter putatively confined to the
neuronal synapse. While recent advances have been made for imaging GABA with
exogenous fluorophores in FRET-based biosensors (41, 42), and a genetically-
encoded GABA biosensor (43), comparatively few tools have been developed for
GABA imaging. To test the capability of SRS microscopy to image the uptake and
stimulated release of a deuterated GABA isotopologue we used dissociated fetal
hippocampal neurons from rat.

The isotopologue used, d6-GABA, was fully substituted for deuterium at all
available carbon bonded hydrogens (figure 5a). Fetal hippocampal neurons,
dissociated from day 19 Sprague Dawley rat embryos, were incubated to a final
concentration of 50 $\mu$M d6-GABA prior to imaging. d6-GABA has a different
spectral signature in the cell-silent window, so images were acquired instead
at the new d6-GABA maximum of 2149 cm-1 (figure 5b) for its visualization.
Images were again acquired in the 2950 cm-1 channel to provide contrast of the
cell body.

Hippocampal neurons displayed a strong signal at 2149 cm-1 when incubated with
the deuterated neurotransmitter (figure 6, middle row). In contrast to d4-DA,
signal was visible across the entire cell. This whole-cell d6-GABA
biodistribution may be explained by the higher cytoplasmic concentration of
GABA, compared to dopamine, which is maintained in the 5 mM range (44, 45).
d6-GABA signal was absent from images acquired from hippocampal neurons not
incubated with d6-GABA but otherwise cultured under identical conditions
(figure 6, top row). Across many neurons, we observe a significant separation
between the average signal intensity in d6-GABA incubated vs unincubated cells
(figure 7a; p $<$ 0.001, $\Delta$ $>$ 5.58). In contrast to the case of PC12
cells, however, a population of cells do not uptake d6-GABA, reflected in a
substantial overlap between the incubated and unincubated distributions. This
large distribution in d6-GABA uptake is likely due to the many neuronal
subtypes that exist in the hippocampus where glutamatergic neurons may be more
prevalent than GABAergic neurons (46). Evidence also suggests that glutamine
and GABA may coexist and may be co-released in certain neurons, which could
affect relative GABA uptake (46, 47). For non-GABAergic, or neurons containing
a mixture of multiple neurotransmitters, no or decreased uptake of d6-GABA is
expected.

Fig. 6 SRS imaging of a deuterated GABA isotopologue primary hippocampal
neurons.
(a) SRS images of primary hippocampal neurons without incubation with the
deuterated GABA isotopologue (top), with incubation with the GABA isotopologue
(middle), and with incubation with the GABA isotopologue, but two minutes post
stimulation to release neurotransmitter with 50 mM potassium (bottom). The
left column displays images at 2950 cm-1, the C-H stretching frequency range,
broadly corresponding to protein signal and allowing cell localization. The
right column displays images at 2149 cm-1, the C-D stretching frequency range,
corresponding to the signal originating from the deuterated GABA compound.

To examine whether the signal measured at 2149 cm-1 did arise from
internalized d6-GABA, we repeated the above dopamine-specific experiments by
incubating neurons with 50 $\mu$M d6-GABA for one hour. The neurons were then
stimulated with 50 mM potassium. Images were acquired in PBS two minutes after
stimulation, after first washing away the potassium and d6-GABA containing
medium. Neurons imaged after potassium stimulation showed little to no
observable signal at 2149 cm-1 (figure 6, bottom row). This 55% decrease in
d6-GABA signal following potassium stimulation was found to be significant
across many cells (figure 7a; p $<$ 0.001, $\Delta$ $>$ 0.65). When
considering that no significant difference was observed between d6-GABA
neurons imaged post potassium stimulation, and unincubated neurons, this
result suggests successful and near complete release of internalized d6-GABA.
Together these data suggest the ability to visualize uptake and release of a
GABA isotopologue within cultured primary neurons. To the best of our
knowledge, these represent the first optical micrographs of the
neurotransmitter GABA itself within a a neurobiological system.

Fig. 7 Cell-averaged SRS intensity distributions for cells incubated with
deuterated GABA isotopologue.
(a) The cell-averaged SRS intensity for cells that were not incubated with
deuterated GABA (left column), cells that were incubated with deuterated GABA
(middle column), and cells that were incubated with deuterated GABA,
stimulated to release neurotransmitter with 50 mM potassium, and imaged two
minutes later (right column). * p $<$ 0.001, $\Delta$ $>$ 5.58 (incubated vs
negative control), $\Delta$ $>$ 0.65 (incubated vs incubated post
stimulation). N(negative control) = 1840, N(Incubated, pre-stimulation) =
2110, N(Incubated, post-stimulation) = 2537.

## Discussion

####

Isotopes have long a history of use in neuroscience, which the present report
has leveraged and extended to the method of vibrational microscopy. Among the
first techniques developed to measure concentrations of neurotransmitters were
based on radioactive isotopes (48, 49). Positron emission tomography (PET)
continues to make frequent use of radiolabels to study the proteins of the
nervous system (50, 51). Powerful mass spectrometry imaging methods use
deuterated analogues as calibration standards for quantitative spatial mapping
of neurotransmitters (52–54). Herein, we demonstrate that by combining the use
of isotopologues with vibrational microscopy, we can to image the
biodistribution, and stimulated release, of deuterated dopamine and GABA
isotopologues in cultured cells and primary neurons. Our results suggest that
commercially-available deuterated neurotransmitters and neuromodulators could
be applied to a broad range of studies of neurotransmitter uptake and release
kinetics in neuronal cells, tissue, and potentially in living animals. Recent
developments in the field of vibrational microscopy also provide opportunities
to further enrich the information this technique can provide through
integration of orthogonal methodologies that could simultaneously measure
neurotransmitter release kinetics and electrical activity, or follow uptake
and release of multiple neurotransmitters simultaneously.

