[table]capposition=top

# MCL-1 promiscuity and the structural resilience of its binding partners

Philipp J. Heckmeier*, Jeannette Ruf, Brankica G. Janković, and Peter Hamm
Department of Chemistry, University of Zurich, Zurich, Switzerland
<EMAIL_ADDRESS>
###### Abstract

MCL-1 and its natural inhibitors, the BH3-only proteins PUMA, BIM, and NOXA
regulate apoptosis by interacting promiscuously within an entangled binding
network. Little is known about the transient processes and dynamic
conformational fluctuations that are the basis for the formation and stability
of the MCL-1/BH3-only complex. In this study, we designed photoswitchable
versions of MCL-1/PUMA and MCL-1/NOXA, and investigated the protein response
after an ultrafast photo-perturbation with transient infrared spectroscopy. We
observed partial $\alpha$-helical unfolding in all cases, albeit on strongly
varying timescales (1.6 ns for PUMA, 9.7 ns for the previously studied BIM,
and 85 ns for NOXA). These differences are interpreted as a BH3-only-specific
“structural resilience” to defy the perturbation while remaining in MCL-1’s
binding pocket. Thus, the presented insights could help to better understand
the differences between PUMA, BIM, and NOXA, the promiscuity of MCL-1 in
general, and the role of the proteins in the apoptotic network.

Protein-protein interactions are the fundamental driving force for a majority
of cellular processes1, 2. Understanding the molecular mechanisms behind this
protein-protein interplay is of highest scientific interest3. For numerous
protein complexes, it is not clear how they form, how small conformation
fluctuations contribute to the complex function and stability, and whether
there are intertwined intermediate states of altered conformation 4, 5.
Illuminating the nuances of these dynamic processes is particularly essential
for complexes formed by intrinsically disordered proteins6. These proteins do
not assume an ordered structure in their isolated form but only when they are
bound to their complex partner. For intrinsically disordered proteins, the
process of complex formation can be explained with models7, 8 such as the
induced fit model9, the conformational selection model4, or a hybrid version
of both theories10. Intrinsic disorder plays a significant role in promiscuous
protein networks, as it enables the complex formation with numerous binding
partners11, 12. Being on “the edge of chaos”13 ensures structural and
functional flexibility and provides an ideal basis for diverse protein-protein
interactions, for instance in a network of activator, inhibitor, and effector
proteins.

The BCL-2 protein family is a paramount example for an intricate protein
network, which is driven by promiscuous interactions of several intrinsically
disordered protein domains. In this protein family, categorized in subfamilies
based on the type and the number of their BCL-2 homology (BH) domains, the
disordered binding domains of so-called BH3-only proteins – they solely have a
BH3 domain – form complexes with numerous other BCL-2 proteins, thereby
controlling apoptosis in a finely-balanced manner14, 15, 16, 17. The BH3-only
proteins, abundant when cells suffer cytotoxic stress18, 19, 20, either
directly activate the pro-apoptotic effector proteins (BAK and BAX) or inhibit
the anti-apoptotic effector-inhibitors such as the Myeloid Cell Leukemia 1
protein (MCL-1), one of the key players in apoptosis regulation15 (Fig. 1).
MCL-1 is overexpressed in various tumor variants, which makes it of high
interest in therapeutic application 21, 18, 22, 23, 24, 25. At its canonical
binding site, MCL-1 promiscuously binds the $\alpha$-helical binding domain of
the BH3-only proteins PUMA, BIM, and NOXA26, 27, most likely by induced fit28,
29, 30, 31, 32. PUMA and BIM inhibit MCL-1 by occupying its binding pocket
with affinities in the sub-nanomolar range, but additionally bind other anti-
apoptotic factors and the pro-apoptotic effector proteins BAK and BAX17, 33,
26. In contrast, NOXA binds MCL-1 specifically and with a weaker affinity,
however effects the degradation of the whole complex34.

Figure 1: The protein MCL-1 promiscuously binds BH3-only peptides NOXA, PUMA,
and BIM, and has a central position in an entangled binding network,
regulating apoptotic effector proteins BAK and BAX.

In their intriguing review15, Kale, Osterlund, and Andrews describe this
interplay of different pro- and anti-apoptotic factors very figuratively as a
“dance” of various partners – an interesting and vivid depiction of protein
promiscuity. In this “apoptotic dance”, the protein affinity and stability
determine the complex formation and whether cell death will be initiated or
not.

To date, little is known about the protein dynamics behind the MCL-1/BH3-only
complex stability and formation, let alone intermediate states of BH3-only
folding and unfolding at MCL-1’s binding groove35. Investigating subtle,
dynamical rearrangements inside proteins like the MCL-1/BH3-only complex is
connected to a fundamental challenge: the ability to resolve structural
flexibility and small conformational fluctuations in a reasonable time
frame36, 37. In the past, the dynamics of protein ensembles – intrinsically
disordered or folded – have been experimentally studied via single-molecule
FRET spectroscopy 38, 39, NMR spectroscopy 40, 41, as well as transient
infrared (IR) spectroscopy 42, 43. IR spectroscopy allows the differentiation
of very small conformational differences44 and, in its transient form, the
sensitive detection of non-equilibrium processes45, making it a highly
suitable method to study MCL-1/BH3-only complexes.

To selectively trigger a dynamical process inside a protein for transient IR
spectroscopy, a fast and precisely induced perturbation of the proteins is
required, ideally initiated by short light pulses. In this regard, a plethora
of photoreceptor proteins, i.e., light-sensitive or fluorescent, have been
investigated in the past 46, 47, 48, 49, 50, 51, 52. Beyond proteins that show
natural photo-activity, linking azobenzene photoswitches covalently to
selected protein domains – most prominently $\alpha$-helical structures –
defines a potent strategy to introduce photo-sensitivity in molecules, which
are otherwise “blind” to light. The light-induced isomerization of the cross-
linked azobenzene moiety leads to a fast perturbation of the secondary
structure53, and in turn to a slower protein response, both of which can be
detected via transient IR spectroscopy 54, 55, 56. With this technique, the
allosteric signal propagation in PDZ domains and the unbinding in the RNase S
complex were investigated 42, 57, 43. Similarly, we recently revealed the
signal propagation inside the MCL-1/BIM complex.32

Figure 2: Structures of MCL-1 (purple) binding BIM (orange), PUMA (red), or
NOXA (brown), adapted from PDB entries 2NL926, 2ROC27 and 2ROD27,
respectively. The peptides were covalently crosslinked with an azobenzene
photoswitch (blue) at introduced Cys residues (cyan). Illumination with 450 nm
light promotes the isomerization of the photoswitch from the cis\- to the
trans-state, exemplified here for NOXA. The opposite direction can be induced
with 375 nm light. The $\alpha$-helical BH3-only peptides are destabilized in
the trans-state. Aligned peptide sequences are also shown together with the a
and d positions of their heptad pattern. The residue positions that are known
to form the contact interface with the binding groove of MCL-1 are marked in
yellow 58.

In this study, we apply transient IR spectroscopy to explore the protein
dynamics of the intrinsically disordered binding domains of PUMA and NOXA in
complex with MCL-1. An azobenzene photoswitch has been covalently linked to
the short PUMA and NOXA peptides. The induced isomerization of the photoswitch
leads to a subsequent destabilization of the secondary structure of the linked
peptides, yet remaining bound to MCL-1 (Fig. 2). Together with data from the
previously analyzed MCL-1/BIM complex32, we classify the kinetic response of
these three MCL-1/BH3-only complexes and discuss how the observed protein
dynamics integrate into the promiscuous nature of the MCL-1/BH3-only system.

We designed photoswitchable variants of the bindings domains of PUMA and NOXA
in complex with MCL-1, to compare them to the previously generated
photoswitchable MCL-1/BIM variant 32. The domains, from here on pars-pro-toto
referred to as PUMA, NOXA and BIM, are 25 to 29 amino acid long intrinsically
disordered peptides that become $\alpha$-helical when complexing with MCL-1
16. They interact with a classical heptad pattern with hydrophobic side chains
at the a/d positions of the helix (Fig. 2, sequences). On the solvent-exposed
side of PUMA and NOXA, we introduced two cysteine residues, which were used to
covalently bind the photoswitch 3,3’-bis(sulfonato)-4,4’-bis-(chloroacetamido)
azobenzene (BSBCA)59 to the peptide. In a previous study with BIM, we
identified positions 16 and 23 (two consecutive c positions of the heptad
pattern) as anchoring points for the azobenzene moiety. For the newly
generated complexes, we introduced cysteines at the corresponding positions
(Fig. 2, in cyan).

Figure 3: (a) CD spectra of MCL-1 and PUMA (both 20 µM) in isolation and when
forming a complex. The PUMA spectra are shown for both the cis\- and the
trans-state of the photoswitch. (b) Ellipticity at 220 nm of the MCL-1/peptide
complexes relative to that of MCL-1 without peptide (dashed line). The data
for BIM are adapted from Heckmeier et al.32.

Circular Dichroism (CD) spectroscopy of the isolated BH3-only peptides reveal
a random coil structure in both the cis and the trans-state of the photoswitch
(exemplified for PUMA in Fig. 3a, red and blue). The CD spectrum of MCL-1
alone displays the classical response for an $\alpha$-helical structure (Fig.
3a, black). The $\alpha$-helical content is increased by a factor of
$\approx$1.2 when the BH3-only peptides BIM, PUMA, and NOXA are added in
equivalent amounts in the dark, in which case the photoswitch is in the trans-
state (Fig. 3a, purple). This proves that the photoswitchable BH3-only
peptides assume an $\alpha$-helical structure in the presence of their natural
binding partner MCL-1. By illuminating the MCL-1/BH3-only complexes with 375
nm laser light, we uniformly switched the photoswitchable BH3-only peptides to
the cis-state and could detect a further increase in $\alpha$-helical content
for NOXA, PUMA, and BIM (see Fig. 3b), as anticipated from the spacing of 7
amino acids between the two anchoring points of the photoswitch. Previous
results on BIM showed that the slight destabilization of the $\alpha$-helix in
the trans-state is not sufficient to result in unbinding of the peptide in the
concentration range needed for IR spectroscopy32.