SRS microscopy, for instance, has been shown to be sensitive to membrane
potential (55), a property recently exploited to track neuron depolarization
in response to action potentials (56). In combination with isotopologue
imaging, this offers the potential for a single optical system to image both
neurochemical and electrical information simultaneously. The narrow linewidths
of vibrational transitions ( 10s cm-1), compared to electronic transitions (
100s cm-1), raise the additional possibility of simultaneous observation of
multiple neurotransmitters. Acquisition of hyperspectral image stacks, i.e.
multiple images of the same field of view at many wavenumbers, allows for the
spectral unmixing of the contribution of distinct neurotransmitter
isotopologues. A multiplexed imaging strategy could help answer questions
surrounding release and uptake of multiple neurotransmitters in single
neurons, as in the case of GABA and glutamate, two neurotransmitters that
canonically have opposing activity (46, 47).

The current study was limited to cells and primary neurons which could be
cultured and prepared for imaging under tight geometric constraints, a
consequence of the transmission-based SRS imaging modality. In order to extend
the use of neurotransmitter isotopologues to tissue samples, or even live
animals, an epi-SRS microscope could be used, with a geometry analogous to
more familiar fluorescence-based microscopes. The use of epi-SRS has
previously been demonstrated to work in brain tissue (57, 58). In fact, SRS
imaging is rather unique in that regard, as signal collection in the epi-
direction increases with increasing sample thickness. For samples more than 2
mm thick, it has been found to be preferable to image in this back scattering
geometry (59). Additionally, other vibrational imaging techniques are not
subject to the same constraints. For instance, CARS, another coherent Raman
technique, is frequently implemented in either a transmission or epi geometry,
and confocal spontaneous Raman and SERS microscopy are almost exclusively
implemented in epi geometries.

Similarly, exclusively intracellular populations of dopamine and GABA were
measured using our current imaging setup, since neither spontaneous Raman nor
SRS microscopy are likely to achieve the sensitivity required to observe the
lower concentrations of neurotransmitter present extracellularly. For
neuromodulators like dopamine, for which the study of volume transmission is
of interest, their extracellular concentrations can vary widely but are often
present at 20 nM concentrations extracellularly (60). Isotopologue imaging for
intracellular studies combined with existing probes to image volume
transmission would present a powerful apporach to track neurotransmitter
uptake, release, and volume transmission. As such, the current combination of
neurotransmitter isotopologues and SRS microscopy is best implemented to study
intracellular neurotransmitter populations and can serve as an orthogonal
measure, complementing other tools available to study neuromodulator volume
transmission. These include direct extracellular concentration sensors, such
as those based on carbon nanotubes or genetically engineered proteins, or
techniques like magnetic resonance spectroscopic imaging (MRSI), which is used
to study neurotransmitter concentrations under a variety of paradigms. While
powerful, MRSI is unable to distinguish intracellular and extracellular
contributions to the signal (61), due to lower spatiotemporal resolution, a
void which this technique may help to fill when questions of mechanism or
attribution arise. Additionally signal enhancement techniques may also be used
to further increase the sensitivity of vibrational microscopy. Recent work
suggests that surface enhanced SRS (SESRS) can be used to image at the single
molecule level under certain conditions, as demonstrated with bacterial cells
in (62). Similar observations have been made for surface enhanced CARS
(SECARS) (63–65). Furthermore, a recent study (66)describes a widefield
implementation of SECARS which could extend that level of sensitivity to the
ms temporal resolution of more conventional widefield microscopies. While the
use of plasmonic nanoparticles, or culturing on plasmonic nanostructure
surfaces, introduces an additional level of complexity, the SERS field has
frequently demonstrated its applicability to and utility in neuroimaging (67,
68), even in intraoperative/clinical settings (69, 70).

In this report we have presented a practical strategy to image
neurotransmitters under native biological conditions using vibrational
microscopy. This strategy generalizes to any class of neurotransmitter, many
of which are already commercially available in isotope-labeled form, opening
new avenues for fundamental investigations into the biology of
neurotransmitters. Notably, many commericially-available isotopologes, such as
GABA as demonstrated here, are neurotransmitters or neuromodulators that lack
cellular-scale imaging tools. While this work represents the first
demonstration of the use of isotopologues and vibrational microscopy in this
context, it makes use of a technique, i.e. isotope labeling, which is broadly
familiar to both the microscopy and neuroscience communities. Furthermore, a
growing demand for new tools to study the nervous system has led to increasing
adoption of vibrational imaging methods in neuroscience, and a focus on
adapting existing imaging techniques to tackle the unique problems that poses
in the imaging community. As such, this positions this strategy as a tool with
the potential for wide adoption to further the understanding of
neurotransmission.

## Materials and Methods

### PC12 cell culture

####

PC12 rat chromaffin cells (UC Berkeley cell culture facillity) were
continuously cultured in RPMI-1640 medium with L-glutamine, containing 10%
goat serum, 10% FBS, and 1% penicillin/streptomycin. For d4-DA imaging, cells
were passaged onto 25 mm diameter round coverslips. For FFN102 imaging, cells
were passaged into 24 well-plates. Upon achieving 50-70% confluency on the
coverslips, the media was exchanged to a serum deprived DMEM media, containing
0.5% FBS, 1% penicillin/streptomycin, and 100 ng/mL NGF-$\beta$. The
coverslips were imaged 5 days post media exchange. The cells were maintained
in an incubator at 37 °C and 5% CO2.

### Flow cell construction and PC12 cell culture on device

####

A consequence of the chosen imaging modality for d4-DA is a set of strict
geometric constraints for sample placement on the microscope. This is due to
the fact that the SRS signal is, generally, collected in a transmission
geometry, requiring the sample be placed between an objective and collecting
lens. SRS microscopy can also be subject to non-specific signals arising from
the optical Kerr effect in the form of cross-phase modulation (XPM). The most
straight-forward way to reduce or eliminate this effect is to use a collecting
lens with a numerical aperture equal to or larger than that of the objective
lens (71), necessitating a 1.4 NA oil immersion condenser lens, for use with
the 1.2 NA water immersion objective used in this study. Introduction of
potassium, for the stimulation of a sample that has just been imaged, is
complicated by the presence of these two immersion fluids, and the need to
keep the sample sealed off from both. In order to make measurements of d4-DA
signal intensity on the same PC12 cells both before and after potassium
stimulation we constructed a simple flow cell.

The flow cell was constructed using two 48 x 60 mm coverslips (Gold Seal,
Thermo scientific). RTV 615 PDMS was first mixed in a 10:1 ratio of monomer to
cross-linker. PEEK tubing (1/32”x0.010, IDEX Health and Science) was cut into
lengths of 2” to serve as inlets into the device, and to control the height of
the device. Two lengths of tubing were placed in the center of one short edge
of the coverslip, while another two were placed along the entire length of
each long edge. The PDMS mixture was then applied using a brush along the two
long edges and the short edge with tube inlets. The PDMS was applied in a
thickness equal to the height of the PEEK tubing extending up from the
coverslip. The second coverslip was then placed on top, enclosing the device.
The device was baked for 10 minutes at 70 °C, then flushed with ethanol and
further sterilized under UV-C light for 2 minutes. The final height of the
devices was 1.1 mm.

PC12 cells were introduced to the device by extracting 2 mL of trypsanized
cells from the passaging culture using a syringe, sterilized as above using
ethanol and UV-C light, and then affixing the syringe to one of the available
inlets. Cells were allowed to grow for 2 days before the medium was exchanged
for the serum deprived medium. Cells were grown for a further 5 days before
imaging, the same as those on coverslips. The devices were maintained within
separate sterile petri dishes in the same incubator until imaging.

### Primary hippocampal neuron dissociation and culture

####

Neurons were dissociated from extracted hippocampi of day 19 embryos of
Sprague Dawley rats, as previously described (72). Briefly, cells are
dissociated for 15 minutes from excised hippocampi using a trypsinated buffer.
The neurons are then plated onto 25 mm coverslips and incubated at 37 °C in
MEM medium with 5% FBS, 2% B-27, 2% 1M dextrose and 1% GlutaMax. One day after
dissociation, half the medium is exchanged for neurobasal medium with 2% B-27
supplement and 1% GlutaMax, to prevent proliferation of glial cells.
Coverslips are then imaged using SRS microscopy two weeks post dissociation.

### SRS microscope optical setup

####

The optical setup has been described previously (73)and is shown in figure S2.
Briefly, the synchronized dual output from a commercial femtosecond laser/OPO
system is used for the pump and Stokes beams. The Stokes beam was fixed at
1040 nm, while the pump was tunable. For imaging at 2950 cm-1, 2149 cm-1, and
2135 cm-1, the pump was tuned to 796 nm, 850 nm, and 851 nm respectively. The
Stokes beam was intensity modulated at 10.28 MHz using a quarter waveplate
(Thorlabs), a resonant electro-optic modulator (Thorlabs), driven by a
function generator (Hewlett Packard) and power amplifier (Minicircuits), and a
polarizer (Thorlabs). The pump and stokes were combined colinearly using a
1000 nm short pass dichroic mirror. Temporal coincidence of the pulses was
controlled using an optical delay line (Newport). A commercial scanning
microscope (FV1200, Olympus) was used to raster scan the beams across the
sample. A 60x 1.2 NA water immersion objective (Olympus) was used for
excitation, and a 1.4 NA oil immersion condenser (Thorlabs) was used to
collect the transmitted light. The Stokes beam was filtered by a 1000 nm
shortpass filter, before the pump was detected on a photodiode (Hamatasu)
reverse biased at 61.425. The stimulated Raman loss signal was extracted from
the photodiode signal using a lock-in amplifier (Zurich Instruments).