Figure 4: IR spectroscopic analysis of PUMA (a,b) and NOXA (c,d) in complex
with MCL-1. (a,c) Steady-state cis-to-trans difference spectrum (black) and
the last kinetic trace at 42 $\mu$s (red). (b,d) Transient cis-to-trans
difference spectra as a function of pump-probe delay time. The triangles mark
the blue shift of the amide I band from $\approx$1640 cm-1 to $\approx$1660
cm-1.

For the transient IR experiments, the sample was first prepared in the cis-
state with the help of a cw-LED at 375 nm. Upon subsequent switching of the
azobenzene moiety from the cis\- to the trans-state by the irradiation with an
ultrashort UV/VIS laser pulse at 420 nm, the abrupt isomerization process
perturbs the secondary structure of the peptide, and in consequence also that
of its binding partner MCL-1. To monitor these structural changes, we set our
focus on the C=O stretch vibrations of the protein backbone, i.e. the amide I
band in the region around 1650 cm-1. The amide I band serves as an indicator
for rearrangements and alterations in the protein structure60. We performed
these experiments both in a steady-state manner with the help of a Bruker
Tensor 27 FTIR spectrometer (Fig. 4a,c), as well as transiently with the pump-
probe delay time ranging from picoseconds to 42 $\mu$s (Fig. 4b,d). The late
time (42 $\mu$s) transient spectra are in essence the same as the steady-state
difference spectra (Fig. 4a,c red vs black lines), indicating that most of the
structural changes have already found their end at this time point.

In order to extract the dynamical processes contained in the transient
spectra, we performed global multiexponential fitting, assuming
interconverting discrete states with time-invariant spectra61, 62, 63:

$d(\omega_{i},t_{j})=a_{0}(\omega_{i})+\sum_{k}a(\omega_{i},\tau_{k})e^{-t_{j}/{\tau_{k}}}.$
(1)

Here, we treated the amplitudes $a(\omega_{i},\tau_{k})$, as well as a common
set of time constants $\tau_{k}$ as the free fitting parameters, with the
number of exponential terms being restrained to a minimum64, 47. We fitted the
experimental data of PUMA with four states, S1, S2, S3 and a terminal state St
and three time constants connecting them ($\tau_{12}$=1.6 ns, $\tau_{23}$=18
ns, $\tau_{t}$=1.6 $\mu$s). For NOXA, three states S1, S2, and St and two time
constants were sufficient ($\tau_{12}$=85 ns and $\tau_{t}$=1.4 $\mu$s). The
corresponding time constants in BIM are $\tau_{12}$=9.7 ns, $\tau_{23}$=150
ns, and $\tau_{t}$=3.6 $\mu$s, see Ref. 32.

Figure 5: Timescales of dynamical activity for MCL-1/PUMA, MCL-1/BIM, and
MCL-1/NOXA upon photo-perturbation. For PUMA and NOXA, the time constants were
determined from the transient spectra in Fig. 4, while the data for BIM were
taken from Heckmeier et al.32. Time constant $\tau_{12}$, associated with the
spectral blue shift (and thus marked by triangles), is underlined in blue,
time constant $\tau_{23}$ in red, and the terminal time constant $\tau_{t}$ in
purple. For NOXA, $\tau_{23}$ cannot be resolved. In Heckmeier et al.32, we
reported an additional time constant prior to 100 ps, which however was
related to the pump-pulse duration and hence does not reflect a kinetic
process.

The observed timescales are summarized in Fig. 5. While the last timescale
($\tau_{t}$) is more or less the same in all three samples, the preceding two
processes ($\tau_{12}$ and $\tau_{23}$) vary by almost a factor 100, with PUMA
being the fastest and NOXA the slowest. In the case of NOXA, $\tau_{23}$ is
not resolved, presumably since it coincides with the terminal process
$\tau_{t}$.

Figure 6: Evolution associated difference spectra (EADS) of photo-perturbed
(a) MCL-1/PUMA, (b) MCL-1/BIM (adapted from previous study32), and (c)
MCL-1/NOXA. The triangles mark the blue shift of the amide I band from
$\approx$1640 cm-1 to $\approx$1660 cm-1.

Assuming a sequential, unidirectional process (see Fig. 6, top), we also
calculated evolution associated difference spectra (EADS) according to:

$d(\omega_{i},t_{j})=\sum_{k}C_{k}(t_{j})A_{k}(\omega_{i}),$ (2)

where $C_{k}(t_{j})$ is the concentration profile of component $k$ as a
function of time $t_{j}$, and $A_{k}(\omega_{i})$ its spectrum at probe
frequency $\omega_{i}$. The spectra $A_{k}(\omega_{i})$ are a linear
combination of the fitting amplitudes $a(\omega_{i},\tau_{k})$64.

For all investigated peptides, the first EADS, revealing the response
immediately after the pump pulse is over, shows in essence a bleach of the
amide I band (Fig. 6, red). The subsequent state (Fig. 6, yellow), populated
with time constant $\tau_{12}$, reveals an additional positive band on the
higher-frequency side, and hence a blue-shift of the amide I band. This blue
shift has been similarly detected for MCL-1/BIM previously32 and can be
interpreted as a partial $\alpha$-helical unfolding44, 65. In Ref. 32, we had
also investigated the response a localized vibrational mode directly
associated with the photoswitch, supporting this interpretation. The blue-
shift can be identified in the raw data of Fig. 4 as well, where it is marked
with triangles. While the blue-shifted band is observed for all three samples
with very similar spectroscopic characteristics, the timescale with which it
appears, $\tau_{12}$, varies by almost a factor 100.

Based on isotope labelling experiments,32 the subsequent spectral changes
occurring with $\tau_{23}$ and $\tau_{t}$ have been attributed to mostly the
protein MCL-1 responding to the structural perturbation of its binding
partner. We assume the same for PUMA and NOXA.

MCL-1 promiscuously binds to numerous intrinsically disordered inhibitors,
BH3-only peptides, at its binding groove 26, 27. MCL-1’s centrality in the
cancer-related apoptotic networks and its promiscuous nature makes it of high
interest for biochemical and pharmaceutical research 22, 21. In this study, we
characterized the promiscuity of MCL-1 on an atomistic level. To that end, we
explored the protein response of MCL-1/PUMA and MCL-1/NOXA upon ultrafast
photo-perturbation. Together with the recently published data on a
photoswitchable MCL-1/BIM variant, we could draw on dynamical information for
three of MCL-1’s natural inhibitors in a pico- to microsecond time window.

Figure 7: Correlation of (a) the binding affinity of the peptide, (b) its
polarity66, (c) its hydropathicity67, and (d) the loss in $\alpha$-helicity
upon photoswitching (cis-to-trans) against time constant $\tau_{12}$ for PUMA
(red), BIM (orange), and NOXA (brown). The Kd values were taken from Dahal et
al.,17, the scores in (b) and (c) were computed from amino acid sequences (see
Methods for details), and the data points in (d) from Fig. 3.

After the abrupt photo-isomerization of the azobenzene moiety, the three
complexes reveal very similar spectroscopic responses, both regarding the late
time (steady-state) response (Fig. 4a,b), as well as that of the transient
intermediates (Fig. 6). There is, however, one crucial difference between the
three investigated MCL-1/BH3-only complexes: Partial $\alpha$-helical
unfolding, occurring with $\tau_{12}$, happens at strongly divergent time
points for every BH3-only peptide, see blue lines in Fig. 5. The fastest
response was detected for PUMA ($\tau_{12}$=1.6 ns), succeeded by BIM
($\tau_{12}$=9.7 ns), and finally by NOXA ($\tau_{12}$=85 ns). Interestingly,
the time point of partial $\alpha$-helical unfolding does not correlate with
the binding affinity of the peptide17, but significantly with computed scores
for its polarity66 and hydropathicity67, as well as with the decline of
$\alpha$-helical content due to photoswitching (Fig. 7). The stronger the
decrease in $\alpha$-helicity is upon cis-to-trans isomerization, the faster
is its response. The same is true for increasing hydrophilicity as well as
polarity.

Disrupting PUMA, which reveals the fastest response, with the trans-state
azobenzene moiety may result in unfavorable contacts between hydrophilic side
chains of PUMA and the hydrophobic interface of the binding groove of MCL-1.
This strong tension could be dissolved by a rapid partial $\alpha$-helical
unfolding. Interestingly, a computational study solely on PUMA highlighted the
increased $\alpha$-helical propensity at the C-terminal relative to the rest
of the peptide68, the same region that we destabilize in our experiments.
Apparently, the perturbation of this region which is innately more likely to
form ordered structures, leads to conformational tension and thus to the fast
protein response. In contrast, the slow opponent NOXA – inherently more
hydrophobic and less polar – is not confronted with comparable tensions, thus
it is not forced to rearrange quickly and thus unfolds later. The
discrepancies between the peptides may also arise from breaking peptide-
exclusive electrostatic interactions at the putative contact interface (Fig.
2, yellow residues) upon photoswitching. Electrostatic interactions at the
contact interface are known to support the complex formation of MCL-1 and
BH3-only peptides, namely those formed by MCL-1 Lys234 and Glu9 of NOXA or
PUMA (numbering according to the alignment in Fig. 2)26, 69, MCL-1
Asp256/peptide Arg1570, 69, and MCL-1 Arg263/peptide Asp1970, 69.

These observations pose the question of how the peptide-specific differences
correlate with the promiscuous nature of the MCL-1/BH3-only complexes on a
cellular level. Pro-apoptotic BH3-only proteins are “damage sensors” and
abundantly expressed when cells suffer cytotoxic stress 18, 19, 20. Although
their binding domains share the same $\alpha$-helical structure of similar
length and identical hydrophobic heptad pattern, as well as the same binding
site at MCL-1, BH3-only proteins differ substantially in their relationship
inside the BCL-2 family and their interaction pattern (Fig. 1). BIM and PUMA
bind the full spectrum of the anti-apoptotic BCL-2 family, and also activate
the effector proteins BAK and BAX, whereas NOXA selectively inhibits MCL-1 and
BCL-2, another member of the same protein family15. BIM and PUMA stabilize
MCL-1, while NOXA promotes its degradation 26, 34.

Studies with truncated or mutated versions of NOXA demonstrated that the
C-terminal region of the binding domain regulates the stability of the
MCL-1/NOXA complex and therefore is important to control MCL-1/NOXA
degradation34, 26, 71 (Fig. 1, left). For its biological function, this region
of NOXA has to bind to MCL-1, which requires a certain “structural resilience”
for this region, even when the whole complex is destabilized in the
proteolytic process. We delimit this “structural resilience” from the already
established terms “stability”72 and “structural plasticity”73. The former is
classically used in a thermodynamic context to describe how partners in a
protein complex form and maintain folded conformations 72, 74, 75, 76. The
latter is frequently used to characterize the ability of promiscuous proteins,
e.g. BCL-2-type77, chaperones78, 79, Trypsin80, or protein kinases 81, to
flexibly bind various, different binding partners at the same interface. With
“structural resilience”, we have in mind a kinetic stabilisation, to contrast
to a thermodynamics one.

Our results show that even after photo-induced destabilization, the photo-
perturbed C-terminus of the NOXA peptide remains folded more than 10 times
longer than the equivalent region in other peptides (Fig. 5). While that is
definitely speculative, the high structural resilience of NOXA’s C-terminus
inside the binding pocket of MCL-1 could help to remain in place, even when
the whole complex is confronted with major rearrangements leading to the
potent proteolysis of MCL-1. Future experiments could test this hypothesis by
connecting a mutational analysis of NOXA and in vivo screening of MCL-1
proteolysis with monitoring the peptide’s structural resilience by IR
spectroscopy.

Different to NOXA, PUMA and BIM increase the stability of MCL-133, 26 and
block the binding groove of MCL-1 (Fig. 1, right). This in turn limits MCL-1’s
ability to bind the effector proteins BAK and BAX. The competition between
BIM, PUMA, and the effector proteins for MCL-1 manifests in the extremely high
affinities of BIM (Kd=25 pM) and PUMA (Kd=180 pM) in comparison to the already
high affinities for the effector proteins BAK (Kd=1.4 nM) and BAX (Kd=22
nM)17. Transient IR spectroscopy demonstrated that the structural resilience
of PUMA and BIM is smaller than that of NOXA, presumably because their natural
role is different, binding the partner in a highly-stable complex with little
structural flexibility. Any disturbance seems to result in a fast adaptation,
i.e. the partial unfolding that we observed in our experiments.

In summary, our study reveals insights into the promiscuity of the anti-
apoptotic MCL-1 for the intrinsically disordered binding domains of BH3-only
proteins. By using transient IR spectroscopy in combination with photo-
switchable protein complexes, we quantified the dynamic response of BIM, PUMA,
and NOXA in the binding pocket of MCL-1 upon photo-perturbation. All peptides
show partial $\alpha$-helical unfolding, however on very different time-
scales. The correlations in Fig. 7 are indicating that the speed of protein
response in our system is coupled to the extent of structural rearrangements
(Fig. 7d) and possibly linked to favourable or unfavourable interactions,
caused by the perturbation. Computational studies could test this hypothesis.

NOXA is structurally more resilient than BIM and PUMA. This finding reveals a
new viewpoint on the nature of BH3-only peptides, which could help to better
understand promiscuous protein-protein interactions in general, as well as the
design of novel molecules and peptides to therapeutically manipulate the
oncologically relevant MCL-1/BH3-only complex.

## Materials and Methods

###  .1 Peptide preparation

The BH3 domains of PUMA (EEQWAREIGAQLRCMADDLNCQYER) and NOXA
(RAELPPEFAAQLRCIGDKVYCTWSAP), both containing cysteine mutations with a
spacing of 7 amino acids, were synthesized using solid state peptide synthesis
on a Liberty 1 peptide synthesizer (CEM corporation, Matthews, NC, USA). These
peptides were purified analogously to the BIM variant
(GGSGRPEIWIAQELRCIGDEFNCYYARRV), which was investigated in a preceding study
32. The watersoluble photoswitch
3,3’-bis(sulfonato)-4,4’-bis(chloroacetamido)azobenzene (BSBCA) was
subsequently covalently linked to the cysteine residues, as described before
59, 82. The successful linkage, purity, as well as the integrity of the
peptide was controlled via mass spectrometry.

###  .2 Protein preparation

Human MCL-1 (hMCL-1$\Delta$N-$\Delta$C, 171-327, C286S83) was expressed in
Escherichia coli BL21. The cells were grown until they reached OD600=0.6 and
induced with 0.75 mM isopropyl $\beta$-D-1-thiogalactopyranoside. After the
induction, the cells were incubated for 20 hours at 30° C until they were
harvested end lysed using sonication. The proteins were purified under native
conditions via Ni-affinity chromatography and a His6-Tag at the N-terminus of
the protein. The N-terminal His6-Tag was removed by 3C protease cleavage. All
experiments were performed in 50 mM Tris (pH 8) and 125 mM NaCl. The
integrity, as well as the purity of the protein sample was controlled via mass
spectrometry. For IR spectroscopy, H2O containing sample buffer was exchanged
against D2O based buffer via spin column centrifugation. To circumvent the
contamination with atmospheric H2O, the sample was kept in a water-vapor free
nitrogen environment.

###  .3 Transient IR spectroscopy

For pump-probe measurements, we used two electronically synchronized 2.5 kHz
Ti:sapphire oscillator/regenerative amplifier femtosecond laser systems
(Spectra Physics), allowing a delay of maximally 45 $\mu$s 84. For the pump
pulses, one laser system was tuned to 840 nm and then brought to 420 nm pulses
via second harmonic generation in a $\beta$-BaBO4 crystal, later needed for
cis-to-trans-isomerization of the photoswitch. After light amplification, the
compressor has been bypassed, resulting in $\approx$60 ps stretched pulses, in
order to reduce sample deposition on the sample cell windows. At the sample
cell, the power was 3 $\mu$J per pulse, focused to a $\approx$140 $\mu$m beam
diameter. The second laser system was used to generate midIR probe pulses in
an optical parametric amplifier (100 fs, spot size 110 $\mu$m, center
wavenumber 1625 cm-1)85. To prepare the investigated samples for the
spectroscopic experiments, the crosslinked peptides and MCL-1 were mixed in an
1:1 ratio in D2O with a total complex concentration of 1 mM. The protein
samples were constantly circulated in a closed-cycle flow cell system,
comprising a CaF2 measurement cell (50 $\mu$m optical path length) and a
reservoir. The investigated sample was irradiated with a 375 nm continuous
wave diode laser (90 mW, CrystaLaser) before entering the measurement cell, in
order to prepare $>$85% of the sample in cis-state.

###  .4 Peptide parameter computation

To better understand the differences of the various BH3-only peptides, we
calculated parameters for the polarity according to Grantham,66 and the
hydrophobicity/hydropathicity according to Kyte & Doolitle67 from the
sequences of PUMA, BIM, and NOXA, using the Expasy/ProtScale server
(http://web.expasy.org/protscale/). The calculations were executed with a
window size of 7 residues (interval length for the computation) and the same
weight for every residue in that interval (relative weight = 100$\%$)86.

## Abbreviations

BH, BCL-2 homology; BSBCA, 3,3’-bis(sulfonato)-4,4’-bis-(chloroacetamido)
azobenzene; CD, Circular Dichroism; EADS, evolution associated difference
spectra; IR, Infrared; MCL-1, Myeloid Cell Leukemia 1.

## Author Information

Corresponding Author
* E-mail<EMAIL_ADDRESS>

ORCID identifiers
Philipp J. Heckmeier 0000-0002-7311-977X Jeannette Ruf 0000-0003-0495-6459
Brankica G. Janković 0000-0002-8073-3014 Peter Hamm 0000-0003-1106-6032
Notes
The authors declare no competing financial interest.

Acknowledgement We thank Markus B. Glutz for the synthesis of the peptides and
Functional Genomics Center Zurich for their work on the mass spectrometry
analysis. We thank Roland Zehnder for the technical support and Kerstin T.
Oppelt, Gerhard Stock, and Steffen Wolf for fruitful discussions. The work has
been supported by the Swiss National Science Foundation (SNF) through Grant
200020B_188694/1.

Data Availability: The data that support the findings of this study are openly
available in Zenodo (the link will be provided at the proof stage.)