### Deuterated neurotransmitter imaging

####

PC12 cells or primary neurons cultured on coverslips were incubated for one
hour at 37 °C with a final media concentration of 50 $\mu$M d4-DA or d6-GABA,
respectively. After one hour the coverslips were washed three times with DPBS,
before being placed on 25 mm diameter, low-profile 600 $\mu$m culture dish
(Grace coverwells), filled with DPBS. The edges of the culture dish contain an
adherent which fixed the coverslips in place. PC12 cells were then imaged at
2950 cm-1 and 2135 cm-1 using the SRS microscope. The pump and Stokes power
were both set to 20 mW. Images were acquired at a size of 512 x 512 pixels,
using a 10 $\mu$s pixel dwell time and a lock-in constant of 3 $\mu$s. Neurons
were imaged using a pump and Stokes power of 15 mW, and at 2950 cm-1 and 2149
cm-1, but otherwise identical imaging conditions. For the negative control
samples for both PC12 cells and primary neurons, the media was exchanged for
fresh media not containing the deuterated compound. The cells were then left
to rest for one hour before being washed three times with DPBS and prepared
identically to the cells incubated with the deuterated isotopologues, and
imaged using the same conditions. For the post stimulation condition PC12
cells and primary neurons were incubated with 50 $\mu$M d4-DA or d6-GABA,
respectively. After a one hour incubation, the media was exchanged for media
containing 50 mM potassium. After two minutes, the coverslips were washed
twice with DPBS, and prepared and imaged analogously to the previously
described conditions.

PC12 cells which were cultured on the flow cell devices were incubated with 50
$\mu$M d4-DA by first emptying the device of media through the open side and
then introducing new media with the deuterated compound. After one hour the
device was then washed twice with DPBS, before being filled with fresh media.
In contrast to the PC12 cells cultured on coverslips, those cultured on the
device were imaged in media not DPBS, in order to enable stimulated release of
the d4-DA while on the microscope. The devices were then imaged with the same
settings as those used for the cells cultured on coverslips. One image was
taken at 2950 cm-1, then a pre stimulation image at 2135 cm-1, an image at
2135 cm-1 following influx of media not containing supplemental potassium, and
finally a post-stimulation image 2 minutes after an influx of 50 mM potassium.
This set of images was also repeated for PC12 cells cultured on devices which
were not incubated with d4-DA.

### Two-photon excited fluorescence imaging of FFN102

####

For imaging of FFN fluorescence in PC12 cells, the cells were first incubated
to a final media concentration of 10 $\mu$M FFN102 for one hour at 37 °C.
After one hour, samples were washed three times with DPBS. For imaging, the
pump wavelength on the SRS microscope was tuned to 800 nm for two-photon
excitation of FFN102, with the power set to 20 mW. Images were acquired in the
epi-direction at a size of 512 x 512 pixels, using a 10 $\mu$s pixel dwell
time and non-descan PMT available on the Olympus microscope. For post
potassium stimulation measurements, FFN102 was either imaged in cells 2
minutes after exposure to potassium in adjacent wells on the well plate that
had not been previously imaged, or for measurements on the same cells, 2
minutes after exposure to potassium in a well that had been previously imaged.

### Image processing

####

All statistics reported are calculated over the intensity distribution of the
average signal in individual cells. Statistical significance was tested using
a Welch’s T-test assuming unequal variance. Effect sizes are displayed as
Glass’ $\Delta$. For calculations involving SRS images, all images in both the
2950 cm-1 and 2135 cm-1 or 2149 cm-1 are pseudo-flatfield corrected using a
replica of the respective image, convolved with Gaussian kernel of radius 200
pixels, as the flatfield. The 2950 cm-1 channel images are then used for
thresholding and segmentation. First a Gaussian blur of radius 5 pxiels is
applied, followed by a contrast limited adaptive histogram equilization with
of radius 15. Otsu’s method is then applied to produce a binary thresholded
image. A hierarchical contour tracing algorithm is then used to determine the
outlines of the thresholded image. Internal contours, arising from organelles
or low-intensity regions within a cell, are discarded, leaving the highest
level outlines of the cell. Any contours with a total length of less than 100
pixels are also discarded. Each of the remaining contours is used to determine
a mask over which the signal inside the contour is averaged in the
corresponding 2135 cm-1 or 2149 cm-1 channel. Beyond flatfield corrections the
channels corresponding to deuterated compounds undergo no additional
processing.

Fluorescence images of PC12 cells incubated with FFN102 are processed
similarly; however, as the signal is strong enough to see cell boundaries in
the fluorescence channel, no additional channel is needed to serve as the
mask. Instead, after flatfield correction, a second copy of each image is used
for masking, and undergoes the same processing as the SRS images acquired in
the 2950 cm-1. The discovered contours are then used as masks on the
unprocessed copy.

## Acknowledgements

####

We would like to acknowledge Professor Evan Miller and his students Pavel
Klier and Deshka Lynn Niell for providing primary hippocampal cells, and
advice on maintenance of their culture conditions.

## Funding

####

We acknowledge support of a Burroughs Wellcome Fund Career Award at the
Scientific Interface (CASI) (to M.P.L.), a Dreyfus foundation award (to
M.P.L.), the Philomathia foundation (to M.P.L.), an NIH MIRA award (to
M.P.L.), an NIH R03 award (to Mp.P.L), an NSF CAREER award (to M.P.L), an NSF
CBET award (to M.P.L.), an NSF CGEM award (to M.P.L.), a CZI imaging award (to
M.P.L), a Sloan Foundation Award (to M.P.L.), a USDA BBT EAGER award (to
M.P.L), a Moore Foundation Award (to M.P.L.), an NSF CAREER Award (to M.P.L),
a DOE office of Science grant with award number DE-SC0020366 (to M.P.L.), an
NSF award with award number 1845623 (to A.S.). M.P.L. is a Chan Zuckerberg
Biohub investigator, a Hellen Wills Neuroscience Institute Investigator, and
an IGI Investigator. A.S. is a Chan Zuckerberg Biohub investigator, and a Pew
Biomedical scholar. Work at the Molecular Foundry was supported by the Office
of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy
under Contract No. DE-AC02-05CH11231. Co-corresponding authors contributed
equally and may modify the ordering of listed corresponding authors when
disseminating this work.