References:

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Thermodynamics and the Joule-Thomson expansion of dilaton black holes in 2 +1
dimensions

Leonardo<EMAIL_ADDRESS>
Departamento de Ciencias Físicas,
Facultad de Ingeniería y Ciencias
Universidad de La Frontera, Casilla 54-D
Temuco, Chile.
Sharmanthie<EMAIL_ADDRESS>
Department of Physics, Geology & Engineering Technology
Northern Kentucky University
Highland Heights, Kentucky 41099, U.S.A.

Abstract

In this paper, we study thermodynamics and its applications to the static
charged dilaton black hole in 2+1 dimensions. We compute the first law and the
Smarr relations for the black hole and introduce a new thermodynamical
parameter in order to satisfy the first law. We compute specific heat
capacities, internal energy and free energy for the black hole and study local
and global stability of the black hole. For $N=1$, the black hole is locally
stable for all values of the horizon radius and does not go through phase
transitions. Maxwell’s relations are presented for this case. For $N=6/7$,
small black holes are locally stable and large black holes are not. There is a
first order phase transition between small black holes and the thermal AdS
space. We have also studied the Joule-Thomson expansion for black holes with
$N=1$. It is also noted that unlike the charged BTZ black hole, the charged
dilaton black hole does not violate the Reverse Isoperimetric Inequality for
certain values of parameters of the theory.

Key words: static, dilaton, black hole, anti-de Sitter space, first law, Smarr
relations, Joule-Thomson expansion, Hawking-Page phase transition

##  1 Introduction

Black hole thermodynamics has been a very active area of research for the past
50 years or so since the seminal work of Bekenstein and Hawking where the
relation of entropy to the area at the horizon and temperature at the horizon
were discovered [2, 3]. Most interestingly, the work of Hawking and Page on
anti-de Sitter black holes, where a first order phase transition between the
thermal anti-de Sitter space and the Schwarzschild anti-de Sitter black hole
was demonstrated inspired many works on thermodynamics of anti-de Sitter black
holes [4]. Another interesting property of anti-de Sitter black holes is that
they have shown to have Van der Waals type phase transitions between small and
large black holes. In this case, the cosmological constant is treated as the
thermodynamic pressure. Hence the first law of thermodynamics has an extra
$VdP$ term and the mass is considered as the enthalpy rather than the internal
energy [5, 6]. Many of the AdS black holes has Van der Waals type phase
transitions between small and large black holes in this extended phase space.
Due to lack of space, we will only mention few examples here: PV criticality
of black holes in massive gravity was explored by Fernando in [7]; phase
transitions and thermodynamic volume of thermodynamics of rotating black holes
and black rings were studies by Altamirano et.al [8]. Exhaustive study of
thermodynamics of Born-Infeld black holes were done in [9]. Thermodynamic
phase transition of Euler-Heisenberg-AdS black hole on free energy landscape
was studied by Dai et.al. in [10]. Few other works along those lines are [11,
12, 13, 14, 15, 16, 17, 18]. The reader may read the nice review on
thermodynamics of AdS black holes by Kubiznak et.al. [19] to see other
examples related to this subject.

Within the framework of the string theory there arises a certain type of
scalar field called the dilaton field, which is related to the coupling
constant of the string. This field has been coupled with the neutral or
electrically (or magnetically) charged BTZ solution [20], which has allowed
the construction of various more general solutions [21, 22, 23, 24, 25, 26,
27, 28, 29, 30]. Also considering 2+1 dimensional gravity with nonlinear
electrodynamics models, Born-Infeld [31], Coulomb-like field [32] and power
Maxwell [33], with dilaton fields, new generalizations of the BTZ solution
have been obtained in Refs. [34, 35, 36], respectively. Particularly in Ref.
[21] a one-parameter $N$ family of 2+1 dimensional black hole solutions with a
dilaton field minimally coupled is obtained, where the dilaton is coupled
exponentially to the electric field and the cosmological constant. One of the
members of this family ($N=1$) corresponds to the MSW black hole solution
[37], one that arises naturally in string theory as a 1+1 dimensional model of
dilaton gravity, and from which, by its product with a unit circle, the MSW
black hole in 2+1 dimensions [21, 38] can be obtained. The goal of this paper
is to study thermodynamics of the charged dilaton black hole developed by Chan
and Mann in [21].

The paper is organized as follows: in section 2, the charged dilaton black
hole in 2+1 dimensions is introduced. Thermodynamics for $N=1$ case is
presented in section 3. Joule Thomson expansion for $N=1$ case is discussed in
section 4. Thermodyanmics for a general value of $N$ is discussed in section
5. Finally the conclusion is given in section 6.

##  2 Introduction to static charged dilaton black hole in 2+1 dimensions

We consider the Einstein-Maxwell-dilaton action for 2+1-dimensions given by
Chan and Mann in Ref. [21]

$S=\int
d^{3}x\sqrt{-g}\,\left(R-\frac{B}{2}(\nabla\phi)^{2}-e^{-4a\phi}F_{\mu\nu}F^{\mu\nu}-V(\phi)\right)\,,$
(1)

where $R$ is the Ricci scalar, the third term in the parentheses represents
the coupling between electrodynamics and the dilaton field $\phi$,
$F_{\mu\nu}$ is the Maxwell field strength, the dilaton potential is
$V(\phi)=2e^{b\phi}\Lambda$, $\Lambda$ is the cosmological constant and $a$,
$b$ and $B$ are coupling constant. Here, when $\Lambda<0$ corresponds to anti-
de Sitter space and $\Lambda>0$ corresponds to the de Sitter space. Notice
that in the paper by Chan and Mann [21], $\Lambda>0$ is treated as the anti-De
Sitter case and $\Lambda<0$ is considered as the de Sitter case. To align our
work with most literature on anti-de Sitter black holes, we will consider
$\Lambda<0$. Hence, when $\Lambda$ appears in our work, it will be a
$-\Lambda$ in Chan and Manns work [21].

We consider the following line element

$ds^{2}=-f(r)dt^{2}+f^{-1}(r)dr^{2}+r^{2}R^{2}(r)d\theta^{2}\,,$ (2)

where

$R(r)=\gamma\,r^{\frac{N}{2}-1}$ (3)

and

$f(r)=-\frac{2mr^{1-\frac{N}{2}}}{N}-\frac{8\Lambda\beta^{2-N}r^{N}}{(3N-2)N}+\frac{8Q^{2}}{(2-N)N}\,.$
(4)

Here $\gamma$ is a integration constant. Note that the dimensions are:
$[r]=L$, $[m]=L^{\frac{N}{2}-1}$, $[\Lambda]=L^{-2}$, $[\beta]=L$, $[Q]=L^{0}$
and $[\gamma]=L^{1-\frac{N}{2}}$, where $L$ represents units of length.

From Eqs. (1), (2) and (10) in Ref. [21] we get

$\frac{1}{rR(r)}\frac{d(rR(r))}{dr}=-\frac{B}{2}\left(\frac{d\phi}{dr}\right)^{2}\,,$
(5)

which implies the equation

$\frac{d\phi}{dr}=\frac{k}{r}\,,$ (6)

where

$k=\pm\sqrt{\frac{N(2-N)}{2B}}\,.$ (7)

Then

$\phi=k\ln\left(\frac{r}{\beta}\right)\,.$ (8)

Here $\beta$ is a constant of integration. Later in the paper $\beta$ will be
presented as a thermodynamic quantity. We will choose $\beta>0$ since it
appears in the $\ln(r/\beta)$ for $\phi$ in eq.$(\ref{phi})$.

##  3 Thermodynamics of the black holes for $N=1$ case

When $N=1$, the black hole solution is given by following metric:

$ds^{2}=-f(r)dt^{2}+f^{-1}(r)dr^{2}+r^{2}R^{2}(r)d\theta^{2}\,,$ (9)

where

$R(r)=\gamma r^{-1/2}$ (10)

and

$f(r)=(-2m\sqrt{r}-8\Lambda\beta r+8Q^{2})\,.$ (11)

Here $\gamma$ is a integration constant. Note that the dimensions are:
$[r]=L$, $[m]=L^{-1/2}$, $[\Lambda]=L^{-2}$, $[\beta]=L$, $[Q]=L^{0}$ and
$[\gamma]=L^{1/2}$, where $L$ represents units of length.

Following Ref. [39] we can obtain the entropy of the above black hole as,

$S=\frac{2\pi r_{+}R(r_{+})}{4}=\frac{\pi}{2}\gamma\sqrt{r_{+}}\,$ (12)

and the total mass of the black hole as,

$M=\gamma\frac{m}{8}\,.$ (13)

Here $r_{+}$ is the event horizon radius. Since $f(r)=0$ at the horizons, one
can obtain two horizon radius as

$r_{\pm}=\left(\frac{M\pm\sqrt{M^{2}+\beta\gamma^{2}Q^{2}\Lambda}}{\beta\Lambda\gamma}\right)^{2}\,.$
(14)

Notice that $r_{+}$ is the event horizon and $r_{-}$ is the Cauchy horizon. We
have substituted for $m$ in this expression. There will be an extreme black
hole when,

$M=\gamma Q\sqrt{-\beta\Lambda}$ (15)

Hence, black hole exists only when $M\geq\gamma Q\sqrt{-\beta\Lambda}$.
Horizon radius of the extreme black hole is,

$r_{ex}=-\frac{Q^{2}}{\beta\Lambda}\,.$ (16)

One can obtain an expression for the mass $M$ as,

$M=-\frac{\Lambda\beta\gamma}{2}\sqrt{r_{+}}+\frac{Q^{2}\gamma}{2}\frac{1}{\sqrt{r_{+}}}\,.$
(17)

Note that we have substituted for $m$ from eq.$(\ref{mass})$.

In addition to the thermodynamic quantities $S$ and $Q$, we consider the
cosmological constant as a thermodynamic pressure given by [40],

$P=-\frac{\Lambda}{8\pi}\,.$ (18)

###  3.1 First law of black hole thermodynamics and the Smarr formula for
$N=1$

If we rewrite the mass $M$ in terms of $Q,S$ and $P$ as,

$M(S,P,Q,R)=\frac{\pi\gamma^{2}Q^{2}}{4S}+8PS\beta\,.$ (19)

We then obtain the following expression for the first law of thermodynamics

$dM=\left(\frac{\partial M}{\partial
S}\right)_{P,Q,\beta}dS+\left(\frac{\partial M}{\partial
P}\right)_{S,Q,\beta}dP+\left(\frac{\partial M}{\partial
q}\right)_{S,P,\beta}dQ+\left(\frac{\partial
M}{\partial\beta}\right)_{S,P,Q}d\beta\,\,\,,$ (20)

or

$dM=TdS+VdP+\Phi dQ+Bd\beta\,\,\,,$ (21)

where $\beta$ is taken as a new thermodynamic quantity. Note that we have
considered $M$ as the enthalpy $H=U+PV$ in the extended phase space in
deriving the first law for the black hole.

Then the temperature can be calculated from the first law of black hole
thermodynamics as

$T=\left(\frac{\partial M}{\partial
S}\right)_{P,Q,\beta}=-\frac{Q^{2}}{\pi\,r_{+}}-\frac{\Lambda\beta}{\pi}\,.$
(22)

The Hawking temperature obtained using the surface gravity is

$T_{H}=\frac{1}{4\pi}\left|\frac{dg_{tt}}{dt}\sqrt{-g^{tt}g^{rr}}\right|_{r_{+}}=-\frac{2M}{\pi\gamma\sqrt{r_{+}}}-\frac{2}{\pi}\Lambda\beta=-\frac{Q^{2}}{\pi\,r_{+}}-\frac{\Lambda\beta}{\pi}\,.$
(23)

It is clear that $T_{H}=T$. Notice that when the black hole is extreme, then
$T=0$. Also when $r_{+}\rightarrow\infty$,
$T\rightarrow-\frac{\Lambda\beta}{\pi}$. There are no local minima or maxima
for $T$. Hence black holes cannot exist for $T<0$ or
$T>-\frac{\Lambda\beta}{\pi}$. Fig.$(\ref{cpt})$ shows the behavior of $T$ vs
$r_{+}$ for $N=1$ case.

If $P=-\Lambda/(8\pi)$ then one can obtain the thermodynamic volume from the
first law given in eq.$(\ref{firstlaw})$ as,

$V=\left(\frac{\partial M}{\partial
P}\right)_{S,Q,\beta}=4\pi\beta\gamma\sqrt{r_{+}}\,,$ (24)

which is different from the geometric volume, $V_{geo}=\pi r_{+}^{2}$.

The electric potential at the horizon is given by

$\Phi=\left(\frac{\partial M}{\partial
Q}\right)_{S,P,\beta}=\frac{\gamma\,Q}{\sqrt{r_{+}}}\,.$ (25)

Where $Q$ is related to the electric charge $Q_{BTZ}$ of the BTZ black hole
[20, 41] as follows from Ref. [39]

$Q_{BTZ}=\frac{\gamma\,Q}{2}\,.$ (26)

Finally we get the thermodynamic quantity $\cal{B}$ conjugate to the parameter
$\beta$

${\cal{B}}=\left(\frac{\partial
M}{\partial\beta}\right)_{S,P,Q}=-\frac{\gamma\Lambda\sqrt{r_{+}}}{2}\,.$ (27)

Now to obtain the Smarr formula we follow the scaling arguments of Ref. [42]

$\left(-\frac{1}{2}\right)M=\left(\frac{1}{2}\right)\left(\frac{\partial
M}{\partial S}\right)S+(-2)\left(\frac{\partial M}{\partial
P}\right)P+(0)\left(\frac{\partial M}{\partial
Q}\right)Q+(1)\left(\frac{\partial M}{\partial\beta}\right)\beta\,\,\,,$ (28)

namely

$-\frac{M}{2}=\frac{1}{2}TS-2VP+\cal{B}\beta\,\,\,.$ (29)

Now, some clarification and justification is in order as to why we added the
${\cal{B}}d\beta$ term to the first law of thermodynamics in
eq.$(\ref{firstlaw})$. If we had not done so, then the dimensional analysis of
the right side of the Smarr formula will look like $\frac{1}{2}TS-2VP$ which
is not equal to $-\frac{M}{2}$. Hence the only way one can develop the first
law which is consistent with the Smarr formula is to add the ${\cal{B}}d\beta$
to the first law of thermodynamics and to include the $\cal{B}\beta$ term in
the Smarr formula analysis.

It may be surprising that the factor that accompanies the term $TS$ in the
Smarr formula is $1/2$ in eq.$(\ref{smarr})$. Let us consider the following
toy model in 3+1-dimensions

$ds^{2}=-f(r)dt^{2}+f^{-1}(r)dr^{2}+r^{2}R(r)^{2}\,d\Omega^{2}\,,$ (30)

where $R(r)=\gamma\,r^{-1/2}$, and,

$f(r)=1-\frac{2M}{r}\,,$ (31)

The entropy in this case is given by [43],

$S=\pi r^{2}_{+}R(r_{+})^{2}=\pi\gamma^{2}\,r_{+}\,.$ (32)

Hence,

$M=\frac{r_{+}}{2}=\frac{S}{2\pi\gamma^{2}}\,.$ (33)

Here the dimensions are: $[r]=L$, $[M]=L^{1}$ and $[\gamma]=L^{1/2}$.

Then the temperature is obtained via the first law as,

$T=\left(\frac{\partial M}{\partial S}\right)=\frac{1}{2\pi\gamma^{2}}\,.$
(34)

By combining eq.$(\ref{entropy2})$ and eq.$(\ref{mass2})$, a relation between
$M,T$ and $S$ can be obtained as,

$(1)M=(1)TS\,\,\,.$ (35)

This is the correct Smarr formula with a factor of 1 instead of 2 for the $TS$
term.

The same factor 1 must be considered even if the metric function (together
with metric (30)) is

$f(r)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\,.$ (36)

Here the Smarr formula is

$M=(1)TS+(2)\Phi Q\,.$ (37)

###  3.2 Other thermodynamical quantities

We will calculate few more thermodynamical quantities, $C_{P,Q},C_{V,Q}$ and
internal energy $U$ of the black hole as follows:

$C_{P,Q}=T\left(\frac{\partial S}{\partial
T}\right)_{P}=\frac{\pi\sqrt{r_{+}}(-Q^{2}+8\pi\beta Pr_{+})}{4Q^{2}}$ (38)
$C_{V,Q}=T\left(\frac{\partial S}{\partial T}\right)_{V}=0,$ (39)

since when volume is kept constant, i.e. $r_{+}$ is kept constant, $S$ does
not change. Since the enthalpy is given by $M$ which is equal to $U+PV$, one
can compute $U$ as follows:

$U=M-PV=\frac{\gamma Q^{2}}{2\sqrt{r_{+}}}\,.$ (40)

If we write $U$ as a function of temperature $T$, volume $V$ and pressure $P$,
it is,

$U=-\frac{VT}{8\beta}+PV\,.$ (41)

If we write $U$ as a function of $V,Q,\beta$, then,

$U=\frac{2\pi\beta\gamma^{2}Q^{2}}{V}\,.$ (42)

As it can be observed, the internal energy depends on both the temperature and
the pressure. In an ideal gas, the internal energy only depends on
temperature.

Helmholtz free energy is

$F=U-TS=\frac{16\pi\beta\gamma^{2}Q^{2}}{V}-\frac{TV}{\beta}\,.$ (43)

One can substitute $\Lambda=-8\pi P$ in eq.$(\ref{temp1})$ to obtain the state
equation as,

$P=\frac{T}{8\beta}+\frac{Q^{2}}{8\pi\beta r_{+}}\,.$ (44)

The pressure $P$ is plotted vs $r_{+}$ in Fig.$(\ref{pvsr})$ below to
demonstrate that there are no critical points.

Figure 1: The figure shows $P$ vs $r_{+}$ for the dilaton black hole for
constant temperature. Here, $Q=1,\beta=0.1$. and $T=0.28.$

###  3.3 Thermodynamic stability

For the black hole to be locally stable, $C_{P,Q}>0$. From eq.$(\ref{cp})$
this implies,

$P>\frac{Q^{2}}{8\pi\beta r_{+}}$ (45)

or,

$-\Lambda>\frac{Q^{2}}{\beta r_{+}}\,.$ (46)

Hence it is clear that if the black hole to be locally stable, then $\Lambda$
has to be negative or has or be in anti-de Sitter space. This is very similar
to the Schwarzschild anti-de Sitter black hole where it was demonstrated that
$-\Lambda r_{+}^{2}>1$ for $C_{p}$ to be positive [6]. When $C_{P,Q}$ and $T$
are plotted against $r_{+}$, it is clear that both are positive for the same
values of $r_{+}$ as demonstrated in Fig.$(\ref{cpt})$. Both $C_{P,Q}$ and $T$
are zero when the black hole is extreme with
$r_{ex}=-\frac{Q^{2}}{\beta\Lambda}$. Hence, provided $\Lambda<0$, all black
holes are locally stable.