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## Supplementary Material

####

Fig. S1 Relative intensity change of d4-DA and FFN102 due to potassium
stimulation versus initial intensity.
(a) The two-dimensional histogram displaying the relative change in SRS
intensity of d4-DA at 2135 cm-1 due to potassium stimulation, versus the
initial SRS intensity at 2135 cm-1 per cell. The one-dimensional histograms
for the relative change distribution, and initial intensity distribution, are
displayed on the right and top axes, respectively. The distribution on the
right axis corresponds to the violin plot depicted in figure 4b. (b) The
corresponding two-dimensional histogram for the relative change in FFN102
fluorescence intensity due to potassium stimulation versus the initial
fluorescence intensity per cell. The one-dimensional histogram on the right
axis corresponds to the violin plot in figure 4c. There is an additional third
population of cells among those incubated with FFN102, not present in the PC12
cells incubated with d4-DA, which display an high initial fluorescence
intensity but little to no change in intensity in response to potassium
stimulation.

Fig. S2 Schematic of optical setup.
Solid black lines represent mirrors, black lines with ridges are reflective
gratings, with the translucent variant being a transmission grating. Lenses
are doublets depicted as unlabeled white components outlined in black. EOM:
Electro-optic modulator; $\lambda$/2: half-wave plate; Pol: polarizer;
$\lambda$/4: quarter-wave plate; LP: long-pass filter; DM: dichroic mirror;
SP: short-pass filter.


# Electromagnetic Energy, Momentum, and Angular Momentum in an Inhomogeneous
Linear Dielectric

Michael E. Crenshaw and Thomas B. Bahder  US Army Aviation and Missile
Research, Development, and Engineering Center, Redstone Arsenal, AL 35898, USA

###### Abstract

In a previous work, Optics Communications 284 (2011) 2460–2465, we considered
a dielectric medium with an anti-reflection coating and a spatially uniform
index of refraction illuminated at normal incidence by a quasimonochromatic
field. Using the continuity equations for the electromagnetic energy density
and the Gordon momentum density, we constructed a traceless, symmetric
energy–momentum tensor for the closed system. In this work, we relax the
condition of a uniform index of refraction and consider a dielectric medium
with a spatially varying index of refraction that is independent of time,
which essentially represents a mechanically rigid dielectric medium due to
external constraints. Using continuity equations for energy density and for
Gordon momentum density, we construct a symmetric energy–momentum matrix,
whose four-divergence is equal to a generalized Helmholtz force density four-
vector. Assuming that the energy-momentum matrix has tensor transformation
properties under a symmetry group of space-time coordinate transformations, we
derive the global conservation laws for the total energy, momentum, and
angular momentum.

##  I Introduction

Starting with the hydrodynamic continuity equation, Umov BIUmov  obtained an
expression for energy continuity in a continuous spatial flow of an
electromagnetic field in 1874 BIHeaviside ; BIPoynting ; BIMicrophysics . A
decade later, Poynting BIPoynting  derived a similar energy continuity
equation as a general theorem of the macroscopic Maxwell equations. Poynting’s
theorem is generally preferred because it can be derived directly from the
macroscopic Maxwell equations

$\nabla\times{\bf B}-\frac{n^{2}}{c}\frac{\partial{\bf E}}{\partial t}=0$ (1)
$\nabla\times{\bf E}+\frac{1}{c}\frac{\partial{\bf B}}{\partial t}=0$ (2)
$\nabla\cdot({n^{2}\bf{E}})=0$ (3) $\nabla\cdot{\bf{B}}=0$ (4)

of classical continuum electrodynamics. Here, the Maxwell equations are
written in Heaviside–Lorentz units for a nonmagnetic linear medium in the
absence of free charges and currents. We have assumed that the index of
refraction, $n=n({\bf r})$, depends on position, occupies a finite region of
3-dimensional space, and is independent of time. Poynting’s theorem can be
derived by subtracting the scalar product of the Maxwell–Ampère law, Eq. (1),
with ${\bf E}$ from the scalar product of Faraday’s law, Eq. (2), with ${\bf
B}$ to obtain

$\frac{n^{2}}{c}\frac{\partial{\bf E}}{\partial t}\cdot{\bf
E}+\frac{1}{c}\frac{\partial{\bf B}}{\partial t}\cdot{\bf B}$
$+(\nabla\times{\bf E})\cdot{\bf B}-(\nabla\times{\bf B})\cdot{\bf E}=0.$ (5)

Upon application of a vector identity and the definition of the energy
density,

$\rho_{e}=\frac{1}{2}(n^{2}{\bf E}^{2}+{\bf B}^{2}),$ (6)

the preceding equation becomes Poynting’s theorem

$\frac{\partial\rho_{e}}{\partial t}+\nabla\cdot c({\bf E}\times{\bf B})=0$
(7)

and defines Poynting’s energy flux vector ${\bf S}_{P}=c({\bf E}\times{\bf
B})$.

In a previous work BICB , we considered a dielectric medium with an index of
refraction that was time-independent, spatially uniform, and covered with a
thin gradient-index antireflection coating. For this case, we showed that the
energy

$U=\int_{\sigma}\frac{1}{2}\left(n^{2}{\bf E}^{2}+{\bf B}^{2}\right)dv$ (8)

and the Gordon BIGord  momentum

${\bf G}_{G}=\int_{\sigma}\frac{1}{c}(n{\bf E}\times{\bf B})dv$ (9)

are the conserved electromagnetic quantities with integration performed over
all space $\sigma$. We constructed the corresponding traceless, symmetric
energy–momentum tensor, whose four–divergence provided the continuity
equations for the energy and momentum densities of the closed system. This
result incorporated a condition of $\nabla n/n\ll 1/\lambda$ corresponding to
unimpeded flow in the absence of external forces.

In the current work, we consider the case of an inhomogeneous dielectric
medium in which the condition $\nabla n/n\ll 1/\lambda$ no longer holds. The
assumption that the index of refraction is independent of time necessarily
implies that the dielectric medium is mechanically rigid RigidFootnote  and
that no momentum is transferred to the dielectric medium. In other words, at
each spatial point, there is effectively an external field that acts as a
constraint that holds the dielectric in place. Therefore, the system that we
are considering is not a closed system and the continuity equations and global
conservation equations will reflect this feature. The external field that acts
as a constraint is not a field that we impose, instead it arises as a result
of the way in which we arrange terms in the continuity equations. We construct
the energy and momentum continuity equations and find that the spatial
gradient of the refractive index appears in the continuity equations in a
generalized Helmholtz force density. This generalized Helmholtz force density
is a spatial and time-varying field that provides the constraint for the
dielectric medium to be mechanically rigid. We then write the continuity
equation as a four divergence of a symmetric energy–momentum matrix. Assuming
that the energy-momentum matrix has tensor transformation properties under a
symmetry group of space-time coordinate transformations, we apply the four-
dimensional divergence theorem to derive global conservation laws for the
total energy, momentum, and angular momentum.

##  II Energy and Momentum Continuity Equations

The energy–momentum tensor is a concise way to represent the local continuity
of energy and momentum of a field. While simple in concept, the form of the
energy–momentum tensor for the electromagnetic field in a dielectric has been
at the center of the century-long Abraham-Minkowski controversy BIPfeifer .
The tensor that was proposed by Minkowski BIMin  in 1908 is not diagonally
symmetric, a fact that is adverse to conservation of angular momentum BILL .
Abraham BIAbr  subsequently proposed a symmetric tensor at the expense of a
phenomenological force. The disagreeable properties of the Minkowski and
Abraham energy-momentum tensors are manifestations of underlying conservation
issues: neither the Minkowski momentum nor the Abraham momentum is conserved.
In this section, we derive the energy and momentum continuity equations for an
inhomogeneous dielectric and construct the corresponding tensor continuity
equation and energy–momentum tensor. In the following section, we will obtain
the globally conserved quantities—the total energy, the total momentum, and
the total angular momentum system, from the energy momentum tensor.

Starting from the Maxwell Eqs.(1)–(4), we use the continuity of the
electromagnetic energy density and the Gordon momentum to obtain a symmetric
stress-energy tensor. We begin with the temporal derivatives of the energy
density and the Gordon momentum density,

$\frac{n}{c}\frac{\partial}{\partial t}\frac{1}{2}\left(n^{2}{\bf E}^{2}+{\bf
B}^{2}\right)=\frac{n}{c}\frac{\partial(n{\bf E})}{\partial t}\cdot n{\bf
E}+\frac{n}{c}\frac{\partial{\bf B}}{\partial t}\cdot{\bf B}$ (10)
$\frac{n}{c}\frac{\partial}{\partial t}(n{\bf E}\times{\bf
B})=\frac{n}{c}\frac{\partial(n{\bf E})}{\partial t}\times{\bf B}+n{\bf
E}\times\frac{n}{c}\frac{\partial{\bf B}}{\partial t}.$ (11)

The Gordon momentum in Eq. (11) has been scaled by $c$ so that the equations
are in the same units. Next, we apply the vector identity

$\nabla\times(\psi{\bf a})=\nabla\psi\times{\bf a}+\psi\nabla\times{\bf a}$
(12)

to Faraday’s law and write the macroscopic Maxwell equations, Eqs. (1) and
(2), as

$\frac{n}{c}\frac{\partial(n{\bf E})}{\partial t}=\nabla\times{\bf B}$ (13)
$\frac{n}{c}\frac{\partial{\bf B}}{\partial t}=-\nabla\times(n{\bf
E})+\frac{\nabla n}{n}\times n{\bf E}.$ (14)

The variant form of Maxwell’s equations, Eqs. (13) and (14), are
mathematically equivalent to the original versions, Eqs. (1) and (2),
respectively. Substituting the Maxwell equations, Eqs. (13) and (14), into
Eqs. (10) and (11), we produce the energy continuity equation

$\frac{n}{c}\frac{\partial\rho_{e}}{\partial t}+\nabla\cdot(n{\bf E}\times{\bf
B})=\frac{\nabla n}{n}\cdot(n{\bf E}\times{\bf B})$ (15)

and the (Gordon) momentum continuity equation

$\frac{n}{c}\frac{\partial}{\partial t}(n{\bf E}\times{\bf B})+\nabla\cdot{\bf
W}+n{\bf E}\left(\nabla\cdot n{\bf E}\right)=$ $\left(n{\bf
E}\times\frac{\nabla n}{n}\right)\times n{\bf E}.$ (16)