Figure 2: The figure shows the specific heat capacity $C_{P,Q}$ and Hawking
temperature $T$ vs the horizon $r_{+}$ for $N=1$ and fixed charge. Here
$Q=0.5,$ $\Lambda=-0.2,\gamma=1$ and $\beta=0.5,1,2$ (green, red, blue).

In order to study global stability of a black hole, one has to study the Gibbs
free energy G. Note that the Gibbs free energy $G=M-TS$ for $N=1$ does not
depend on $\Lambda$ nor on $\beta$ as follows:

$G=\frac{\gamma Q^{2}}{\sqrt{r_{+}}}$ (47) $\frac{dG}{dr_{+}}=-\frac{\gamma
Q^{2}}{2(r_{+})^{3/2}}<0\,.$ (48)

Since $\frac{dG}{dr_{+}}$ is decreasing, it does not have a local minimum.
Hence we can conclude that the black hole does not admit Van der Waals type
behavior for $N=1$ black holes. Also, $G>0$ for all black holes and will
approach zero when $T\rightarrow-\frac{\Lambda\beta}{\pi}$.
Fig.$(\ref{gvst1})$ represent G vs T graph for N= 1.

Figure 3: Gibbs free energy $G$ versus $T$ for $N=1$, $Q=0.5$, $\gamma=1$,
$\beta=3$ and $\Lambda=-1.0$.

###  3.4 Maxwells relations

We can find the Maxwells’ relation for thermodynamical quantities from the
Helmholtz free energy $F$. Since $F=U-TS$ and $H=M=U+PV$, one can obtain $F$
as,

$F=M-PV-TS\,.$ (49)

Combining $dM$ in eq.$(\ref{firstlaw})$ with $dF$ from eq.$(\ref{helm})$, we
obtain,

$dF=-PdV-SdT+\Phi dQ+{\cal{B}}d\beta$ (50)

which is equivalent to,

$dF=\left(\frac{\partial F}{\partial
V}\right)_{T,Q,\beta}dV+\left(\frac{\partial F}{\partial
T}\right)_{V,Q,\beta}dT+\left(\frac{\partial F}{\partial
Q}\right)_{V,T,\beta}dQ+\left(\frac{\partial
F}{\partial\beta}\right)_{V,T,Q}d\beta\,\,\,.$ (51)

From the above differential, following relations are obtained:

$S=-\left(\frac{\partial F}{\partial T}\right)_{Q,\beta,V}$ (52)
$P=-\left(\frac{\partial F}{\partial V}\right)_{T,Q,\beta}$ (53)
$\Phi=\left(\frac{\partial F}{\partial Q}\right)_{T,\beta,V}$ (54)
${\cal{B}}=\left(\frac{\partial F}{\partial\beta}\right)_{T,V,Q}$ (55)

From above four relations, one can differentiate to obtain the following
additional 5 relations:

$-\left(\frac{\partial P}{\partial
Q}\right)_{T,\beta,V}=\left(\frac{\partial\Phi}{\partial
V}\right)_{T,Q,\beta}$ (56) $-\left(\frac{\partial S}{\partial
Q}\right)_{T,\beta,V}=\left(\frac{\partial\Phi}{\partial
T}\right)_{Q,\beta,V}$ (57) $\left(\frac{\partial S}{\partial
V}\right)_{T,Q,\beta}=\left(\frac{\partial P}{\partial T}\right)_{Q,\beta,V}$
(58) $-\left(\frac{\partial
P}{\partial\beta}\right)_{Q,T,V}=\left(\frac{\partial\cal{B}}{\partial
V}\right)_{T,Q,\beta}$ (59) $-\left(\frac{\partial
S}{\partial\beta}\right)_{Q,T,V}=\left(\frac{\partial\cal{B}}{\partial
T}\right)_{V,Q,\beta}$ (60)

###  3.5 Reverse Isoperimetric Inequality

There is a conjecture that the isoperimteric ratio $\cal{R}$ given by,

${\cal{R}}=\left(\frac{(D-1)V}{\omega_{D-2}}\right)^{\frac{1}{D-1}}\left(\frac{\omega_{D-2}}{A}\right)^{\frac{1}{D-2}}$
(61)

will always satisfy ${\cal{R}}\geq 1$ when $V$ is the thermodynamical volume
and $A$ is the area of the horizon [44]. Here, $\omega_{d}$ is the area of the
$d$-dimensional unit sphere. For the given black hole,

$A=2\pi\gamma\sqrt{r_{+}}\,\,;\hskip
14.22636ptV=4\pi\beta\gamma\sqrt{r_{+}}\,\,;\hskip
14.22636pt\omega_{d}=\frac{2\pi^{(d+1)/2}}{\Gamma(\frac{d+1}{2})}$ (62)

leading to

${\cal{R}}=\sqrt{\frac{4\pi\beta}{\gamma\sqrt{r_{+}}}}\,.$ (63)

Since $\beta$ and $\gamma$ are integrating constants, and their values are not
determined a priori, it is not possible to say whether ${\cal{R}}\geq 1$. One
may recall that in charged BTZ black holes, ${\cal{R}}<1$ leading to a
violation of the Reverse Isoperimetric Inequality [45]. Hence, inclusion of
the dilaton field leads to the possibility of charged black holes in 2+1
dimensions to satisfy the Reverse Isoperimetric Inequality.

##  4 Joule-Thomson expansion

First, let us describe the Joule-Thomson expansion for a thermodynamical
system as follows: In a Joule-Thomson experiment, gas will flow constantly
along a thermally insulated tube which is divided into two compartments. There
would be a porus plug in between the compartments. The gas will be at higher
pressure and will move into a low pressure section. The enthalpy will remain
constant along the expansion process; hence it is a iso-enthalpic process. The
purpose of this experiment is to find how the temperature $T$ varies with
pressure $P$ at constant enthalpy $H$, which is given by $\left(\frac{\partial
T}{\partial P}\right)_{H}$ and is known as Joule-Thomson coefficient $\mu$.

We will omit the derivation of the Joule-Thomson coefficient since detailed
derivations are given in [46]. $\mu$ is given by,

$\mu=\left(\frac{\partial T}{\partial
P}\right)_{H}=\frac{1}{C_{P,Q}}\left[T\left(\frac{\partial V}{\partial
T}\right)_{P}-V\right]\,.$ (64)

Here, $C_{P,Q}$ is the heat capacity at constant pressure and is given in
eq.$(\ref{cp})$. One can determine if the gas will cool or heat after the
adiabatic expansion by evaluating the coefficient $\mu$. If $\mu>0$ then the
temperature decreases and if $\mu<0$ then the temperature increases. Note that
in the Joule-Thomson expansion, change of pressure is negative.

For the dilaton black hole considered in this paper, we will keep charge $Q$
constant when computing $\mu$. Hence $\mu$ is given by,

$\mu=\left(\frac{\partial T}{\partial
P}\right)_{H}=\frac{1}{C_{P,Q}}\left[T\left(\frac{\partial V}{\partial
T}\right)_{P}-V\right]$ $=\frac{1}{C_{P,Q}}\left[T\frac{\left(\frac{\partial
V}{\partial r_{+}}\right)_{P}}{\left(\frac{\partial T}{\partial
r_{+}}\right)_{P}}-V\right]$ $=8\beta\left(\frac{-3Q^{2}+8\pi\beta
Pr_{+}}{-Q^{2}+8\pi\beta Pr_{+}}\right)\,.$ (65)

In Fig.$(\ref{mutemp})$, the Joule-Thomson coefficient $\mu$ and the
temperature $T$ are plotted against the horizon radius $r_{+}$ for different
values of charge $Q$. One can observe that there is a divergent point and a
zero point for $\mu$. For the $T$ vs $r_{+}$ graph, it is clear that the
divergent point of $\mu$ coincide with the zero point of temperature. $T=0$
for extreme black holes where the horizon is given by
$r_{+}=r_{-}=\frac{M}{8\Lambda}$.

One can define a quantity called the inversion temperature, $T_{i}$ and
inversion pressure, $P_{i}$ when $\mu=0$. $T_{i}$ is given by,

$T_{i}=V\left(\frac{\partial T}{\partial V}\right)_{P}\,.$ (66)

The gas will heat if one starts above the inversion temperature and will cool
if one starts below the inversion temperature. The inversion pressure $P_{i}$
is calculated from $\mu=0$ via the eq.$(\ref{muc})$ as,

$P_{i}=\frac{3Q^{2}}{8\pi\beta r_{+}}\,.$ (67)

Substituting $P_{i}$ to $T$ in eq.$(\ref{pi})$, the inversion temperature is
obtained as,

$T_{i}=\frac{2Q^{2}}{\pi r_{+}}$ (68)

By eliminating $\frac{Q^{2}}{r_{+}}$ from both $P_{i}$ and $T_{i}$, one can
obtain a relation between the inversion temperature and inversion pressure as,

$T_{i}=\frac{16\beta}{3}P_{i}\,.$ (69)

It can be observed that the relation is independent of the charge $Q$. It only
depends on $\beta$. The inversion temperature increases monotonically with the
inversion pressure $P_{i}$. There is only one inversion curve for the dilaton
black hole. In contrast, for the Van der Waals fluid, there are two inversion
curves as shown in [46]. The relation between $T_{i}$ and $P_{i}$ is given in
the Fig.$(\ref{tipi})$. Furthermore, the dilaton black hole does not have
critical behavior such as black holes that have demonstrated behavior as the
van der Waals fluids as shown in section (3.1). Hence a ratio between $T_{i}$
and $T_{c}$ does not exists. This behavior is similar to the BTZ black hole as
shown in [47].

The minimum inversion temperature, $T^{min}_{i}$ is obtained at $P_{i}=0$
which is,

$T^{min}_{i}=0\,.$ (70)

Hence, $T_{i}^{min}$ occurs when the black hole is extreme. This is similar to
the case for the BTZ black hole [47]

Figure 4: The figure shows the Joule-Thomson coefficient $\mu$ and Hawking
temperature $T$ vs the horizon $r_{+}$ for fixed charge. Here
$Q=2.97,3.97,4.82$ and $P=1,\beta=0.1,\gamma=1$.  Figure 5: The figure shows
the inversion curves, $T_{i}$ vs $P_{i}$ for the dilaton black hole. Here
$\beta=1.00,1.50,2.00$ and $\gamma=1$.

Since the Joule-Thomson expansion is an isenthalpic process, we have plotted
the isenthalpic curves in Fig.$(\ref{iso})$. When $\mu=0$, i.e.
$\left(\frac{\partial T}{\partial P}\right)_{H}=0$, the isenthalpic curves and
inversion curves intersect at $\mu=0$ point. From Fig.$(\ref{iso})$, it is
clear that the slope of the isenthalpic curve is positive above the inversion
curve. Hence, if the black hole starts above the inversion curve, it will cool
under this expansion. On the other hand, since the slope of the isenthalpic
curve is is negative below the inversion curve, the temperature of the black
hole will increase if it starts below the inversion curve.

Figure 6: The figure shows isenthalpic and inversion curves for the dilaton
black hole. The isenthalpic curve is given for mass $M=1.42,1.29,1.14$. The
purple line is for inversion curve. Here, $Q=1,\beta=0.2$ and $\gamma=1$.

##  5 Thermodynamics of the black hole for the general N case

In this section we will address thermodynamics of the dilaton black holes for
general $N$ values for the metric in eq.$(\ref{metric1})$,
eq.$(\ref{metric2})$ and eq.$(\ref{metric3})$. First we will replace the mass
$m$ with mass $M$ in $f(r)$ with the following relations as given in [39],

$M=\frac{\gamma\,m}{8}\,$ (71)

$f(r)=-\frac{16Mr^{1-\frac{N}{2}}}{N\gamma}-\frac{8\Lambda\beta^{2-N}r^{N}}{(3N-2)N}+\frac{8Q^{2}}{(2-N)N}\,.$
(72)

In order to address thermodynamics of black holes, some comments for $N\neq 1$
cases are in order: There is no general method to find black hole event
horizons for $N\neq 1$ case as discussed in [21]. In [21] it was demonstrated
that black hole event horizons exists for $2>N>\frac{2}{3}$ for $\Lambda<0$.
Since it is not possible to find $r_{+}$ values for a general $N$ value even
within $2>N>\frac{2}{3}$, Chan and Mann [21] discussed examples for
$N=\frac{6}{5},\frac{6}{7},\frac{4}{3},\frac{4}{5}$. In this paper, we will
choose $N=\frac{6}{7}$ to discuss specific values of thermodynamical
quantities. First, we will present thermodynamical quantities for a general
$N$ value and later will present detailed analysis of thermodynamics of the
black hole for $N=\frac{6}{7}$.

Following Ref. [39] we can obtain the entropy

$S=\frac{2\pi r_{+}R(r_{+})}{4}=\frac{\pi}{2}\gamma\,r_{+}^{N/2}\,.$ (73)

Considering the fact that $f(r_{+})=0$ we get

$M=\frac{\gamma}{2}r_{+}^{\frac{N}{2}-1}\left(-\frac{\Lambda\beta^{2-N}r_{+}^{N}}{3N-2}-\frac{Q^{2}}{N-2}\right)\,.$
(74)

###  5.1 First law of black hole thermodynamics

If we rewrite the mass $M$ in terms of $S,P,Q$, we obtain,

$M(S,P,Q,\beta)=\frac{2^{5-\frac{2}{N}}\pi^{-2+\frac{2}{N}}P\beta^{2-N}\gamma^{\frac{2}{N}-2}}{(3N-2)S^{\frac{2}{N}-3}}+\frac{\pi^{\frac{2}{N}-1}Q^{2}\gamma^{2/N}}{(2-N)4^{1/N}S^{\frac{2}{N}-1}}\,.$
(75)

We can write the following expression for the first law of thermodynamics as,

$dM=\left(\frac{\partial M}{\partial
S}\right)_{P,Q,\beta}dS+\left(\frac{\partial M}{\partial
P}\right)_{S,Q,\beta}dP+\left(\frac{\partial M}{\partial
q}\right)_{S,P,\beta}dQ+\left(\frac{\partial
M}{\partial\beta}\right)_{S,P,Q}d\beta\,\,\,,$ (76)

or

$dM=TdS+VdP+\Phi dQ+{\cal{B}}d\beta\,\,\,.$ (77)

Then the temperature can be calculated from the first law of black hole
thermodynamics as

$T=\left(\frac{\partial M}{\partial
S}\right)_{P,Q,\beta}=\frac{-\Lambda\beta^{2-N}r_{+}^{N}-Q^{2}}{N\pi
r_{+}}\,.$ (78)

The Hawking temperature obtained using the surface gravity is

$T_{H}=\frac{1}{4\pi}\left|\frac{dg_{tt}}{dt}\sqrt{-g^{tt}g^{rr}}\right|_{r_{+}}=\frac{2M(N-2)r_{+}^{-N/2}}{\pi
N\gamma}-\frac{2\Lambda\beta^{2-N}r_{+}^{N-1}}{\pi(3N-2)}=\frac{-\Lambda\beta^{2-N}r_{+}^{N}-Q^{2}}{N\pi
r_{+}}\,,$ (79)

which is equal to $T$.

If $P=-\frac{\Lambda}{8\pi}$, then one can obtain the corresponding
thermodynamic volume as,

$V=\left(\frac{\partial M}{\partial
P}\right)_{S,Q,\beta}=\frac{4\pi\beta^{2-N}\gamma\,r_{+}^{\frac{3N}{2}-1}}{3N-2}\,,$
(80)

which is different from the geometric volume. The electric potential at the
horizon is given by

$\Phi=\left(\frac{\partial M}{\partial
Q}\right)_{S,P,\beta}=\frac{Q\gamma}{(2-N)r_{+}^{1-\frac{N}{2}}}\,.$ (81)

Finally we get the thermodynamic quantity $\cal{B}$ conjugate to the parameter
$\beta$

${\cal{B}}=\left(\frac{\partial
M}{\partial\beta}\right)_{S,P,Q}=-\frac{(N-2)\Lambda\beta^{1-N}\gamma\,r_{+}^{\frac{3N}{2}-1}}{2(2-3N)}\,.$
(82)

###  5.2 Smarr formula

Now to obtain the Smarr formula we follow the scaling arguments of Ref. [42]

$\left(\frac{N}{2}-1\right)M=\left(\frac{N}{2}\right)\left(\frac{\partial
M}{\partial S}\right)S+(-2)\left(\frac{\partial M}{\partial
P}\right)P+(0)\left(\frac{\partial M}{\partial
Q}\right)Q+(1)\left(\frac{\partial M}{\partial\beta}\right)\beta\,\,\,,$ (83)

namely

$\left(\frac{N}{2}-1\right)M=\frac{N}{2}TS-2VP+{\cal{B}}\beta\,\,\,.$ (84)

###  5.3 Thermodynamical quantities

We will calculate few more thermodynamical quantities for $N\neq 1$ case as
follows:

$C_{{P,Q}}=\frac{-N\pi\gamma r^{N/2}_{+}\left(\beta^{2}\Lambda
r^{N}_{+}+Q^{2}\beta^{N}\right)}{4Q^{2}\beta^{N}-4(N-1)\beta^{2}\Lambda
r^{N}_{+}}\,.$ (85)

Since the enthalpy is given by $M$, which is equal to $U+PV$, one can compute
$U$ as follows:

$U=M-PV=\frac{\gamma Q^{2}r_{+}^{\frac{N}{2}-1}}{4-2N}\,.$ (86)

If we write $U$ as a function of temperature $T$ and pressure $P$, it is,

$U=-\frac{\pi\gamma Nr_{+}^{\frac{N}{2}}T}{4-2N}+\frac{8\pi\beta^{2-N}\gamma
r_{+}^{\frac{3N}{2}-1}P}{4-2N}$ (87)

Combining eq.$(\ref{tempN})$ and $\Lambda=-8\pi P$, one can obtain the state
equation as,

$P=\frac{NT}{8\beta^{2-N}r_{+}^{N-1}}+\frac{Q^{2}}{8\pi
r_{+}^{N}\beta^{2-N}}\,.$ (88)

For the general case with $N\neq 1$, the Gibbs free energy is given by the
following expression

$G=-\frac{\beta^{-N}\gamma\,r^{\frac{N}{2}-1}_{+}\left[(3N-2)\beta^{N}Q^{2}-\left(N^{2}-3N+2\right)\beta^{2}\Lambda
r^{N}_{+}\right]}{N\left(3N^{2}-8N+4\right)}\,\,\,.$ (89)

###  5.4 Thermodynamics for the black hole with $N=\frac{6}{7}$

Here we will analyze details of thermodynamics for the black hole with
$N=\frac{6}{7}$. Notice that $\Lambda<0$ for black hole horizons to exist. As
discussed in [21], it is possible to choose $M$ and $Q$ such that there are
two horizons, degenerate horizons or none. In the extreme case,

$Q^{2}=\frac{4096}{9261}\left(\frac{M^{3}}{\beta^{16/7}\gamma^{3}\Lambda^{2}}\right)=aM^{3}\,.$
(90)

Here,

$a=\frac{4096}{9261}\left(\frac{1}{\beta^{16/7}\gamma^{3}\Lambda^{2}}\right)\,.$
(91)

If $Q^{2}<aM^{3}$, one will have a black hole solution and if $Q^{2}>aM^{3}$,
then one will have a naked singularity. In Fig.$(\ref{frr})$ we have plotted
$f(r)$ for three values of $M$ for fixed $Q$. Notice that the relation given
for extreme black hole is different from the one given by Chan and Mann [21]
since we have redefined mass $M$ in this paper.