Here, we have used Eq. (3) and the definition of the Maxwell stress tensor
BIJackson ; BILL

$W_{ij}=\left(-nE_{i}nE_{j}-B_{i}B_{j}+\frac{1}{2}\left(n{\bf E}\cdot n{\bf
E}+{\bf B}\cdot{\bf B}\right)\delta_{ij}\right).$ (17)

We can write the energy continuity equation, Eq. (15), and momentum continuity
equation, Eq. (16), as the matrix differential equation

$\bar{\partial}_{\beta}T^{\alpha\beta}=f^{\alpha}$ (18)

with summation over repeated indices. The quantities that appear in Eq. (18)
are a four-divergence operator

$\bar{\partial}_{\alpha}=\left(\frac{n}{c}\frac{\partial}{\partial
t},\partial_{x},\partial_{y},\partial_{z}\right),$ (19)

the array

$T^{\alpha\beta}=\left[\begin{matrix}\rho_{e}&c{g}_{{\rm G}_{1}}&c{g}_{{\rm
G}_{2}}&c{g}_{{\rm G}_{3}}\cr c{g}_{{\rm G}_{1}}&W_{11}&W_{12}&W_{13}\cr
c{g}_{{\rm G}_{2}}&W_{21}&W_{22}&W_{23}\cr c{g}_{{\rm
G}_{3}}&W_{31}&W_{32}&W_{33}\cr\end{matrix}\right],$ (20)

and a generalized force density four-vector

$f^{\alpha}=\left(\nabla n\cdot({\bf E}\times{\bf B}),-{\bf f}_{H}\right),$
(21)

where

${\bf f}_{H}=-nE^{2}\nabla n+2n{\bf E}\left({\bf E}\times\nabla
n\right)+n^{2}{\bf E}(\nabla\cdot{\bf E}).$ (22)

Using Eq. (3), the last two terms on the right side of Eq. (22) cancel, so
that

${\bf f}_{H}=-n{E^{2}}\,\nabla{n}$ (23)

in the absence of free charges.

The array in Eq. (20) appears to have the properties of an energy–momentum
tensor. By construction, the operator defined in Eq. (19) applied to the rows
of the array in Eq. (20) generates continuity equations for the
electromagnetic energy and the momentum. The same operation applied to the
columns

$\bar{\partial}_{\alpha}T^{\alpha\beta}=f^{\beta},$ (24)

generates the same continuity equations by symmetry

$T^{\alpha\beta}=T^{\beta\alpha}.$ (25)

The array has a vanishing trace

$T^{\alpha}_{\alpha}=0$ (26)

corresponding to massless particles BIJackson ; BILL .

The new feature of this result is the appearance of the four-vector
$f^{\alpha}$ in the continuity equation (18). The appearance of the four-
vector $f^{\alpha}$ in the divergence of the stress energy tensor in Eq. (18)
is a result of the fact that the system is not closed. When the gradient of
the index of refraction is non-zero, there is a back-action on the field
altering its spatial properties, so the field does not experience “unimpeded
flow”. It is possible to define an effective stress-energy tensor that takes
into account the back-action on the field by introducing

$\partial_{\alpha}t^{\alpha\beta}=f^{\beta}.$ (27)

The continuity of energy and momentum can then be expressed by

$\partial_{\alpha}(T^{\alpha\beta}+t^{\alpha\beta})=0$ (28)

where the tensor $t^{\alpha\beta}$ contains the influence of the inhomogeneity
of the material and is zero for homogeneous dielectrics.

##  III Global Conservation Equations

If we assume that $T^{\alpha\beta}$ transforms as a tensor under some symmetry
group of space-time coordinate transformations, then we can apply the four-
dimensional divergence theorem to obtain global conservation equations.
Integrating Eq. (18) over a four-volume, $d\Omega=cd{\bar{t}}\;d^{3}x\equiv
cd{\bar{t}}\;dv$ (where ${\bar{t}}=t/n$), between hypersurfaces BILL  of
constant time at $\bar{t}_{1}=t_{1}/n$ and $\bar{t}_{2}=t_{2}/n$, we have

$\int_{\bar{t}_{1}}^{\bar{t}_{2}}\bar{\partial}_{\beta}T^{\alpha
0}\;cd\bar{t}\;dv=\int_{\bar{t}_{1}}^{\bar{t}_{2}}f^{\alpha}\;cd\bar{t}\;dv,$
(29)

where the integrals over $dv=d^{3}x$ are 3-dimensional volume integrals over
the volume containing the field. Applying the four-divergence theorem results
in

$\int\left(T^{\alpha 0}(t_{2})-T^{\alpha
0}(t_{1})\right)\,dv=c\int\;\frac{1}{n}f^{\alpha}dt\;dv.$ (30)

When $\alpha=1,2,3$, Eq. (30) reduces to

${\bf G}(t_{2})-{\bf G}(t_{1})=\int\,E^{2}\,\nabla n\,dt\;dv$ (31)

where ${\bf G}(t)$ is the Gordon momentum at time $t$, and the time
integration is between $t=t_{1}$ and $t=t_{2}$. Equation (31) shows that the
Gordon momentum of the field is not constant, i.e., there is a source or sink
of momentum provided by the external constraint field ${\bf f}_{H}$. For the
case $\alpha=0$, Eq. (30) gives

$\int\left(\rho_{e}(t_{2})-\rho_{e}(t_{1})\right)dv=\int\,\frac{\nabla
n}{n}\cdot({\bf E}\times{\bf B})\,dt\;dv$ (32)

where $\rho_{e}$ is given by Eq. (6). The left side of Eq. (32) is the
difference in total energy at two different times, $U(t_{2})-U(t_{1})$. We see
that a non-zero spatial gradient in the index means that the field can gain or
loose energy. The results given in Eqs. (31) and (32) assume that the matrix
$T^{\alpha\beta}$ in Eq. (18) transforms as a tensor under some symmetry group
of space-time coordinate transformations. Note that when $\nabla n=0$, Eqs.
(31) and (32) show that the Gordon momentum ${\bf G}(t)$ and the energy $U(t)$
are constants.