Figure 7: The figure shows $f(r)$ vs $r$ for the dilaton black hole for
$N=\frac{6}{7}$. Here $Q=0.376,\beta=0.1,\Lambda=-1,\gamma=1$. The blue curve
corresponds $M=0.144$ and is a black hole solution with two horizons. The red
curve corresponds to $M=0.104$ and does not have horizons. The green curve
corresponds to an extreme black hole with M = 0.1183.

The pressure $P$ vs $r_{+}$ is plotted in Fig.$(\ref{pvsrn})$. As is clear
from the figure, there is a minima for pressure at $r_{p}=\frac{7Q^{2}}{\pi
T}$. Hence for a given black hole,

$P>\frac{1}{\beta^{8/7}}\left(\frac{7}{\pi}\right)^{1/7}\left(\frac{Q^{2}}{T}\right)^{1/7}\,.$
(92)

Figure 8: The figure shows the pressure $P$ vs the horizon radius $r_{+}$ for
fixed charge. Here $N=6/7,Q=0.259,T=0.406$ and $\beta=0.1$.

####  5.4.1  Thermodynamic stability

For the black hole to be locally stable, $C_{P,Q}>0$. We have plotted
$C_{P,Q}$ vs $r_{+}$ and $T$ vs $r_{+}$ in Fig.$(\ref{cptn})$ and
Fig.$(\ref{cptn2})$. Both $T$ and $C_{P,Q}$ are zero when the black hole is
extreme at

$r_{ex}=\left(\frac{-Q^{2}}{\beta^{8/7}\Lambda}\right)^{7/6}$ (93)

Temperature $T$ has a local maxima at $r_{o}$ given by,

$r_{o}=7^{(7/6)}\left(\frac{-Q^{2}}{\beta^{8/7}\Lambda}\right)^{7/6}$ (94)

$C_{P,Q}\rightarrow\infty$ when $r=r_{o}$ and beyond that $C_{P,Q}<0$. Hence,
$C_{P,Q}>0$ between $r_{ex}$ and $r_{o}$ and black holes are locally stable in
this region.

Figure 9: The figure shows the specific heat capacity $C_{P,Q}$ and Hawking
temperature $T$ vs the horizon $r_{+}$ for fixed charge. Here $N=6/7,Q=0.5$,
$\Lambda=-1$, $\gamma=1$, and $\beta=$ 0.97, 0.644, 0.486 (green, red and
blue). Figure 10: The figure shows the specific heat capacity $C_{P,Q}$ and
Hawking temperature $T$ vs the horizon $r_{+}$ with amplified scale close to
$r_{+}=0$ for the same cases as in Fig.$(\ref{cptn}).$

To understand global stability of a black hole one has to study Gibbs free
energy. In the case for $N=\frac{6}{7}$, we have plotted $G$ vs $T$ in
Fig.$(\ref{gvstn})$. We have plotted $G$ vs $r_{+}$ and $T$ vs $r_{+}$ in
Fig.$(\ref{gandtn})$. As demonstared in Fig.$(\ref{cptn})$ and
eq.$(\ref{tmax})$, the temperature $T$ has a maximum, $T_{max}$. When
$T>T_{max}$, no black hole can exist. Hence, for $T>T_{max}$, thermal AdS
space is preferred thermodynamically. For $T<T_{max}$ there are two branches
of black holes: one corresponds to small black hole with smaller (positive)
Gibbs free energy and the other corresponds to large black hole with large
still positive free energy. Smaller black holes are locally stale and larger
black holes are locally unstable. Given the smaller black hole has lower Gibbs
free energy it is the preferred thermodynamical state for $T<T_{max}$. There
is a first order phase transition between the AdS space and the smaller black
holes at $T=T_{max}$. This is very similar to the Hawking-Page phase
transition occurred in the Schwarzschild-AdS black hole[4].

Figure 11: The figure shows the Gibbs free energy $G$ vs the temperature $T$
for fixed charge. Here $N=\frac{6}{7},Q=0.5$, $\Lambda=-1$, $\gamma=1$, and
$\beta=1$.  Figure 12: The figure shows the temperature $T$ vs $r$ and Gibbs
free energy $G$ vs $r$ for fixed charge. Here $N=\frac{6}{7},Q=0.5$,
$\Lambda=-1$, $\gamma=1$, and $\beta=1$.

##  6 Conclusion

In this paper we have studied the static charged dilaton black hole in 2+1
dimensions. We have studied thermodynamics in the extended phase space where
$P=-\Lambda/8\pi$. We have separated the paper into two major groups: $N=1$
case and $N=6/7$ case. We will discuss those two cases below.

###  6.1  $N=1$ case

For $N=1$ case, the black hole has two horizons given the mass $M>\gamma
Q\sqrt{-\beta\Lambda}$. It can have extreme black holes or naked
singularities. The first law for the black hole is derived with $\beta$ as a
thermodynamical quantity. A new thermodynamical quantity $\cal{B}$ is
introduced as the conjugate quantity to $\beta$. Introduction of $\beta$ as a
thermodynamical quantity is essential to have the first law to be satisfied.
Thermodynamic volume is not the geometric volume. For $N=1$ case, the
temperature $T$ approaches $-\frac{\Lambda\beta}{\pi}$ and does not have a
local maxima or a minima. Other thermodynamical quantities such as specific
heat, internal energy, Helmholtz free energy, Gibbs free energy are derived
and Maxwells relations are derived. Here, black holes are locally stable and
there are no phase transitions. Reverse Isoperimetric Inequality is satisfied
for certain values of the parameters in the theory. This behavior is unlike
the charged BTZ black hole where the Reverse Isoperimetric Inequality is
violated. Joule Thomson expansion is studied in detail and Joule Thomson
coefficient $\mu$ is calculated. It is clear that $\mu$ diverges at $T=0$
where the black hole is extreme. Inversion temperature is calculated and is
observed that it is proportional to the inversion pressure $P_{i}$.
Isenthalpic curves are plotted and we have discussed how the black hole heat
or cool with respect to the inversion curve.

###  6.2  $N=\frac{6}{7}$ case

We have derived thermodynamical quantities such as temperature, entropy,
thermodynamic volume, electric potential, specific heats, internal energy, and
Gibbs free energy for a general $N$ value. The first law and the Smarr
relations are presented. $\beta$ is treated as a thermodynamical quantity in
order to have the first law to be satisfied.

There is no general method to compute black hole horizons for a general $N$
value. According to the authors in [21]. It was demonstrated that black holes
existed for $2>N>\frac{2}{3}$ for $\Lambda<0$ and black holes with
$N=\frac{6}{5},\frac{6}{7},\frac{4}{3},\frac{4}{5}$ were discussed in [21]. In
this paper, we have chosen $N=\frac{6}{7}$ as an example for $N\neq 1$ case to
analyze thermodynamics. Black hols with such an $N$ value has the possibility
to have two horizons, degenerate or no horizons depending on the mass and
charge. In the extended phase space where $P=-\Lambda/(8\pi)$, the pressure
has a minimum when plotted against $r_{+}$. Hence black holes can only exist
for the pressure greater than a certain value. The temperature has a maxima
($T_{max})$ and black holes cannot exist with temperature greater than
$T_{max}$. Black holes are locally stable for a range of $r_{+}$. The range
lies in between when the black holes are extreme and at when the temperature
is extreme. In order to understand the global stability we have studied the
Gibbs free energy. Small black holes are preferred state for smaller
temperature values and the thermal AdS space is preferred after $T_{max}$.
Hence, there is a first order phase transition between small black holes and
the thermal AdS space. This is very similar to the phase transition for the
Schwarzschild AdS black hole sin 4 dimensions referred to as hawking-Page
phase transition.

Hence the charged dilaton black hole has presented with interesting behavior
as opposed to the charged BTZ black hole. In extending this work, it would be
interesting how adding spin to these black hole will change the
thermodynamical behavior. Hence it would be interesting to study the spinning
dilaton black holes presented by Chan and Mann in [22].

## Acknowledgements

S. Fernando was on Sabbatical leave from Northern Kentucky University when
part of this work was completed. L. Balart is supported by DIUFRO through the
project: DI22-0026.

## Appendix: Relation between mass $M$ and $m$ in $f(r)$

In this section we will present the relation for $M$ and $m$ given in
eq.$(\ref{mass-16})$.

Considering eq. (5.13) from Ref. [39] namely

$M_{d}=\frac{N_{d}+1}{8}r_{0}^{-N_{d}}\,m_{d}\,,$ (95)

where the subscript $d$ indicates that it is a quantity defined in Ref. [39].
In addition the equivalence with our notation is

$N_{d}=\frac{N}{2}-1\,,$ (96) $r_{0}^{-N_{d}}=\gamma\,,$ (97) $M_{d}=M\,.$
(98)

Consequently

$M=\gamma\frac{m}{8}\,.$ (99)

Note that $(N_{d}+1)m_{d}=m_{c}=m$ where $m_{c}$ refers to the mass parameter
of Ref. [21].

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