In recent years, there is intense interest in angular momentum carried by the
electromagnetic field, see the recent review and references cited therein
TwistedPhotons ; BIJHL . However, the angular momentum carried by the
electromagnetic field in a dielectric environment is no less unsettled than
the linear momentum case BIang . We can define the four-tensor of angular
momentum density in terms of our energy–momentum tensor as BILL

$m^{\alpha\beta\gamma}=\frac{1}{c}\left(x^{\alpha}\,T^{\gamma\beta}-x^{\beta}\,T^{\gamma\alpha}\right)=-m^{\beta\alpha\gamma}.$
(33)

Continuity of angular momentum is given by

$\bar{\partial}_{\gamma}m^{\alpha\beta\gamma}=\frac{1}{c}\left(x^{\alpha}\,f^{\beta}-x^{\beta}\,f^{\alpha}\right).$
(34)

The divergence of $m^{\alpha\beta\gamma}$ is not zero, thereby indicating that
there is a source or sink of angular momentum density, due to the gradient of
the index of refraction.

Once again, we assume that there exists a symmetry group of space-time
coordinate transformations, so that $m^{\alpha\beta\gamma}$ is a tensor. As
above, we can then use the four-divergence theorem to obtain

$\int\left(m^{\alpha\beta 0}(t_{2})-m^{\alpha\beta
0}(t_{1})\right)\,dv=\int\frac{1}{n}\left(x^{\alpha}f^{\beta}-x^{\beta}f^{\alpha}\right)dt\,dv$
(35)

where the time integral is between the two times $t_{1}$ and $t_{2}$ and where
the volume integral $dv$ is over the portion of three-dimensional space
containing the field and includes the region where $n(\bf r)>1$. When $\alpha$
and $\beta$ take values $i,j=1,2,3$, we have

$m^{ij0}(t)=x^{i}f^{j}-x^{j}f^{i}$ (36)

where $f^{i}$ are the components defined in Eq. (23). Equation (35) gives the
change in total angular momentum of the field, $\Delta{\bf M}$, between time
$t_{1}$ and $t_{2}$, and can be written as

$\Delta{\bf M}=\int\left({\bf m}(t_{2})-{\bf m}(t_{1})\right)\,dv=\int
E^{2}\left({\bf r}\times\nabla n\right)dt\,dv$ (37)

where the ${\bf m}(t)$ is the angular momentum density at position ${\bf r}$
at time $t$, defined by

${\bf m}(t)={\bf r}\times{\bf g_{G}}(t)$ (38)

and is defined in terms of the Gordon momentum density

${\bf g_{G}}(t)=n{\bf E}\times{\bf B}/c.$ (39)

Equation (37) shows that when ${\bf r}\times\nabla n$ is non-zero, then the
total angular momentum of the field (as expressed through the angular momentum
density) can change. Once again, we remind the reader that we have assumed the
index of refraction as isotropic, constant in time, but varying in position.
Equation (37) essentially shows that a particular spatial distribution of
${\bf r}\times\nabla n$ can lead to a back-action on the field that can alter
the field angular momentum. Indeed this has been exploited in a number of
recent experiments TwistedPhotons .

##  IV Summary

In a previous work, we considered a dielectric medium with an index of
refraction that was time-independent and spatially uniform, and the dielectric
was covered with a thin gradient-index antireflection coating BICB . We found
that the total energy and the Gordon momentum were conserved quantities
(constant in time). In this work we have relaxed these conditions to include a
dielectric medium that has a spatially varying index of refraction that is
constant in time. The fact that the index of refraction is constant in time
essentially means that the dielectric medium is mechanically rigid and subject
to an external constraint. This constraint means that we are not dealing with
a closed system. Using the continuity equations for energy density and
momentum density, we derived a symmetric energy-momentum tensor for the
electromagnetic field. Due to the fact that the system is not closed, the
divergence of the stress-energy tensor is not zero, but equal to a generalized
Helmholtz force vector that represents the constraint. Similarly, the
divergence of the angular momentum density is not zero, due to the external
constraint of a time independent index of refraction. We found that the total
energy, Gordon momentum, and total angular momentum are not conserved because
of the external constraint on time independence of the index of refraction.
However, the time dependence of the total energy, Gordon momentum, and total
angular momentum is related to the constraint, which is proportional to the
gradient of the index of refraction.

Finally, we note that time $t$ has been renormalized to ${\bar{t}}=t/n({\bf
r})$ in the intermediate steps used to obtain the symmetric energy-momentum
tensor and its continuity relation given by its four-divergence in Eq. (18).
However, the final conservation laws of total energy, total momentum and total
angular momentum, given by Eqs. (32), (31), and (37), respectively, are
expressed in terms of the unrenormalized time $t$.

## References

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